The art of fortification, or architecture militaire as vvell offensiue as defensiue, compiled & set forth, by Samuell Marolois revievved, augmented and corrected by Albert Girard mathematician: & translated out of French into English by Henry Hexam

THE FIRST PART OF REGVLAR FORTIFICATION, DEMONSTRATED BY THE DECEASED SAMVEL MAROLOIS, And Reviewed, corrected & augmented BY ALBERT GIRARD, SANMIELOIS. (Book 1)

GOd having found good to create Man, did with all prepare his affection for societie, that by this meanes he might the longer preserue him, and so make him become more stronge & able. According to which & to the same end, there were first built howses, and lodgings, to keepe men from the injurie of the aire, as windes, stormes, raine, excessiue heate, and colde, as the situation of their dwellings, and the commodiousnesse of those places did afford. And as by divine blessing, and their owne industrie, they attayned unto many Earthly riches, as famillies, cattell, and other goods: so by the instinct of nature, they sought alwaies to strengthen & defend themselues with the best fortifications they could, that thereby they might not onely preserue their bodies against the violence of their Adversaires; but also their goods, kindred, and Associates. To this end (as we may obserue as well in ancient histories, as also at this day in some East and West Indian Nations) they droue in piles of timber, and made Empale∣ments closse one to an other, that enclossing themselues so narrowly together, they might be the better able to defend and resist the furie of their Ennemies. But in

processe of time, the defects of such enclossure being well marked, they began after∣wards to make their Fortifications of a more solid, firme, and durable matter, as stone, brick, earth and such like. And though experience taught them, that these their Empalements of wood, were not able to ••esist fyre, & diuerse other accidents, which might befall them: yet it is certaine, they used these and the like fortifica∣tions, till they found the harme & conveniencie, which they receiued by them. But necessitie, the Inventrix of Arts, found that walls built with stones and bricks were better. These were in use and traine, till such time also, as their imperfections were knowne, and that by the force of man, and Engines they were diuerse wayes attem∣pted; yea many times beaten downe & overthrowne: which made them seeke from time to time to remedie their defects: for perceiving that their walls were hereto∣fore subject to be beaten downe (by Rames) and other ancient Engines; they made them (upon the right line of the wall) Spures, and roundles to hinder the beating of them downe, and to preserue them the better from a breach, and tumbling downe, they gaue them the Talude, that is, a slooping towards the inside of the place, that they might be the more able to resist the violence & force of these Engines. Finally these walls were made after diverse manners: first in a round forme, which, as on the one side, it was held the strongest; because these Engines beating against the wall crushed the stones, & made them stick the closser together, in regard the circle exteriour was greater then the interiour, and that the Engines could not unfasten, loose and breake them but with great difficultie. So on the other side, it was impos∣sible to defend such walls, when approches was made unto them; because no part of them could be discouvered, or flancked: so that afterward they built thē in a square forme, with small squares in their angles, for the defense of their Curtaines. Then made them also Demycircles with angles interiour, and exteriour, and lastly trian∣gular wise, on the one side to resist the force of the furious Canon, used at this day; and on the other side, that they might be the more capable to defend themselues by discovering euery part of the wall, euen to the very foundation. And as invention of attempting is growne to the highest degree in these dayes, by reason of the longe experience of the warrs in these parts (which is the Schole of all military actions) and the abilitie, and capacitie of the Assaillants & Defendants: So am I of the opinion, that the Fortifications made in these Low Countryes, are the strongest, exactest, & perfectest, which can be invented, and which haue bene made and practized, not by a simple Generall, but by one of the greatest Captaines of the World, endowed with a singular courage and spirit, an excellent Mathematician, and is not onely a Prince of a great howse; but also experienced & beaten in all militaire actions, and strata∣gems; aswell offensiue, as defensiue: of which Fortification, we haue undertaken to treate briefly at this present, and as succinctly as possibly may be.

Of the definitions.

I.

Forasmuch then as the definitions of Fortification are by the dayly use of armes growne so common: it were in vaine for me (in my opinion) to make any further explication thereof: yet to satisfie the ignorant, we will marke out the angles, and sides of a Fortresse, by the Letters of the Alphabeth, and opposite to the said letters, ye shall finde their names, and appellations, as we may note by the figures 1. and 2, following.

Icnographie, or ground-markeing.

1. Figure.

• N. O. The side of the Polygone, that is many angles.
• N. D. The line of the gorge,
• D. C. The line of the flanke,
• B. N. The Capitall line.
• B. C. Q. R. The Moate.
• P. The Raveline, or halfe mone.
• Q. S. The covert way.
• T. S. The Parapett thereof.
• B. I. The line of defence.
• D. K. The Curtaine.
• K. F. The Parapett.
• K. M. The Rampart.
• A. N. The semy-diameter.
• V. C. The flanke lengthned.
• C. N. D. The angle forming the flank.
• B. C. D. The angle of the shoulder.

Orthographie, or the Profile.

2. Figure.

• A. B. The foote or basis of the Rampart.
• G. H. The heigth of the Rampart.
• H. B. The talud (or slooping) of the infiel of the Rampart.
• A. Y. The Talud on the outside of the Rampart, or scharfe.
• Z. D. The foote of the Parapett.
• Z. E. The Parapett itselfe.
• D. F. The foote-banke.
• F. G. The Terra-plaine or bredth of the Rampart.
• K. A. The way for the round, or the false¦bray.
• I. K. The foote banke thereof.
• I. M. The Parapett of the falsebray.
• M. N. The Scharfe.
• P. O. N. M. The moate.
• P. O. The Counterscharfe.
• P. Q. The Covert way.
• R. Q. The foote banke thereof.
• T. S. R. The Parapett of the Covert way.

The other names, which haue neede of explanation shalbe declared in their due places.

Before we come to instruct yow particularly in the Art of Fortification; we will briefely treate of the calculation thereof: In which supputation, ye shall haue first sett downe the knowne termes, and under them the disposition of the Characters or Letters: beginning with a square fortresse with foure angles, or Bulwarks, and proceede on to a Dodecagone, a fortresse with twelue angles, or Bulwarks, makeing upon every Polygone, 3 or 4 Trialls, that afterwards one may choose the best of them: and because the angles will not be much altered by the diversitie of the trials: I haue thought good to giue this generall rule for them following.

It is a thing generally received of all men, that a square fortresse with foure Bul∣warks, is not so good as a Pentagonall with fiue angles, nor a Pentagonall so stronge as an Exagonall with sixe, and so consequently of the rest. If the cause thereof be sought out; one may obserue that this proceedes from the smallnesse of their an∣gles; as not being able to beare such a body of a Bastion, as the subsequent Polygones: so that a square fortresse for this reason, wilbe more defective, then the Pentagonall,

and this lesse defensive then the Hexagonall, and so well the rest following even to a Dodecagone, which hath the angle of the Bastion right, which is the cause, that constraines one to make the angles flanked lesser, then the reason of building well doth require, that is, the flanks too litle, the Gorge too narrow, and the line of de∣fence too longe. To encrease then proportionally the angles of Fortresses, accor∣ding as the angle of their Polygone augmenteth; we will take then the halfe of their angles, and adding thereunto 15 degrees, the summe wilbe the angle of the Bul∣warke, which we terme the angle-flanked, and if the angle-flanked be substracted from the angle of the Polygone, there will remayne the double of the angle flanking interiour, which double being substracted from 180 degrees, `then will remayne the angle flanking exteriour, called the Tenaille, and if ye add to the angle flanking interiour 90 degrees, then the summe wil be the angle of the Shoulder.

To finde out the angle of the Polygone from the number of its substract, namely 2, the remaynder must be multiplyed by 2. and the product wil be the number of the right angles, which such a Polygone containeth, as ye maye see by this exemple following.〈 math 〉〈 math 〉

Or thus.〈 math 〉〈 math 〉

And by the same rule ye shall finde the angles of the Subsequent Polygones, be∣ginning from a square Fortresse to a Dodecagone.

4.	5.	6.	7.	8.	9.	10.	11.	12
90.	72.	60.	51 3/7.	45.	40.	36.	32 8/••••.	30 the angl. of the center.
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 halfe.
15.	15.	15.	15.	15.	15.	15.	15.	15.
Sum.	60.	69.	75.	79 2/7.	82 1/2.	85.	87.	88 7/11.	90 The angle flanked.
Remaines 30.	39.	45.	49 2/7.	52 1/2.	55.	57.	58 7/11.	60 the double of the ang.
180.	180.	180.	180.	180.	180.	180.	180	180 The flanke interiour.
150.	141.	135.	130 5/7.	127 1/2.	125.	123.	121 4/11.	120 The flank exteriour.
15.	19 1/2.	22 1/2.	24 9/14.	26 1/4.	27 1/2.	28 1/2.	29 7/22.	30 The flank interiour.
90.	90.	90.	90.	90.	90.	90.	90.	90
105.	109 1/2.	112 1/2.	114 9/14.	116 1/4.	117 1/2.	118 1/2.	119 1/22.	120 the ang. of the should.

And seing the angle flanked of a Dodecagone is right, and able to resist a batterie, which is also made alwayes with right angles to shake the more the face of the Bul∣warke; one must fortifie the Polygones, which are aboue the Dodecagone, with a right angle, that the line of defence may come the more into the curtaine, that one may giue the more fire upon it: but the Polygones, which are under a Dodecagone, must be fortified, according to the precedent Table, and the calculation thereof shalbe made hereafter.

Some times we augment, aswell the angles of the Bulwarkes, as the Octogone, a Fortresse with 8 angles or Bulwarks, with a right angle, and those aboue are alwayes right, & those under diminishing to the square fortresse (which hath the angle of its Bulwarke onely of 60 degrees.) According to which the Bulwarks are somewhat larger, and the gorges and flanke greater then the former, but the second flanks lesser. Now for the finding out of every angle, ye must doe this following, where ye may observe, that in the manner aboue sayd, the angles flanking interiour are the fourth part of the angle flanked, or the 1/•• of the angle of the Polygone.

IIII.	V.	VI.	VII.	VIII	the angle
90.	108.	120.	128 4/7.	135.	the angle	of the Polygone.
60.	72.	80.	85 5/7.	90.	the angle	Flanked.
90.	72.	60.	51 3/7.	45.	the angle	of the center added thereunto.
150.	144.	140.	137 1/7.	135.	the angle	Flanking exteriour.
30.	36.	40.	42 ••/7.	45.	the angle	double of the angl. flank. interiour.
15.	18.	20.	21 3/7.	22 ••/2.	the angle	Flanking interiour,
90.	90.	90.	90.	90.	the angle	which is the flanke alwayes
105.	108.	110.	111 3/7.	112 1/2.	the angle	of the shoulder.

In the same manner also may be made right the angle flanked of the Decagone, a Fortresse of ten Bulwarks, where ye must note also, that before wee proceede

further, we will make use in the supputation following of the tenths or decinall numbers; which though it gives some imperfection: yet seing the things, which we omit therein are of no great consequence, it were ridiculous to make any further search thereof; considering likewise that the tables of Sines, tangents, and secants, are one and the same; I thought it good therfore to make use of these following.

THE I. QUESTION. The 1. Figure, & 1. Plate.

Let there be made upon a Square a fortification of foure Bulwarkes, so that the line of the Gorge be 7 parts: DI the curtaine 21, and IF, which is the flanke of 5, and from the angle of the flanke is drawne the line of defence by the angle of the Shoulder, to giue it a face. The question is how many the angles will be, and every line of the same fortresse, when as the line of defence will take up 600 feete. Now the length of a foote is sett downe in the 25 modell of Geometrie, noted by 1, and is divided into 12 ynches, & the ynch into 10 equall parts, and is the same foote, whereof 12 makes a rodd, which his Excell, useth in all his Fortifications.

ALBERT GIRARD.

THe Authour hath here aboue so disposed his calculations, that in stead of explaining them briefly, he confounds them teadiously, as if heretofore there had bene noe certaine rule sett dovvne in vvriting, to calculate lines and angles as is ordinarily done by the Tri∣gometrie of plaine Triangles, although there haue bene a number of Authours, vvhich haue treated of them, th' one after one manner, th' other after an other; and the most part of them commixed vvith longe discourses, vvhich moved me not longe since, to putt into light some tables of Sines in a portable volume, vvith the most succinct method that possibly could, touching the supputation of such plaine triangles, reducing them into foure diverse cases, vvhere I did insert in their places, some of my ovvne inventions to the purpose, knovvne as I suppose to none others heretofore; so that it cannot be, but having 3 knovvne termes, in a triangle, but that one maye knovve the other three, or one of them onely, vvhich one desires, as the Reader may knovve it; the manner & order thereof being much more facile, & easier to conceiue, then the reading of our Authour in his former editions, being obscure, trouble∣some, and hard to be attayned unto. For this reason the Learners of this science, are required to be forevvarned before they come to the reading of this booke; & are advertized hereby, that vvhen it is sayd, a triangle hath three termes, he must understand knovvne, or given, and that vvhen I say that a triangle right-angle hath three termes, that then I am bound but to shevv tvvo of them; seing that this vvord (right-angle) presupposeth that the triangle hath a right angle, to vveet of 90 degrees: moreover, that touching this present question, these 7, 21, 5 parts, demonstrate the reason of the lines ND, DI, IF, and not the quantitie of the same in feete, as is necessary to finde in the manner follovving.

To finde out the Angles.

IN the triangle right angle CDI, the side CD, is to DI, as 5 to 21, (for CD being of 5 parts; then DI wilbe the 21 thereof) therefore the angle DIC, wilbe found for 13:24 its double CLD 26:48, and its adjunct CLF flanking exteriour, wilbe of 153:12. Now in the figure quadrilatere, or of foure-sides, ABLH; the angle exteriour BLH is alwayes equall to the 3 interiours B, A & H; now A 90, the angle of the center, being taken from BLH, 153:12, there will remaine the two Demy-flankes ABL, & LHA 63:12. for one entire flank ZBC: & seing that the angle of the Shoul∣der BCD, is exteriour in the triangle CDI, it wilbe equall to the two interiours D.90. and 1. 13; 24, and therefore the Shoulder C, wilbe 103. 24.

To finde out the Lengths.

IN a triangle ambligone BNI, the side BI is 600 feete, & the angles are found, there∣fore the other termes shall be found, to wit BN the Capitall line 196, 64, & NI, 444, 62, but ND is the third of DI (for ND being 7, DI is 21 by the Hypotheses) there∣fore the fourth part of the found NI, wilbe ND, 111, 15 the Gorge, & the rest wilbe 333, 46, for DI, the Curtaine, now DI to DC is as 21, to 5: therefore one may say, if 21 giues me 5, how many DI 333, 46, giue me? facit for CD 79, 40. & so CI wilbe 343, 84, which taken from BI 600, there will remaine BC 257, 16, for the face.

Finally, the triangle right-angle VDH hath three termes, DH 600, and the angle H equall to HDI. 13:24 (because of the parallels DI; VH) then yow shall know VD 139, 05, & VH 583, 67, from which side substract VP (equall to Demy DI) 166, 73, there will remaine PH 416, 94, whereof the double wilbe for BH 833, 88, and PA wilbe 416, 94, for it is equall to PH, and thus must ye doe with the rest following.

The 3. Figure, 1. Plate.

Let there be a square Fortresse, whereof the curtaine DI being 4 parts; the flanck CD shall haue 1 of them, and also the Gorge 1: The defence running from the angle of the flanke, formeth the face, & the length BI is of 600 feete, one requires the greatnesse of the other lines of the same Fortresse.

To finde out the Angles.

THe triangle right-angle CDI hath three termes, as the reason of the sides CD, 1, to DI, 4, therefore the flancking interiour CDI wilbe 14 degrees & 2 mi∣nutes, whereunto add D 90 degrees, then the Shoulder BCD wilbe 104: 2; now if ye double the angle CID, it wilbe 28:4, CMD, and its adjunct CMF, 151: 56 for the angle flancking exteriour, from which substract the angle of the center A 90: there will remaine 61: 56 for the angle flancked entire B, and thus much for the angles.

For the sides.

THe triangle BNI, hath three termes, to witt, BI 600 feete, and the angles B, 30: 58. & N, 135. I, 14. 2, so one shall finde the lines BN 205, 76 & NI 436, 59; but in regard that ND to DI is as 1, to 4, according to the Hypotese, then ND will be the fifth of NI, where will follow that ND wilbe 87, 32 (and DC as much) and DI the rest 349, 27. Now seing that in the triangle right-angle CDI, the sides CD, DI are notified, one shall finde the Hypotenuse CI of 360, which taken from BI 600. there remaines the face BC 240. Finally, in the triangle right-angle VDH, which hath 3 termes DH 600, the angle H, equall to CID 14.2, ye shall haue VD 145, 49: also VH 582. 09, from which take VP 174, 64. (equall to the demy-curtaine DI) there will remaine PH 407, 45, also PA its equall: then in the triangle right-angle APH, ye shall finde AH 576, 22 from whence take BN or HO, the capitall line 205, 76, there will remaine AO 370, 46, & in takeing VD, from AP, there will remaine the perpendicular of the Center, A upon the middest of the curtaine 261, 96. Now BH wilbe 114, 90. as being the double of PH.

4. Figure. & 1. Plate.

In this square Fortresse, the defence BI is 600 feete, and the angle flanked 60 degrees, whereof DBC is the fourth part, which is 15 de∣grees, the question is what the quantity of the parts of such a Fortresse wilbe?

For the angles.

SEing that the line flanked is 60, then in the triangle BNI the angle B wilbe 30, & N 135, (the adjunct of ANI 45:) whereof the angle remayning BIN wilbe 15 degrees: therefore the triangle BDI wilbe Isosceles, that is, two sides a like, such that DBI is aswell 15; as BID; The Shoulder C wilbe 105, and in the triangle DMI, the angles upon the basis are each of them 15 degrees, the remaynder M will be then 150 for the flanking exteriour BMH.

For the sides.

THe triangle Isoceles BDI hath three termes, the defence BI 600 feete, and the pointed angles every of them 15 degrees. Therefore DI the Curtaine wilbe 310, 584. Also the triangle CDI, hath three termes, the angle D right; I 15. and the curtaine DI, then CD the flanke 83, 217, & CI 321, 535, which being taken from BI 600, there will remaine BC the face 278, 465. Moreover in the triangle IBN, the angles B 30, I, 15, & the defence 600: so then BN the Capitall line wilbe 219, 623, & NI 424. 268. from whence DI being taken there will remaine the gorge ND, 113. 684. Moreover, the triangle BIL having three termes, L right, B equall to CID 15. and BI 600, then IL wilbe 155, 292, & BL 579, 558, and takeing from it PL 155. 292. (which is the halfe of the curtaine) there will remaine BP 424, 266 for AP; also its double being BH 848, 532, one may also easely knowe BA 600.

The 5. Figure. & 2. Plate.

ALBERT GIRARD.

Heitherto of the defence dravvne from the angle of the flanke, but in those follovving, there is a second flank, and the distance of the angle of the flank, euen to the angle flanked, vvhich is called Fichant.

The 5 figure is the dessigne or draught of a square Fortresse, whereof the line of defence fichant DH maketh 600 feete, the angle flanked 60 degrees, the line HB is divided into 7 equall parts; whereof the one which is betweene the Characters 1. 2. is subdiuided into 5 equall parts, and from the center B, is made the Arch, 4 N cutting the capitall line at N, from which point is drawne NDZ parallel to BH, and from the point V (which is in the character 2) the perpendicular VD, the question is how many these lines, and the angles of such a Fortresse, will make?

For the lines in parts indetermined.

FOrasmuch as BH contayneth 35 such parts as BN 9; BV 10, VH 25, ye shall haue in the triangle right-angle BTN termes enough to knowe BT, or TN 6, 364, and TV or, ND 3, 636: Also the triangle right-angle DVH, had three termes, VD equall to TN, 6364, & VH 25, then DII wilbe 25, 797 parts or the same DH is 600 foote. We must then calculate according to this reason, the lines aboue mentioned to bring them into feete, saying, as followeth.

For the lines in parts determined, to vvit, brought into feete.

THe 25, 797 parts, make 600 foote, how many then will BH 35. come to? There wilbe for BH 814, 04 foote, likewise VL, or DZ the curtaine may be made soe; BH 7 parts make 814, 04 foote, how many VL 3? it will come for the curtaine to 348, 87, & for BN the Capitall line ye shall say, if 35 giues 814, 04: how many 9? ye shall haue for BN 209.32: and so of the other: for TN, or BT, or VD wilbe 148. 02, & TV, or the Gorge ND 84 57. and seing that BH is knowne, also VL the halfe of the remainder wilbe BV 232, 58. In the triangle right-angle BVC, the angle B wilbe 15 degrees: (for PBO is 45, & CBO 30:) and BV being knowne, therefore BC the face wilbe 240, 78, and VC 62, 32, which being taken from VD, there will remaine CD the flanck 85. 7; and therfore the triangle right-angle CDI, shall haue 3 termes, the angle C 75, & CD 85, 7. soe DI wilbe 319, 84, CI 331, 15, likewise BI flancking 571, 93. & I, Z 29. 03 for the second flancke. Now OP is knowne as being the halfe of BH.

Now of the Pentagons, or Fortresses with fiue angles or Bulwarks.

6. Figure. & 2. Plate.

In the Pentagone, a Fortresse with fiue Bulwarks, KFBDL, let the

line, KL be 63 rods, & divided into 7 equall parts, whereof the Capitall line KA is 2 of them, also LE, and from the Character 2 (or G) let GB be the perpendicular. Also the angle flancked is 69 degrees, according to the precedent table, how many then wilbe the lines and the angles of such a Fortresse?

ALB. GIRARD.

Before vvee come to the Construction hereof, take notice, that KA in this figure is not ansvverable to the length of KG, yet ye must suppose them to be equall, asvvell as the cipher 1, and the letter H, are tvvo differing points, vvhich vvould haue bene the better discerned, if the figure had bin vvell made, vvhich may serue as a fore-vvarning for some figures follovving, upon vvhich one ought not to stand so much as upon the Suppositions, or Hypo∣teses of the Propositions.

The triangle right-angle KHA hath three termes, KA 18 rodd, the angle AKH 54 degrees, which being of KH wilbe 10, 5802, & HA or GB 14, 5623. Now if ye take away KH from KG, there will remaine for AB the Gorge, 7. 4198. Item the curtaine BD wilbe 27 rodd, being equall to GM. And because the angle flancked is 69 degrees, if ye add thereunto the angle of the Center, which is 72, ye shall haue the angle flancking exteriour, 141 degrees, the halfe of its adjunct for the interiour 19 1/2 degrees. The triangle right-angle KGF, hauing 3 termes, KG 18, GKF 19 1/2: the face KF wilbe 19, 0953, and FG 6. 3741, which being taken from GB 14, 5623. there will remayne FB (the flanck) 〈…〉〈…〉, then the triangle right-angle FBI may be knowne, seing that FB is found, and the angle 〈…〉〈…〉 1/2 the complement of 1; therefore BI wilbe 23.1227, which taken from BD 27, there will remayne ID for the second flanck 3, 8773, also FI wilbe found to be 24, 5297, whereunto add KF 19. 0953, and ye shall haue KI, the defence flancking 43. 6250, finally the triangle right-angle LGB, hauing three termes, to wit LG 45 rodd, and GB 14, 5623; one may easely know BL, and the distance of the Center of the Fortresse by K, because that KP is 31 1/2 rodd, and the angle AKP 54 degrees.

The 7. Figure, & 2. Plate.

In a Fortresse Pentagonall, BOV, let AB the deffence flancking be 50 rodd, the Flanck ED 9 rodd, and the angle of the Bulwarke 72 de∣grees. How many then will the other parts of the same make, when the second flanck AG makes fiue rodd.

THe angle of the Polygone is 108, the halfe is 54 for FBQ, to which add CBF 36 degrees by the Hypoteses, you shall haue 90 for CBQ, but CF is paralell to BQ therefore BCA right, the triangle BCA wilbe then the right-angle, and BA of 50, and the angle B of 72 degrees, then ye shall finde BC, or QD to be of 15, 451, and CA 47, 553, and the angle A 18 degrees. Also in the triangle right-angle EDA, the angle A is giuen, and the side ED 9 rodd, by the Hypotese, then EA wilbe 29, 12463, & DA 27, 69912; now if ye take EA from DA, there will remaine BE the face 20, 87537, and if ye substract DA from AC, there will remaine CD, or BQ, 19. 854. Also the

triangle right-angle BCF, hath 3 termes, BC and the angles; in fine BF the capitall line wilbe 19, 097 & CF, 11, 226, which being taken from CD, there will remaine FD 8, 628, the Gorge; now if to DA, ye add AG 5 rodd, the second flanck, ye shall haue the curtaine DG 32. 69912, to which add twise BQ, and ye shall haue BV 72, 40711 its half BK 36, 20355, and so the triangle right-angle BKO, hauing 3 termes, the angle KBO wilbe 54 degrees, ye shall finde BO 61, 593099, and KO 49, 82984. Finally VB, BQ being knowne, then the triangle right angle DQV, hauing 3 termes giuen VQ, & QD; ye shall finde the fichant DV easely.

ALB. GIRARD.

IF one vvill consider the difference vvhich is betvveene this operation, and that of our Au∣thour published in his former Editions, he shall finde that this question is defectiue, for I haue added thereunto (that the second flanke AG is rodd) seing that in all such questions, vvhere there is a second flank, he ought to haue set dovvne 5 knovvne termes & neither more nor lesse, vvithout the one depending on the other, as in this present, the name of the Fortresse, to vvit, Pentagonall is a terme, in the second place there is the defence; then the flanck, the angle flancked, and the second flancke, vvhich are fiue termes: vvhere ye must note, that vvhen there are reasons in the proposition, and that though a reason hath tvvo numbers, neverthelesse it is but one terme; but vvhere there is no second flanck (as in the figures of the first plate, then 4 termes vvill suffice.) Finally, my Authour had so ordained his Supputation, that in stead of makeing an addition, or a substraction; he made the rules of three very great, so that ye must imagine it vvas easier for me to change all, then to recorrect it, hauing no other respect, but to the explication of the figures, and as much as is lavvfull and possible to shevv his intention, vvhich I am bound to doe: Moreouer, if peradventure same one should finde this manner of operation strange, vvhich I 〈…〉〈…〉 heitherto, it is requisite for him to knovv also (be it spoken under ••••••••ection) that he understandeth not much, if he doth not practize it soe himselfe in other Subjects, yea though I had not sett dovvne one onely number; vvhich I should haue done already heretofore, if I had not had regard to the obscuritie, vvhich students might pretend to finde, vvhich hauing cleared heitherto, ye may goe on vvith the more assurance in the rest follovving: and must also knovv, that the 8 figure vvas left vvithout explication by the Authour.

The 8. 9. 10 Figure. & 3. Plate.

In this figure Pentagonall, the angle ELO is divided into 2 equall parts by LD, the flanck FB 9 rodd, the angle flancked 69 degrees, and the curtaine 30 rodd, how many then will the other lines & angles be?

THe angle of the Polygone is parted in the midst by KA, and the angle AKF is knowne, also the rest shalbe FKG, or its equall FB, whence followes that the triangle right-angle FBI shall haue 3 termes FB, and the angles; therefore the other 3 termes wilbe knowne, also the whole line DB, and the part BI, ergo the rest ID is the second flanke. The triangle KBI hath three termes, the angle found, BI & K (the fourth part of the flanck by consequence the 3 termes remayning are knowne, and the difference of the found lines KI, IF wilbe for the face KF; which makes that in the triangle right-angle KGF wilbe knowne KG. GF; and if to KG doubled, ye add the curtaine, ye shall haue KL; also KG to the curtaine is for GL, or XB if to GF ye add FB, ye shall haue LX, and therefore in the triangle right-angle LB wilbe

knowne; seing that LX, XB are found. Moreouer, in the triangle right-angle LXE, the angle E is a demy Polygone, and the line LX being knowne, ye shall haue LE the Capitall line, and XE taken from XD (equall to KG) there will remayne DE the Gorge.

• BI 25, 4151
• FI 26, 9617
• ID 4, 5848
• KI 51, 2795
• KF 24, 3178
• KG 22, 9229
• GF 8, 1175
• KL 75, 8459
• XB 52, 9233
• LX 17, 1175
• LB 55, 6227
• LE 21, 1583
• XE 12, 4366
• DE 10, 4863

The 9. Figure, vvhich is the 8, 9. 10. & 3 Plate.

In the former figure Pentagonall, there are other Hypoteses BL, fichant 60 rodd, the angle DLM 36 degrees, and 45 minutes, DM 17 rodd, and the angle OLE parted in the midst by LM.

THe angle MLE is 54: from which take DLM 36, 45; there will remaine 17, 15; for DLE its quadruple or fourefould for the angle flancked 69: ye must first calculate the traingles right-angles MDL, QEL, whereby is found DE the Gorge, LE the Capitall line, and ML in the triangle right-angle LGB; the line of defence LB 600, and GB 17; therefore ye shall finde GL, which added to KG equall to ML, ye shall haue 〈…〉〈…〉 take ML, from it, there will remaine GM for the curtaine BD, in the triangle MLO, ye shall find the face OL, & MO, which taken from 17, there remaines OD for the flanck, & the triangle right-angle ODC may be knowne, because the angle C, is equall to MLO 19 1/2, then CO & CD being notified, ye shall find CL & BC, and from the Center of the Fortresse wilbe knowne the distances towards K, P, A.

• DE 10, 41454
• LE 21, 01319
• ML 22, 76572
• GL 57, 54128
• KL 80, 30700
• BD 34, 77556
• OL 24, 15101
• MO 8, 06179
• OD 8, 93821
• OC 26, 77657
• CD 25, 24075
• CL 50, 92758
• CB 9, 53481
• A (†) 47, 30000

The 10. Figure, vvhich is the 8, 9. 10.

In the figure Pentagonall, let LB fichant be 60 rodd, and LC the de∣fence flancking 50, 92758 fifth-parts. Also BC the second flancke 9, 53481, and the angle OLE parted in the midst by LD.

THe triangle BCL hauing the three sides giuen, then ye shall finde the angle C, or the lengthning of the basis BC unto the perpendicular X, as followeth here, the basis BC, 9, 53481 giueth me the Summe of both the other sides, 110, 92758,

how much then will their difference 9, 07242 giue? it will come to 105, 54816, from which take the basis (because that the perpendicular falls without, which is seene, when the sayd quotient is more then the basis) there will remaine one number, whereof the halfe is CX 48, 00668, this being done ye must calculate the triangle right-angle CLX, by which ye shall know the angle C, or LMO, which taken from LME, the remainder wilbe OLE, 34 1/2, and the angle flancked wilbe the double of it 69; also ye shall haue LX, or MD, and then shall ye calculate the triangle MDL (for the angle MLD is knowne) seing it is equall to MLO 19 1/2, and OLD 17 1/2: Also the triangle LQE, whereby ye shall finde the Gorge, the Capitall line, & CD, then by the triangle ODC is found the flanke, & after that the face: & if to CX aboue sayd, ye put BC being giuen, there will come out BX, or GL, to which add ML by addition, then by substraction ye shall haue KL, and the curtaine BD, whereof the numbers agree with those of the precedent question.

The 11. Figure. 3. Plate.

In this figure Hexagone, that is, a Fortresse with sixe angles or Bulwarks, let the Curtaine be sesquialtere, or halfe as much againe to the face, & the face to the flancke doubled-sesquialtere, the angle flancked 75 degrees, and DP distant from the points 70 rodd. Wee must finde out the rest.

SEsquialtere, or one and a halfe, is as 3 in respect of 2; twise sesquialtere is as 5 to 2: so that BH, DC, CB wilbe as 15, 10, 4. whereunto if yee add 4 cifphers to each of them, DC wilbe 100000 sines of the right-angle. Now ADF is 60, and ADC 37 1/2, then CDF wilbe 22 1/2 degrees; therefore CF 38268, and DF 92388 also KP; but BH, or FK is 150000, then DB wilbe 334776 parts, which make 70 rodd by the Hypoteses, wee will bring then these lines into Rodds, according to this reason following; because 334776 parts make 70 Rodd.

How many partes. they will come to.

• 150000 31, 364 BH
• 100000 20, 909 DC
• 40000 8, 364 CB
• 38268 8, 002 CE
• 92388 19, 318 DF

And seing that PD, DF are knowne, FP wilbe also; and by FC, CB, wilbe knowne FB, or AE, 16. 366, now by the 47 pro.1 of Euclide BP fichant wilbe 53, 259: And if one calculates the triangles CBG, DEA, ye shall finde DG, 42, 765; and DE, 9, 448, ergo EF, or AB the Gorge wilbe 9, 87; the distance from the Center to D, is equall to DP, 70 rodd.

The 12. Figure. & 4. Plate.

Let DP be 72 rod, the angles of the Bulwarks are 72 degrees, the curtaine BH 32 rod, the flancks of 8, 3638, fourth of rodds, how many then wilbe the other parts of this Fortresse Hexagonall?

Seing that DP is 72, & BH, or FK 32, then the halfe of the rest wilbe 20 for DF, and also the angle ABF being 60 degrees, and ADC 37 1/2, then CDF wilbe 22 1/2 de∣grees: whereby ye shall know then the triangle right-angle CDF. Secondly, FC found, with CB giuen, will make knowne FB, but PF is 52, then the triangle right-angle BFP hath the sides BF, FP knowne, & BP wilbe knowne in the triangles right-angles ADE, CBG, their termes will suffice to make the rest to be understood. when ye haue DA, its halfe wilbe for DE, which taken from DF, there will remaine AB.

• DG 21, 64780
• FB 16, 64800
• BP 54, 60000
• AD 19, 22344
• DE 9, 61172
• AB 10, 38828
• GH 11, 80804
• DG 43, 50349

The 13. Figure.

In this Figure Hexagonall, the second flanke is in the flanke as 6 to 7, the flanke hath its gorge as 7 to 10, the Gorge to the line of the Polygone, as 2 to 9. the question is how many they, & the other dimensions wilbe, when as DH makes 60 rodd.

IF ye make GH 6, then HO wilbe 7, & HI 10, the Curtaine wilbe 25; Then ye shall calculate the triangles CBG, DAG by which ye haue the angle G, or his equall 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 sett downe more easie, in takeing but 7, 48926, which make 1000000 parts; Now if ye had rather worke it by multiplication, without division, then ye must take but 100000 parts, which make 13352 rodd, and so ye shall finde

• DC 25, 38302
• CB 9, 34672
• BA 13, 35246
• DP 81, 01484
• BH 33, 38115
• The angle flancked 79. 33.

The 14. Figure, the 1. of the tvvo.

In this Fortresse Hexagonall, let EH be 60 rodd, and EI, BC, KH equall, the flanke KC 8 rodd, it behoues us to finde out the rest.

BY the help of the rule of Algeber, ye shall finde that the angle EIF wilbe 69 de∣grees, 4 1/2 minuts; then in the triangle IBC, the angle I, & the side IB 8 rodd being giuen, ye shall finde BC 20, 9229 for the face EI; also IC wilbe 22, 4002, then EC will make 43, 3231, the rest is easie, without the second flanck.

The 15. Figure.

This figure hath for the flanke 10 rodd; the rest being like to the former they shalbe put in order in the Tables following.

The 16. Figure. & 5. Plate.

The flanke of this Heptagone, that is, A Fortresse with seuen angles, or Bulwarks, maketh 10 rodd, the distance of the points 72 rodd, and the angle flanked 80 degrees, how many then will the rest make, when as the second flanke is 10 rodd.

SEing that the angle of the Bulwarke makes 80 degrees, and the angle of the Poly∣gone 128:34, 17 the difference divided in the halfe wilbe the angle flanking inte∣riour 24, 17, 18 1/2, the flanck CB 10 rodd, whence the triangle CBG wilbe knowne, & by adding GH giuen to BG, ye shall haue the curtaine 32, 1623, which taken from DP 72 rod, and then take the halfe of the rest, ye shall finde DF, and also DC. CF, for the face DC 21, 8525, & DG 46, 1666, also in the triangle DAE; ye shall haue DE, & consequently EF, or AB. Moreouer ye shall finde DH in the triangle DKH, to be 55, 4345.

The 17. Figure, & 5. Plate.

This Heptagone hath the flanke of 9 rodd, the angle flanked 79 degrees 25. 43, DP ••2 rodd, and the choice of the second flanke.

ALB. GIRARD.

HAuing calculated the angles as in the former. OM wilbe 21, 6436, and HM, 19 6836, let us make the second flanke 7, 3164, and then the curtaine wilbe 26; DF, 23; the face OP 25, 2901. MP 46, 9337. Marolois had giuen here 80 degrees, but tooke no more then 79, 25, 43, for the angle flanked, which he did without pre∣meditation.

The 18. Figure. & 5. Plate.

In this present Heptagone the angle flanked is 79 degrees, the gorge 12 rodd, & the curtaine 32 rodd, how much will the other lines & angles make of this Fortresse.

TO resolue this question, ye must suppute from the triangle CBG, the lines CG, GB, and ye shall haue GB the second flanke; now if to BG, ye add AB, ye shall haue AG, then the triangle DAG wilbe knowne; afterward the triangle DAE, and finally the triangles DCF, and DHK.

ALB. GIRARD.

NOte that in this Question Marolois had sett downe the angle flanked 79 and 2 seuenths, yet did not followe this number, but 79 degrees: the same errour also was committed in his supputation of the former, and in the 14 figure. One cannot gesse well his Supposition: neuerthelesse that needes not to stay the reader; for I haue sett downe the question, as it seemes he would haue propounded it, but

the worst is, he made 2 figures, and one cannot understand well of which of either of them he would speake, but we will speake more thereof hereafter; howsoeuer those which are most intricate (aswell by reason of the faults escaped in the impression of the former Editions, as by the errour of his disciples, which did calculate them) shalbe partly omitted, and shall giue no impediment, but that the rest may giue con∣tentment to the Readers, for that which they are desirous to finde out in this booke: (in the 20 figure following the Letters ye were sent unto were wholly repugnant.)

The 19. Figure.

In this Heptagone let the angle flanked be 80 degrees, & the angle from the Capitall line, & from the imagined DB (to wit ADB) let be 22 ½ de∣grees, DP 82 rodd, & the flank CB 10 rodd.

FRom the angle of the Polygone 128, 34, 17, take the flanked 80, the halfe of there∣mainder wilbe 24, 17, 9: for the angle flanking interiour DGA: beginn then the triangles CBG, DBG. (whereof the angle in D maketh 17 ½) ye shall finde BG 22, 1621, CG 24, 3138; & BD 30, 3118, then GD wilbe 49, 1096, and DC the face 24, 7958, afterward the triangle BDF, the line BF wilbe found to be 20, 1982, and DF 22, 6014, whereof the double taken from DP 82, there will remaine for the curtaine BH 36, 7972, and consequently for the second flanke GH, 14, 6350. Now knowing the triangle DEA, ye shall finde the Capitall, also the gorge AB 12, 8744. If in the triangle DKH, yee seeke DH, then ye shall finde it to be 62, 7388. Now the reason of the Demy-diameter or middle line in the side of this Heptagone, is as 1000000000, to 867767478.

The 20. Figure.

This present Octogone, that is a Fortresse of 8 angles or Bulwarks, hath the distance DP divided into 7 equall parts, whereof DF, & FB make each of them 2, & BP fichant is 60 rod, the angle flanked 82 ½ degrees.

THe line BF being 2, and FP 5, then the square of BP wilbe 29, and seing that BP is 60 rodd, its square wilbe 3600 rod, by which reason ye may knowe one part of the 7 of DP, saying, if 29 giues me 3600, how many will 1 giue me (the square of an other part?) it will come to 124, 1379310345, whereof the square roote substracted wilbe 11, 14172, and its double 22, 28344 for DF, or FB, the triple is 33, 42516 for the curtaine BH, and the seventhfould 77, 99204 for DP. Moreouer, the angle flanking interiour, or its equall FDC wilbe 26, 15; so that DC wilbe 24, 84581, also FC 10, 98907, the flanke then wilbe 11, 29437 (for FB was knowne) in the triangle right∣angle DRG, the angle D is also 26. 15; and RC equall to FB, then DG the line of defence flanking, wilbe 50, 38218; finally, the Capitall wilbe found 24, 11937, and the gorge AB 13, 05342, if ye calculate the triangle DEA.

The 21. Figure.

In this Octogone let the face be 24 rod, the flanke 12, & the curtaine 36 rod, & the angle flanked, right, it is required how many the other lines & angles wilbe?

The angles ADF 67 ½, & ADC 45, will make knowne CDF to be 22 ½ Now DC is 24 rod, by consequence DF, FC, DP, FB wilbe knowne; also seing that FB is found (or EA) ye haue also the triangle DEA, by which ye shall finde DA, AB, & shall finde DG in the triangle CBG: for hauing CG, ye add unto it DC) finally DH in the triangle DKH.

• DP 80, 34624
• DG 55, 35756
• DH 61, 91032
• GH 7, 02948.
• AB 13, 39837.
• AI 62, 39837.

The 22. Figure.

In this present Octogone, the flanke is 11 rod, the angle flanked 82 ½ de∣grees, the line DP 76 rod, & the second flanke to be choosen, how many then wilbe the other parts of this Fortresse?

THe angle of the Octogone is knowne, and the angle flanked, also their halfe ADF, ADC, therefore the rest, and the triangle CBG wilbe knowne, and BG wilbe found 22, 3058, suppose the second flanke GH is 9, 6942, then BH 32 rod, likewise as much for FK; but DP is 76, therefore DF makes 22, and after ye haue calculated the triangles DCF, DAE, ye shall finde DC, 24, 52978, DG 49, 40045, AB 12, 94981, and finally in the triangle DKH the fichant DH, 58, 25283.

The 23. Figure.

Let the angle flanked be 82 ½ of this present Octogone, the line of defence DG 50 rod, DP 76, and CAB the halfe of the angle flanked.

AFter ye haue carryed AC, which was forgetten, if ye count the triangles DAG, ACG, ye shall finde the Capitall, the face, and the gorge, and BG, then the trian∣gles CBG, DCF, afterward ye shall haue CB, CF, DF, and therfore FK, or the cur∣taine, and the second flanke, then DH, 58, 3498.

• The Capitall 23, 93655
• The Gorge 12, 84323
• The Flanke 11, 26362
• The second flanke 9, 15324
• The face 24, 53340
• The Curtaine 31, 99346

The 24. Figure.

Let there be an Enneagone, A fortresse with nine Angles or Bulwarks, (that is with 9 sides) whereof the angle flanked is 85 degrees, the face 24 rod, the flanke 12, and the curtaine 36 rod.

THe angle of the Polygone is 140 degrees, seeke the triangles, DCF, CBG, DEA, DKH, and then ye shall finde the lines GH 12, 94824: DP 78, 57648: AB 12, 88709.

The 25. Figure.

As in the tables aboue let the angle flanked of the Enneagone be 85 degrees, the fichant 60 rod, and DP being 7 parts, then let DF be two of them, afterward FB the perpendicular 2, for the makeing of the flanke CB, wee require the rest.

DK being 5 parts, and KH 2. the square DH wilbe 29. which makes 3600, whereof 1 makes 124, 13793103 for the square of one part, its root substracted wilbe 11, 14172 for one part, whereof the double wilbe for DF, or FB, and the triple for FK, or the curtaine 33, 42516 BH, and after ye haue calculated the triangles DFC, CBG, DEA, ye shall finde the other lines DC 25, 1219, DG, 48, 25855, GH 12, 9026, BC wilbe 10, 68335, the seuen parts of the number aboue sayd wilbe for DP, 77, 99204.

The 26. Figure.

In this present Enneagone, let the angle flanked be 85 degrees, the de∣fence 50 rod, the other 60, & the gorge in the flanke as 4 to 3. the rest is required.

IF ye take away the angle ADC 42 ½ from ADF 70, the angles GDR will remaine, and DG makes 50 rod, then the triangle DGR wilbe knowne, to wit, DR, & RG, or KH 23. 0875, which will make ye knowe the triangle DKH (for DH is 60) then RK, or GH wilbe 11, 0301. Moreouer, in the triangle DEA the side EA is equall to KH, by DE & DR, ye shall haue AG 35, 94735, then in carrying AC, ye shall seeke the angle A from the triangle ABC, setting downe AB 4, and BC 3 parts, according to the Hypatese, ye shall finde then the angle A of 36 degrees, 52 min. & 12. seconds. Then let us goe to the triangle ACG, hauing AG knowne, the angle A & G, equall to CDF 27 ½ degrees, to finde CG afterward the face, and consequently DF, FK, EF or AB the Gorge, which wilbe 14, 72808, therefore if 4 giues 3, how many then AB? ye shall haue BC for the flanke 11, 04606, & DP 78, 51183, DA 24, 56925.

The 27. Figure.

The distance DP being 7 parts, DF & FB each of them 2 parts, the defences 50 & 60 rod, and the angle flanked 85 degrees, how many will the other dimensions be of such a Fortresse non angular?

ALB. GIRARD.

THis question is impossible to resolue being exceeding, seing that there is a condition in it more then one desires, & vvhich is vvorst, repugnant to the others, for vvhich fault the Authour may be excused, seing that in his time, there vvere no such advertisements giuen as vvee haue giuen thereof in the beginning of the Trigonometrie, cited in the first question going before, the vvhich though they may seeme to be of litle consequence to some; yet one must acknovvledg, that those that knovve them shall not fall into the like errours, as these vvhich may be explained thus; There are tvvo reasons giuen DP to DF, and DF to FB (a reason of

equality) tvvo lines of defence the angle flanked, & the name of the figure of nine-side figure, vvhich are sixe termes, yet one needes but 5 (as ye shall finde it noted in the 7 figure, vvhere the question vvas defectiue, and of some others aftervvard) finally, the proofe of this may be seene in makeing comparaison of this vvith the 25 figure, vvhere the same question is pro∣pounded, and vvhere ye shall finde that the defence flanking ought to be 48, 25855, and here he vvill haue it 50, vvhich is absurd, as is said.

The 28. Figure.

In this present Decagone, a Fortresse with ten angles or Bulwarks, let the angle flanked be 87 degrees, the Gorge in the flanke in reason sesquitertia, the defences 50 and 60 rod, it is required how many the other parts thereof will make?

THe reason sesquitertia is, as 4 to 3, for AB to BC, then the imaginall angle BAC wilbe 36 degrees, 52 minutes, 12 seconds. Moreouer, the angle ADC being 43 ½ then CDF wilbe 28 ½, which is an angle of the triangle GDR, and which may be knowne, seing that DG is 50 rod, therefore GR 23, 858, or its equall KH, and for as much as DH is 60, then DK wilbe knowne, and also DR withall, and so ye shall finde RK for the second flanke: likewise the triangle DEA wilbe knowne, then ED and DR will make knowne ER, or AG, & consequently ye shall haue the triangle ACG for the angle A was found aboue, and the angle G is 28 ½ degrees, then the face wilbe 26, 11334, and hauing found out DF, ye shall finde EF, or the gorge 15, 197: Also FK for the curtaine 32, 10821; DP 78, 00619.

The 29. Figure.

In this Decagone the angle flanked maked 87 degrees, the defence fichant 60 rod, the flanke 12 rod, and the gorge 16 rod, it is required how much the other parts make?

THe quadrangle ABCD hauing fiue termes giuen, ye shall finde the other parts; also the triangle DFC, whereby ye haue FB, or KH, & consequently the triangle DKH, and hauing FD & DK, their summe and difference wilbe for DP, 78, 65, and BH 30, 32633; the face shall be 27, 49377: DG 52, 64265.

The 30. Figure.

In this Decagone, let the curtaine be 36 rod, and the flanke 12, the face 24, and the reason of the Bulwarke in the flanking interiour, as 58 to 19.

THe halfe of 58 is 29, then the angles ADG to AGD, or CDF, wilbe as 29 to 19, therefore setting downe ADF to CDF, it wilbe as 48 to 19; but ADF is 72 de∣grees, then CDF wilbe 28 ½ degrees, & then the angle flanked 87 degrees, the rest is easie, for the triangles DCF, CBG will make knowne CF, FD; DG is 49, 14888, DH 61, 68324; DP 78, 18336, AD 24, 52336.

The 31. Figure.

Let there be a Decagone, whereof the face is 24 rod, the flanke 12, and the curtaine 36, how many will those parts make, when the defence flan∣king is doubled to the Capitall.

IF DG be sett downe 2. then DA wilbe 1. Now the angle DAG is 108 degrees, then the angle flanking interiour G wilbe 28. 23. 38, and the angle flanked 87, 12, 44•• the rest is easie, & is found in the same manner, as the end of the former, & therefore Marolois leaues its so for this reason.

The 32. Figure.

In this present Vndecagone, a fortresse with eleuen Bulwarks, let the face be 24 rod, the flanke 12, the curtaine 36, and the capitall DA to AG as 5 to 7, the unknowne parts are required.

SEt downe DA 5, and AG 7, and the angle DAG is 106 degrees, and 4 eleuenths: that is 106, 21, 49, and hence ye shall know the angles remayning ADC, 43, 55, 48, and the other or CDF 29, 42, 23, therefore in the triangle CDF, the angle D is then so, and DC 24 rod, and then DF, DK, DP wilbe knowne, likewise FC, FB, BP, & AD, DE & EF for the gorge. Here is nothing but that which is ill calculated by the Autheur, or rather by his disciples, as from the beginning (without all doubt) seekeing to help themselues with the figure put here under, which was needlesse; supposing that they had skill in Trigonometrie. I will onely set downe here the reason of the raid (or semy-diameter) in the side of the Vndecagone inscribed in the Circle, is

as 100000000000000 to 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 curtaine be as 2 to 3: & the gorge in the flanke as 4 to 3, the distance of the points of the Bastions 75 rod, and the angle flanked 88 degrees, 38 min. 11 seconds, the question is how ye shall finde out the other dimensions?

SEing that ADC, and ADF are knowne, the rest CDF wilbe also, being 29, 19, 6, & setting downe DC 2 parts, (then the curtaine, or FK wilbe 3) the triangle CDF wilbe knowne by the parts, namely DF 1, 74384, as much also is KL, and FK 3, then DP wilbe (in parts) 6, 48768, which make by the Hypotese 75 rod, & by this reason, ye shall finde out the face, and the curtaine, saying, if 6, 48768 parts make 75 rod, how many then will make aswell 2, as 3? DC 23, 12075, & BH 34, 68112 as much makes FK, which taken from 75, the halfe of the rest wilbe for DF, ye shall finde then also FC, afterward AB, being set downe upon 4. BC wilbe 3, then the triangle BAC wilbe 36, 52, 12; which taken from DAB, there will remaine DAC, & so the triangle DAC

shall haue 3 termes, ye shall then seeke DA, or AC, by which ye shall come to haue AB 13, 7969, & BC 10, 34743, then in the triangle CBG, ye shall finde BG, afterward DR, or CG being knowne ye shall haue DG 41, 54579, GH 11, 57496, and finally DH the fichant 58, 96636.

The 34. Figure.

Moreouer for this Vndecagone let the angle flanked be 88 degrees 38, 11, & the face to the curtaine as 2 to 3, and the gorge in the flanke as 8 to 5. & the fichant 60 rod, how many will all these lines make?

YE must first finde out the Quadrangle ABCD, whereof the angles are knowne, & sett downe AB, 8 parts; then BC wilbe 5, & DC being found, ye shall finde DF, FC, also BH, seing that DC is to BH as 2 to 3; hauing DK, and KH ye shall haue DH in parts, which make 60 rod, by which reason ye haue the lines required, AB, 14, 6, CB 9, 127, DG 42, 374, DC 23, 7363, BH 35, 604, & DP 76, 99675.

Vpon the 35. Figure.

ALB. GIRARD.

THis question is defectiue, in regard it hath but foure termes knovvne, DP 70 rod, & CB 10, the reason of DC to BG is as 10 to 9, and the name of the figure Undecagonall, vvhich are but 4 termes, he comes to the 〈◊〉〈◊〉▪ setting dovvne the face 20 rod, to see if there be not in it repugnanci••, as if the question vvere exceeding, so that he in this search committeth the faulth, vvhich is called petitio principij. VVhich is spoken not to defame the Authour, but to shevv, hovv this proceeded from hence, that in this time many thought that Geometrie hath attained to her highest degree; though vvee haue had but the A, B, C of it: till that some undertakeing the restauration thereof, haueing in part made it flourish againe, could neuerthelesse escape the blame of some envious & ignorant men in this divine science, as happens often, haue let passe the most difficult of the Analitica, vvhich ought to adorne those, that make profession thereof, vvho contraryvvise setting dovvne the limits of their pretensions from the beginning of their course; by emulating one an other, content them∣selues to grope together vvithout learning to goe forvvard vvith a solide, and a firme pace in a faire vvay.

The 36. Figure.

In this Dodecagone, let the face be 24, the flanke and the curtaine 36 rod, the angle flanked right, how many will the remayning lines make?

LEt DF be found, also FC, ye shall haue DK, & KH for to haue DH, afterward in the triangle DGR, hauing the angles & RG, ye shall finde DG. Finally ye shall finde DE in the triangle DEA, for to haue EF, or the gorge: Now seing that GCB is 60 degrees, then of necessity GC wilbe 24, as being asmuch againe as CB, then DG wilbe 48, and GR, or KH the halfe is 12, therefore DH wilbe 61, 648, DP, 77, 56944.

The 37. Figure.

In this present Dodecagone, let the angle flanked be right, the defences flanking, & the fichant 45, & 60 rod, the gorge in the flanke is as 4 to 3.

SEing that the angle DCF is 60 degrees. DH 45 wilbe the double of GR 22 ½ rod, DH, HK being knowne, DK wilbe 55, 621, DR 38, 971, then the second flanke 16, 65. Also DE wilbe found by the triangle DAE, & taken from DR, there will remaine AG the basis of the triangle ACG, the angle A thereof wilbe 36, 52, 11, because of the reason giuen, and G 60 degrees, then the sayd triangle ACG will make knowne CG 21, 4933, & also the face 23, 5067 (for DG is 45) the halfe of GC is CB 10, 7466, which multiplyed by 4/3, ye shall haue BA 14, 3289, DP 75, 979, BH 35, 2637.

The 38. Figure.

Let the curtaine be 36 rod, the line of defence flanking 45, the angle BAC 36, 45, & the flanked right. How many will the other parts of this Fortresse Dodecagonall come to?

THe triangle DAG, afterward ACG wilbe knowne, and also CBG, DCF, DKH, whereby DC wilbe found 23, 974, GR 22 ½ (the halfe of DG) 10, 5127, DH 61, 0593. DP, 77, 52516, AB 14, 733.

The 39. Figure.

In this Dodecagone the angle BAC makes 37 degrees, the flanked right, the face 24 rod, the fichant 62, how many makes then the other lines?

THe line DC 24 shalbe the double of CF 12; the angle BAC being giuen, the triangle DAC, wilbe knowne, therefore DA, AC, and the triangles DAE, ACB DGR, DHK wilbe knowne; and consequently the lines required, as AB, 14, 6179. BC, 11, 01525, BH 36, 7851; DP 78, 3547. DA 23, 82709; He failed herein to calcu∣late them a new together: but the reason of this was, that wee agree not, seing that in the construction, in stead of 37 for the angle BAC, (as he sett downe) he tooke 38.

All questions comming after this are defectiue euen to the discourse, which endeth the order of the same questions; but seing I know whither it tends, I will explane them, in adding thereunto the things, that were wanting passing ouer the figures 40, 41, and so we will beginn with the 42, as followeth.

The 42. Figure.

In a square Fortresse with foure Bulwarks, let the angle flanked be 60 degrees, the angle forming the flanke FID 40 degrees, the face 24 rod, hauing the reason to the curtaine as 4 to 5, the other lines are required.

Seeke the triangles DAL: AID, IFD, DFP, ye shall finde also PH, for FH wilbe 30: IO 48, 45532: AK 76, 6464, IF 9, 22766: FD, 7, 74298, IA 19, 73479, the fichant AH 54, 98265.

The 43 Figure.

Is a Pentagone, the angle flanked 69 degrees, the angle forming the flanke ••0 degrees, the curtaine to the face as 5 to 4, the face 24 rod; wee desire to know the other lengths.

Ye must doe as before, and then there wilbe no difference in the operation, Touching the number of the names of the figures in this 9 plate, ye shall finde them marked about the angle flanking exteriour, the reason of the face to the curtaine is marked on the point of the Bastions, the length aswell of the face, as of the curtaine next themselues, and must be proposed as the two former, hauing alwayes the angle forming the flanke of 40 degrees: the opening of the flanked angles, is according to the manner of the first table, to wit 15 degrees more then the demy-angle of the Polygone, sauing that in the figures 2, 43, & 2, 44, afterward an other time the 45, 46, he taketh for the angles flanked the ⅔ of the angle of the Polygone, according to the second table in the beginning placed before: Of these things ye shall finde 2 tables in the end of this booke, setting downe the faces to the curtaines, as 2 to 3, the faces 24 rod, which make the former-flankes 40 degrees, and the flanked according to the two manners abouesaid, where ye must knowe, that the lines of defence fichant are about 60, 61 rod. The said tables are both calculated 〈◊〉〈◊〉; because of the errours escaped in the other editions. Moreouer, after the Dodecagone ye shall haue ouer and aboue following a Fortification upon a right line, which is called upon a right curtaine. Now let us marke what the Authour saith.

Note 1.

NOw not to take so much paines in remembering the diverse proportions of the face to the curtaine, whereof the Fortresse quadrangular, & Pentagonall Fortresse is as 4 to 5, and the Exagonall as 3 to 4, it will not be amisse (seing that the line of de∣fence will beare it) to make also the said Fortresses of the same proportion, as the subsequent, to wit, in reason sesquialtere, as appeareth by the 3 figures of the 9 plate, quoted by the numbers 2, 42. 2, 43. 2, 44. where the lines of defence doe not much exceede 60 rod, which is as farre as a musket can well carry, & therefore one ought not to exceede this number; because that alwayes from the flanke reciprocally the entrance into the moate must be defended, which is often done, and most commo∣diously by the Muskettiers, because that Canon cannot so suddenly (by reason of the weight) be brought thither, many good occasions are neglected, for which cause they are preferred before Canon, and in regard a Musket will but carrie some 700 feete point-blanke, Bulwarkes ought to be made noe further one from another, for otherwise the line of defence would be too long, which should cause imperfection.

Note 2.

IF in stead of takeing the face 24 rod, for the line of the Polygone exteriour take 80 rod, and the rest according to the former proportion, the parts wilbe brought very

neere together, as appeareth by the precedēt Examples, according to the figures of the 10 plate, marked with the numbers 51. 52. 53. 54. 55. 56. 57. 58, 59, whereof we haue made no calculation; because they are easely done by the former supputations.

Note 3.

THat in all our designes and calculations, we are resolued to use but one measure onely, which is in the 25 plate of our Geometrie upon the rule of the instrument marked number 1, which is a foote deuided into 12 equall parts, whereof 12 foote makes a Rheneland rodd in the territoire of Leyden.

Designes & plat formes of diverse Fortifications.

WE might according to the former supputations giue diuerse constructions of the platformes of Fortifications; but considering the great diversity of rules, (which often causes confusion, and that time will not permit us) wee will giue but one simple and generall rule for them, which is this following.

The 9 Plate, and the 42, 43, 44, 45, 46, 47, 48, 49, 50 Figures.

FIrst, if the face be giuen (suppose it to be 24 rod) then ye must doe as followeth: Ye shall draw a right line infinite AC from the point A, the angle CAB shalbe made according to the forme of the Polygone, upon which A, B shalbe sett downe the 24 rod abouesayd, as from A to D, and by D is drawne the line, which formeth the angle of the Center of the Bulwarke 40 degrees, then must ye sett downe upon A, C and A, B, the reason of the curtaine to the face that by the same ye may draw the line infinite A, E, makeing from the point D, a parallell with A, C, cutting the said AE, at G, by which point is drawne the line, that formeth the other face G, K, in such sort that the angle G, K, A, wilbe equall to the angle D, A, K, and so the figure wilbe drawne, according to the said proportion.

The 10. Plate, & the 51, 52, 53, 54, 55, 56, 57, 58, 59, Figures.

SEcondly, if the line from angle to angle be giuen AB (for example 80 rod) ye shall then proportion out the face to the curtaine sesquialtera, as 2 to 3. Then in con∣formitie of the table of angles set downe before, wilbe made the angle CAB, accor∣ding to its forme, to wit, in a square of 15 degrees, in a Pentagone of 89 ½ degrees, in the Exagone 22 ½ degrees, and so consequently of the rest, then for the forming of the abouesaid proportion of the face to the curtaine, which is (as we haue said) as 2 to 3. aswell in a square Fortresse as in a Pentagone, and others following ye shall take upon a scale of a reasonable greatnesse 2. which shalbe sett from A to C, and 3 from A to D, suppose that it be at C & D, then from the point D wilbe made the arch G of the distance of 2, (to wit CA) & from C of the distance of D, A, 3, wilbe made the arches, which cutt betweene each other at G, by which a right line being drawne to A, where the same cutting the line FB, as here at F ye haue the face of the Bulwarke, and by that the curtaine wilbe knowne. To knowe also the line of the gorge, and the flanke, and all the other parts, ye must make the angle forming the gorge, and the flanke of 40 degrees, as is here the angle KIE, cutting the diameter P, A, at I, from which point I a right line parallel being drawne from A, B, as is I, N, ye shall drawe the line perpendicular to be E, K, which shalbe the flanke, and K, I, the line of the gorge of this Fortresse, and then ye haue the thing required.

In like manner ye shall finde, and marke out the other Fortresses, according to their seuerall formes, takeing heede that as here ye take 2 & 3, ye must in the others also take 2 & 3, to place them aswell upon A, B, as upon A, C, and the rest being the same; as is in the former construction it wilbe needelesse, to giue yow here any further particular instruction for them.

The 3, 50. Figure, and the 9. Plate.

ALBERT GIRARD.

THirdly, VVhen upon the line of the Polygone interiour AB, ye desire to construe, or ex∣plane a Fortresse, hauing 5 termes, to vvit, the name of the figure, the angle of the bastion M, The angle forming the flanke GBH, and the reason of the face DG, to the curtaine HL, conceiuing therein that AB is giuen.

This figure is in stead of the 2, 50, being vvorth nothing. I am so constrained to make a nevv figure, and a nevv explanation. Let then NBA be equall to the demy-angle of the Poly∣gone, and let the angle forming the flanke be HBG, LAK, so ye shall haue the intersection or cutting betvveene C, then DBF the halfe equall to the demy-angle flanked, and finding the point F, so that AB, to BF is the reason of the curtaine to the face, and hauing dravvne CF, cutting NB at D, and makeing DG a paralell vvith FB, meeting vvith BC, at G, ye haue finally GH the perpendicular vvith BA, and doing the like also on the other side, ye shall haue all the parts required; the demonstration is manifest, seing that C is the common point of the like figures FBA, DGK: then as FB, to BA: so DG the face, vvilbe to GK, or HL.

The 11. Plate, and 60, 61, 62, 63, 64, 65, 66, ••7, ••••, Figures.

IF there be a question to proportion out the face to the curtaine, and the gorge to the flanke, when the line EF is giuen, ye shall doe as the figures in appearance de∣monstrate to the eye, namely, ye shall first proportion out the face to the curtaine by the former rules, and make it soe that the curtaine of a square Fortresse be 300 foote, and the face HF, 250 foote, hauing reason as 5 to 6, according to which ye shall finde the point G, which is the angle of the shoulder. To finde then afterward the gorge & the flanke, ye shall make the couert line GH, which must be diuided into 4 equall parts (in foure, because ye would haue the reason of the gorge in the flanke as 4 to 3) whereof ye must place three of them from H to I, and then drawe I, G, cutting through the semy-Diameter of the Polygone, which is here squared into A, and then the line AB being drawne it wilbe parallell with EF, and so ye haue the side of the Polygone interiour, upon which the 2 perpendiculars being drawne GC, HD, ye haue the 2 flankes, and consequently the essentiall parts of a square fortification knowne: ye must also obserue the same in the other figures following.

If the line EF, which is the side of the Polygone exteriour (or the distance of the angles of the Bulwarks) be not giuen, but onely the line AB, the side of the Polygone interiour, ye shall seeke out in the tables a Fortresse of such a forme, whereof the face to the curtaine is as 5 to 6, and shall worke it by the rule of proportion, or else let it be made after the former manner, vvhich is much more easie, then all the figures of the 11. Plate.

If such a side of a Polygone interiour, giueth such a side of a Polygone exteriour, what will such a side interiour giue? & ye shall haue the thing required; to wit, the Polygone

exteriour, whereby one may easely know the other parts of the Fortresse, when the line AB of the Polygone interiour is giuen. But if in the tables the proportion be not found, make first upon the line AB the triangle ARB, whereof the halfe of the basis hath the like reason to the perpendicular, as ye will haue proportioned out the gorge of the flanke, demonstrated by this exemple as 4 to 3, which wilbe done in setting downe upon the said line AB foure equall parts (4 because ye desire to haue proportioned out the gorge to the flanke, as 4 to 3) to wit from A to D, raising out of the point D a perpendicular, and put 3 of the said parts from D to I, then doing the like on the other side, by this meanes wilbe formed the triangle ARB, and from the points A & B, the angles flanking interiour wilbe made according to the forme of the Polygone, (which is a Pentagone) conformable to the table of angles described heretofore, as appeareth by the letters SBA and TAB, which cutt the lines infinite A, L. L, B at the points S, T, and the line S, & T being drawne, the angles T, S, B, and S, T, A wilbe equall to the angles S, B, A, and T, A, B, as appeareth by the 28 proposi∣tion of the first part of Euclide.

According to which the reason of the face being putt to the curtaine, upon T, S, and T, A, namely, the face upon TA, and the curtaine upon T, S; ye haue the like distances, and from the points I & X, the arches which cuts betweene them in a certaine place, by which intersection the line T, V, shalbe drawne cutting the line S, B, at V, then the right line V, L. being drawne passing through the center L, & cut∣ting A, R at G, ye shall haue all the parts of this Fortresse: for hauing sett downe the distance R, G, from R to H, from the same point G is drawne a paralell to S, B, as E, G, cutting the Diagonall line L, A, at E, the like ye shall finde for F, and consequently all the other parts of the said Fortresse Pentagonall. The same must ye also understand of the other figures following, from the figure 60, to the figure 68. Also ye may finde the point V, in setting downe upon T, A, 2 equall parts, such as ye require, and from the extremity or utmost end, hauing made it a parallell to S, T, and upon the same, 3 of those parts, and drawne from the extremity the line TV, the said point will con∣sequently be knowne thereby.

NOte that the proportion giuen here, betweene the face and the curtaine is not so much to relye fast upon it, as to show that the generall rule sett downe here before in the 9 and 10 plates, take place also in all other reasons, which may be pro∣pounded. For otherwise wee are of the opinion; that the former figures would rather be accepted then these here, because that the reason of the curtaine to the face is (as wee haue said aboue, sesquialtera as well in square Fortresse, as of Pentagonall, and others following. Which for the facility, & simplicity together with the goodnesse of them ought to be preferred before the figures 60, 61, 62, 63, 64, 65, 66, 67, and 68, of the 11 Plate aboue said. Whereof the face to the curtaine is (as the exemples shew) of diuerse reasons; in such sort, that hence forward one ought to resolue, that the reason of the curtaine to the face ought in all formes of Fortification to be sesqui∣altera, and the face 24 rod, each rod containing 12 foote, the length whereof is sett downe in the 25 plate of our Geometrie, noted with the character or the figure 1, & is diuided into 12 ynches, which rod is used in the Fortifications of the United Pro∣vinces, to the end that the line of defence may not exceede much aboue 60 rod, which is about as farre as a Musket can well beare, the gorge to the flanke may be made by the rule giuen in the 11 plate, according to the reason giuen, but forasmuch as it is more convenient to make the angle forming the flanke GAC (plate the 11) of 40 degrees which giueth the reason almost as 6 to 7, in my opinion one ought to rest thereupon, and so ye shall haue a generall rule for all manner of Fortifications,

whither they be Quadrialtera, Pentagonall, Hexagonall, or the others following, as ye may perfectly understand by that which shalbe said hereafter. In the meane while note that I giue here the reason of the gorge to the flanke as 7 to 6, which ought to be understood in Fortresses without Casemates, but if it were my intention to make some of them, I would then alter somewhat of the said proportion.

The manner how to describe succinctly the designes or Plots of some regular Fortifications.

The 12. Plate, and the 69. Figure.

ALthough one may (by that which we haue taught) sufficiently now understand the order, and Method requisite to be held in all manner of Fortifications: yet I haue thought it good to trace out here some from among the rest, to make the louers of this Art to conceiue the better my intention, and will begin with a regular Pentagone, whereof the angle of the center maketh 72 degrees, and the angle of the Polygone 108 degrees, and seing the angle flanked according to the table before mentioned, maketh 69 degrees: It wilbe easie to finde the angle C, A, D, which is alwayes equall to the angle flanking interiour, and shall finde the same to be 19 ½ de∣grees, setting then your graduate Instrument upon 19 ½ degrees, it makes the angle CAD, and drawing first the covert line infinite AB, takeing upon the scale 24 parts, or rod: and sett then from A to C, drawing from the point C a perpendicular upon A, B, as is CD afterward hauing sett on DE the length of the curtaine, which is here onely 34 rod (because the false brayes makes the lines of defence too great) & then the distance AD, from B to E, raysing from the E the 〈◊〉〈◊〉cular EF, being equall to D, C, and FB drawne out, the two faces wilbe knowne; for to knowe the center of the Bulwarke, shalbe made of 2 lines GA, & GB the Demy-diameter of the Polygone, then your instrument being sett upon 40 degrees, ye shall finde the angle HKA, cutting the said demy-diameter at the same point H: from which point the line HI being drawne; ye shall haue the side of the Polygone interiour upon which out of the points C, and F, shalbe drawne the perpendiculars CL & FM, which will forme the flanks, and the gorges of the said Fortresse Pentagonall, in the inside of the side of the Polygone interiour shalbe drawne a parallell of 5 ½ rod for the thicknesse of the rampart, as N, O N, R, and RS, and ye must draw for the parapett of the rampart a parallell of 20 feete (also in the inside of the said side of the Polygone HI) and on the outside thereof ye shall drawe a parallell of 20 foote for the falsebray as LX, and yet more outward 20 other foote for its parapet, & so all the other parts wilbe traced out, which are within the moate; for seing that there is here noe Case∣mates, the falsebray is carried not onely about the curtaines; but also about the flankes, and the faces of the Bulwarks, in such sort, that the falsebray serueth as a Casemats to the said flankes, the entrance or sallies ought in my opinion to be made in the midst of the curtaines, as covertly as possibly may be; the Bulwarkes are made either massiue, or hollow from Earth; at this present wee haue drawne them out as being hollow from earth, & so the superficies, or plaine in the midst of the bulwarke N, R, S, H. is of the same heigth, as the rest of the enclosure of the Pentagone is, the moate shalbe 10 rod broad, which is traced out as followeth: In the point A, or the angle of the Bulwarke, shalbe made an arch of the distance of 10 rod, & in the point V make an other arch of somewhat a lesser extention then the former about 10 or 12 foote, then drawe upon the back thereof a covert line infinite; the like must be done on the otherside of the Bulwarke, and so from place to place, with covert lines being

drawne as abouesaid, the intersections of them will bring forth the angles of the said moate, aswell on the interiour as on the exteriour parts, ye must make upon the brinke of the moate exteriour a parallell of 20 or 24 foote; for the bredth of the covert way, and besides somewhat more towards the exteriour part, ye shall make a parallell of 50 foote for the thicknesse of the parapet thereof, which shall goe downe sloaping being onely raised towards the covert way, or the Coridor of 6 foote, and diminishing litle by litle to the furthest part of the 50 foote, & so ye shall haue finished the Icnographia, or ground-draught of this Fortresse Pentagonall, the streets betweene the ramparts and the housen ought to be 30 foote, that with the more convenience they may be entrenched in the time of neede, and that the souldiers may stand there in battle-ray. The streets ought to be made of 24 or 30 foote, and for the market place every side shalbe made 12 rod, and of the same forme, as the Polygone is, which is fortified, being a Pentagone, in which the towne-house is to be made, & other publick buildings, & thus we haue finished the forme thereof.

The 70. Figure.

THe Profile or draught shalbe made in this māner: first a privie line is drawne from the utmost part thereof, then ye shall beginne to lay out the breath of the ram∣parts, moates and wayes, & as wee haue sett downe in our Icnographia for the bredth of the rampart 66 foote, which make 5 ½ rod, we will take upon the skale joyned to the Profile 66 foote, for to place them upon the said covert line as from A to B: then as the way of the rounds, which we otherwise terme the falsebray, is counted 20 foote, which one may take upon the said skale, & remoue them upon the said covert line, as from B to I, and the same 20 foote from I to M, for the bredth of the parapet, then ye shall make an edge MN of 6 foote, which is called in flemish the Teen, or the toe of the foote of the parapet, the better to prevent the falling of the same into the moate: from whence ye must take the bredth of the moate, which shalbe 120 foote; for so much makes the 10 rod as from N to Q, from Q to R, ye must drawe out 20 or 24 foote for the covert way, and from R to T 50 foote for the parapet; and so ye haue all the dimensions required. For their heigths aboue AB ye shall make a parallell of 14 foote, which shalbe the height of the rampart, and underneath a parallell of 10 foote for the depth of the moate, the talud or slooping interiour of the rampart A, V, ought to be 14 foote, and the talud exteriour XB, must be the halfe of its heigth, which is 7 foote, and the trapeze, or unequall sides giuen A, D, C, B, being the profile of the rampart: for its parapet ye shall take 20 foote from C to E, and from the point E draw a perpendicular of 6 foote, which shalbe the height of the said parapet, and upon the said perpendicular EG shalbe layd out 4 foote to drawe a parallell to DC, cutting the line B, H, at H, which shalbe the heigth of the said parapet in this place, and for the superiour part ye shall draw the line GH. Your foote banke shalbe made as followeth: from E is drawne the line FE of 3 foote, and 1 ½ foote 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 heigth then 2 foote, and shall haue the talud 1 foote, which shalbe the talud interiour of the said parapet. The parapet of the falsebray IM shalbe made as is aboue said; to wit, from the point I, is raised the perpendicular IK of 6 foote, which shalbe the height of the said parapet, and from I towards B draw 3 foote for the footebanke, being a foote and a halfe high, and so drawe out the line K2, that the bredth of the upper part of the footebanke be not (as wee haue said) aboue 2 foote, in such sort that the Talud interiour (after ye haue drawne out the said line K 2) of the said parapet shalbe but a foote: for from the point I, the foote

banke is 3 foote broad, but in the point 3 it is found to be but 2 foote broad. For the moate first draw out NO, which is the brinke interiour thereof, hauing the talud, as much as is the depth of the moate, to wit, of 10 foote, likewise the line Q, P, is taken, which is the brinke exteriour of the said moate, the parapet of the covert way Q, P, is made by rising from the point R, the perpēdicular line RS of 6 foote, takeing from the said point R towards the left hand 3 foote for the bredth of the footbanke, and to make it a foote and a halfe high; To giue the more firmenesse to the said parapet, there shalbe made from the upper part of the point S an oblique line upon the said footebanke, which is (as is before said) three foote broad; in such sorte, that the said line declining a foote upon the said footebanke, maketh the said footebanke to be noe broader then 2 foote. Finally a right line is drawne from S to T, which makes the slooping of the said parapet.

Note.

WE haue made the rampare no higher then 14 foote, upon the upper part whereof is made a parapet of 6 foot high, all upon a plaine field, which is a convenient heith for a place, that hath noe hils or heigths about it, but in case there be any hils, that doe commaund the said place, it wilbe then necessarie to raise the rampart somewhat higher, then is here described, answerable to the heigth of the said hill, which otherwise would commaund the Fortresse, & for the better blinding of such places, sometimes one makes upon the upper part of the rampart an other parapet about 7 foote broad, or there abouts & 6 foote high, which Parapet serueth so much the better to blinde the said place, and to discouver the fields about it, but when an Ennemy hath approched neere unto such places, then the said parapet can doe them no further seruice, so that necessity will compell them to take it away for their more safegard, and to lye under the said parapet of 20 foote broad: for Canon shott comming every time to pierce the said parapet being but 7 footebroade; they cannot safely lodge behinde it, without extreame danger; and therefore is necessary to be razed, to make use of the other, which is able to resist the force of Canon.

Wee haue not made here in this 79 figure any Ravelins or halfe moones, which are loose peeces on the further side of a moate, neither any Catts, or high plat∣formes to plant ordinance upon, which are usually made in divers places within Forts, and upon Bulwarks; because we intend to speake of them hereafter. In the meane while it wilbe good to note, that the said Cats, Ravelins, and other workes, may be applyed to this present Fortification, but in this place is omitted for the avoyding of prolixity, because we are minded to treate of them briefly in an other place.

The description of the designe or the plate forme of a Fortresse Hexagonall.

The 13. Plate, and 71. Figure.

LEt there be giuen a Fortresse Hexagonall to be fortified, whereof the face AC ma∣keth 24 rod, and the angle flanked 80 degrees, according to which the angle flanking interiour will make 20 degrees, and the exteriour 140 degrees, and let the curtaine be 30 rod, which giueth the reason of the face to the curtaine, as 3 to 4. To doe this, ye shall draw the covert line infinite, A, B, & by the helpe of an instrument

graduate, the other angle CAD of 20 degrees (of 20, because that the angle flan∣king interiour, which is alwayes equall to it maketh here 20 degrees) by meanes of the line indefinite AC, upon which ye make the length of the face 24 rod, as from A to C, from which point C, the perpendicular CD being drawne upon the line A, B, shalbe placed from D, the length of the curtaine, which is here 32 rod as from D to E, finally the distance AD from E to B, and the perpendicular EF the distance of CD as from E to F, drawing the line FB, ye haue the other face, so that all the parts of the reason giuen are described; and for to finde the curtaine, ye must make first the angles GAB, & GBA of 60 degr. seing the whole angle of the Polygone maketh 120 degr. by the lines AG, GB, which cutting through on an other at G, is the center of the Polygone, and seing the gorge, when as ye would make casemates in them or in a bulwarke where Cats or mounts are raised, they had neede of a larger extent, then otherwise; we suppose that it were requisite, to make Casemates in them, to that end we make the angle HKA, which otherwise might be but 40 de∣grees onely of 35, according to which the gorge in the flanke wilbe almost as 4 to 3, or somewhat more by reason of the line H, K, cutting the line Diagonall AG at H, from which point H the line HN being drawne, parallell to AB, ye shall haue the Polygone interiour, upon which the lines CL, and FM, being drawne in length the lines DC to L, and EF to M; in so doing all the essentiall parts of the said Fortresse, wilbe described; Now to continew the same draught or platforme in every place ye shall make from the center G a privie circle from the distance GB, and shalbe sett upon the privie circumference the distance AB, which being the 6 part of the said figure Hexagonall, the said circumference will contayne justly still fiue parts, which finally will come to end at A. In the like manner is the privie circle made out of the same center G, and from the distance G, N, upon the circumference thereof shalbe set the line of the Polygone interiour HN, afterward ye must onely coppie out the rest. Moreouer, the parapet wilbet 20 foote, which shalbe made in the inside of ACL. MFB, and as the way of the rounds, or the falsebray, is on the outside of the body of the Fortresse, we will make parallells towards the moate of 20 foote broad, & then 20 foote more for the parapet thereof, on the outside of the parapet, before ye come to the moate, is made an edge, or a toe of 6 or 8 foote, to keepe the said para∣pet from falling into the moate, which is made of 140, or 150 foote or thereabouts, as necessitie, and the bottome requires; for it must be fitted according to this con∣sideration, because the ground lying lowe, it will not beare much digging before ye come to water, and therefore in this place one is forced to make the moate to gett earth enough for the rampart, but when the ground lies high, then ye may digg very deepe before ye come to the water, & so by this meanes ye may haue asmuch earth, as possibly ye can well use, & seing without it cannot be but hurtfull, it wilbe better not to make the moate so broad, that ye may receiue noe hindrance or hurt by the store of Earth that it affordeth; which may be cast upon the outside: For to beleeue that the ramparts being raised higher then the dimension giuen in the former figure to wit of 14; or at least 15 foote, that they were better as some haue mainteyned, yet experience hath showne many times the contrarie, for an ennemie hauing once ap∣proched to the brinke of the moate, the bredth of the parapet hinders one from makeing any defence upon it, which notwithstanding is most necessarie: because the neerer an ennemie comes with his approches towards us, we ought the more to defend our selues, and this being committed by our default, it is too manifest that one ought to take heede of it, and to remedie this errour by the former way: namely in makeing the rampart of the heigth abouesaid. Some are of the opinion by a sim∣ple cōmaund that one ought to heigthen the ramparts aboue the said 14 or 15 foote,

and not the Bulwarks, for which they haue some reason; for indeede the Bulwarkes make the greatest defence, and when an ennemy is become master of the said bul∣warke, there were a meanes to cōmaund the said bulwarke more absolutely, so that an ennemy could not hinder the use of them, if he raised not his works aboue the said rampart, to hinder the defences of the besieged, which they might make with the said ramparts so raised by a simple commaund, which in some manner might hinder the intents & approches of the besiegers, the more easie would the entren∣chements be made, and seing the more the bulwarks are raised, the more one ought to use diligence about the entrenchment; hence would follow this difficulty, that the comming to the rampart or the bulwarke, would not be so easie, as otherwise, neither could one make any great defence from such ramparts by reason of their too much heigth, in such sort that one bulwarke ought to defend an other, & in such a case the second flanke would be of noe use after an ennemy is entred into the moate, at which time it is then most needfull to giue the greatest resistance that may be, seing that when they are gotten ouer the moate, and taken in the foot of the bul∣warke, then the courage of the besiegers begins to encrease, & the besieged to faile them, for it is too apparant, that many times resistance failes in such and the like accidents.

Betweene two bulwarkes are commonly made ravelins or halfe moones, which are Quadrangular figures, euery face containing 12, 15, or sometimes 20 rod, which beginn at the brinke of the moate: so that its angle interiour, or the tenaille O, lies just upon the brinke and angle of the moate, the reason why their faces P, Q, and Q, R, are made of a greater distance, is that the deffence may be the better: for otherwise their could be noe defence made but from one part of the face of the bulwarke, which otherwise would not be sufficient to hinder the assaults of an enne∣mie, which ravelins or loose peeces li•• so open and are raised aboue the plaine field some foure foote, that one may the easier discouer the said field, and so also to pre∣vent the inconveniences which might happen, when one would giue fire from them upon the besiegers, (if the said ravelin had not bene raised) which would haue annoyed those which were under the covert way. Upon the said Ravelin being so raised, is made a parapet of 20 foote broad, and 6 foote high, which wilbe able to resist Canon, the moate whereof ought to be made 50, or 60 foote, and as deepe, as possibly may be about the said moate, ye must make also a cover way of some 20 foote broade, as hath bene noted before; Afterward ye must make a parapet of 50 foote broade & six foote high, ending at the end of the 50 foote towards the plaine field; as the profile doth show in the 72 figure, and to the intent ye may the better understand my meaning, we haue made the said Hexagone in perspectiue, with the falsebrayes, Ravelins, covert wayes, and their parapets apart, as appeareth by the second 71 figure, but ye must note, that the said Ravelins must be in respect of Ram∣parts, which haue a double height, to see in them the more perfection, also a double height, to make them show the better: for otherwise they would seeme too little for the reason abouesaid against our intention.

The 72. Figure.

THe Profile shalbe made as in the 12 Plate of the former figure 70, by drawing a privie line infinite, and takeing upon it all the dimensions, as first the bredth of the ramparts, the parapets, the falsebrayes, its parapet, the bredth of the moate, the covert way, and the other parts of the said profile; and seing wee haue here ordained Ravelins, wee will make this profile from the midst of the rampart, passing also

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through the midst of the Raveline, that ye may the better understand our meaning. According to which ye shall take the bredth of the Rampart 68 foote from A to C, from C to D 20 foote, from D to E 20 foote, and from E to F 6 foote, the bredth of the moate shalbe 150 foote, as from F to G, which ought to be broader, but seing the distance falls out too litle, it shalbe made but of 150 foote, the Raveline ought to haue in this place 180 foote, but seing the ground faile us, wee haue made the dimen∣sion thereof 180 foote, but this hindrance would not permit us, to make it of that competent greatnesse as it ought, which I speake to giue noe stop to the Reader, which Ravelin is raised 4 foote aboue the plaine field from 6 to T, upon which a parapet is made, whereof the basis shalbe 20 foote broad, and 6 foote high, to wit its bredth from Q to R, and the height from Q to S, the footebankes are made of the same bredth & heigth, & therefore I will make here no mention of them, neither of the taluds or sloopings of Ramparts, Parapets, & moates; aswell the interiour, as the exteriour parts are made according to the nature of the soile; for by how much the ground is leane & sandie, by so much ye ought to giue it the talud, & by that meanes ye shall hinder the falling downe of the workes, for which reason often times ye must giue on the exteriour side, being a sandie ground, as much talude as heigth, and if the said Ravelin be also raised 4 foote aboue the plaine field, me thinks it will not be amisse, to raise the rampart thereof, which is 15 foote, some 6 foote higher to com∣maund the better the said Raueline, & leaue the Bulwarks of the height of 15 foote, that all inconveniences may be prevented: on the outside of the Raveline is made the edge HI of 6 foote, and a moate of 50 foote, then the covert way 20 foote, & the parapet LN 60 foote, with a footebanke of the ordinary bredth of 3 foote in that manner, as the 72 figure demonstrateth.

The description of the plote of a Fortresse Heptagonall.

The 14. Plate, and 73. Figure.

LEt there be giuen to be fortified a Heptagone, whereof the side of the Polygone ex∣teriour maketh 63 rod, and the angle flanked 80 degrees. To doe this, ye shall first divide the said side AB into 7 equall parts, as ye see by the points 1. 2. 3. 4. 5. 6. 7. and from the points of the 5 and 2 parts the perpendiculars D, N, and E, O shalbe drawne being equall to A, D, and E, B, to wit, either of them 2 seuenth parts of the line A, B, then ye shall draw from the extremities, or the utmost parts of these per∣pendiculars, the privie line infinite H, I. which shalbe the side of the Polygone inte∣riour, and by consequence N, O the curtaine. The angles C, A, B, and C, B, A, shalbe made by the helpe of a graduate-instrument, as our compasse is, or an other; either of 64 ½ degrees; in such sort that the intersection of the lines A, C. C, B, which is C, shalbe the center of the said Fortresse Heptagonall. Afterward I, B, 8, of the 40 degr. which is the halfe of the angle flanked, that ye may haue the whole angle of 80 degr. according to the Hypotheses, and where the said lines cutt the said perpendiculars D, N, and E, O, in the points 2. & 8, ye shall haue the faces A 2. & B 8, the flanks N. 2. & O. 8, and also the gorges H, N. & O, I, the Parapets, Ramparts, Falsebrayes & for the other parts of this Fortresse, ye shall doe the like as hath ben taught in the for∣mer figure Hexagonall, the faces in this present Raveline shalbe made of 15 rod, the moate of 10 rod, and the moate to the Ravelin of 15 foote, then about it shalbe made a covert way of 20 foote, & a parapet of 60 foote broad descending (as we haue said before) sloopingly. According to this forme the flanke wilbe then made of a rea∣sonable greatnesse, but the gorge so much the lesser, which may be made so, seing

ye intend not to make any flankes in the Casemates, but in case ye would, ye must then make the flanke the lesser, and consequently the gorge the greater, to the end ye may haue space enough for to make within them the said Casemates. The like is to be understood when ye meane to make a Catt upon the Bulwarke, for the gorge being so litle (besides other inconveniences) it is impossible to make there such a bodie, in reserving sufficient space for the flanks, which me thinks in such places, that haue such narrow gorges, it were better to make the said Cats upon the Cur∣taine, yet so that they be noe hindrance to the Rounds going along the Ramparts, for which reason they ought to be made more inward, and so that they may lie as neere the rampart as possibly may be, the better to discouver, and commaund the fields about them.

The 74. 75. Figures.

THe figure 74 is the profile of the Rampart, Falsebray, its Parapet, moate, covert way, and its parapet, whereof the length, bredth, heigth and depth are marked out in the said figure 74, which is a section of the Bulwarck, as the figure 75, the section of the midst of the Rampart, and because all the profile cannot commo∣diously be drawne out, we haue represented but a part of the moate, the forme of the Raveline, its heigth, and the forme of its parapet, the bredth and the depth of its side, with their taluds or sloopings, the covert way with its parapet & footebanke, as ye may exactly note by the figures hereunto annexed.

The 2. of the 14. Plate.

ANd for the better facilitating of that 〈◊〉〈◊〉 we haue spoken of this present figure 73. we haue added hereūto the platforme of the Fortresse of Coverden in Frizland, whereby ye may the better understand our intention, being accounted the master peece, and the most regular, and royall Fort in the Low-Countries, hauing a False∣braye, Rauelins, or half moones, covertwayes, with which wee will put an end to the description of our regular Fortresses.

A succint description of some other works in the said Heptagone, which are of an other manner of makeing then the former.

The 15. Plate, & 76. Figure.

LEt the side of the Polygone interiour be B, C, upon which ye would haue a part of a Fortresse Heptagonall made, whereof the face to the curtaine is as 3 to 4. and the gorge to the flanke as 13 to 9. To doe this, ye must doe as we haue taught yow in the 11. Plate, or in the 9. Plate, and Figure 3. 50, where ye shall finde the proportion requi∣site: on the outside of the same face, shalbe made the parallells each of 20 foote, aswell for the parapet, as for the falsebray, the like is to be done in the flanks, and curtaines, and on the inside a parallell of 20 foote, with an other parallell of 72 foote for the bredth of the Rampart in the basis, in the same manner as this 76 figure sheweth, on the outside ye shall make a parallell of 10 rod for the bredth of the moate, and upon the brinke of it to the tenaille, ye shall lay out the Ravelins, as we haue said in the for∣mer figure 73, in the angles of the Bulwarks ye shall make also Ravelins, such as ye see here marked with I, K, L, M, which shalbe defended by the Ravelins E, F, G, H, & the better to defend the said ravelins, which are in the angles of the Bulwarks, ye

shall make the other works N, O, P, & Q, which are called Horne-works, which are made in such a manner, that the moates of these Horneworks comes to answere to the Falsebray, which is in the flanks of the Bulwarks: so that their bredth S, T, ma∣keth about 32 rod, of which distance the two demy-Bulwarks V, X, are made (whereof the face, and the curtaine are of a like greatnesse) according to the rule prescribed in the 10. Plate, the flanks wilbe found, makeing from the angle of the shoulder, two lines perpendicular upon the curtaine, which is directly opposite to it. euen as the figures 77, 78 demonstrate. The utmost angles of the said Horneworks ought not to be further from the curtaine of the Fortresse, then a Musket can carry point-blanke, which is 60 rod or there abouts, and if the Rauelins EF, GH, are not made in the tenailles, then ye may cut the said Hornworke, as is here marked out in the 78 figure: the bredth of the moate shalbe 24 or 25 foote: The Rampart here is noe otherwise then the parapet of the same bredth.

The depth of the moate may be made 6 foote in case the ground lies low; but otherwise the deeper it is made it wilbe so much the better, the bredth will not hinder it, though it were made but 36 foote; for according to the same bredth & depth ye may enlarge the Rampart, aswell in height as in bredth, upon which afterward ye may make a parapet as great as it can well beare: but when haste requireth, I would make noe Rampart, but onely a Parapet of 24 foote thick, as wee haue made here being 6 foote high, and a footbanke of 3 or 4 foote.

The figure 79 marketh out a Profile, which is a section or cutt drawne through the midst of a curtaine passing through the midst of the Ravelines G, and traversing through the Horneworke P, and the Rampart and the moate. The true dimen∣sions thereof are clearely expressed in the said Profile, by the helpe of the scale; joy∣ned to it: as also by the meanes of the Alphabeticall letters, shewing the feete to be 12 ynches, whereof the length is 〈…〉〈…〉 in the 25 Plate of our Geometrie.

The benefit of such workes are well knowne, when they are made in places of ad∣vantage for deffence, or where noe men, victualls, and amunition of warre are wan∣ting, as also where the ground is of a reasonable largenesse, to wit at the least 32 rod, that the faces of the said horneworkes may be about 12 rod, which is the least length that one can giue to such workes, against the attempt, which may be giuen by an Army: & as a great assault, cannot be resisted with a few men, but up a small roome: so is it manifest that the greatnesse & largenesse of such a place, must be answerable. Therefore in my opinion, it is a thing repugnāt to the rule of Fortification, to make such Horneworkes in the angles of Bulwarkes, where they are so straightned. For Demy-bulwarkes cannot (by reason of their smallnesse) be vvell mainteyned and defended, and on the other side, their tvvo vvings cannot be defended from the maine Fortresse, but vvith great disadvantage, as vvee intend to discusse: thereof more at large hereafter in the plateforme of Gulick.

How Citadels, or Castles may be joyned to Townes, or Townes to Castles,

The 16. Plate, and the 81. 82. Figures.

FOr to build a Castle (or a Citadell) to a Towne, either to help to defend it the better, or to curbe it: ye must finde out first the most advantagious place: in case there be a river, then ye shall build your Castle upon the side of it, where it may best

commaund, with all the advantage that possibly may be, also takeing heede that it be built in such a place, that the Castle may receiue noe disadvantage thereby, but that the Rampart of the Towne where the letters A, G are, be the weakest part; to the end that by this meanes, ye may frustrate the desire of the inhabitans of the said towne, from opposing themselues against the said Castle: but if ye feare noe such inconvenience, ye may make the places A, G, as strong as the other: saving that the Ramparts in the said angles be noe thicker in the angles A, G (which are angles of imperfect Bulwarkes) then in other places, and not massiue or full of earth, as the Bulwarkes B, & C, D, are. Let there be a towne then contayning 8 angles, whereof 7 of them are marked out with these Alphabeticall letters A, B, C, D, E, F, & G, and that you are desirous to make a Pentagonall Citadell. For the doing whereof ye must first make a mapp of the said towne Octogonall: in case it be not made, according to the methode giuen heretofore in our Geometrie, which is here marked out by these letters A, B, C, D, E, F, & G, with the Falsebrays, Parapets, Moates, Covert way, & its Parapet. Then upon a paper of the same scale, whereof the mapp of the said towne is made, being traced out in a Pentagone with all the outworkes, such as the figure marked H, I, K, L, M, demonstrats, according to the method giuen yow before in the 10 and 11 Plates, then the same figure Pentagonall must be cutt perfectly, and so fitted to the towne, remouing it so longe till yow haue at length found out the most ad∣vantagious place to make there your Castle Pentagone, takeing heede that the tenailles N and O, fall out right, to answere the curtaines P, O, and Q, N, to the end that the entrance into the towne may be the more commodious, & that the towne in the places N, R, S, T, O, be not discouered without, whereof one ought to haue a singular care: for there where the said angles of the tenaille, be not well joyned to the Ramparts, those places will alwayes be badly defended in the said joynings, as ye may obserue in the Castle of Gulick, which platforme followes next. It is true; that when ye make these angles in the curtaines, that the parts N, R, S, T, O, take up a great deale of the towne, which may be prevented in setting the Castle more outward, and makeing the said curtaines meete neerer the faces of the Bulwarks of the Castle. But the side of the towne N, R, S, T, O, will in noe wise be guarded, as the Platforme of the Castle of Gulick sheweth: so that necessity will require, for the preventing of such an inconveniencie, to make some Ravelins, or some other works without the said place, as the commodiousnesse of the ground may afforde, which would make mee resolue rather to approue of the forme of the 80 figure, then of the other, were it not for some cause more urgent, then is here alledged, which reason would moue mee to change my opinion.

If the towne or Cittie A, B, C, D, E, F, G, were greater, then were it vnnecessary to take so much care for the place, which the Castle taketh up: for then ye might make the curtaines P, O, & Q, N, longer, to wit some 50 rod, that the flanks V, X, and Y, Z, might not come so neere the Bulwarks E, M: from whence one may easely discouer the Falsebray of the Castle, for the preventing of which inconvenience, it wilbe good to raise the Bulwarkes of the Castle, which looks into the towne, somewhat higher aboue the Ramparts thereof, but the towne being of a small circuit (as wee haue said) one shalbe constrained to use for it the same figures as are demonstrated by the figures 80. & 81.

If one would built a towne to a Castle, first ye must obserue diligently the site of the place, and if it may be made regular, to take care, how many Inhabitants this place will containe, the nature and quality of them, and whether the towne may in processe of time be enlarged: which ye shalbe able to judge, when ye see the countrie that lies about it, what wayes, avenues and passages there are to come to it, as also to

consider the goodnesse of the aire, and of the soile of all which things (as Vitru∣vius saith) one ought to haue a most speciall care. According to all which one ought to gouerne themselues: for to enclosse a great peece of ground, where there is none, or little appearance of Inhabitants, this were rather to ruinate, then to edifie, to which takeing heede then, ye must resolue of the Circuit, or Capa∣city, which ye would giue to the said towne, and suppose ye would make the forme thereof Octogonall, and regular: ye must trace out then an Octogonall Fortresse regu∣lar, according to the former method, with covert wayes, and what ye require more upon the same scale, as you made your Castle Pentagonall, then ye must cutt the said forme Octogonall, and joyne it to the said Castle, removing it so long, till yow haue found out & chosen out (with all circumspection requisite) the most convenientest place, and staying there ye may trace out the said towne, as the figures 80 and 81 de∣monstrate, joyned to the Castle. But as we haue said, it will loose much of its capa∣city, because the curtaines of the towne will come to meete 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 tye your selfe so precisely to this forme Octogonall, ye may make the towne much better by the meanes which wee will now speake of. Let first be drawne the lines D, D, and A, A. &c. right perpendiculars upon the midst of the curtaines 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 curtaine D, E; in which ye haue the center of the circle, which makes the circle of all the figure, and suppose ye desire to make a forme regular of nine Bulwarks, whereof the angles flanked, are distant one from an other 80 rod, from the utmost end Q, shalbe 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 center 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 interiour flanking, according to their forme, that ye may haue 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 skilfull, and diligent man M. Iohn Errard of Barleduke, Ingenier to the King of France hath obserued well, and to know how much the said angle flanking will make, ye must substract the halfe of the angle from the Bulwarke, or flanked from the halfe of the angle of the Polygone, the rest shalbe the angle flanking interiour. But if the angle of the Polygone be lesser then 150 degrees, which is the angle of a Dodecagone, ye shall then governe your selfe according to the table giuen before, takeing heede that the angle of the Polygone agreeth most with the angle to be fortified, and in its columne is found the angle flanking interiour, necessary to the said angle, by which ye may come to the angle of the tenaille, & seing in this exemple wee haue set downe the distāce frō angle to angle, as wel of the Pentag. as of the towne 80 rod, wee wil set downe upon the lines, which make the tenailles 2 equall parts, & from the said 2 parts in the line which formeth the angle flanking, a parallell line being drawne to that of 80 rod, & placed upō the 3 equall parts, a privie line infinite, must be drawne through the said 3 point, & through the angle, and where the same comes to cutt the other side of the tenaille in the point O, ye shall haue the shoulder, through which the line A, B, being drawne, makeing the angle forming the flanke 40 degrees, ye shall haue the center of the Bulwarks justas wee haue taught heretofore in the 10 & 11 plates. If the Castle had bin made with foure angles, the construction would haue ben the

same, and if the lines D, D, Q, and A, A, &c. had bene somewhat more then 60 rod, the forme would haue bene better, for though we haue 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 Bulwarkes Q, &c. advance themselues towards the angles Q, &c. from whence the said curtaines Q, E, E. & B, B &c. maye be sufficiently de∣fended, when they doe not exceede aboue 70 rod, the moates, covert wayes, false∣brayes and other workes, must be made in that forme as we haue sett them downe before in the 16. plate.

How to describe the order which is obserued in makeing of a platforme, and of some other appendances.

THe draught being made upon a paper, ere ye make a platforme of it, ye ought first to ouercast, aswell the lines, as the body of the Fortresse, in that manner as we haue showne yow before in our Geometrie, that ye may know pertinently all the particular dimensions, & so committ noe errour, in makeing a plateforme thereof: Before ye lett it out, according to our Method giuen in the conclusiō of the practize of Geometrie, which being done before the Inginier begins the worke, he shall de∣clare the particular conditions, which he would haue obserued in the makeing and finishing of this worke, as the time when they are to beginn, how long it wilbe before it be finished, how many men he must ordinarily haue to worke, what instruments they must use for the driving in of piles, and the laying of the foundation, how many foote, and what edge they are to giue to the wall, or the parapet of the falsebray, the thicknesse 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 goodnesse or weaknesse 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 bredth and depth of the moate, & its slooping, and generally and specially all things touching the makeing of such a Castle, whereof he hath made his platforme. And some dayes before they come to the place, or that the forme be drawne out, they shall sett up billets in the next adjoyning townes to giue notice to the Worke masters, that upon such and such a day, one is resolued to hire and let out such & such works, to him that will take them on upon the least price: and before the day pre∣fixed the said Inginier shall come thither himself with some one of the Deputies, or some other that hath the managing of that worke, to draw a draught of the said Fortresse, and to prepare all things necessarie against that day assigned. Then tho day being come, all the Worke-masters & Undertakers shall come together, either upon the place where the worke is to be made, or else in some other place where the conditions and couenants are read, according as they will haue the worke hastned: this done, they shall demaund, who will take it on for the least money, then one of the Workemasters will say for so much, an other for so much lesser, which is done till noe man will take it on at a lesser rate, then underneath they signe unto the articles of the conditions, that I N, N, haue undertaken the worke upon such and such conditions, for the summe of so much money: sometimes ye shall haue two or three Undertakers joyne together and undertake the whole worke, all of them signing to the articles, as also the Commissioners, and Inginier must doe on their part: then they beginn the worke, and commonly the Undertakers are bound by the said articles and Contracts, to deliuer all the materialls necessarie for the said worke, which they receiue from the Master of the Magazin there or else, where

giving him a note under their hands for the restitution of them back againe, when the worke is finished, and to make good whatsoeuer they either breake or loose.

Then the Workemaster placeth his men in order, as he knowes the quality of the worke requireth best, some to digg, and cast up the Earth, others to fill Karrs, and Wheele-barrowes, others to levell the ground which is pricked out; for in the be∣ginning it wilbe necessary to carry the Earth away, with a Horse and a Karre, which is cast up on the outside of the moate upon the basis of the Rampart, & not with Wheele-barrowes as they must doe at the last, when the worke growes high, and that ye are come to the depth of the moate, for then ye cannot so well use Horse and Karrs; by reason of the difficulties knowne to all men, namely, that the treading of the horses and the Karrs spoyle the groundworke, and cannot be laid so well, as when men brings the Earth up in Wheele-barrowes upō away ascending with plan∣cies, as hath bene found by experience often times by the undertakers of such workes.

Of the Foundations.

2. The 17. Plate.

WHen the foundation is bad, it wilbe your best course to driue in piles, along the foote of the Rampart, just where the said Rampart beginneth, some piles 25, or 30 foote, or else yong Alders-trees, about foure or fiue yeares growth along the worke about 18 or 20 foote long, closse one to an other, and the closser the better, and though this be a great charge, and a long worke; yet this is the best foundation ye can choose: But as we haue said oftentimes, seing it hapneth that one cannot stay so long, ye must then take bundles of brush, and binde them up in fagots, & lay them crossewise, driving them into the Earth with stakes to fasten them together, that they may lie firme, and if ye lay the foundation in a ditch or a quagmire, ye must make use of such fagots bound and tyed up together with cords, & if it be in a river, then ye must binde brickbats, or stones betweene two bundles, to make them sinke to the botome: ye may make them as long, or as short as ye please, and so necessity requires, till ye haue filled up the water with these sinkers, upon which yow must lay your foundation, then ye must also driue in some piles about 18 or 20 foote longe, (according to the depth) to make them lie the more firme and solid together. Now these bundles are made in the forme of Saucidges, as here followeth.

First, ye driue into the ground stakes of the heigth & bredth as ye desire to make these Saucidges, either great or small, the one a foote high, the other a foote and a halfe, and sometimes two foote, then ye lay betweene these stakes bulrushes, boughs or small bundles of brush bound fast together, and in the midst of them ye must lay brick-bats, when ye sinke them in a river, but of Earth if ye will lay them as a foun∣dation in a moate all along the said worke, then ye binde the said bundles with stones, or Earth in them, fastning them with withs in the same manner, as ye binde up a fagot, or a bavin, and in this forme 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 perceiue by the letter B in the 2. 17. Plate.

Then ye sinke these downe to the bottome, that ye may afterward build your Fortresse upon them, according to the former platforme, whereof a Plate is made before, which groundworks among the Ingeniers 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 then the rest of the building it self,

yea in diuerse places of these Countries, the Inhabitants are driuen to keepe their grounds from the violence of the Sea and Rivers, by such and such like meanes and remedies, to their great and unspeakeable charges, for one would haue thought it had ben impossible for them so many yeares to doe it, since the times that these Provinces were united, had not the superintendants, & Dike∣graues used great care and diligence in continuing the same preservation for the good of the inhabitants, and common weale, whereof wee will speake more at large hereafter.

When the worke is raised aboue the water, and that the Earth in the moate di∣minisheth, the Worck-master shall then beginn to lay his edge, or border upon the line drawne out for the Rampart. Now the line being drawne out upon the ground, as ye see in the line C. by the 2. of the 17. Plate, being fiue or six ynches broad, and as deepe, which is done by the Ingenier himself to take away all excuse from the Work-masters, that he may likewise carrie a right line, aswell on the inside as on the outside of the Rampart, as also for the moate, and the other parts of the said For∣tresse, represented in the plat-forme: Besides if the condition of the undertaker containes, 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 betweene the said rowes of sodds, one may lay a litle earth to settle them the better together to the bodie of the Rampart. Moreouer, they are so laid together that the rowes aboue must lie euen in the midst of the joynture of the rowes beneath, makeing them settle together according as the condition beares, and as the articles are which the undertaker hath signed unto. Now for his better ease, he must make use of a triangular Instru∣ment, such a one as the letter F demonstrats, being 〈◊〉〈◊〉 ••••de A, B, about 2, or 3 foote more or lesse, according to the quality of the worke, and the rest so consequently: For as ye desire to haue the Talud, either great or little, so ye must enlarge the line A, G, or lessen it: If it be needefull to lay bundles of brush to strengthen your works of the Fortresse, ye must lay them so that the ends touch the former sodds, to wit the one row half a foote aboue the other, and betweene each of them halfe a foote of Earth, continuing the worke so untill it be finished: upon the top of the Earth of the Rampart, ye shall raise the parapet with such a Talud and bredth, as the condi∣tions imports, in the same manner as before, raising it with sodds, as we haue said. If ye make your Fortresse in a fatt and a clayie soile, then ye may (in stead of sodds) make a cruste of 3, or 4 foote, or more, stamping it well with a stamper made to that end, and of such an inclination as your agreement beares, in which crust ye may sett a certaine kinde of grasse, called in Flemish Queeckcruyt, in Latin Gramen, & in English Meddovv grasse, which roote is of such a nature, that it will spread itfelfe ouer the whole Rampart, and binde the Earth fast together: so that it will make the said crust of a long durance, and almost perpetuall: on the outside of the said brust, ye may sowe Oats or Hay-seed: they use also to sett upon the Bulwarke, roots of seuen-leaued-grasse, which is likewise very good: but the leaues doe not spread themselues ouer the exteriour superficies, or this crust, as the abouesaid grasse doth: for this reason some yeares since his Excellencie hath found it good, to repaire all the Forti∣fications of such a crust without sodds; because experience hath taught us, that the said sodds bindes not the rest of the Earth, so well as the said crust doth, which being moist by that meanes joyneth so much the better the rest of the wall or Ram∣part 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 moate, whereof good store was layd on the outside of the moate, to that purpose: and the said undertaker ought to that end make an exact calcu∣lation of the quantity of Earth, which is necessary for that use. But seing oftentimes the undertakers are not skillfull in casting up such accounts: It were good that the Ingenier himself should make an exact calculation, not onely of this part, but also of the whole worke, which may be done in that manner, as wee haue showne in our Geometrie, or else in this easie manner following.

The second 17. Plate, and Letters G, H, R.

LEt the 12 part of the Fortresse be Dodecangular G, and AB the face 24 rod; the flanke AD 12, and the curtaine 36. and for the Rampart H. Let the talud inte∣riour be 15 foote, and the height asmuch, the talud exteriour 10 foote, the bredth upon the Rampart 50 foote, which makes that the superficies of the Profile of the said Rampart wilbe 933 1/3. Moreouer, for the Profile of the parapet, which is K 99 1/4. the heigth EC 6 foote, HF, 4. AB for the footbanke a foote and a halfe; AC 3. BD 2. CG 20. HG 2 1/2. the solidity of the Rampart & the parapet is required.

Of the Rampart.

THe Icnographie sheweth, that CM 10 foote is the talud exteriour, and HN 15 foote the interiour, 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 BAD shalbe cutt by halues from the lines HB, EA, KD. Also the angle of the Polygone being 150, & the flanked right; the angle flanking interiour wilbe 30 degrees, as it is marked, and the shoulder 120 degrees, which makes that the triangles right-angles are as followes: those which haue their angles pointed of 45 degrees, shalbe Isosceles, or like angles, BM, MC upon the basis each of them 10 foote, the lines C 3. 3 G, each 50 foote, BX & XH, each 75 foote, GN, NH 15 foote, and the like in the triangles hauing their Hypotenouses at KD. Moreouer 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 BA 24 rod, or 288 foote, there will remaine HF 169, 69875. Likewise AD 144 foote, PD, or RV, wilbe 138, 2265, and by adding to it FV. FR will come to 148, 2265, the line A 12 is equall to A 5. taken from AD, there will remaine 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 haue EK 175, 69875.

The demy-curtaine 13 D, makes 216, then 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 haue 646, 453 for CFR 9. the circuit exteriour.

Ye shall finde also GL 193, 359, & LI, 169, 35904: & 10 I, 276. whereof the summe for GLI 10, wilbe 638, 718 the circuit interiour, which being added to the exteriour abouenamed, will amount to 1285, 171, its halfe is 642, 5855, which multiplyed by G3, the distance of the parallells 50, will come to 32129, 275 for the superficies, betweene the abouesaid circuits, and then multiply by 15 the heights, ye shall haue the solidity of the wall, without the taluds 481939, 125. CRF 9. 10. ILGC, which had bene too short as here followeth.

The Authour hauing made this aboue with so many lines, if from the midst of LR, he had drawne a perpendicular line upon the face, and cutt af from it 20, 2071 its double substracted from the summe of the abouesaid perpendicular, and from BA, AD, D 13, which is 683, the remainder wilbe 642, 5856, which multiplyed by 15, it wilbe the same as aboue.

Of the Taluds or slooping vvalls or Ramparts.

THe Taluds without the Pyramids, which are in the angles, shalbe calculated thus, their Profile is a triangle, which multiplyed by the length, or else as Demy-paralleli∣pedes, the Basis multiplyed by the heigth, afterward the halfe taken, add to it then CR 272, 2265, RV 138, 2265, with D 13, which is 216, the summe 626, 453 multiplyed by MC, 10, the superficiall basis wilbe 6264, 53; likewise HE 169, 69875, EY 160, 69875 & I 10, which is 276, makeing together 606, 3975, which being multiplyed by IY 15, will make 9095, 9625 for the other superficiall basis, together with the former will make 15360, 4925 (because they are of a like heigth) by which heigth of 15 being mul∣tiplyed, then the halfe will come to 115203, 69375 for the taluds, aswell interiour, as exteriour besides the Pyramides.

Of Pyramids, and double Pyramids of the Taluds.

THere are 3 Pyramides in the angles convexe B, A, K, and three double Pyramids in the angles concave, H, E, D, which I say because Marolois makeing no distinction of them, taketh two things for one: As for exemple, the body upon VFXD is 1000 foot cubicall, and he 〈…〉〈…〉 but 500, and so of the two others.

The Superficies of the triangle BMC, is 50 from ROAP, is the right-angle RO, OA 57, 735, and YIZK 225, and then the double superficies of GNH 225 (according to that as we haue said before touching the Authours fault) ESLT its double Superfi∣cies is 259, 8057, & the double superficies VFXC, of 200, whereof the summe is 1071, 5425, which multiplyed by the tierce of the common heights, ye shall haue for the abouesaid fixe 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 summe wilbe 602230, 5312, for the soliditie of the Rampart, besides the parapet.

Those which will not take the paines to haue it done soe exactly, but are conten∣ted to come somewhat neere unto it, shall add unto it BAD 13, which is 648, with HEK 11, being 636, 396, and the halfe of the summe multiplyed by the profile of the Rampart 937 1/2 ye shall haue 602060, which comes the neerest unto 602230, found out aboue (being onely 170 too litle) for Marolois found out but 600518, with much adoe, hauing 1712 too litle.

Of a Parapet.

WE haue before sett downe the dimensions of a Parapet as ye may see by the letter K, newly cutt, with its Rampart H, in this plate, & ye must not thinke it strange, that the Parapet is made upon a greater measure, then that of the Rampart, the reason is, because there are noe more dissections, or cutting of, and that wee haue not roome to make that which we would within the some Plate.

Now if ye desire to calculate the Profile as the Authour hath done, the whole as the brevitie of the Rampart showes aboue mentioned, its situation is termined on the outside by CRF 9. and on the inside by a parallell line, betweene the abouesaid & GLI 10, distant from the other 20 foote, and for the avoyding of the confusion which the plurality of lines might cause, we will take for the said paralell, the line GLI 10, and therefore 3 G, or 7 S, will make 20, or else the line LT; ye shall finde by the triangles, that TR wilbe 5, 369. add hereunto C 3, which is 20, and the summe of CR knowne aboue, taken from 272, 2265, there will remaine for GL 246, 8675, then from RF 148, 2265, substracting the length equall to TR, 5, 359, there will remaine 142, 8675, afterward add thereunto 8 V, or its equall 20, ye shall haue 162, 8675 for LI; also F 9, being 226, then I 10 wilbe 246; then the summe of CRF 9, (added to GLI 10) and the halfe taken, ye shall haue 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 DOE 2 1/4, the trapeze, (or unequally side) IFGC 75; and BOCA 4 1/2, the summe is 99 1/4, for the Superficies of the Profile, which multiplyed by the middle length aboue said, ye shall haue 64621, 0795 for the solidity of the parapet, which is much too much; for it ought to be as I haue calculated it with its dissections 63976, 9475, to which add 602230, 5312 from the Rampart ye shall haue 666207, 478, for the whole solidity of the Rampart and Parapet, whereof the double 1332414, 956, wilbe the twelfth part of the Circuit, which multiply by 12, ye shall haue the totall summe 15988979, 472, and diuided by 144, ye shall haue 111034, 68 plyntides, or as Marolois calls them Chevilles, peggs or pynnes, for the whole circuit of the Dodecagone.

Seing then there are 9252, 88 pinnes in the 12 part of this Fortresse Dodecagonall, the Ingenier may conjecture, for how much he may let out the same worke, which is about 16, 20, 25 or 30 pinnes, according to the situation of the place, the ground and other considerations, and after that the undertakers shall offer to make it ac∣cording to the calculation, made thereof, they may take on the said worke upon condition, that they shall finish it in such a time, & with so many men as is specified in the contract, and soe at that instant they beginn the worke, & finish it according to their bargaines and conditions, as the greatnesse of the worke requires, and the time limited, for which also the undertaker is bound to haue as many worke men upon it, as shall be found needfull for the finishing of the said worke: & for the better assurance thereof, it is requisite to know how many pinnes a man may spit, and lay upon a wheele-barrow 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 pinnes in a day, but ordinarily men suppose that one can doe three pinnes euery day, without takeing too much paines, according to which, one may easely make his calculation how many spad∣men he must use to make up that worke in such a time prefixed, for men which shall doe nothing else, but spit and digg up the Earth & the others to carrie it away upon wheele-barrowes to the Rampart, for ye must haue as many men to carry up the Earth in the wheel-barrowes, as that the spiters may be continually kept in worke without any cessation or stay, to the intent the Work-master may finish his worke within the time limited.

Note.

WE say here, that a man may in case of necessity spit in a day 4 1/2 pinnes, and commonly three in a day with leasure, but it is to be understood, this must be in a good and fast ground, which wilbe beare spiting without falling or breaking; but when the Earth is leane and sandie, it wilbe impossible for a man to doe so much, & so by consequence one cannot know any certaine rule.

The 1. Maxime.

ACcording to which plotts and calculations all other places aswell regular as irre∣gular may be fortified, and hold this for a maxime, that the place so fortified, as we haue taught aboue, wilbe better then the others.

The 2. Maxime.

THat the line of defence which comes from the angle of the flanke to the angle of the Bulwarke, called the line of deffence fichant ought not to be much longer then 60 rod, because it is as farre as a Musket can well carry, but if ye will defend the face with Canon, then it may be a 100, or 120 foote longe.

The 3. Maxime.

THat lesser the angle of the tenaille is, it is so much the better, because the faces looke more directly one upon another, therefore ye may see that in the plots of the Dodecagone aboue said: ye must make the angle of the Bulwarke onely of 90 de∣grees, that the angle of the tenaille may be shutt the closser.

The 4. Maxime.

BY how much the flanke and the gorge are greater and spacious, by so much they are the fitter for defence.

The 5. Maxime.

THe further the defence is made in the curtaine, it is the better, because ye may bring the more men to giue fire upon it, for the defence of the faces. Therefore in the plots of the Dodecagone abouesaid, ye must see that the angle of the Bulwarke be made of 90 degrees, to the end ye may giue fire the larger.

The 6. Maxime.

That the angle flanked ought to be noe lesse then 60, and not aboue 90 degrees.

The 7. Maxime.

THat the angle of the Polygone, which ye intend to fortify, ought not to be lesse then 90 degrees, but the greater it is, it wilbe so much the better.

The 8. Maxime.

THat in all Fortresses wee will make the face to be 24 rod, and the curtaine 36, according to which the lines of the Polygones exteriour will make about 60 rod, and the lines of the Polygones interiour 60 rod. In a Heptagone and the Polygones un∣derneath, the Polygones exteriour wilbe more then 80 rod, and aboue lesse then 80. The Polygones interiour which are beneath the said Heptagone, are lesse then 60 rod, and aboue more then 60, the one encreasing, the other decreasing, untill that the

angles of the Polygones come to be 180 degrees, and then ye haue the Polygones exte∣riour, and interiour equall, and each of them 70 rod, the line of defence fichant in the whole about 60 rod, & the angle forming the flanke of 40 degrees.

The 9 Maxime.

THat the angle flanking interiour ought to be noe lesse then 15 degrees, but the greater the better they are.

The 10. Maxime.

THat the angle of the flanke be alwayes right, the angle of the shoulder must be at the least 105 degrees, and the greater the better.

The 11. Maxime.

IF it be needfull to fortifie any Polygone, either regular or irregular, whereof the sides are lesse then is here spoken of, then ye must proportion out the gorges, flankes and faces, according to the forme of the angle, as ye shall se in a table follo∣wing.

Herewith wee will put an end to our first part of Fortification, and will come now to the fortifying of places irregular.

FINIS.