Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

CHAP. XIII. Of FORTIFICATION.

THE Utility of this Mathematical Art called Fortification, or Military Archi∣tecture, is so well known, that it needs not my commendation, and therefore to speak any thing thereto, were but to light a Candle before the Sun.

In the handling of this part of the Mathema∣ticks, I shall be as brief as possible, yet as plain as can be desired: In the prosecution of which, I shall use this Method. As First, I shall give you the most principal Definitions or Terms be∣longing to this Art. Secondly, I shall prescribe the most conducing Maxims or Rules herein observed. Thirdly, I shall shew how to deline∣ate the Ground-line of any Fortification, accor∣ding to the several Proportions, used by the best and most experienc'd Inginiers of Italy, France, Holland and England; Fourthly, I shall describe the Construction of the chief and principal Out∣works now in use; and Lastly, lay down some general Maxims or Rules, by most Modern Au∣thors observed in Irregular Fortifications.

SECT. I. Of the Definitions of the Lines, and Angles, belonging to the Principal Ground work of any Regular Fortification.

• 1. THE Exterior or outward Line, which* 1.1 boundeth the Rampart, at the Foot next the Ditch, is the principal and only Line to be regarded 〈…〉〈…〉 all Regular, or Irregular For∣tifications, being the Basis on which all the o∣ther Lines, and parts of the Fortification doth depend.
• 2. The Exterior Polygon, is the outward side of any Regular Figure, as in the Hexagon (which Figure I shall make use of through this Tract) the side AA, is the Exterior Polygon.* 1.2
• 3. The Interior Polygon, is the inward side of any Regular Figure, as in the Hexagon is noted by any of the sides between P and P.
• 4. The Bastion or Bulwork, is that great work of any Fort, that advanceth its self towards the Campaigne, and here are six all marked with B, the lines which terminate them, are two Gor∣ges, two Flanks, and two Faces.* 1.3
• 5. The Demi-Gorge or Gorge-line PC, is half the Entrance into the Bastion, and terminates the point C, whereby the Flank shall be raised.
• 6 The Flank is another Out-line of the Ba∣stion as CF, which terminateth the Curtain, and Face.

• 7. The Face is the utmost line of the Bastion, as FA, two lines thereof doth form the Angle of the Bastion A, or the Flanked Angle.
• 8. The line forming the Flank FF, is a prick∣ed line, made use of by the Dutch Inginiers, and others.
• 9. The Capital is AP, part of the line com∣ing* 1.4 from the Center ☉, terminated at the point of the Bastion A.
• 10. The Curtain is that part of the Interior Polygon CC, which lieth betwixt the two Basti∣ons B, and B.
• 11. The line of Defence is AC, passing from A, the point of the Ba∣stion, to C the Angle of* 1.5 the Flank, and Curtain, and ought never to ex∣ceed 800 English Feet* 1.6.
• 12. The line Stringent, is the line coming from the point of the Bastion A, and prolonged on the Face AF, to the Curtain D, which shew∣eth that DC, the part of the Curtain, (by some called the second Flank will scour the Face.
• 13. The Diameter of the Interior Polygon, is the line ☉P, coming from the Center thereof ☉.
• 14. The shortest line from the Center unto the Curtain, is ☉ m. These are the Definitions of the principal lines, appertaining to the Ground∣work of any Regular Fortification, the Angles followeth.
• 15. The Angle of the Center of the Polygon* 1.7 is P☉P
• 16. The Angle of the Polygon PPP, is always the Complement of the Angle at the Center, or remainer unto 180 Degrees.

• 17. The Angle of the Triangle PPO is always the one half of the Angle of the Polygon PPP.
• 18. The Angle of the Bastion, or the Flanked Angle FAF, is exposed unto the Batteries of the* 1.8 Besiegers, and formed by the two Faces, FA, and FA, which ought never to be less than 60, nor much above 100 Degrees.
• 19. The Angle of the Espaule, or Shoulder, is formed by the Face, and Flank, as AFC.
• 20. The Angle of the Flank CCF, is formed by the Curtain, and the Flank, and is most com∣monly a Right Angle, but by some later Ingi∣niers, is made Obtuse, or more than a Right Angle, or 90 Degrees.
• 21. The Angle made by the two lines Sitchant, At A is called the Angle of the Tenaile.
• 22. The Angle forming the Flank, is CPF, which Angle is made use of by most of the Dutch Inginiers.* 1.9

SECT. II. Of General Maxims or Rules observed in Fortifications.

• 1. THat all the parts of the Place, be of Cannon Proof flanked, i. e. defended from another place, which place is no farther

• distant than the reach of a Musket-shoot, from the place to be Flanked or defended.* 1.10
• 2. That in all the Place, there may be no part of the Wall, or outside of the Rampire, that is not seen from the top to the bottome of the Mote, or Ditch.
• 3. That the Bastions are large, and full of Earth, and not empty; the bigger they are, they are the more to be esteemed, there being the more room to intrench, in case of necessity: whose Gorge let be at least 35 fadoms, and their Flank at least 18 fadoms.
• 4. That the Angle of the Bastion, or Flanked Angle, be not much above 90, nor much less than 60 Degrees, for in the former it would lie too very Obtuse, and open, at the Point; and in the latter it would be too slender, and so easily to be battered down, by the Ene∣mies Cannon.
• 5. That the Angle of the Flank may be some∣what Obtuse; neither is there any more virtue in a Right-angle, than in any other, for the defence of the Fort.
• 6. That the length of an extended Curtain be not above 135 Fadoms, nor the single above 80 Fadoms, nor be less than 40 Fadoms, to be well defended from two Flanks.

• 7. That the Rampire be so wide, that so a Parapet of Earth Cannon proof may be erected thereon, and a Teraplane left, full wide for the Ordnance to be recoiled.
• 8. That the Mote or Ditch be at least 20 Fadoms broad, and as deep as possible. Now dry Motes in great Cities are to be preferred before others, that are full of Water, to facili∣tate the Sallies, the relief, and retreat of the Besieged; and in small Fortifications the Motes full of Water are the most Esteemable, because in such Sallies are not necessary, and Surprises are very much to be feared.
• 9. that the Parts that are most remote from the Center, be commanded by those which are nearest to it.
• 10. That the Defence of a Face is much stronger, when the Angle made by the Face, and Exterior Polygon is a great Angle; this Maxim is so very essential, that it will try the goodness of any Fortification whatsoever: Thus I have described the 10 chiefest Maxims, ne∣cessary for good Fortifications.

SECT. III. Of the Construction and making of the prin∣cipal Ground-line of a Fort, according to the most Modern ways, used by the Ita∣lian, Dutch, French, or English Ingi∣niers.

I. Of the Italian Fortifications.

GEnnaro Maria, Mathematician to the Ca∣tholick King, wrote at Florence, his Ele∣ments of Military Architecture entituled, Breve Trattato delle Moderne Fortificazioni. This Ita∣lian Author was a very Learned and Skilfull Mathematician, and famous in his Nation. In his said Book Printed 1665, he makes the Interior Polygon 800, and not less than 600 Feet, his De∣mi-Gorge, he makes ⅛ of it, and so for the Flank of the Quadrangle. But for the Pentagon, and all Figures above, he makes the Flanks 1/10 part of the Gorge more, and he placeth his Flank at Right Angles with the Curtain.

Supposing his Interior Polygon 1000 parts, his Gorges will be 125, and in the Quadrangle the Flanks will be 125, but of the Pentagon, and all above, 138 parts. For the Faces, he makes them to fall on the third part of the Curtain, unless in the Square, which he allows no second Flank.

PROP. I. To fortifie a Hexagon according to this Author's Proportion.

First describe the Hexagon PPP, &c, then* 1.11 divide the Interior Polygon PP, into 1000 equal parts, take 125 for the Gorges, and set it from P to C. Then on C raise a Perpendicular, make it equal to 138 parts, for your Flanks CF, then draw the Face AF, falling on the third part of the Curtain CC, at D, and so do on every Ba∣stion, untill the work is compleated.

II. Of the French Fortifications.

Monsieur De la Mont, in his Fortifications Offensive, and Defensive, printed 1671: And Monsieur Manesson Mallet in his late work, in∣tituled Travaux de Mars, printed 1672, assign∣eth these proportions for the laying down the Ground-line of a Fort.

Both these Authors make the Interior Polygon 768 English Feet, which they divide into 5 parts, and taking one for the Gorge 153½ Feet. Both divides it into 3 parts, and takes one for the Capital, that is 256 Feet.

Now our first Author De la Mont, makes the* 1.12 Flank to stand at Right-angles and takes 115½ Feet for it, which is ¼ of the Curtain, and so draws the Bastions, in all save the Quadrangle, and Pentagon, which he makes to have no second Flank.

PROP. II. To fortifie a Hexagon according to the Proportion of De la Mont.

First describe your Hexagon P, P, P, &c. Now supposing your Interior Polygon PP, 1000 parts, the Capital 333, the Gorge 200, and the Flank 150 parts, take out of your Triangular Scale Fig.* 1.13 75, (which is made for the more speedy de∣lineation according to this proportion of De la Mont) PA for the Capital, and prick it off from PA, on all the Bastions. Then take PC, and prick off all the Gorges from P to C. Then take FC and prick it off at Right Angles, from C to F. Lastly draw all the Faces AF, AF, &c. so is your Hexagon compleat, as required.

PROP. III. To fortifie a Hexagon according to Manesson Mallet's Proportion.

Now our Authour Monsieur Manesson Mallet, in his Works intituled Travaux de Mars, de∣viates from our former Authour, only in this: that as De la Mont did place his Flanks at Right Angles, he places them at 98 Degrees with the Curtains, and leaves no second Flank in all his Fortifications.

Therefore having described the Polygon PP,* 1.14 &c. divide PP into 1000 parts, prick off the Capitals PA 333, and the Gorges PC 200, then lay off the Flanks CF, 150 parts, at an Angle

of 98 deg. with the Curtain CC (by prop 5. §. 1. chap. 4.) and draw all the Faces, AF, AF, &c. Falling on C the point of the Flank and Curtain, so shall your Hexagon be fortified as was required.

III. Of the Dutch Fortifications.

The Emperour Ferdinand III. hath learnedly* 1.15 altered the Method of Fritach, Dogen, Gold∣man, and Faulhaberus, all which were Dutch Inginiers, and wrote large Volumes on this Subject; in his Works intituled Amussis Ferdi∣nandea, published 1654; by turning their way of working by Angles, into working by Sides.

Thus he setteth down a Catholick way of delineating the Sides, or Lines of any Fort by his 60 prop. thus, the Interior Polygon to be 66, the Capital 24, the Gorge 15, and the Flank 12. Or in making the Interior Polygon 22, the Capital 8, the Gorge 5, and the Flank 4. Or yet making the Interior Polygon 1000, the Ca∣pital 363, the Gorge 227, and the Flank 181, this is an Epitome of all the Dutch Fortifications, and is general excepting for the Square, which must have no second Flank.* 1.16

PROP. IV. To fortifie a Hexagon according to the Emperour's Proportion.

First describe the Polygon PPP, &c. divide P P, &c. into 22 parts, take 8 for the Capitals

PA, which prick off all round from P to A, take 5 for the Gorges; which prick off all round from C to P, then take 4 for the Flank CF,* 1.17 which prick off all round at Right-angles from C to F, lastly draw the Faces AF, AF, AF, &c. So is the Hexagon compleated as was required.

IV. Of the English Fortifications.

His late Majesty of Great Britain Carolus II. of ever blessed Memory, hath much facilitated the Method of Count Pagan, who in his Fortifications printed at Paris 1645, did place the Flanks at Right-angles with the Line of De∣fence, and he works by the Exterior Polygon. Now His Majesty places the Flank, at Right-angles with the line of defence of the Interior Polygon, and works after another manner: Count Pagan makes the proportion of the Grand Royal Fort. Supposing the Exterior Polygon to be 1000 parts, will make the Perpendicular MT to be 150, and* 1.18 the Complement of the line of Defence TC to be 185, which may serve for a general proportion be the length what it will, only in a Square the proportions must thus be altered in the Grand Royal Fort, the Perpendicular MT must be 162, in the Mean R 144, and in the Petty Royal 126, the Complement of the Line of Defence for the Grand Royal Fort is 228, and for the Mean Royal Fort 198, also for the Petty Royal Fort 198.

PROP. V. How to fortifie a Hexagon according to Count Pagan's Proportion.

To delineate this Work draw a line, about the middle whereof as at M, set off MA, the half of the Exterior Polygon 500 parts, which makes the Exterior Polygon 1000, then on M (by prop. 1. §. 1. chap. 4.) raise the Perpendi∣cular* 1.19 Mm, which make Mt, MT Equal to 150, then draw ATC, and ATC, then take 185, and place it from T to C, and to C, and draw CC for the Curtain, then on the points C raise Perpendiculars CF, to the line of defence CA, for the Flanks, so have you also the Faces FA. Then on the Points A set off half the Angle of the Figure, to wit 60° (as you see in the Ta∣ble in page 38) and draw the lines OA and O A, so shall O be the Center of the Figure, and PC the Gorge, and AP the Capitals: then finish* 1.20 each Bastion at your own discretion, and the Work is finished as required.

PROP. VI. To fortifie a Hexagon according to the way prescri∣bed by His Majesty Carolus II.

His late Majesty C. II. hath much facilitated this Work, as will appear in this following Ex∣ample, by making the line of Defence, stand at Right-angles with the Flank of the Interior Polygon, by this Table, which supposes the Inte∣rior* 1.21 Polygon to be 1000. Then

Polygons	4	5	6	7	8	9	10	Strait∣lines.
Capital	398	437	367	333	312	300	291	233
Gorge-line	155	196	203	242	252	260	263	300

Now describe the Hexagon PP, &c. Then divide the Interior Poligon PP, into 1000 parts, take 367 and prick off all the Capitals PA; Then take 203 and prick off all the Gorges from P to C. Now draw the lines of defence AC and* 1.22 AC, &c. Then at C, set the Flanks at Right Angles with the line of Defence AC, so shall FC be the Flank, and FA the Faces, then finish every Bastion, and your Hexagon is for∣tified as was required.

☞ Thus have I set down the several Ways and Rules, for laying the fundamental Ground∣line, from the most con∣siderable Inginiers of* 1.23 this last Age, out of all which it's most agreeable to those Authors, and to practice, to take ⅓ of the Interior Polygon for the Capital, ⅕ for the Gorge, and Flank, which leaves 6/10 for the Cur∣tain, and let this be taken for a general Rule, where the Flank, and Curtain, stand at Right Angles.

PROP. VII. By the Semicircle to lay down on the Ground, any of the former Fortifications.

Having drawn the Plot of your Fort on Im∣perial* 1.24 paper, or Vellom, and if it be a Regular Fort you need not describe it but two half Ba∣stions from the Center, for that will be sufficient. Having such a Plate whose length is set down on each respective line, and all proper Angles expressed, will not only be usefull for laying down the Work, but for finding the Solidity of the Ramparts, Parapets, and the other Earth Works See Fig 76.

If it be in such a Place, that from the Center* 1.25 of the Fort, all the Angles may be seen, place your Semicircle at Z, and lay off all the An∣gles of the Center, which here is 60°; then mark out the Diametrical lines, and making them their due length, as by your Plate they appear to be, set Piquets, on all the P, P'^s upright with the Plane, Then take up your Instrument and place a Piquet at Z. Then lock-spit out all the Polygons PP. Then mark out the Gorges CP, then set out the Flanks CF, either at Right* 1.26 Angles, or as otherwise required. Then lock∣spit out the Flanks CF, and the Faces AF, ha∣ving first set off the Capital PA, so is the Fort lined out for the Ground-line.

But if there be Houses and Obstacles in the* 1.27 way, that from the Center all may not be seen, then must you mark out any one side and mea∣sure it, and at each End set off the Angles of

the Polygon, (which here is 120°) and draw side after side, untill all be finished: Then fi∣nish the Bastions as before, and here great care must be had, or else you will run into infinite Errours.

☞ But you have liberty Experimentally to alter any of the former proportions, as you have occasion, and as will best serve the Place; as you see by the fortifying a streight lined Figure: Fig. 77. wherein Count* 1.28 Pagan's or in Manesson's way it may not be al∣lowed without some alterations.

SECT. IV. Of the Dimensions, and Measures of the Rampires, Parapets, Mote, Coridor, or Covert-way, and its Esplanade, or Breast∣work.

THE Rampire's thickness and height, must receive its Determination from the Judg∣ment of the Inginiers, and Purse of the Prince.

The Height TS, must not exceed 18 Feet,* 1.29 not be less than Ten; the thickness may be from 50, to 80 RA, in all Royal Works, and ac∣cording as Earth is to be had. The slope of the inward side of the Rampire TR, is commonly a foot for a foot, therefore RS, the Talu, will be equal to the Height TS, so if TS be 18, RS will be 18, if 15 then 15 feet.

The outward Slope QA, is generally pro∣portioned ½ a Foot for a Foot, so if the Height OQ be 18, the Talu OA, will be 9 Feet, &c.

The Height of the Parapet ZD, must always* 1.30 be 6 Feet, the Exterior Height PM must be 4 Feet, the thickness of the Parapet DQ, in light Earth must be 20 Feet, in stiff Earth 16, and in Solid Rough Clay 14 Feet; suppose it be 18 Feet, PM will be 4 Feet, MQ 2 Feet, LD 1 Foot, so will the lower thickness LQ be 21 Feet.

The Height of the Banquet VX is 1½ Foot, and thickness VL, 3 Feet.

The Lizier must be made so wide, as to support the Rampire from slipping into the Ditch, and is taken from 3, to 10 Feet; the Mote or Ditch may be from 70, to 130 Feet broad, that is, from E to G, and the depth IF may be 8, 10, or 12 Foot deep, the little Ditch at the bottom of the Mote represented by c q g, must be as large and deep as the Earth and Work will give leave.

The Coridor and the Esplanade or Breast-work* 1.31 on it, is left about 18 Feet wide, from G to C; on which is placed a Parapet, and Banquet, like that on the Rampire, which Parapet or Espla∣nade, must slope so into the Campaigne, that a streight line drawn from Z, the Top of the Ram∣pire, may terminate OFd, the Slope thereof.

PROP. VIII. How to lay down the Profile of the Work, according to this Table.

Feet.
The Base of the Rampart RA	70
Height TS and QO	16
Interior Talu RS	16
Exterior Talu OA	8
Base of the Parapet LQ	21
Interior Height ZD	6
Exterior Height MP	4
Exterior Talu MQ	2
Interior Talu QD	1
Breadth of the Banquet V, L	3
Height of it V, X	1½
The Terra Plana TV	25
The Lizier AE	3
The Mote's breadth EG	112* 1.32
The Depth of it IF	12
Breadth at its bottom FH	88
The Talus EI, or KG	12
The Breadth of the little Ditch c g.	18
The Depth of it	5
The Coridor GC	18
The Seat of the Esplanade	60
The Height CF	6

Now to lay down this Profile draw a line of a convenient length as RSOACGD for the level or Ground-line, then by your Scale of 20, o•• 30, at most in an Inch, representing Feet. Take out of it 70 for RA, 16 for R S, 8

for OA, 3 for AE, 112 for EG, 18 for GC, and 60 for CD, and mark them off on your Paper (as in Fig. 78.) at S, O, I, K, C, raise or* 1.33 let fall Perpendiculars (by prop. 1, 2, or 3. §. 1. chap. 4.) then take 16 for ST, and OQ, 12 for IF and KH, and 6 for Cf, and draw RT, TQ, QA, EF, HG, cf, fd: Then from Q set off QL 12, LV 3, QM 2, and LD 1, and raise the Perpendiculars MP 4, DZ6, and VX 1½ Then draw VX, XY, YZ, ZP, and PQ, and make the little Ditch by its measure, so is th•• Profile perfected: as for the Faus-Bray, they ar•• now out of use, therefore I omit them.

The Solid Content of those Earth-Works may easily be attained by the former Rules which Content being got in Feet, divide that product by 324, the Quotient shall be the Soli•• Flores contained therein, a Flore being 18 foo•• square and 1 Foot deep.

SECT. V. Of the Dimensions and Construction of Pla〈…〉〈…〉 forms, Caveleers and Cazemats in t〈…〉〈…〉 Flanks.

1. PLatforms are Plantations where G〈…〉〈…〉 are to be placed, and are common〈…〉〈…〉 made of Plank, and Sleepers, there neede•• for one Gun, to be but one Platform, whi〈…〉〈…〉

must be 8 Feet broad next the Parapet, and 14 Feet wide at the other End, and their length should be 18 Feet.

2. An Embrasure is the Port-Hole made in the Parapet, which towards the Gun must be 4 Feet wide, and towards the Campaigne 8 Feet wide, whose height must be proportioned unto the Wheel of the Carriage; and are 16, 18 or 20 Feet assunder.

3. Cavaleers or Mounts are Massy pieces of Earth raised on or near the Rampart, above the Parapet, on which Ordnance and small shot may be planted. As to their Construction I shall follow the Method of Manesson, who pla∣ces them in the Gorge of the Bastion, and gives this Rule for it, [saith he].

Lengthen out* 1.34 the line of defence to E, untill it cut the Capi∣tal, the Center of your Cavaleer shall be the middle point betwixt P and E, to wit at F, then with the distance of 84 Feet on the Center F strike a Circle, which shall be the Base of your Cavaleer:

Now its height ought to be at least 20 Feet; and if the Work be to be faced with Stone, or Brick it needs not not have a Talu above 3½ Feet, so that the Dia∣meter at the top will be about 153 Feet, whereon, set a Parapet of 20 Seat, and high, and other Demensions as aforesaid in the Rampire, and there will be a Terra-plana at the top of above 100 Feet, whereon six pieces of Ordnance may be planted, making Embrasures and Platforms as was last directed.

4. Cazemats are made in the Corners of the Flanks, and are several Platforms for Guns to be planted on, thereby to be hid from the 〈1 page duplicate〉〈1 page duplicate〉

〈1 page duplicate〉〈1 page duplicate〉

Battery of the Enemy: As to the Construction I shall follow Manesson's Directions, first as to the form, and also to the measure: [saith he]

The Caremate shall take up one half of the Flank, and no more; The Grand Caremate D B is about 7, 8 or 9 Feet from the Level of the Plane of the Fort, and hath a passage into it from within the Fort A, C is its Parapet of 20 or 22 Foot Seat, and in it let there be 3 or 4 Embrasures; D is the part thereof most hid from the Enemies Cannon; F is the Magazine* 1.35 for this Battery; H is the second Caremate, G the Ladder, and L the Magazine, and M the Parapet; this is to hold but one Gun; M is the third Caremate on the level of the Bast. which let be all firm, in which let there be no void place.

The Dimensions and Construction accor∣ding to the Method of this our Authour are thus [saith he]

Lengthen the Line of Defence from C to G some 40 Feet, then draw CD, parallel to Cf, (by prop. 4. §. 1. chap. 4.) let CF be half of cf, so that cF may be equal to Ff, then from the middle of the Face opposed, draw KF, and let it cut GD, in I, then make I* 1.36 L, and FM equal to 6 Feet, then make MN 66 Feet, and draw NO parallel to the Flank, which let be 24 Feet: Lastly [saith he] for the Orillon or Blind, prolong the Face FT 36 Feet, and also FV 36 Feet, then joyn TV, and make that part all solid: So is your Caremate finished: Let the height of the lower Caze∣mat, be 6 Feet as before, and let all the rest be compleated as you see in the Figure.

SECT. VI. Of the Dimensions, and Constructions of those Out-Works, called Ravelins, Horn, Crown∣works, &c.

THE Ravelin is a certain Work lying be∣yond the Mote, or Ditch, for the co∣vering the Curtain, Bridge, and Gate; the Angle of the Ravelin must not be less than 60, nor much above 100 deg. the manner of deline∣ating it is thus. Lengthen out the middle line of the Curtain OM unto a convenient length,* 1.37 then take with your Compasses the length of the Curtain CC, and setting on Foot in F, the point of the Face and Flank, cross the middle line in q; then laying your Ruler at q, and to the points F, draw the lines of the Ravelin q R and qS, which shall be the Ground-lines of the Ravelin: The M••te surrounding it must be half the breadth of the Great Mote; the Ram∣pert may be 30 Feet thick, and some 6, 7 or 8 Feet high, on which may stand a Parapet equal to that of the Rampire.

Now if from the points F you raise streight lines into the Campaign, at Right Angles to the Curtain, and from the points F set off FE, and* 1.38 FE 720 or 750 Feet, then may you joyn EE either with, A;

Single Tenaile: which is done by joyning EE, and dividing it into four equal parts, take one and place from D to N, and so draw EN

and EN, so have you a Single Tenail IENEL, which must have a Mote Rampire, and Parapet like the Ravelin. Secondly it may if occasion require be fortified with, A;

Horn-Work: which is done by joyning the points E E, and fortifying the Exterior Polygon EE as is afore taught: Or divide EE into three* 1.39 parts; make ME, and EN equal to MO; then draw N M, which divide so likewise at O and P; then draw E O and E P; then at P and O raise Perpendiculars O Q and P R, so shall M, E, Q, O, P, R, E, N, be the Horn-Work which was desired: which must likewise have small Rampires and Parapets, as afore.

For the Crown-Work: From the Center of the Fort O draw O M B of a convenient length, then from the middle of the Ravelin set off 1000 or 800 Feet to B, then on q, strike the Arch D B E, set off the Curtain, and Demi∣Gorge P C C, from B to F, and G both ways, then draw C F and C E, to terminate the points I and H on the Counterscerp; then toke* 1.40 ⅓ part of B F or B E, and set it from B to M; and srom F to L, and from E to M; then draw L M, and M N; then for your Demi∣Bastions make N P and L O equal to N E, &c. Then for the Demi-Gorges of the whole Bastion in the middle, let them be equal to ⅕ of the In∣teriour Poligon L M or M N, viz. M Y or M X; then finish the Bastions by drawing the lines of Defence, and raising Perpendiculars, or making Angles of 98° at O, X, Y, and P, then the Crown work is finished as desired. You may make Ravelins and other Works (beforementi∣oned) before these Curtains if occasion require.

There are some other Works which are used; as Half Moons, Bonnets, Double Tenails, Counter∣gards, Horseshoes, Priests-Caps, &c. which would be superfluous to speak of in this place.

5. Cittadels, are Castles or Forts of the least sort, and are the Out∣works lastly used, which are* 1.41 commonly of 4 or 5 Bastions, and are pla∣ced in such Order, that there may be two Faces, and a Curtain towards the Town: the Construction whereof is after this manner. Lengthen out the line OM, and therein find the Center of the Cittadel, the In∣terior Polygon of the Pentagon may be ¾ of the Curtain adjoyning, or a little more; the Cen∣ter of the Square may be on P the point of the Interior Polygon, the Center of the Hexagon may be near the outward point of the Bastion of the Town, taken away to make the Cittadel in, which may be delineated as afore: The Motes and other Works in proportion accordingly, and the Rampires as high as those of the City or Town.

SECT. VII. Of some Maxims or Rules necessary to be known in Irregular Fortification.

IRregular Fortifications is when any Town or Place is to be fortified, which lieth in an Irregular form; i. e. whose Sides and Angles are unequal in the forti∣fying of Irregular Fi∣gures* 1.42. I shall here say very little, only I shall lay down some Precepts that are of immediate concern in fortifying of Irregular Figures, and shall refer you to peruse Marlois, Dogen, Fritach, Taurnier, Dilichius, &c. which will greatly satisfie and help you: To this end know,

1. That the same Laws and Maxims for Re∣gular Fortifications stand and be in force for Irregular; i. e. that the line of Defence must not exceed the Port of a Musquet, nor the Angles of the Bastion be less than 60°, nor much above 90°, &c.

2. That no inward Angle of the Place be less than 90°, if it be so it must be altered, and that point may be made the outward point of a Bastion.

3. That between Regular and Irregular For∣tifications, there is no other difference, but by rectifying the sides that are too short, or too long, and altering the Angles that are too little; as for the sides, if they be above 500, and un∣der 1000 Feet, they may be fortified by Ba∣stions placed according to the usual manner, at the extreme points thereof; But if the sides be between 1000 and 1700 Feet, then in the midst you may place a Plat Bastion, and at the Extreme Points, place two Bastions, as before: But if the line be less than 500 Feet, you may lengthen it, by producing it into the Plane: As for the Angles, they are made greater or lesser according as occasion requireth. For the Raising the Rampires, Parapets, and other Out∣works, they are to be as in the Regular, and the Out-work may be placed before the Cur∣tains as was before mentioned.

4. That the Capital, in any Regular or Irre∣gular Bastion, is found by dividing the Angle of the Polygon into two equal parts (by prop. 7. §. 1. chap. 4.) and by producing the line of Angular Division or Separation, on which the due length of the Capital must be placed, which observe for a general Rule.

SECT. VIII. Of the Dimensions and Construction of small Forts, or Scones, which are built for the Defence of some Pass, River, or other place.

WHEN they are made Regular, of 4, 5, or 6 Bastions, then they may be fortified by the precedent Rules, but there are others of smaller Dimensions fit for the same purpose: viz. Triangle with Demi-Bastions, Square with Demi-Bastions, Parallelograms with Demi-Bastions and Tong, Star Redoubts of four, five or six points, and Plain Redoubts.

PROP. IX. To fortifie a Triangle, with Demi-Bastions.

This Triangle may consist and be compre∣hended of three equal or unequal sides in this Example: let it be an Equilaterial Triangle PPP* 1.43 Now divide PP into three parts, then take 1, and prick off the Capitals PA, &c. and the Gorges make equal thereunto, as PC, PC, &c. then make the Flanks FC to stand at Right Angles, and to be ½ of PC or PA, then draw the Faces AF, AF, &c. and the Work is finished as required.

PROP. X. To fortifie a Square with Demi-Bastions.

The sides of the Square may be from 100 to 200 Feet, let PP be 180 Feet, which divide in∣to 3 parts, take one for the Gorges PC, and for* 1.44 the Capitals PA, and prick them off all round as you see, then take ⅙ of PP, and at Right Angles prick off the Flanks CF, then draw the Faces AF, AF, &c. and the Figure is compleated.

PROP. XI. To fortifie a Parallelogram with Demi-Bastions, and Tong.

First describe the Parallelogram, or Long∣Square, PPPP, then divide PP into 6 parts (the side on which the Tong, or Tenaile, is placed) and make MC equal unto ⅙ thereof, and also MG, and MH. Draw CG, GC, and CH, HC,* 1.45 then finish the Demi-Bastions as before, so shall the Work be compleated as was required. A Long Square may also be fortified as Fig. the 77.

PROP. XII. To fortifie a Star Redoubt of 4, 5, or 6 Points

• 1. A Star Redoubt of four points may have his side from 40 to 60 Feet: First describe the Square PPPP, then divide PP into two parts at M, take ¼ of PM, (and by prop. 1. §. 1. ch. 4.)* 1.46

• raise Perpendiculars round at M, make MA equal to ¼ of PM, and draw all as in the Fi∣gure.
• 2. A Star Redoubt of five points is thus fortified. Describe the Pentagon PP, &c. then divide PP into halves at M, raise the Perpendiculars MA, make MA equal to ⅓ of PM, and draw the Fort* 1.47 in all respects as the Figure representeth.
• 3. A Star Redoubt of six points is thus fortified. Describe the Hexagon PPP, &c. divide PP into two equal parts at M, then raise Perpendiculars at the M'^s, then make MA equal to ½ of PM,* 1.48 or ¼ of PP, and draw every respective line as you see in the Figure.

PROP. XIII. To Delineate a Plain Redoubt

Plain Redoubts are called Grand Redoubts, which are used as Batteries in Approaches, whose side may be from 60 to 80 Feet, or Petit Redoubt,* 1.49 which are used for a Court of Guards in the Trenches, and may be from 20 to 50 Feet, and are framed and delineated in all respects as you see in Fig 90.

The Profile's to be set on these several Works, and the Motes, are alterable and uncertain, for they being sometimes used in Approaches; then they do require the Breast-work at the Bottom to be 7 or 8 Foot wide, and the Interior Height 6, and the Exterior 5 Feet, and the Mote to be either 8, 10 or 12 Feet, sometimes 14 or 20 Feet wide at the bottom, and the height of 7, 8 or 9 Feet, to have two, or three

ascents to rise to the Parapet. There are many other things belonging to this Art, which the limitation I am bound to, will not permit here to be treated of.

Notes

• * 1.1

  Defin.
• * 1.2

  Fig. 73.
• * 1.3

  Fig. 73.
• * 1.4

  Fig. 73.
• * 1.5

  Fig. 73.
• * 1.6

  Because the length of the part of a Musket doth not much exceed that Mèasure.
• * 1.7

  Fig. 73.
• * 1.8

  Fig. 73.
• * 1.9

  Fig. 73.
• * 1.10

  Because the Defence ought to be easie, quick, certain, and of little charge, all which qualities the Musket hath and the Can∣non hath not, therefore the Defence of Fortification ought to be measured by the Port of a Musket, and not by that of a Cannon.
• * 1.11

  Fig. 74.
• * 1.12

  Fig. 74.
• * 1.13

  Fig. 74.
• * 1.14

  Fig. 74.
• * 1.15

  Fig. 74.
• * 1.16

  Fig. 74.
• * 1.17

  Fig. 74.
• * 1.18

  Fig. 74.
• * 1.19

  Fig. 74.
• * 1.20

  Fig. 74.
• * 1.21

  Fig. 74.
• * 1.22

  Fig. 74.
• * 1.23

  Observe this for a ge∣neral Rule in Regular Fortification.
• * 1.24

  Fig. 76.
• * 1.25

  Case 1.
• * 1.26

  Fig. 76.
• * 1.27

  Case 2.
• * 1.28

  Fig. 77.
• * 1.29

  Fig. 78.
• * 1.30

  Fig. 78.
• * 1.31

  Fig. 78.
• * 1.32

  Fig. 78.
• * 1.33

  Fig. 78.
• * 1.34

  Fig. 79.
• * 1.35

  Fig. 79.
• * 1.36

  Fig. 79.
• * 1.37

  Fig. 80.
• * 1.38

  Fig. 81.
• * 1.39

  Fig. 83.
• * 1.40

  Fig. 84.
• * 1.41

  Built to bridle the Town or the Place, left the Burghers should be rebellious, and to be the last refuge or place of retreat.
• * 1.42

  The Inginier must first form a Map of the Town or Place, with all the Ways, Passages, Old Walls, Rivers, Pools, Enclosures, and all o∣ther matters fit to be known in the draught, and then he is to design what Works he findeth most agreeing to the place to be Fortified.
• * 1.43

  Fig. 84.
• * 1.44

  Fig. 85.
• * 1.45

  Fig. 86.
• * 1.46

  Fig. 87.
• * 1.47

  Fig. 88.
• * 1.48

  Fig. 89.
• * 1.49

  Fig. 90.