The Calculation of my great Fortifications, con­sisting in the little, mean, and great Royal.

TO do this, I state first my exterior Polygones, after which I regulate the proportions of the whole Work. For the exterior Polygone of the little Royal, I take 80 ⓪, for that of mean Royal 90 ⓪, and for the exterior Polygone of the great Royal 100 ⓪; and I follow the same proportions in the Cal­culating of my Lines, which Fryday hath, that is, making the Courtine to the Face, as 36 to 24, or as 3 to 2; but the diffe­rence is, that I take my parts greater, and redress them, after the first Position is done. This Calculation is done onely by the Rule of Three, thus: Admit there is a place to be Fortified with a Regular Fortification of four, five, six, or more Angles, and that after the first, second, or third of my Methods; I take then the Lines of such a Polygone, as the Fortress should be, that I intend to make, out of Fryday's Tables. For example, I would make a Square after my first Method; I take then out of the Tables the length of the Lines; as 24 ⓪ for the Faces; 36 the Courtines; for the Gorges 9.23 ②; for the Capital 19.73 ②; for the Flanque 7.74 ②; and so for all the Lines: these Lines I lay before me, and say, by the Rule of Three, If the exterior Polygone of a Square of Fryday's, being 82.36 ②, giveth me the Courtine 36 ⓪, what does my exterior Polygone of 80 ⓪ after the first Method in the great Royal, give me for my Courtine? and the answer is, 34.97 ②.

And so observing the same Rule for a Square, whose exterior Polygone is 80 Rods (as in the Figure of Numb. 1.) there will arise the following proportions of the Lines, viz.

Of the Fausse-bray.

THe Modern Engineers draw the Fausse-bray parallel, round about the foot of the Wall, at such a distance, that between the Parapet of the Fausse-bray and the Wall is left a hidden way of between 15 and 25 Foot. This Parapet is made to defend the Moat, when the Enemy is passed the Counter-scarp, and intendeth to carry over the Gallery. I make also Fausse-brays, but after another manner than the Modern, viz. thus: I draw a Parallel-Line with the Flanques, as in Numb. 1. X W, parallel with the Flanque 6 V, at the di­stance of five Rods; so that the place may be capable to hold Canon; afterwards I lengthen the Faces of the Bulwork, as B V, till they cut the parallel X, W; this done, I draw a parallel with the Courtine, at the distance of 3 or 4 ⓪, as X Y; so that the Lines V, W, X, Y, 2, 3, make the Fausse-bray; and this I do round about the Fortress, onely before the Flanque and Courtine, and not before the Faces. The reasons hereof are [Page 15]these: First, It is found by experience, that the Fausse-brays are more hurtful than advantagious before the Faces; because they are not of service, before the Enemy cometh to Lodge up­on the Esplanade or Parapet of the hidden way, and then they are to oppose him, which they cannot possibly do: And this so much the less, because the Enemy may plant his Canon there for the Destroying and Dismounting of them. Which is easily done, seeing it is but a bare Parapet, which is soon thrown down; so that in this part before the Faces, which are Attacqued, no body can remain for Defence, by reason of the playing of the great Guns from the Batteries, which may be made as need requireth; as here Fig. 11. Numb. 1. is intimated by the Letters A, B, and C; where from the Battery A, the Face of the Bulwork D, and its Fausse-bray is pierced Perpendicu­larly, and from the Battery B Obliquely; but the Battery C cutteth and cleareth the Earth away, which is shaken by the two others; so that by these Batteries and Musquet-shots, coming from the Lines and Cors de Guards, the Faces, as well of the Fausse-bray, as of the great Wall, are soon ru­ined and left without Defence, so that no body may be seen upon these Lines without being hit; whence it followeth, That they cannot hinder the coming over of the Gallery, but it must of necessity be done from the opposite Flanque. Item, That so soon as this part of the Fausse-bray, lying before the Face, is discovered, it is presently made useless to the Defen­ders; and not onely so, but it does them this hurt and disad­vantage, that so soon as the Gallery cometh over, the Miners are able to Undermine the Faces of the Bulwork, where they please, and at several places together. The Mines being blown up, the Enemy may set Footing upon the space where the Fausse-bray lies, not onely to Assault the Breach, but he may run along the Foot of the Wall through the Fausse-bray, to fall on at more places at the same time. To this it might be repli­ed, That the Fausse-bray ought to be Cut with Trenches, to hinder this running about. But every one who has experience in these Actions, knows very well, what slender Oppositi­on these Trenches are able to make, especially, when the Earth, which is raised by the Mine, falls down, and some­times buries the Defenders alive. This place is also of Ad­vantage for the Enemy, when the Breach is Retrenched, and he will make new Attacques upon it, and by that means [Page 16]will lodge himself conveniently in the Fausse-bray, and fur­ther in the Breach; whereas otherwise, where there is no Fausse-bray before the Faces, he findeth very inconvenient Lodging against the slooping of the Wall, seeing no other place is found to fix upon. The Miners are forced to make their Mines about the place, where the Gallery cometh on; besides that the Faces, which are surrounded with these Fausse-brays, being Attacqued by Canon and Mines, all the Ruines of them remain lying in the same, and afford an easier Ascent unto the Enemy; whereas otherwise, where they are not, the Ruines rowl down into the Moat, and the Breach remaineth clear, and the Enemy findeth little Earth to Retrench himself. By all which it is easie to understand, That this part bringeth more advantage than hurt unto the Enemy. To what end is it then, to be at such Charges to our Prejudice, and to cum­ber the Moat before the Bulworks with this needless Parapet? Therefore to conclude, I say, That the Faces of a Bul­work being Attacqued, can make little Opposition of their own; but ought to be Defended from the Flanques and parts of the Courtine; and the more eaven and neatly the Faces lie, the better may they be Flanqued. That part of the Flanque of the Fausse-bray D, hinders also, that the Face a G from being defended at the bottom from the opposite Flanque. Some Engineers are of opini­on, That the Fausse-brays ought to run out after the shape of Ravelins in the middle, before the Courtine. I think this needless, where the Lines of Defence are of a just length, not much exceeding 60 Rods; but where they be longer, this must be practised. At Breda, it is done before some long Courtines. And this is enough as to what concerns the Ground-Lines of the Fausse-bray, in the Profiles: I shall shew hereafter, how the Work is to be raised. It remaineth now, that I speak of the Ground-Lines of the Counter-scarp and hidden way, and how it may be made, that it may defend it self with great Advantage, without the great Wall; and that these Counter-scarps are as strong, yea stronger, than the Mo­dern Fortresses themselves.

Of the Counter-scarp, or hidden Way.

THe Modern Engineers do strengthen the outward Brink of the Moat (or Counter-scarp in Italian,) with Ravelins and Half-moons, leaving a hidden way with a Ditch about it, and without this hidden way they make sometimes another, without a Ditch; as may be seen at Breda: Which latter I hold to be very good, because not onely great De­fence may be made from thence, but they are also very con­venient for Sallying out, both on Foot and on Horse-back; for the Moat doth not hinder it, and the Enemy doth not know from what side he must expect the Sally: Those who thus Sally out, being driven back, may always retire hither, and present­ly be in a posture of Defence again behind this Parapet; whereas otherwise they are many times driven into the Ditch of the hid­den way, with great loss and disorder. This hidden Way (or Esplanade) costeth little, for in places where the ground is some­thing dry, it is cut in above half its height under the Horizon, that is, three or four Foot, that it may be the better flanqued from the first Esplanade which is behind it; as also, that the Shots may flie more Horizontally, &c. I do Fortifie my Moat with another kind of Work, and I shall shew first the manner how I do it, and afterwards give reasons why it is so much [Page 18]better than the Modern Works. And at the same time I shall discover the Defects, which the Modern Works are sub­ject unto.

See Fig. 4. Numb. 3. which is part of a Regular Seven-an­gled Figure after my little Royal, where the exterior Polygone is 80 ② long: Now to draw my Counter-Guards (for so I call them instead of Counter-Scarps, this name being impro­per) about it, and to give them such a form, as may fit them for that service I expect from them; I lengthen the Half-diame­ter K, O to A, or further at pleasure; then I measure upon the lengthened Half-diameter from the brink of the Moat O to I, thirty Rod or more, according as I would have the Flanques N 5 and 4 R short or long. Then I draw from ⅓ of the Face, as from K the Line K, G 5, as also on th'other side the Line L, R, M, from the Point where these two Lines meet, which should be before the middle of the Courtine; as here from G, I draw the Perpendicular G H, which is the Capital-Line of the Ravelin W, G, P, H; then I measure from N to L, as also from G to P, upon the Lines K, G, on th'other side L, G, such a di­stance as I need for my retired Flanques: from these Points, viz. from N and 4, I draw N 5 Perpendicular to L, M, and 4 R, Perpendicular to K 5, as is clearly seen in the Figure, and want­eth no further explanation as to the Delineation of it. I make further upon the Line 5 R an equilateral Triangle, as 5 R, R, which giveth a Ravelin, whose Angle at the Point R is 60 de­grees: He who desireth the Angle to be greater, may make it so, without prejudice to th'other Work. About this Work I draw a Parallel-Line at the distance of five Rods, viz. M R G 5 2; which is the Ditch about the Counter-Guard, so that the Rave­lins lying before, become little Islands. At the brink of this Ditch I make a hidden way or Esplanade, whose sloping is regu­lated according to the Wall of the Counter-Guard, without Ditches, with small cost, which becomes a great benefit. This I continue about the whole Fortress; and all is clearly seen in the Figure, how the Work must lie according to the scale and measures. These are all the Outworks, which I look upon as necessary for a good Defence; and I make no Hornworks, &c. unless extream necessity do require it, when perhaps there is some height; and then I make them after another manner than the Modern, as the nature of the place does require.

Of the Vertues and Defects of the Modern Outworks.

THe Modern Outworks are Ravelins, Half-moons, hid­den Ways, Crownworks, Hornworks, Tongues, Ket­tles, &c. and are made about the Fortresses, as Fig. 5. Num. 3. sheweth; where the Figures marked with G are called Half-moons, those with B Ravelins, D is a Crownwork, such as are laid before the Courtines, before which there lieth a single Tongue; E is a Crownwork lying before a Bulwork, being strengthned with a double Tongue, and at the side of it there is represented the form of a Hornwork. This Figure is part of a Regular Seven-angled Work, according to the most Modern practice, as well the Outworks as the Royal Work. The Vertues and Defects of the Royal Work are in part manifested already. But now I shall shew those of the Outworks; and first

Of Ravelins.

THe Ravelins, marked with B, I count very necessary, see­ing they multiply very much the Defence, especially of the Outworks: But because they do in a manner stand instead of a Royal Work in Modern Fortresses, they have this Defect, that their Flanqued Angles are too small, and the Points may be easily broken, as also the Faces, as may be seen in Fig. 5. where the Ravelin B being Attacqued from the Batteries A, A, is easily made useless. There may be more Batteries placed to Attacque these Ravelins, as it is required, because it is seen from the whole Field I A L. These Works being Attacqued, it is very diffi­cult to succour them, and no less difficult for the Defenders to retire when it is taken; and being once taken, it becomes, as it were, a Knife to cut the Throat of the whole Fortress.

Of the Modern Half-Moons.

THese Half-Moons marked Fig. 5. Numb. 3. with C, I look upon as Works of little use, because they can make but poor Defence, and are Defended but weakly; since they have no other Flanque, than from the Faces of the next situated Ravelins; which Defence being broken (which is easily done, as I have shewed by the Ravelins, and appeareth by the Fi­gure) they remain to the use of the Enemy, to make in it the Battery H, whereby the opposite Flanques of the Bulworks are ruined, and the Galleries are brought over with surety; all which is clearly shewed.

Of the Modern hidden Way, or Counter-scarp, as they call it.

THese hidden Ways are very advantagious before Fortres­ses; for lying low by the Horizon, they make all the Shots that come from thence, to Mow and Clear the Horizon: neither can they well be broken by the Canon, but must for the most part be taken by storm, and they are commanded from the great Wall: This is a work which covereth much the defects of the Modern Fortifications. They favour also the Sally­ing out, when they have no Ditches; but here in these United Provinces they have almost all of them their Ditches; which fault being discovered, it hath been remedied by another hidden Way, running about the Ditch of the first, and being cut in, part of it, according to the nature of the ground, under the Ho­rizon, and the surface of the Parapet of this hidden Way, loseth it self, sloping into the Field, without a Ditch.

Of Crown-works, and Horn-works.

IT is many times necessary to make these or the like Works to take in any heights or suburbs, according as the place re­quireth it. They are made after the manner and practice of the great Fortresses, and according to the same proportion, except­ing onely, that their parts are smaller, and they have but half-Bulworks on either side, which are very weak; and yet their strength may be much increased by an understanding Engineer. They ought, in my judgement, to be made so, that they need not to relie upon the Defence of the great Wall; but that they be sufficient to defend themselves, especially those that lie be­fore the Courtines, as in Fig. 5. may be seen by the Crown­work D, whose Wings have a very weak and oblique defence from the Faces of the Bulworks, and being Attacqued by night, the Lines E E cannot be well defended from the Faces of the Bulworks. Yea, in such occasions the Defence of the great Wall is but troublesome to these or the like Works; for the Musqueteers of the great Wall, shooting by night onely upon the Alarm of the Assaulters of the Outworks, do hit as well within as without, and wound as well the Defenders as the Adversaries. They ought nevertheless to lie open towards the City; and to be made so, that being taken, they may be lit­tle able to hurt it.

Of single and double Tongues.

THese are very slight Works, and never ought to have o­ther Profiles, then those of ordinary Trenches; for else, if the Wall should be high, they were not able to defend their own Front. Neither are they but seldom made otherwise than Trenches, or to hinder the Approaches; as also the Kettels mark­ed with H. Hence it may be seen, wherein the strength of the Outworks now in use doth consist, which the Modern Engi­neers make such boast of. Let us now examine the Outworks according to my Practice.

First of the Counter-Guard.

OF these Outworks it might be said, That they take in much place, which is a truth; but when we shall have shewed what benefit cometh by them, I doubt not but that will outweigh any inconvenience that shall be occasioned be the great space and place which these Works take in more than other Works usually do. Neither is the difference so very great: For admit, that about the Modern Seven-angled Work Fig. 5. Numb. III. there should be made another hidden Way about that which is there already, or that it should be strengthened with two hidden ways; the Semi­diameter of my Heptagone Fig. 4. is but five Rod longer then that of th'other. In recompence whereof it is to be considered, That my exterior Polygones of the Heptagone are 80 Rods; whereas those of th'other (or Modern) Heptagone are but 70 Rod long, which in the whole maketh 70 Rod difference, which is an entire Polygone. Wherefore to include the place of my Heptagone after the Modern manner, there would be required eight Bulworks; so that I do include and Fortifie more place with fewer Bulworks than now is usual, which under­standing men will easily conceive, by comparing the one with th'other. To shew then, what good this Counter-guard doth, let us suppose, That the Fortress Fig. 4. is also Attacqued by the Approaches O O. Now skilful Artists know, That it is best for the Enemy to Attacque the Point D, and after he hath gained the Counter-scarp, to pass the Ditch of the Counter-guards of 5 Rods. To do this, it is needful for him to render first the Flanque of this Work D, and the Face of the Ravelin F uncapable of Defence: but if he would break the Flanque of the Counter-guard D, he findeth no other place for his Bat­tery than by the letter C, where but five Pieces can be placed, [Page] [Page]

[Page] [Page] [Page 23]to Dismount the lowest Flanque; and surther in the Field about G there might be planted 3 or 4, to throw down the uppermost Parapets of the Retired Flanques, so that nine Pieces of Canon may be used; whereas those of the Retired Flanques may bring 17 or 18 Pieces in Counter-battery, and from the part H, 4 or 5 Pieces, for the Defence of the Face of the Coun­ter-guard; therefore the Defenders are able with more ease to break the Battery of the Assaulters, than they to break the Flanques of the Defenders. That which hinders, that no greater place is found for the Battery of the Assaulters, is the Ravelin F, and the Angle of the Flanque of the Coun­ter-guard I. Let us see, whether there be no where else a place to make a Battery, to break the Retired Flanques withall, and to Dismount the Canon standing upon the same. Suppose there be a Battery by B, from thence with the Line B one might boar in between the Ravelin F, and the Angle I, but not so, as to touch the lowest and first Flanque; yet this shot (which cannot be done by more than one Piece of Canon) would come obliquely against the last Shooting-hole of the second Retired Flanque, and cannot Disinount the Canon that standeth there, because the shot hitteth the Parapet obliquely in the forepart. No more can it be done from the Battery A, against which Batteries upon the Line D E, every where Counter-batteries may be made. Put the case, That the Enemy not regarding all these difficulties, do with great loss pass the Ditch by D, and undermine the Point of the Counter-guard, (which would also be very hard to do, seeing the Walls are made so, that one cannot well put a Spade into the Ground without being discovered) and make a Breach. In this case, I can first with small labour cut off this Point by the Retrenchments 1 and 2, till the great Retrenchments by 3 and 4 be finish't, which are duly defen­ded from the Flanques 6 and 7, as also from the Bulworks­point 8; besides that they succour and defend one another. And in case the Besiegers mean to pass between the Retrench­ments towards the Bulworks-point, without gaining before the two Lines marked with 3 and 4, they will find hot work. But suppose they force one of these Lines, then there is within, two or three Rods after them, another ready, and those that lie before are Undermined, which Mines are blown up, when the Enemy once thinks to lodge there; and [Page 24]this is a thing that may be done with great advantage. There needs not one Rod of ground be lost, but what is gained with abundance of bloud, and also with great toile and labour. Being forced, we have alway a free retirement into the Ravelins marked with K. and L. And in case there be but men within a Fortification made after my Method, who understand the Work, it seems impossible to gain it any other way, than by length of time, and with the loss of many men. This Work helpeth it self, nothing being in the way that hindereth; and all parts are so contrived, that they are provided against all inconveniencies. Hence the un­derstanding Reader may easily conceive, what advantage these Outworks have before the Modern. They are Flanqued within and without with Canon and Musquets from the great Wall; there may be made in haste, and with small pains, as many Retrenchments as is desired, which are all enfiled with­in and without from the great Wall; and the more obsta­cles are cast in the way, the more the strength of these Works is discovered. This is enough as to the form and advantage of my Outworks, which each according to his judgement and place may apply: In my opinion, it would be needless to propound these things in a plainer manner, seeing all particularities may be observed by the Figures. It might be said, That these Outworks would cost more, than those that are now in use: the contrary of which is true; for the Pa­rapet of the Modern hidden Way having a Base of 60 or 70 Foot, costeth as much as my Counter-guards-wall, whose Profile I do make so, that it is just Canon-proof. I shall not add here the charges of the Modern Crownworks, Hornworks, &c. but I put my two Ravelins against the Mo­dern Half-moon and Ravelin; the Counter-guard against the Counter-scarp now in use, my outward hidden Way with­out Ditches (which is made with small charges) against the Modern second hidden Way, being also without Ditches. And although this Work should cost much more, yet the charge is sufficiently balanced by the benefit that is had thereby: For the Fortresses made after this manner, have quite another Defence, and are, besides their Out and In­works, provided against all Occurrences, with 2, 3, a 4, double Flanques, made as well for Canon, as for Musquets. If any body thinketh, that I propound new and strange [Page 25]things, as doubtless those will do, who are possessed with the Modern practice of Fortifying, and never did search af­ter more convenient means, or examine the strength of eve­ry part of it distinctly; or what may be the cause, why Strong-Holds are lost so soon, neither understand the force of the Maxims of War now in use: To those I reply, That if they will consider these new things judiciously, they will find, that the practices of War now in use, ought to be preferred before those old things; and that the Work is ve­ry proper and capable of making a good Defence; and that these things were not practised conjecturally in a Closet, but in occasions of Besieging, or being Besieged; and that in se­veral countreys and grounds, and in such places where in­deed flew swift Bullets, not Sugar-plums, or roasted Capons. Many things may be projected upon Paper; but we ought to look, whether they be practicable in the open Fields. The Latin proverb, Facile est inventis addere, might be ap­plied here, I do confess, but it hath not been done as yet by others. I have been about this a long time; but alway observing more and more, whether any thing might hinder it, which perhaps I had not considered of; and possibly there may be some such thing, but whatever it be, it may with ease be rectified by time and practice. All things devised by men, do not immediately come to perfection. Let them who please to use this way, first consider the Work well; and if there be any thing that they do not like, let them practice the Latin proverb abovesaid. There are yet many Particularities in this VVork, which I do not describe here, but keep them for my use, and propound onely the Gene­rals. It is true, That the Art of Fortifying is highly impro­ved in these United Provinces, by reason of the continual exercise in it: But here have been but few alterations or diversities of Attacques, seeing they are almost all of them made after the same manner; as also the Defences of the Spaniards, who were furnished most commonly with ignorant Engineers, and their places ill Fortified; so that being Be­sieged, and much pressed, they wanted many things that were most necessary. Here in these parts they have been a while at a stand, without practising any thing at all, which might serve to rectifie their Errors; although understanding men saw well enough, that there was something wanting, and [Page 26]that the Work did want another Form; and this proceeds chiefly from the impression which many had, and still have, that the Fortifications were at perfection, contrary to expe­rience and matter of Fact in the late Wars. In this opi­nion they were confirmed by the curious searches, experi­ences and amendments, made by His Excellency Prince Mau­rice, and His Highness Friderick Henrick, Princes of Orange, of glorious Memory: These indeed did in those days re­form many things exceeding well; but the manner of At­tacquing being much altered since, the Fortifications ought also to be strengthened against them. The Practitioners have onely joyned to the first Inventions, those things which serve to establish Fortifications upon sure Rules, according to the Mathematicks, to which we ought not to be tied too close; however it is necessary to take their assistance in this Science. The question is not, Whether the Fortifica­tions do lie exactly according to the Rules of Mathematicks; but whether the Lines be well Flanqued, and all things or­dered for most conveniency. I confine my self indeed to these Rules and Calculations, for so much as the first Ground-Lines do require; but upon these I make my Observations afterwards: so that my Work receives no small benefit from thence, by reason of the firm and well-contrived Flanques, and the giving of little advantage and place to the Enemy to erect his Batteries against them.

Many places (though well Fortified) are also lost in a short time, for want of skilfull Officers and Governours. For although a man had the best Sword he could wish for, and yet did not know how to use it for his advantage, the worst would be as good for him as the best. Let the best Fortress be defended by a Governour, who knows it not, that Fortress shall be gained sooner than a mean place Commanded by an experienced Officer. This hath been sufficiently made good in these late Wars, which al­so Histories do testifie abundantly. For, Retrenchments, Cuts, Counter-approaches, Counter-mines, besides the ordering of Sallies at due times; as also the managing of Fire-works, con­trived by an expert person and skilfull Architect, are all very advantagious for Places, and detain the Enemy long. And yet they cannot well be known, before there is Attacque made; and then they may easily be so ordered, as to be made [Page 27]more advantagious for the Enemy, then for the Place it self: Hereof there are examples enough, and these Works can­not have a good success, unless they be directed by a skil­full Architect, who has not onely a perfect knowledge in For­tification, but understands also the power of Artillery and Fire-works, as also the manner of managing them, and how Mines ought to be made upon all occasions. And this I think sufficient for the designing of Ground-lines, and have described onely three Figures more of three Regular Hexa­gones Fortified; see Num. IIII. one of them, after the great Royal, whose exterior Polygone is 100 ⓪; another after the mean Royal, whose exterior Polygone is 90 ⓪; and a third after the little Royal, the exterior Polygone being 80 ⓪.

Lengths of the Lines according to the first Calculation.

The Intelligent Reader will see all things better in the three Figures, Numb. IIII. and may measure it by the Scale that is joyned to them, by which all three of them are made; and it is to be observed, that these Lines serve onely for an Hexagone. But if any one would For­tifie another Figure, he must first calculate the Lines, and afterwards alter the work according to the directi­ons heretofore mentioned. By the first Figure are shewed three several manners of Counter-guards, being all three very good and commodious, and may be very fitly apply­ed to places, that are already Fortified; as also the Royal Work, and the great Wall may be redressed in such places, with little charge, after this way. The second and third manner of Counter-guards I count better, when the place and occasion permitteth it, to make them so large, for they are extraordinary Advantagious; but for brevity sake, I shall not rehearse here all their Advantages and Conveniencies. Some will say perhaps, That the Ground which these Out­works take in, being so large, the same would cost very much to buy it from the owners, in such places where it is required. To these I answer, That in this case there is not [Page] [Page]

[Page] [Page] [Page 29]any part, which after the Fortification is finished, cannot as well as before, either be sown, or used for other conveni­encies, except the Wall and the Moat; and these parts may also be used in time of Peace, and the Walls may be plan­ted with fruitfull Trees, which may so soon as a War shall break out, be cut down if need be: But they may be plant­ed so, that they may as well be left standing, and the Moats may be used for Fish-ponds, &c. By the first Figure I do also shew, how I joyn my Hornworks, when there is need of them, viz. as followeth: The Capital of the Ravelin, (or properly the outstanding Point Q,) I lengthen towards the Fields 60 Rod to R, and upon this I draw on both sides a Perpendicular; as R M, and R N, each of 20 or 25 or more Rods, as need requireth; and this I take for the out­ward Polygone of the Hornwork: From M and N, I draw the Lines M L, and N O, to the ⅔ of the Faces of the Counter­guards. After this manner the Angles of the Half-bulworks come to be much greater, than if the Wings L M, and N O, were drawn parallel to one another, &c. All may be eon­sidered better in the Figure. There might also instead of the Hornwork another be made, as is marked with the Letters L P O, which is also good, and well Flanqued. Here it is to be observed, That to these and the like Works no great Profiles must be given.

Thus I finish my Discourse, as concerning the Ground-lines of my Fortification, as well of the Inworks as of the Outworks. Now I come to the Profiles, and shall first treat of those that must be made in wet places, where the Ditches are full of water: And first of all,

Of the Walls of the great Strong-Holds.

THe Walls ought not to be too high, nor too low: For when they are very high, the Moat can hardly be defended from them; and the nearer the Enemy getteth under them, the more he is sheltered. They have also this fault, that they are very chargeable; for they ought also to have a thickness proportionable to their height; and the thick­er the Walls be, the better they may be Undermined, and the Mines that are made in them, do better Execution than in lesser Walls; for the more the Powder is inclosed, and [Page 30]the more it sindeth resistance, the greater force it hath, which is evident. Some are of opinion, That the Walls ought to be of such a height, as to cover the Houses from those Canon-shots that come from without: which is ridi­culous: For although Walls were made so high, that no houses might be seen; yet by all this we cannot hinder the shots from flying through the Houses; neither are the Mortar-pieces hindred by it, which do the most hurt to houses; besides that, I being Besieged, could rather wish, that the Enemy would busie himself with shooting through the houses, than against the Wall, in case it were a Fron­tier-town, where more Souldiers lodge, than Citizens; for this shooting through the houses is onely to frighten the Inhabi­tants or Citizens, to compel them the sooner to compositi­on. But Souldiers count this their pastime: they had rather that the tyles and laths should flie from the houses, than that the earth should be torn as raggs from the Parapet. To shew that the high Walls cannot hinder this nevertheless, See Fig. 6. Numb. V. Let there be a Battery by A, from whence out of the Canon A is made the shot A B F, so that it go­eth just over the Parapet of the Wall; this shot offendeth the roof of the houses more or less according as the Wall is low or high; as is seen by the shot A C. Suppose the Wall to be as high again as B C, and that the shot be A C G, this goeth indeed far over the houses; but let the Canon D be elevated some degrees, and it will make the shot D E H, against which the height of the Wall availeth nothing, al­though it were five times higher; for the Bullet cometh down with a Circular-line. It might be said, That the Canon would carry the Bullet over and beyond the City: which is true, in case the Piece standeth near by it; but it can be placed further off, at a convenient distance, which is easily found by a good Canoneer: which I intend to shew clearly in my Treatise of Artillery, which, God permitting, I shall commu­nicate to the Lovers of this Art. Such Pieces may be pla­ced here and there in the Fields, or even in the Quarters, where they please, without making any Beds for them, and so be used; for they ought to stand without the true Line or ordinary Canon-shot. Besides that the houses may be troubled after this manner by the Canon; the same may be done more conveniently by Fire-Mortars, out of which Gra­nadoes, [Page] [Page]

[Page] [Page] [Page 31] and Fire-balls are cast; as may be seen by the F I. In case the Walls be too low, the Enemy can heighten the outward Batteries, from whence he may enfile the Wall of the Fortress. It is true indeed, That the Ca­non or Musquet-shots, the nearer they are shot the Horizon, the more they sweep and clear the same, and the greater Execution they do over the Fields against the Assaulters: For a shot coming from alost, can hit the Horizon but in one place; whereas contrariwise a shot being made about the height of a man parallel with the Horizon, doth sweep the same off continually; as is seen in the Fig. 7. Numb. V. where the shot made by the Musqueteer 4, standing at the top of the high Wall, hitteth and sinketh into the Horizon but in one place, as in 1: but the Musqueteer standing in the Fausse-bray A, shooteth the Line 3 2 6, parallel with his height; whence it is manifest, that in such manner it has a greater effect. To help these deficiencies, I betake my self to mediocrity, that is, to a mean height, and make the Wall in my great Fortification with Parapet, and all from 18 to 24 foot high, and the upper breadth of it, not reckoning the Parapet, as much as a Demy-Canon, having discharged an ordinary load, will run back. I take it from 20 to 30 foot broad, the Base is regulated according to the sloping of the Wall: but I do not make my Walls alike every where, but make the Profiles lighter or stronger according as the part is; for I make the Profiles of the Faces of Bulworks much lighter, than those of the Courtines or Flanques, and that with great reason, as I shall shew in its due place.

Of the Faces of Bulworks.

THe Faces are the weakest parts of the Fortification, see­ing they lie nearest to the Enemy, and cannot defend themselves; but their Defence must come from the next situated Flanques, which are made for that purpose. The Faces are always first Attacqued by the Enemy, unless the Courtines be too long. When the Enemy is past the Moat, he Undermineth this part, to make a Breach fit for an As­sault. To prevent this, as far as it is possible, there is no better remedy, than to hinder the coming over of the Gallery [Page 32]by well ordered Flanques: But being weakened by time, and the Enemy bringing, as it were with leaden shoes, his Work as far as to the Wall of the Faces; then it is time to know the way of making Counter-mines in wet places: for where the Moats are dry, there is made another kind of Attacque. To make these Counter-mines very hastily ready, as it ought to be done, the Wall of the Faces must be no thicker than onely to be Canon-proof, and that there be onely place for some Canon, to be used there, when occasion requires. For in case it be very thick, the Miners can the more securely accomplish their work, and the Mines do greater execution than otherwise. The Canon which is placed upon the Fa­ces, is commonly set upon the point of the Bulwork, where the Lines meet; and where this is done, there is after this manner place enough at the top of the Wall near the Point of the Bulwork, though the Wall be very thin. I make it in my great Fortifications from 16 to 24 foot broad, to be reckoned from the Parapet, whose top or surface I extend to the breadth of from 12, to 18 foot.

Of the Flanques.

I Make the Wall of this part of several heights, see Num. II. I divide first the Flanque 5, 4 into two equally, as in 8; then I draw from ⅓ of the opposite Face the right Line A 6, 8, 7, by the Point 8, so long, till it touch the Perpen­dicular upon A, 9 as 9, 7. This done, the Lines of the re­tired Flanque are made. The Profile of the Flanque I de­figne thus, following the Fig. 8. Numb. V. Let here the Line K, H, G, D, A be taken for Horizontal, beginning from A, which is the inner brink of the Moat; I mark from A to B, 5 or 6 foot for the berme; from B to B, 24 foot, or so thick as I desire to have the Base of the Parapet of the Fausse-bray; from B I dig under the Horizon 2, 3 or 4 foot, according as the water standeth high or low in the Ditches. At this depth I make an even place from B to D, 30 or 35 foot broad, fit for the Canon or Musqueteers. Before this place I or­dain the Parapet of the Fausse-bray, 7 or 9 foot high, as from B to L: in case one would lodge Musqueteers there, Benches are made behind, and besides a convenient form is [Page 33]given to the Parapet; then the Flanque of the Fausse-bray is made, which I make as low, as the place will permit, yea so deep, that the Canon-shots are even with the Horizontal Line, and lower when the Moats be dry. After this I measure from D to E 27 foot, from E, I draw a Perpen­dicular E, 8 foot high from the Horizon. At this height I make the first Flanque of the great Wall, with its Parapet of 8 foot high, the breadth, as need requireth; as is seen clearly in the Figure: this place or Bedding for the Canon is 10 foot higher than the lowermost. After the same man­ner I proceed with the last, second and third Flanques: But observe always, that one place with its Parapet may come so much lower than th'other, as that the shots coming from the one or other, be not interrupted; and the smoak and blow of the upper Canon do not much trouble those, that are in the lower Flanques. But for these inconveniencies, I shall pre­scribe sufficient remedies hereafter.

There are many more Considerations, required by the pla­ces and situations, which I leave to the understanding Ar­chitect. The Figure 8 sheweth the particularities something near. Against these raised fourfold Flanques, one might ob­ject and say, That the Canon standing upon the two hinder­most places, being Discharged, would let fall its stops, and Fire into the two foremost, and kindle the Powder, which is used for the Artillery standing below, and so make dumb the Canoneers that are ordered to be below. Concerning the stops, I have seen many a time, and observed, that they flie 20 or 30 Rod from the mouth of the Pieces, especial­ly being wrapped close together, and sometimes much further; so I did not see where they fell. Here it is to be observed, That those Pieces that stand upon the last and highest Flanque, are distant but 13 Rod from the first or Fausse-brays Parapet, from whence it appears, that such stops can do no harm: Put the case it were so, yet this obstacle cannot hinder the advantage, which I reap by these places; for in time of Attacques, I make in the Courtines little hou­ses supported with Wood, like Mines, wherein the Powder being in Casks, and other things necessary for the great Guns are put. Those Pieces that are Charged and Primed, I co­ver with a small Leaden Plate, till they are to be Dischar­ged. It is also to be remembred, That all the Pieces of these [Page 34]Flanques must not be discharged at once; but first from the uppermost or lowermost, and then from the others; and those in the lower Flanques may be advertised to retire them­selves into the said little Houses, or in the entry of the raised Flanque, in case this precious Musick do too much offend their ears.

Of the Moats.

THere are two kinds of Moats, viz. Wet and Dry. Of the Dry I shall speak afterwards, because they require another kind of Prosile; but now onely of the Wet Moats. What they serve, and are made for, every one knoweth: Concern­ing their fashion and making, I give them a convenient wide­ness, yet according to the nature of places, broader or nar­rower: where there is good Ground, I make them ordina­ry as broad, as the length of the Flanque, which falleth Per­pendicular upon the Courtine, or 10 Rods; and as deep, as I have need of earth for the Wall. He who desires to cal­culate, may cast up their depth according to the Profiles; I esteem the deepest for the best, as well in dry as wet pla­ces. To design then the Ground-lines of the Moats, I draw Parallel-lines with the Faces, at such a distance, as their breadth should be; and where the Parallel-lines intersect one another (which in Regular places is before the midst of the Cour­tines, and Bulwork-points) there are the extremities of the Ground-lines; as is seen in the Figures Numb. III. and 4. Just before the Bulworks-point I make the circular Ark, 1, 2, 3, at the distance of the breadth of the Moat; see the Figure, Num. II. by Y. And this is sufficient as concerning the wet Moats.

The Construction of the Profile of the Walls for Ravelins, Counter-guards, Hidden Ways, and what dependeth on it.

TThe Walls of the Ravelins joyning to the Counter­guards, I make something higher and stronger; and those of the Counter-guards, lower and weaker. Likewise those of the Ravelin before the Counter-guard, still some­thing [Page] [Page]

[Page] [Page] [Page 35]weaker and lower, and I order the Profiles as follows: The great Wall with its Parapet being 20 foot high, I make the joyning Ravelin with its Parapet 13 foot high; the top of the Wall, to be measured from the Parapet 20 foot broad; the Surface of the Parapet 12 or 15 foot, and the rest proportionally; the Wall of the Counter-guard with its Parapet is made 11 foot high; the top of it 16 or 18 foot broad, to be reckoned from the Parapet; the top of the Parapet from 10 to 15 foot; the rest accordingly. The Wall of the uttermost Ravelin with the Parapet I make 9½ foot high; the upper breadth of the Parapet also from 10 to 15 foot; the top of the Wall from 12 to 16 foot; the Ditches from 50 to 60 foot wide, and the deeper the better; the Hidden Way from 15 to 25 foot; its Parapet 6 foot high; the sloping (or Esplanade) is regulated after the Wall of the Counter-guard. To have a clearer declaration of this, see the Profiles Fig. 8. which are both made after the Scale ad­joyned; and the uppermost sheweth the Profile of the Wall of the Courtines, with the Moat before it; besides the whole Capital-line with the Profile of the innermost Ravelin, or Counter-Guards-Ravelin and its Ditches, which continueth by the letter A, A; the remaining space is the Capital of the outmost Ravelin with its Profile and Ditches, as also its Hid­den Way and Esplanade, which in the Print Numb. IIII. by the III. manner is shewed. The lowermost Profile shew­eth the thickness of the Wall upon the Faces, the wideness of the Moat, the distance of the Counter-guards Wall, with the Profile of it; and the wideness of the same Ditch, which continueth by the letters B, B, on which the Hidden Way and Esplanade is also described. This is enough as touching the Profiles of the wet Moats, for such as are skill'd in this Art. Where dry Moats are, I make quite other Profiles, for they are to be designed according to the nature of pla­ces. In the Print Numb. VI. there are described Profiles for two several dry Moats. In the Moat of the first by the let­ter A is the Cunette, or false Moat, strengthened with a thin Parapet and Palisadoes, which serveth for Enterprises, and is also very good in Sieges; whose use I shall describe in the second part of Defences and Attacques. The Ditches of the Counter-guards and Ravelins, may also be sunk to the half, or above one third part, to be measured from the [Page 36]VVall, and strengthened with Palisadoes against Invasions, as may be observed by the letters C and D: the inward sloping of the great dry Moat must be raised of a Brick-wall; as is seen by E E. These Brick-walls must be made thick and high, according to the nature of places, and the Princes or Lords purse. At the top of these Walls, there is plant­ed a Thorn-hedge, which is either bound, as by F and D; or else planted wild, and left to grow close through one another, as by C and G. These are also very good, for they make the Earth stand sirm with their roots. The Profile A continueth with the letters H, H, where the cutting in of the two hidden Ways without Ditches are described. In the Architecture or Building of dry Profiles, there is requi­red great experience in the practice of War; for they must be made with great skill; and being so made, I prefer them before those that are fill'd with water, of which I shall give reason in my Discourse of the business of Defence.

The Profile of the Retired Flanques is seen Numb. V. Fig. 8. according to its Scale. In the same Print is also seen a Bulwork with part of the Courtines on both sides; as it ap­peareth from within Fig. 12. Numb. V. Fig. 11. is a Bulwork, looking against the Flanques, which Ground of two Bulworks is designed, Fig. 10. Fig. 9. is a perspective of the Counter-guard, with its two Ravelins, viz. that which joyns to it, and the other, which after the manner of an Island lieth before it. In the Print Numb. VII. is seen the Perspective of this whole Work. First the Ground-lines of two Bulworks of an Hende­cagone according to the middle Royal, whose outward Polygone is 90 Rods, wherein the disposition of the whole VVork is perfectly seen, with that of the Counter-guard, besides their Ravelins and Counter-scarp. In the second is seen, how the whole VVork lieth in Profile; yet this forelying Coun­ter-guard hath another disposition, and is without Flanques of its own; but it is Flanqued from the joyning Ravelin, and from that, which lieth in the Island. This manner of Coun­ter-guards is also very good, and doth not take in much space. In the Figures all this is clearly seen, and therefore it requires no further explanation for the understanding thereof. I have yet many other speculations and particularities, concerning these manners of Fortifying; which I shall keep to my self, till I find a fit opportunity to put them in practice. If God give [Page] [Page]

[Page] [Page] [Page 37]me health, till I bring to light the other parts (viz. of Attacques and Defence, of Undermining, Counter-mining, Fire-works and Artillery, when it may be done most conveniently) then the advantages of this manner of Fortifying will be better con­ceived and understood; for the quality of every part of a Forti­fication shall there be declared one by one separately and di­stinctly.

Of Irregular Fortresses.

I Shall say onely this, That it would be too long to shew here particularly the manner of working, which I esteem as need­less, seeing that Irregular Fortresses are made up out of parts of Regular ones: The calculation thereof may be done after the Modern practice, and the parts afterwards reformed according to my way; as hath been shewed in the Regular. There are many Engineers, who think it an Art to Fortifie any places altogether Irregular, although they might well enough alter them. I do not glory in this; but am very well content, when I meet with such an unfit Figure, to reform it after the Regular, as near as the place will permit. A Regular place, besides that it hath its De­fence more equal, is also the fitter, and taketh in with the same Lines more space, which I prove thus: Let there be a Right­angled Square, each of whose sides is 12 Rod, and the four together 48 Rod, whose content is 144 square Rod: Let there be another Irregular Square or Rhomboides, having al­so four streight Lines each of 12 Rod, that is, the four of 48 Rod; let this Rhomboides be made of two Equilateral Triangles, then the shortest Diameter is 12 Rod, and the longest 20 Rod, and 8 foot, whose content is therefore 124 square Rods, and 8 square Feet; From whence it appeareth, That the Rhomboides contain­eth 19 Rod, and 2 square Feet less than the Quadrat; and the more the Rhomboide recedeth from Regularity, the less is the content of it, &c. But this is true of Irregular Fortifications, That the Rules and Maxims of the Regular being known, the good or bad Form, as also the strength or weakness dependeth wholly on the knowledge and experience of the Engineers, &c.

I.

II.

III.

IV.

Arithmetique in Decimals.

THis Calculation doth not differ from the ordinary Me­thod, onely in the understanding of the Signs of the Fractions, and how to place them rightly, which is as fol­lows:

First take a Rod, (which is the usual measure of the Coun­trey where you are,) or what measure you please, and call that an Integer; this divide into ten equal parts, and take one part for a Foot; this Foot divide again into ten parts, and one of them take for an Inch; the Inch divide again into ten parts, and call one of them a Grain; the Grain divide again into ten, one of which call a first Scruple; and that divide again into ten, and let one of those parts be called a second Scruple; and so on to the tenth Scruple; which tenth Scruple is 1/10000000000 part of a Rod, and is so small, that it cannot be compre­hended otherwise, than by understanding the following Signs:

• A Rod, or Integer, ⓪
• Foot, ①
• Inch, ②
• Grain, ③
• First Scruple, ④
• Second Scruple, ⑤
• Third Scruple, ⑥
• Fourth Scruple, ⑦
• Fifth Scruple, ⑧
• Sixth Scruple, ⑨

and so on to the Ninth Scruple.

Addition in Decimals.

OBserve here the same manner, as in the Addition of Inte­gers; but have a care, that the Signs of the Fractions of one sort, do stand under one another; as Rods under Rods, Inches under Inches, &c.

Substraction in Decimals.

PUt here also the Signs of one sort under one another; and where they fail, fill up the place with 0.

Multiplication in Decimals.

HEre follows also the common rule; but you must add the Signs of the Fractions together.

Division in Decimals.

THis is also done the ordinary way; but here the Sign of the Divisor must be substracted from the Signs of the Dividend, or the Number to be divided: and in case the Num­ber to be divided, be less than that by which it must be di­vided, joyn so many 0 to it, till it come to be equal to it.

The Rule of Three in Decimals.

THis is performed after the same manner, as in the Integers: and forasmuch as it consisteth of Multiplying and Divi­ding, observe the foregoing Rules.

To extract the Square Root in Decimals.

TO sind the Root of a Number given, even in its smaller parts, put after the Number given towards the right hand, as often two Cyphers, or 00, as you desire parts, and half the Number of the Cyphers is the Sign of the Fractions that belongs to the Root; as may be seen in the Example.

To extract the Cube-root in Decimals, or Fractions.

OPerate in the same manner, as in Integers, and look how many Fractions you desire, so often put 3 Cyphers be­hind it; and this wanteth no further declaration, &c. Having thus shewed the Arithmetick in Decimals, I proceed to shew some Draughts and Operations with the Compasses, which any one may use at pleasure.

How a given Circle is to be divided into equal parts, and the Polygones of all Regular Fi­gures, necessary for Fortification, are to be inscribed within it.

LEt the given Circle be A, B, F, E, Numb. VIII. Fig. 1. through which draw the Diameter A B; and where it toucheth the Circumference, as in A and B, put one foot of the Compasses, and make the two Intersections, C and D; and from the Points, where they cut, draw a streight Line C D, falling Perpendicular upon the Diameter A, B: these two Lines A, B and C, D divide the Circle into four equally, and A, E is a side of a Square.

Taking F for the Center, and drawing at the distance F, G an Ark H, G, I, and then a right Line from H to I; where this Ark cutteth the Circumferencc, as H, I, this is a side of an Equi­lateral Triangle, which may be inscribed within this Circle. The half of this Line, as H, K, is a Polygone of an Heptagone, and the distance I B is the side of a Dodecagone; Take K for the Center, and at the distance A, K describe the Ark A, L, the Line drawn from the Section L to A is a side of a Pentagone, the Line G, L is the side of a Decagone; G, N of an Hexagone; E and B are the Centers of the Arks G, M, which Line divideth the fourth part of the Circumference into two, and N, B is the side of an Octogone; divide the Ark H, F, I into three equal parts, and one third, as H, O is the side of an Enneagone; from L to P is the side of an Hendecagone; E, L is the Polygone of a Figure of 16 sides.

Another way to find the Sides of every Regular Figure, that may be inscribed within a Circle.

THis is done thus; Divide the Circumference into four equal parts, and then one fourth part of the Circle into so ma­ny equal parts, as the Figure should have sides; and take four of those parts, and that will be the length of the Polygone of such a Figure as you seek.

To divide a Line with the first opening of the Compasses, into as many parts as you desire.

LEt the given Line be D, E, which you desire to divide into ten equal parts; therefore draw an undetermined Line B, H, and open the Compasses at pleasure, and with that opening set off as many parts upon the Line as you desire, (as here ten from B, to H;) so the tenth part reacheth to H; upon this Line B, H make an equilateral Triangle B, A, H, and from A draw the Lines AI, AK, AL, AM, &c. Thus you have by the parts of the [Page 51]Line B, H, a Triangle, which is divided proportionally: then take with the Compasses the given Line D, E, and set it off from A to G, and also from A to P, and draw the Line P, G, equal to D, E, which by the intersecting Lines is divided into 10 equal parts; as also the Line F, G, if it be required, and so with all other Lines whatsoever.

To let fall a Perpendicular upon a given Line.

THe given Line is G, F, and upon the point G we let fall the Perpendicular G, H; therefore set one foot of the Com­passes in G, and open th'other at pleasure, as here to I; from thence make the Circumference H, G, F; and where this cutteth the Line G, F, (as in F) from thence draw a Line just by the Center, till it touch the Circumference in H, and from the Point H draw the Line H, G, which standeth right-angled or Perpendicular to G, F. There are many other ways, which I here omit for brevity sake.

From a given Point in a Line, to raise a Perpendicular.

THe given point is A, and the Line B, C, in Fig. 5. Set one foot of the Compasses in the point A, making with th'other the Ark B, C; upon these Points set one foot of the Compasses, opening th'other at pleasure, as to D, and make from these Points the intersection of the Arks in D, at the distance D, C, and draw a Line from the intersection by the given point A, to the Line B, C, which is Perpendicular to B, G.

To make an Equilateral Triangle.

UPon the Line B, H, in Fig. 6. set one foot of the Compas­ses in B, and with th'other take the distance B, H, and with that make an Ark about A; as also out of H: where these Arks intersect one another, there is the third Point of the Tri­angle; and draw from thence the Lines A B, and A H, and so the Triangle is made.

To make a Triangle of three right Lines given, whatever they be, provided that two of them be longer than the third.

THe Lines are AB, CD, EF; take with the Compasses the length of one of the three, which you would have to be the Base, (which is here A, B) and mark two Points, one in H, and th'other in L; from the Point L describe the length of the Line C, D with an Ark in K, and from H the length of the Line E, F, and cut the former Ark in K, and there is the third Point: from K draw the Line K, H, equal to E, F; K, L, equal to C, D; and L, H, equal to A, B, &c.

Three Points being given, and not standing in a right Line, to draw a Circumference, which shall pass through all three.

THe Points are O, P, Q; make the equal sections on both sides of the Center, as from P and also from Q, which is done in V; as likewise from O, which falleth in 5, Perpendicular upon Q, P, and P, O. Where these touch one another, there is the Center (as is seen in Fig. 7. Numb. VIII.) of the Circular Ark O, P, Q, which being drawn, &c.

The same manner observe in the finding of the Center of a Circular Ark, or a whole Circle.

To divide a given Circle into 360 degrees, and to make an instrument from it which Geome­metricians call a Theodolit, or Astrolab.

THis Instrument is of great use in the Mathematicks; for by the same all inaccessible distances, heights and depths are measured, the use of which I shall declare in its own place in the following parts, and it is divided thus:

The Circle A, B, C, D being first divided into four equal parts by the two Diameters A, B and C, D, take with your Compas­ses the exact length of the half Diameter; as A, E or E D, by [Page 53]which the Circle is divided into six equal parts, and after the manner following into 12; with this opening of the Compas­ses, set one foot in C, and with th'other make upon the Circum­ference marks on both sides in F and G, likewise from D in H and L, &c. Thus the Circumference is divided exactly into 12 equal parts. The part D, A, or a quarter of a Circle, containeth 90 de­grees, as a quarter of 360 degrees; F, C being a sixth part, 60 de­grees; and D H as a twelfth part, 30 degrees; then divide a twelfth part, as D, H or D L into two, as in O and P, and such a part holdeth 15 degrees; then divide one of these parts into 3, and one of them will hold 5 degrees, of which one being exact­ly divided into 5, one of these is a degree, of which 360 make up the whole Circumference of every Circle, whether it be great or small, and are always proportional, viz. the least Cir­cle that can be made, to the greatest Circle of the Sun, or any other Astronomical Circle: For if you draw several Circles out of one Center, and from the same Center several Semidiameters to the utmost Circle, the Arks intercepted in the innermost Circles contain as many degrees, as those in the outmost, &c.

Each degree containeth 60 Minutes, a Minute 60 Seconds, a Second 60 Thirds, a Third 60 Fourths, and so on; and they are written thus, 4.° 5. 6. 7. 8. which is as much, as 4 De­grees, 5 Min. 6 Seconds, 7 Thirds, 8 Fourths.

How all kind of Ovals may be drawn by the Compasses, is to be seen by the marks 1, 2, 3, 4, 5, and needeth in my opinion no further declaration, seeing the manner of working appear­eth enough by the Figures. By 6 is an Oval, which can be made by a Cord and 2 or 3 Nails: This manner is of much use in Ar­chitectura civili, to make oblique Vaults. By 7 and 8 are seen two sorts of spiral Lines, whose structure may also be perceived by the Figures.

And thus I have shewed the principal Draughts with Com­passes, serving for Geometrical use.