A TREATISE OF ARTIFICIAL FIREWORKS Both for Wars and Recreation: with diverse pleasant Geometrical observations, Fortifications, and Arithmetical Examples.  
In favour of Mathematical Students.  
Newly written in FRENCH, and Englished by the Author THO: MALTHUS.  
Printed for RICHARD HAWKINS, and are to be sold at his Shop in Chancery lane near to Sergeant's Inn. 1629.   

A TREATISE OF ARTIFICIAL FIREWORKS by F Malthus LONDON for Richard Hawkins in Chancery Lane. 1629   

THE PREFACE Apologetique to the Reader.  
ALthough that Wars have been authorized by the divine powers, as witnesseth the new and old Testament, yet me thinks I hear many nice, peevish, and ignorant brains blaming me for setting down to the view of the world (at a few leisurely hours) so many ways and inventions how to corrupt and destroy what nature and art strive to make, edify and conserve, to whom I answer they have the same reason to exclaim against justice which hath found and ordained many means to abolish and exterminate wicked offenders: for my intention is here but only to find inventions to confound and ruinated Rebels and their habitations; that afterward Empires, Kingdoms, and Commonwealths may the better live in peace and tranquillity: and what more is I have set down perspicuously this Treatise of Fireworks, which many heretofore have written with confusion and great danger, having penned large Volumes with most ample and idle discourses touching this subject, whose names for modesty's sake I will not here set down with an infamous character to posterity (as Master Robert Norton hath done some few slight offenders names in the Preface to his translation of Vianos' works, which he calleth his own, having only added seven figures taken out of Thybourels book, and transposing the Authors works to disguise it) that I be not accounted amongst the calumniators of this age, who write with serpent's tongues spitting their venom upon the silent and sleeping innocents gone before us. But I will only show their errors in general, that the learners may the better avoid them: For in some you shall see prescribed all sorts of Apothecary's drugs for the compounding of Fireworks, which are no more capable of fire then stones, earth, or mettle, as Adamant-stone, Verdigrease, Crystal, Vitriol, Salarmoniacke, Sublime, Mercury, with a number of the like, which in the judgement of ignorants may produce in Fireworks wonderful effects. In others you shall find the conjunctions of combustible materials in such a sort or manner disposed and ordered to such strange ends, that you would say they were set down to make the Reader laugh, rather than for service of Fireworks either for Wars or Recreation, as the means how to poison an Army in the wide Field by the smoke of an artificial Ball, is it not to be thought a very pretty imagination, or else that the Ball ought to be somewhat big; and afterward which is more ridiculous, you shall find in the same Authors the same ingredients and mixtures prescribed for Fireworks for Pleasure and Recreation: which to me seems strange, that the Fireworks whose smokes are so venomous even as to poison Armies in the open Fields, shall be also fit to burn in the middle of a City or Town amongst thousands of people for Recreations: now all these idle conceits with a number of the like are printed in many new and recent books, the Authors whereof doubtless have set down all these ridiculous inventions by ignorance, (thinking them good) and not of purpose for spite; and therefore they do deserve pity rather than blame; as well as Master ROBERT NORTON, (who hath erred only in attributing another man's works, and faults to be his own) and what is it that gives me cause to think that all these have erred by ignorance, is because that all their works are alike where that they do speak of Fireworks. But reading these absurdities with a great quantity of others, which made me take resolution to set my pen to paper, and show to posterity how and what way to avoid their errors, having here set down the true rules and mixtures required for Fireworks, both for Wars and Recreation, with some few Geometrical observations necessary for the practice of Fireworks in wars; and also I have added unto it certain rules and maxims of Fortifications, very requisite for Enginiers to have in their memory, with a compendious Arithmetic, having written the whole first in French at some broken hours whilst I followed the wars; and bringing certain copies from the Press to my special and very good friends; at whose requests I have translated the same into English, that others might not translating it, mistake my meaning; and the work being already but of small commendations, they to make it of lesser. Adieu. London the 22. of May. 1629.   

To the Author.  

THy Archimedean hand hath learned to frame  
Celestial Meteors out of Nitrous flame:  
And represents strange fires of different sorts  
Suited to Martial use, & Courtly sports:  
So pleasing that great Kings have spared some hours  
To be spectators of thy golden showers.   

The Thames and Seyne have seen thy Balons fly,  
From their affrighted bosoms to the sky:  
Swift beyond all belief; as if thy aim  
Were, to restore Prometheus' theft again.  
Were, when their force seems spent, breaking asunder  
They rain down stars, or else outroar the thunder.   

The Roman Apotheosis I deem,  
Would still have had a reverend esteem,  
If they had used these miracles of fire,  
To mount their Caesar's souls, & bear them higher  
Than Eagles flight, but now they serve to raise,  
The Pyramid and Trophy of thy praise.   

And sure thou hast attained sufficient glory,  
In perfecting the Pyrotecnique story:  
Which some took upon trust, & by concealing  
Their creditors; made borrowing to seem stealing.  
But their fire's out, now thine hath passed the Press.  
For greater fires do extinguish less.   
WILLIAM BASTIAN studious in Mathematics.   

A TREATISE OF FIREWORKS. for War.  

CHAP. I.  
TO imitate nature, our first and chiefest teacher, who far from error seems to have produced and brought forth all things ●y a curious and special order; without whose beautiful disposition on the whole world had still continued in the most prodigious con●●sion of Chaos, which displeased the ●●●ure aspect of the highest, eternal and divine powers, being but a cloud or mixture of darkness. My intention is here to institute and observe an order, treating of every thing one after another in his proper and convenient place: beginning with artificial Fireworks, as the chiefest and principal cause which hath urged me to take my pen in hand, the description of those which belong to wars, shall obtain the first place, and next the Fireworks for recreation or pleasure, passing from these to many pleasing Geometrical observations, aswell of Mecanical observation, as by the sins, tangents, & succents, joining to them a most easy & succent method to fortify all places regularly and irregularly. And to conclude I will add many brief and instructive Arithmetical examples.  
Now to begin with the Fireworks for wars, I will first describe the Morterpeece, and the use thereof: that being an instrument, the most noble, the most universal, the best of greatest effect, and of most wonderful operation of all the instruments, the practice whereof may be used amongst Fireworks for wars, for this instrument may serve for a Petard to split, break and hurl down doors, gates or walls, likewise to massacre, tear in pieces, overthrow and confound assailants of any place or breach: and for diverse other most worthy offices, and accidental occasions, which happen in the troubles of wars: whereof I will not here make a long, weary, and tedious discourse, but in few words clearly set down the use of this instrument for the shooting of diverse sorts of granads, stones, or other weighty burdens to ruin rebels, their habitations and dwelling places. Then first of all I will treat of the mettle, whereof this instrument ought to be made, and measures appertaining thereunto.   

CHAP. II. The manner how to make the Morterpeece.  
THis Instrument may be made of diverse sorts of mettle or stuff, according to the means, times and occasions which shall offer themselves to those who would make them, or cause them to be made. The first and chiefest matter of all, is red copper, brass and tin, but very little of the two last, such as Canons are made of, without any other brickle or harsh met●●● mingled with it, and being made of this stuff, let the measures following be observed. If the diameter of the caliber or bore be one foot, let the mortar be two foot of length, & let the sack or hole for the powder be the third part of a foot broad, and half a foot deep, and let the mortar bear in thickness an inch and a half, about one foot high, and the rest only one inch thick, the foot shall be one inch and a half thick, and made square whether the instrument be for the service of wars, or recreation, as doth plainly demonstrate the Figure A following. The second mettle is iron, such as commonly Ordnance for ships are made of: and being cast of this stuff, the rules prescribed shall be found fit, only let the mettle be somewhat ●hicker in every place. The third ●nd fourth, and most common materials are past-boards and can●as, or pastboords and chord, and either or these materials must be joined together with glue, and being to be made of either of these materials there must be had a wooden foot, with the powder sack or hole made hollow below in the wood, as is represented by the Figure B. the precedent measures being observed which is the third part of the diametre, etc.  
But if the instrument be to be made of either of the two first metals, that belongs to the Founder's office, but of the two last materials every man at his own pleasure may compose it of what size he please, and to begin he must have a wooden rowler of such bigness as he shall desire to make the diametre of his Morterpeece, and upon that rowler let the pastbords and canvas with good store of glue be rolled: which being done, let them dry a while upon the rowler, and another while off from the ●owler, and when this kind of trunk is very dry, let the wooden foot be joined to the one end with glue and nails very fast; and then cover the whole with chord and glue again; which being well dried, the instrument may be of long service, provided that it be not overcharged, and as for the length, the longest carry furthest, and the thickest dure longest, by the figure C. precedent the instrument is represented perfect, with the touchhole in the right place required.   

CHAP. III. The manner how to make granades or mettle for the mortar, or hand.  
ALl granads being made to break, aught to be composed of the most brickle mettle that may be found, as of brass, adding the third part of Tin to it. (as the Founders know right well) they may be also made of iron, or of glass to cast out of men's hands, so that the glass be made very thick, and these will work wondrous effects, especially in any throng of horsemen or footmen. And as for the thickness of those which are made of brass, if the diameter be one foot, let the mettle be one inch thick, and let them be somewhat longer than they are round as doth demonstrate the figure A. leaving at the one end, as it were a handle, and on the other, a screwed hole; by which means the granad may be charged, also, let there be made a hollow vice fitting for the former screw, which vice shall be filled with a slow composition made with gunpowder well ●ruised and culled, and then made ●nto passed with oil of petrole: but if the powder be excellent good and strong, then mingle with a pound of powder, one ounce of Charcoal dust, to make it weaker, then let ●he composition be well beaten ●nto the hollow vice, which is represented by the figure B.  
And the granad shall be filled with fine gunpowder, which being full, let the vice be screwed into the hollow hole of the granade, only it is to be noted that the vice must not be open at the lower end, but have a bottom sodered strongly, wherein shall be made three or ●●ure little holes about the bigness of a tag of a point to give fire to ●●e powder when the granad hath f●owne the space required. For the performance whereof it is needful that you know the quickness o● slowness of the mixture wherewith the hollow screw is to be filled. And for the better experience, it shall be convenient to make of all of one with a granade made 〈◊〉 wood, cloth, pastboord, or a●y other stuff filled with earth, ●ut let it be near to the weight of ●●ose which you desire to make use of, afterward, and by that means you may know how long ●ou aught to make your hollow vices, which are called by their ●roper terms Portfires, and ●hus much of your granade being performed, now let it be all covered over, with either Chord, Canvas, or Pastboord, and dipped in glue or in pitch; this covering may be near half an inch thick: to the end that the granade going out of the Mortar with violence it break not, and work his execution upon yourself or your company instead of your enemies; of what mettle soever the granad is made, these rules are to be observed, and by the figure C, is represented a granade quite finished; and dipped into pitch, by which means it may be conserved many years: these sorts of granads work great effects falling upon houses; they cast down the walls and cover, likewise they work wondrous operations amongst either horsemen or footmen, tearing both man and beast in pieces, sparing nothing.   

CHAP. IU. The manner how to make Granads of Canvas for the Mortar.  
THe operation or execution of these sorts of granads made of Canvas is quite contrary to those prescribed: these are only fit to set a Town on fire, the houses whereof are most covered with reed, straw, or broom, they are not of so violent execution as the precedent, yet notwithstanding of as great cost, and for the making of them, you must have a wooden ●owler, which is represented by the ●igure A, of the bigness you desire ●o make the granad (which always ought to be less than the caliber or bore of your mortar, to the end ●hey may be covered afterward) ●nd upon the rowler make a sack of ●uch cloth as you please, as doth represent the figure B, cutting a piece to ●ouer the upper end, when the sack ●hall be full of the composition following. Take four pound of saltpetre, two pound of gunpowder-●ust, two pound of brimstone, all ●hese being well pulverised, let them ●e moistened with oil of petrole, ●nd then fill your sacks with the ●ame, and cover them with chord, which done, pierce the sack full of holes with a great bodkin, as representeth the figure E, and in every hole, place a little iron barrel charged like a pistol barrel, these little barrels are represented by D, which must be driven into the sack up to the head, and the granad being thus disposed, let there be made at the one end a hole about one inch deep, which shall serve to prime it with powder-dust, moistened with oil of petrole, only it is to be noted that the touch-holes of the little barrels be made somewhat large, to the end ●hat the rust stop them not, being ●ong time kept amongst the salt-peter, and so they may be conserved many years, and ready for service ●n all occasions, the figure C. doth ●emonstrate the granad perfect and ●urning.   

CHAP. V. How to make fiery arrows.  
TO avoid all confusion, I will treat first of all the fires which are cast with violence, and afterward of those which may be cast out of men's hands. And first it is to be observed, that fiery arrows are of great effects and noisome at sea, specially if the ships approach any thing near, so that they may be 〈◊〉 or cast into the sails chording 〈◊〉 of the ships, either by crossbows, long bows, or any other means, for so much as a ship being once a fire, hardly can it be extinguished. And thus the arrows ought to be made: first a long shaft of wood, ●nd joined to it an iron head made after the manner of the figure A, and about the middle of that head, make fast a linen bag in the form of an olive, leaving open a hole at ●he end before, as may be seen by the figure B, that it may be filled with the composition following. ●ake one pound of Saltpetre, half a ●ound of gunpowder-dust, and ●alfe a pound of brimstone in ●owder: all these ingredients ●eeing well mingled, and min●ed with oil of petrole, then fill ●e bag round about the arrow ●ead, noted by B, and then let all be well bound about with wire. And ●r the priming of these arrows, dip cotton week into gunpowder, wet with water, but let the cotton be well dried again before it be applied to the arrow head. Now for the joining of your arrow head, to the wooden shaft it ought to be so slightly fastened to it, that being shaken into any sail, cordage, or wood, so that if any one would pluck it back, the shaft should slip out of the head, and so continue burning in the place pretended, and to hinder that one may not pluck out the head with their hands, there may be made ● small hole quite through backward, and so the fire will hinder a man from touching any part thereof, although it should stick in other of his fellow's clothes. There may ●e made little arrows of the same fashion, to be cast out of men's hands in ●ny meetings, assaults, breaches, or ●ther occasions; and if there shall be ●reat haste for the making of them, ●esse fashion may serve, having only ●●ttle sticks of wood, about a foot ●nd a half long, and in stead of the ●ead prescribed, a pike (like a ●reat nail) and in lieu of feathers, pieces of past-boord, stuck into the ends of the wood being slit, but the former composition shall be required.   

CHAP. VI How the granads are to be charged into the mortar.  
ALthough it seem to be but a small difficulty to charge the granads into the mortar, yet it is the greatest mastery and most curious work which hath been found amongst fireworks, and the most industrious Engineers that ever I yet saw, have after been deceived therein, and shor ●ut their granads, they not taking ●ire, which is absolutely one of the greatest faults that may be committed. Then to avoid such great errors it shall be needful to note ma●y observations following, whereof the first is that you put not too much powder in your mortar, marmarked A, but only the powder-●ckefull; secondly, that the priming ●f your granad be firm, not spoilt with dust or dyrtie moistness, or other like thing, that it be not also too dry, lest all break about yourself going out of the Mortar taking ●re too quick, which all fireworks ●re very subject to do, then take ●eed that the granad enter not too hardly into the mortar, which will cause it to break in the discharging, also it is not needful that it enter too loosely. All these things being well observed, you must have always ready portfires for your mortar, which may be made about the bigness of your little finger, and hollowed within about the bigness of a quill, even to the bottom, noted B, this portfire is to enter with a vice into the touchhole of the mortar, about half ●n inch, and to turn the vice without any key: there may be made 4, buttons at the outside of the upper ●nd, and let this portfire be charged with a slow composition, mingling 4. ounces of charcoal with a pound of powder-dust, this being done, and your granad placed in your mortar, ●o that it shake not, close up the ●hinks round about your granade, within the mortar, with grease, wax, ●itch & grease, or any such like thing you may so cover the granad with ●ome such stuff; that in any weather wet or dry, you shall find no difficulty to shoot off the granad in any place wheresoever you please.   

CHAP. VII. The manner how to shoot the granads.  
THere is no less difficulty well to discharge the granads then to charge them, but the contrary will easily appear, for it is there where the hazards and dangers most great, do meet together. And the first of all are the adversary's Canons from whose dangers the Engineer shall easily conserve himself, as followeth, causing to be made a trench like unto an half moon, the connexity thereof being towards the enemy, as appeareth by the figure A, this trench shall be of such bigness ●s the Engineer may conveniently place all such things as shall be needful for his present use. The second difficulty following, is how to shoot ●hem right to the places of any Fort ●or Town desired, being not possible ●o see the places to take levelly at them without eminent danger of their Canon and musket-shot: which difficulty shall be thus avoided, and ●et levelly right, and send the granads ●o the places required: let there be ●e two staffs set upright, between ●he Town, Fort, or Citadel and the Trench, the one of these staffs shall be somewhat long, the other may be ●hort, and that, planted upon the ●rim of the trench; but the longest somewhat distant in a right line towards the place pretended, as doth demonstrate the figures BC, these staffs shall be so disposed that you may see the extremities of each, being in the trench out of the enemy's sight, and by the sight of these two extremities, you may direct your granads at your pleasure inclining your ●orter-piece upon either side, as shall 〈◊〉 needful without danger of the e●●mie, having only known the di●●●nce betwixt the trench and the twne, or place of the town which ●●u shall desire to destroy. The third ●●d last difficulty is to know the true length or distance that your mortar ●oth bear, of which to be certain, ●●u must try it before, shooting gra●●ds of the same weight, & with one ●●rt of powder, and with a level, take ●●vers sorts of elevations, & so may ●ou know such an elevation doth oa●e so far, and such carry so far with ●●ch powder such a weight, being charged, the manner of taking these ●●●vations is represented by the figure 〈◊〉, & the level in a bigger volume by ●●e figure E, this level being placed ●pon the mouth of the mortar piece, under the foot thereof: wedges, yo● may give to the mortar what inclinnation shall be required: and to b● sure from all harm of your own● granads, which by chance may brea● going out of the Morter-piece, le● there be made in the trench a little corner hollow into, or behind the which you may always retire, halving given fire to the Mortar. It shall be also needful that some one watch where the granad doth fall whethe● too short or too far; that you may the better direct the next shot: thi● method is to be observed in shooting all sorts of granads, stones, o● other like things.   

CHAP. VIII. A most violent method to set a Town on fire.  
HItherto there hath not been found any means or invention more certain or violent, whereby one might convey fire for the destruction of the enemy's habitations then by the help of the precedent granads, which very hardly cause any disorder or combustion, unless amongst houses covered with straw, whereof few are to be found in goodly towns. And therefore I will here prescribe a ●ew way much different from the others before written▪ against which hardly or not at all may be made any resistance or possibility for defence, the violence, rage, and order being of such a fearful power and execution, provided that the Engineer do but meanly effect his purpose & design, and for the practice hereof, let there be made a kind of well in the ground but sloping, as doth demonstrate the figure A, of such deepness as shall be thought fit, but it is to be noted, that the nearer you are to the town, the lesser slooping will be best, and most easy for the performance. The hollow pit or well being made, let there be placed in the bottom thereof one or two barrels of gunpowder, as shall be seen needful, according to the proximite or ●●stance between the place and ●●wne, & these barrels may be places ●●●ry conveniently placed with good ●ore of straw about them to keep ●●em from the moistures of the ●●rth. And in one, or else in both, 〈◊〉 shall place a gunlocke, making 〈◊〉 to the clicket thereof a chain of sufficient length to reach out of the hole upon the plain ground, but this chain shall be curiouslyy lodged in a small gutter, which afterward shall be covered with earth, to the end that the fire do not over-heate it, then cover the barrels of powder with strong wooden planks, or broad stones, covering them again with earth, and then lay good store of fresh straw and bavins, and faggots, and in the midst of these let there be placed a great faggot, made of pieces of timber, of a reasonable length, being well bound together with chains of iron in three or four places, as doth appear by the figure B, and then set fire on the straw wood, and all that shall be in the pit▪ giving the great wood time to kindle, and this great faggot being in a great arder, let the chains be drawn which are tied to the latches of the gun locks, and doubtless you shall see brave sports; and if there be many pieces of Ordnance about the town or place thus besieged, let them be all disposed ready to shoot where this great faggot falls, and whosoever would but this invention in practice, aught to make many of these pats about a town, giving fire to them all at once: but the Inginier to know justly how far these great flaming faggots may fly, the hollow pit declining so many degrees, he ought to make trial before in some large place, where he may have room for his experience: now resteth only to explicate how a man may give fire to the stra● in the pit without danger, for so much that every man would not lest all should take fire together. Then to be free from this peril, it may be fired by a train of powder, by a Gunlocke, a piece of match, a candle's end, or other like invention: Peradventure, you may think that there may be danger in drawing the chain when all is on fire; to which I answer these may be joined to it, a chord of such length as shall seem necessary; The Engineer who shall put this in practice, may add of his invention what he thinks fit, Facilius est enim addere invento quam in venire.   

CHAP. IX. How to make Granads to cast with men's hands.  
HAving taught how to make Granads for the Mortar, which by the violence of that most wonderful instrument may be transported very far over waters, walls, or bulwarks, working strange and rude executions or effects, I will now show how it shall be easy to make two or three sorts of Granads to be cast out of men's hands, the effects whereof are no less to be esteemed then the other, chiefly in assaults, whether it be for the offendants or defendants. The chiefest and of greatest operation is made of any kind of brickle mettle as the precedents▪ and their fabrication is not much different from the other, only more rounder and lesser, and the single ones having a long neck to seem for the Port fire, as doth appear by the figure A, and first of all the granad is to be filled with fine gunpowder, and afterward fill the neck with a slow composition as followeth. Take 1. pound of gunpowder dust, two ounces of charcoal d●ust three ounces of saltpetre, all these well mingled together, fill the portfire required, but this mixture must be well beaten into the port-fire, and the granads being thus finished, they may be kept a year, ten years, yea twenty if need be. It is to be noted that ●hese granads must not be freed until a man is ready to cast them away even in the same instant, because the port-fire being consumed, the granad breaks in diverse pieces, and each piece sufficient to kill either man or beast, and therefore the handling of them is somewhat dangerous. The second manner of Granad to be cast out of the hand, is made also of brickle mettle, but hath no neck for a Portfire, but is only round, having in one side a hole as appears by the figure B, and at this hole it is to be filled with dry powder, and then is to be put into this hole to stop it, cotton wieke made as followeth. Take good Gunpowder and steep it in fair water, and being dissolved, take the cotton wieke and steep it in the Gunpowder dissolved, which done, hang it out to dry; but let it be doubled eight or ten times, to the end that it may the better receive the powder, and being dry, cut it in such pieces as you shall need, putting the one end into the Granad at the hole, and then put that Granad into another pot a little bigger than the Granad, as appeareth by the figure C, so that you may put a little dry powder round about the lesser Granad between the pots and Granad, then cover the mouth of the last earthen pot either with leather or such like stuff; and bind it round about with wire, but first put the wire through pieces of match about half a foot of length as may appear by D, and when you desire to make use of this sort of granad, set on fire every end of these pieces of match, and cast the pot away, which falling either upon the ground, or any soldier's Armour, you shall see the earth-pot break, the Granad split, and work such furious effects, either at a breach or any approach. The third sort of Granad is made much like to either of those precedent, except only it may be made double or single, of glass or of earth only shall be needful to be made very thick, and if it be to be made single, the portfire ought to be either of wood or of pastboord, and well glued in, and this last sort may work great effects. Now having somewhat largely treated of the mortar, granads, and of their use, I will speak a little of other oily fireworks very pernicious and noisome to all assailants.   

CHAP. X. How to make fiery wheels to be cast with men's hands.  
FOr as much as every soldier will not meddle with making nor casting out of his hands granads, the handling of them being somewhat dangerous, I will here teach another manner of fireworks, which is not much less offensive, which being well compounded and compacted, and for the making thereof, these are more ingredients required, then for the precedents, and is more difficulty in making of them as followeth, Take four pound of gunpowder in dust, one pound of charcoal dust, two pound of tar, two pound of salt-peter, and one pound of Rosin. All these ingredients being well mingled and heat over the fire, steep tow or flax in the same, ●n● then wrap the tow or flax about ●● whope, which is represented by A, and then cover all this again with Gunpowder dust, & in time of need give fire to them, and cast 'em amongst your enemies; only is to be observed that the whops or circles ought not to be to large, but falling upon a many of soldiers they may there stay and stick, and to make them more troublesome and intangling, two may be tied a cross on another, and then falling upon any man, he cannot choose but be much astonished with such a fearful element, and put his company in great disorder.   

CHAP. XI. How to make a Ship of wild Fire.  
AFter a sufficient discourse of all manner of Granads, fiery arrows and burning wheels, I cannot think it extravagant or amiss to lay open to the industrious or curious, a method how to make a fiery ship at sea, which shall not begin to burn till it hit against some other vessel, rock, or such like obstacle, or be hooked or grappled on the inside to be drawn a shore. He that will undertake to build one of these must spare no reasonable cost to put his intentions or designs in execution, and first of all must be had one ship or many, according to the times and occasions wherein they shall be required; and within the ship shall be disposed great store of broken Canons well charged with bullets, and from one Canon to another shall be joined a port-fire, so that the first, having taken fire, all the rest must need do the same; these Canons shall have their mouths put out of the Gun-ports ready to be discharged; this done, let there be good store of dry straw, small dry wood, and tow and flax steeped in pitch, turpentine, rosine, and tar; mingle and wrap the tow thus steeped, about and amongst the wood and straw, and place it between all the Canons; the straw and small wood shall be so disposed, that it may take fire when the best part of the Canons are discharged, and the piece of Canon which is to give fire to the straw, shall be set with the mouth upward to blow part of the deck away, and at the same time give fire to great store of granads of all sorts, to add musket-barrells to barrels of Gunpowder ranged in order above the deck, to blow out these things by order, now one, than another by the help of slow portfires, but the beginning of this sport ought to be very violent and furious, to frighten and also tear in pieces the enemies at once, & that the rage hereof be the more furious, let the greatest part of the Canons ●●oot off thick one after another e●●n with the water, and at the same 〈◊〉 the Granads shall fly the one ●●re off, the others fall near, now a ●●rrell of gunpowder blows out a ●●rt of the flaming ship here, and a●n another barrel cast out a quar●●● of the vessel ardent there, casting ●●uers sorts of fire here and there as ●ls, arrows, granads, double and ●●gle ones, made of the compositi●● prescribed in the 5. and 10. chap●●●s, but to hinder all the barrels of ●●wder from taking fire together, ●hich is no small difficulty, every ●●rrell ought to be plastered over, ●●d then covered with glue and a ●●re-cloth, and then put into another ●●rell; but to make them take fire e●●●y one at your pleasure, put a Port●● unto them, of such length, & slow composition, as you shall think fit●ting. Now to begin this mad spor● fasten three or four good strong gunlockes within the ship, at one en● of the train, which is to give fire to the first Canon, and to every click or tackit of the locks, fasten an yro● rod, and let the other end of the 〈◊〉 rod pass through the ship, and ●ee made fast to pieces of strong wood, in the manner of a girdle on ●he outside of the ship, which is represented in the figure by the points; ●o that touched upon the outside a●y thing rudely, the locks must needs strike fire, and set going all the rest; ●nd to make it give fire being hooked in the inside, to draw it ashore, 〈◊〉 out of the way, let there be made ●ast to each clicket of the locks, long wires, and the other ends of the wires may be fastened to pieces of wood in the inside of the ship, round about the edge, so that the first thing that shall touch it, sets all going as upon the outside, but the pieces of wood upon the outside ought to be very near the water, as doth demonstrate the figure with the points, and so to conduct this ship as near to t●● place as may be, without dang●● there may be fast joined to the stern piece of timber noted A, B, to the en● whereof may be fastened two lon● ropes, and to them two little boats, i● which men shall be for the condu●cting thereof. This here is but a spar● of invention, to which the industri●ous Engineer shall add of his wha● he please, and take this but for an entry of such works; for although have here set down gunlockes▪ t● give fire to the Canons, my meaning is that they be made like gunlocks but ten times stronger, and harder i● going off, leaving to the judgement of the discreet Engineer, the true disposition of his own designs.   

CHAP. XII. How to make a Petard.  
DIego Vsan, a Spaniard, Francis Tibourele a Lorraine, and Master Robert Norton an English man, having all written of fireworks, and neither of them all understanding how to charge a simple Petard, I thought fit to end these fireworks for wars by the description of the same, that being of great violence to make entries & breaches into towns castles, or houses, the mortar prescribed, may serve to petard a place as hath been said already: but whosoever would make exactly a Petard, aught to cast a Mortar much like unto an Apothecary's mortar, as doth represent the figure A, observing the rules following, if you make it, to weigh six pound of mettle, let the caliber or bore be of such bigness to contain one pound of powder▪ or one pound and a half: if you add or diminish more or less mettle, augment or diminish the caliber likewise, to hold the fourth part of powder which the mettle doth weigh; and for the charging of the Petard, fill it only with the best gunpowder you can, almost to the brim; and then cover it with a round board made fit for the purpose, leaving aside all frivolous directions, written by others aforenamed; and for the ●riming of the Petard, make a port-fire of slow composition, as for the precedent fireworks of what length you please; and to break open the place you desire if it be accessible, then with the heel or breach of the Petard upon the ground or some great stone or piece of wood, and the mouth against the part of the door, gate, or elsewhere, which you shall judge fittest, but if the place be unaccessible, then make a kind of a little Cart with two or four wheels, as doth appear by the figure B with a long fork very strong to bear the Petard, and also support the requile of the Petard shooting off: This fork is represented by C, but the backer end of this fork must be stayed either in some hole, or against a stake, or other means; Now here in this treaty not intending to imitate these late Authors, who writing of artifi●iall Fireworks; have prescrbed natures and compositions of almost all manner of drugs, the means whereof may be found in any Apothecaryes' shopppe, doubtless either to persuade the curious Readers that they had profound knowledge, many rare secrets, or else the better to hide their ignorance, by that great confusion and expense, whereunto few or none would extend and use their purses to make proof and experience of. I will finish this treaty of artificial Fireworks for Wars, and go forward to the second treaty of Fireworks for pleasure and recreation, and explain in the Preface apologitike, for what use all these unknown dregges are fit for, which seems to those that are ignorant of such rare and wonderful effects. As Salarmoniake, Antemonie, Arseneeke, Vitriol, Stonelime, Thutie, Adamant stone, not forgetting quick silver.    

THE SECOND TREATISE OF Artificial Fireworks for pleasure.  

CHAP. I.  
PErspicuous & plain shall be the method to make all manner of Fireworks for pleasure, (which hear I will set down for the contentment of all curious and ingenious Artists) without such a number of unknown dregges for mixtures, as many heretofore have prescribed more fittest for some prodigious actions, then to mingle for artificial fireworks; and principally for those which are invented for pleasure, for these have no need of venomous smokes to poison the spectators, making mirth turn to mischief (which notwithstanding cannot be done in an open air) neither have they need of such a continuing, ardent and violent flame, as to consume Cities or habitations, but only of a gentle and pleasing flame to the eyes of the spectactors; and thus they are divided into three sorts; The first are those which ascend and mount into the air; The second are such as consume upon the earth; The third and last sort, are those which swim and burn in the water. Those which work their effects in the air, are ●●kewise divided into three sorts, the chiefest and most noble of all are the ●allouns; The second the rockets; ●nd the third are flying saucissouns, ●hose which stand fixed upon the ●round are also distinguished into ●hree sorts; The first are the rockets ●or the ground, the second the fiery ●nces and the third saucissouns: The ●res for the water, may have their ●iple division; globes or balls make ●he first, double rocket; the second, ●nd single ones the last; and to treat ●f every one in particular, I will begin ●ith the rockets for the air; and ●●rst of all describe their moulds, and ●he measures which must be observed in making of them.   

CHAP. II. A method to make moulds for rocket for the Air.  
ALthough that ballou● are absolutely th● most noble sort of fir● works, yet for so muc● as all great fireworks are compose● rather of rockets, then of balloun● I think it convenient, to begin th●● second treatise with these: first of a●● making the description of the moulds, which may be made either of brass or wood, the one being 〈◊〉 good as the other for use, if the mea●sures following be observed; whic● ●●all be convenient for all sorts of highness', because they depend one●● upon the caliber or boar, and that ●eing enlarged, the other parts will ●e all enlarged also, that being diminished, all the rest will be diminished ●●kewise. Then to make a mould, the calliber whereof shall be suppored an ●●ch of diamer, and is represented 〈◊〉 the figure A, than the mould ●ight to be six inches in length; 〈◊〉 the breech which is represented ●y the figure C, must enter into the ●ould one inch and a half, and the ●och noted with O, should be three ●●ches and a half long, and a quar●●r of an inch in bigness, and the ●●wler D, which is to roll the cartoush or coffins of paper upon, ●●all be three quarters of an inch in dignesse, & the rammer E, to charge and ram the rocket, shall be some what less than the rowler, to the en● that it may enter easily into the ca●●toush or coffins.  
Also it shall be made hollow, 〈◊〉 deep as the broach is long, as dot● appear at P, because the rocket mu●● be loaded, the broach being in it, 〈◊〉 ●●r the massif marked with F, shall be 〈◊〉 the same bigness as the charger. ●he 2. other ●owlers noted G, G, or 〈◊〉 make serpents, as shall be instructed hereafter, and the bodkin H, is 〈◊〉 make holes through your stars, 〈◊〉 shall follow, the Lantern I, is pro●er to charge the composition into 〈◊〉 rockets. The figure noted K, is 〈◊〉 cartoush or coffin ready to be charged, and by the figure L, is represented a rocket quite made. This method prescribed may be observed 〈◊〉 all sorts of moulds, for if you ●ould make one lesser, as of half an ●●ch, the caliber or boar, than the length must be 3. inches, which is 6. ●alfe inches, the breech shall enter 3. ●uarters of an inch, the broach shall be 〈◊〉 inch 3. quarters long, & the rowler ●●r the cartoush shall be half a quarter of an inch; but if you make the caliber of two inches of diameter, 〈◊〉 the mould be 12. inches in length the breech shall be 3 inches, and th● broach 7. inches of length; and so fo● all the other parts appertaining to th● mould. Now follows the descript●●on and manner how to make a ro●●ket.   

CHAP. III. How to make a flying rocket for the Air.  
FOr to make the mixture or composition of all middle-sized rockets, take a pound of gunpowder finely guised and bolted with two ounces of charcoale-dust, mingle these ●ell together, and try one rocket, ●hich if it break, add more charcoal dust to the composition; but the rocket do not ascend, being ●●eak, add more powder dust bolted as before said; and so shall you make your composition stronger 〈◊〉 weaker according to your desire; ●●stead of charcoal, you may put sea-coals, sawdust, or any other like thing, so that you keep a Medium not putting too much nor too little which thus you shall know if th● rocket mount not, you have adde● too much coledust; but if it breaks you have not put coledust enough for the charcole-dust which you p●● into the powder dust, is only to moderate the violence of the powder and to make the rockets tail appe●● more beautiful; which sea-coa●● doth as well as charchoale, and 〈◊〉 charge the rocket with this composition as it ought to be many rul● are required or to be observed; and 〈◊〉 of all that in charging the rocket, yo● put not too much composition at on●● in the cartoush or coffin; 2., that 〈◊〉 ●e beaten with a mallet 3. or 4. good strokes; and then put new composition thereupon, beating it again with ●●lowes as before; and thus putting 〈◊〉 at several times the quantity of ●ne or two spoonfuls, at each time ●eating it well with a mallet, and let ●ot the cartoush be filled higher ●hen the mould, but being just full ●hereto, then double down half of ●he paper with the bodkin beating it ●as ●he composition before) & at last ●ierce 2. or 3. holes with the same ●odkin to the powder to give fire, to ●ither stars, serpents, or faucessous, as ●hall be taught following; and these ●oles are represented by C, let the rest of the cartoush be cut even with the mould, and thus must be made all ●orts of rockets, great or little ones. ●t is to be noted, that the composition prescribed is fit; for all sorts of rockets, great or little, but only it must be made weaker for great one● then for little ones, adding more charcoal to it, for the whole secret is in making the composition neither to strong, nor to weak, and not as all writers hitherto have thought, following each one the others errors, thinking that the true manner of making excellent rockets, is to put many ingredients into their compositions, wherein they have all erred absurdly; and last of all Master Norton, for absolutely the best composition is, that which is made of fewest ingredients; and the rockets made thereof may be kept many years, and good at all times.   

CHAP. IU. How to make moulds for Rockets for the ground.  
THe mould must be otherwise made then the precedent, because such violence is not required in these, as in those which mount into the air, but only of somewhat longer continuance; and therefore these measures following shall serve at all times, never being made bigger nor lesser, but always the same for all occasions. Let the caliber behalf an inch, the diameter; and 5. or 6. inches of length; and the rowler for the cartoush shall be four lines in diameter which is the third part of an inch, and the rowler to charge withal somewhat lesser to enter without spoiling the cartoush: next let not the broach be longer than 3. quarters of an inch; and the breech shall enter half an inch into the mould; and the cause of this great alteration is, that the rockets which mount into the air, have need of great and violent strength; and by consequence last not long, but we desire that those upon the ground may dure a long while, with a gentle motion, which shall be easily performed, the rules following being observed.   

CHAP. V. How to make the composition for rockets upon the ground.  
THis is the easiest composition of all others, for in this shall be nothing required, but only Gunpowder well beaten and bolted, so that it be as fine as flower, and then fill your rockets by little and little, well beating it into the cartoushes or coffins with a mallet, as the rockets for the air, and being full within an inch of the brim of the mould, double down one quarter of the paper, or cartoush, beating or pressing it with 3. or 4. good strokes of the mallet, and then with your bodkin, pierce a hole to the composition; which done; put into the cartoush about the charge of a Pistol, of good gunpowder, and then double down the one half of the cartoush, giving a gentle blow or two with the mallet, & with a small cord, or big line, choke the rest, and what shall rest above the choking, cut it off, leaving a piked point, as appeareth by the figure E, in the 3. chapter of this Treatise; and your rocket being thus finished, and primed with a little wet powder, and laid a drying, shall be ready for all occasions.   

CHAP. VI The manner how to make Serpents.  
THe Serpents are to be made either of the composition for rockets on the ground, or of that for the air; for being filled with the composition fitting for the ground; they will spread and sparkle lively in the air, but if filled with the other composition, they will fall weaving nearer together; notwithstanding either sort will show diverse pleasing actions in the air, being made as followeth. Let the cartoush be about four inches long and rolled upon a rowler, somewhat bigger than a goose quill, as is represented by the precedent figure G, in the third chapter, the paper ought to go about the rowler nine or ten times, and then choked almost in the middle, yet leaving a little hole to see through, and the longest part shall be filled with the composition, but the shorter with fine grained powder, and choked close; also the longest end must be half choked close; as doth appear by the figure F, chapter third but if you desire not to have them wamble in the air, then let them not be choked after the composition, but as doth represent the figure G, both which figures, F, G, represent Serpents quite finished.   

CHAP. VII. How to make golden rain.  
MAny there are (specially in France) who make rockets, yea and boast that they are perfect therein, who know not what golden rain is, but think it to be some other thing than it is; wherefore to put them out of doubt, and to teach all others who desire the knowledge thereof, I will here set down the description and manner how to make it. Take goose quills, and cut off the hollow ends, leaving them as long as may be, as the figure K, doth demonstrate third chap. and fill these quills with the composition of rockets for the air, at the last stopping every one with a little wet powder to keep in the dry powder, & crowning a rocket with these (as shall be taught following, chap. 12. in its true place) will show a most glorious & pleasing rain, which some having in times past seen, have called it golden rain for the beauty thereof, but of later times it is more commonly called golden hear: many beautiful and strange figures may be represented in the air, with this manner of rain, as shall follow in the 13 chap. treating how to represent many sorts of figures in the air with rockets.   

CHAP. VIII. The manner how to make Stars.  
ALthough that there be many sorts of compositions for stars, yet I will set down here but two of the best; all the rest being nothing worth, but frivolous and expensive: the fi●st and best sort, is to be made of dry powder, and the other of moistened powder as followeth. For the ●●sort take 1. pound of saltpetre, half a pound of brimstone, and a quarter of a pound of gunpowder dust, all these being plu●ri●ed & mingled together, wrap the quantity of a nutmeg in tow in a linen rag, or in paper, and bind it fast, as appeareth by the precedent figure H, chap 3▪ and to prime them, you must pierce them with the bodkin, and put stoupell or cotton wieke dipped in powder through them (which shall be made as followeth in the 11. chap.) and to make the second sort, take 1. pound of salt-peter, and half a pound of powder-dust, and half a pound of brimstone all these well pulverised and mingled together, moisten them with either oil of petrole, or else wi●h fair water only, to make a past of them, whereof make little balls about the bigness of a musket bullet, and whilst they are moist, roll them in dry powder dust, then let them dry, and then may you employ them at your pleasure, without further trouble; for the last powder in which they are rolled, doth serve for their priming. This last sort of stars doth not make so beautiful a show in the air as the others, for falling down, the flame of them takes the form of a lamp, having no force to expel it like wings as the others do, for the flame of the others blowing out of the two sides pierced, make it stretch in length, and by that means show greater in the air.   

CHAP. IX. How to make Stars giving great Reports.  
TO make Stars, that each one shall give a report like a Pistol or bigger gun, you must first make little saucissons (as I taught in the chapter following, but the saucisson need not to be covered with chord) and being made and pierced, take as much of the former dry composition, and bind it to the end of the saucisson which is pierced, making a hole through the composition, and pass a piece of stoupell or cotten-wieke as in the other stars; but if you take of the moist composition, you may only leave the paper hollow at the end of the saucisson, fitting to contain the quantity of composition required, putting a little grained powder before, and prime these stars as the others, of the same composition; these stars are very troublesome and little in use, because that a great rocket can carry but few up into the air, and by consequence work but a small effect, and moreover they are very long in making. One may make Stars in the same manner, which ending, turn to serpents and others as shall please the workman.   

CHAP. X. The manner how to make Saucissons.  
IN this chapter, my intention is not to treat of the saucisson which flies into the air, but only of that which stands firm in great works, or else which is applied to rockets, which thus is made as followeth; you must have a rowler of such bigness as you desire to have the concavity of your saucisson, wherupon roll as much paper as you please, and then choke it at the one end; which done, fill it with grained powder, & choke the ●ther end also, and cover all the saucisson from the one end to the other, with small chord, as doth represent ●he figure I. chap. 3. and glue that ●ord with strong glue all over, and when you would make use of these ●aucissons, pierce them at the end with your bodkin, and put into the ●ole a quill filled with fine powder dust, which shall serve for a portfire, and the other end of the quill shall pass through a board, whereupon you mean to fasten them, and shall enter into a portfire in the other side of the wood, which shall be fastened all along the wood, and so may you fasten what store you please near together, or far asunder, this quill is represented by the figure L, and by this means one end of the 〈◊〉 fire beginning; all the whole ran● of saucissons will give their report one after the other. But if your sa●●cisson is to be applied to a rocket, shall only be pierced at one end, a●● primed with a little grained powder and fasten it to the top of the rock● either with paper, parchment, or an● thing else, so that the rocket, endi● the saucisson, may take fire; so shall yo● not fail of your intent or design.   

CHAP. 11. How to make Stoupel or preparing of your Cotten wieke.  
TO make Stoupell, not the meanest, but the best that may be made to give fire (which in ●per term is called estoupel●●g) to all manner of fireworks. 〈◊〉 Cotten-wieke, and double it ●oft as shall be needful for your 〈◊〉, as if it were to estoupel your fie●ances; then double it eight or ten ●●es, and also for the stoupeling of great rockets; but if it be to 〈◊〉 through your stars, than 4. 〈◊〉 threads shall suffice; so having do● led your cotton wieke fitting for yo● purpose, steep it in fair water, 〈◊〉 then wring it between your han● and take gunpowder dust, without other mixture, and steep it in 〈◊〉 water, not putting too much wat●● nor too little, but only till it app● like dirt, and then put into it 〈◊〉 cotton wieke, and there turning a● winding it till it hath sufficiently 〈◊〉 powder in every place and 〈◊〉 cell, then draw it out, putting a 〈◊〉 dry dust upon it, & hang it a dry● in the sun or elsewhere, and 〈◊〉 being dry, you shall have the 〈◊〉 excellent stoppel that may be ma● for your use at all times, leaving 〈◊〉 all such ingredients which ignorant ●●●●eme so highly of, as Aquavitae, 〈◊〉 wine, strong lee, quicklime, ●ger, urine, and diverse others, for ●ich I would not lose so much 〈◊〉 time as to repeat them, but 〈◊〉 go forward, and show how to 〈◊〉 all the parts of a rocket.   

CHAP. XII. The manner how to assemble and set together the parts of a rocket.  
WHen the rocket is finished and taken out of th● mould noted A, in the figure of the 3 chapter, an● the rocket is noted with the letter ● in the same figure, then must yo● join to the end of the rocket, which is not choked an empty cartoush 〈◊〉 coffin, much bigger than the rocke● is large, which is represented by 〈◊〉 in the same figure, in this cartoush you shall put your serpents, golde● rain, stars, saucissons, or other things at your pleasure, but first of 〈◊〉 you must put in the bottom powder dust; only enough to cover the bottom of the large cartoush; and then put the serpents with the ends downward which are to take fire; and likewise the golden rain in the same manner, but it is the custom to put a little powder dust amongst the stars; this cartoush being thus filled, cover it with a single piece of paper, and afterward passed upon that a picked cap made also of single paper, and to make perfect your rocket you must bind a rod of such length and weight, that being bound to the rocket, it must weigh down the rocket on your finger being near to the rocket, and then stoupel your rocket, ●hat is to say, prime it with cotton-wieke, and it is thus finished either ●o fire or keep.  
This method is fit for all sorts of rockets, great ones or little ones, except that little ones must have the upper cartoush no bigger than the rocket, only to hold half a dozen of stars or serpents, or one saucisson for so much as many being fired together, great confusion would happen, if great rockets should be put together, and not prepared otherwise.   

CHAP. XIII. How to represent diverse sorts of figures in the Air with Rockets.  
THe first and most rarest is a tree or fountain, and is made putting many little rockets upon one great rocket, passing all the roads of the little ones through the large cartoush of the great rocket, and if the little rockets take fire while the great one is mounting up, they will represent a tree: but if they take fire when the great rocket is turning down again towards the ground, than they will be like to a fountain of fire, and if there be two or three little rockets having no rods amongst others they will make diverse motions contrary to the rest most pleasing. The second figure is the golden rain; and that is made when many quills filled (as aforesaid) are put upon a great rocket: for certainly all those quills taking fire will seem like a great shower of fiery rain to those who are under it, but to those who are a side of, like beautiful long hair. The third figure are Stars, which are represented putting only many Stars upon a great rocket. The fourth are serpents which are already described, but to make all these prescribed figures vary, you may tie many little ones together by the ends, which take not fire, the quills, or serpents may be tied in like manner; but the thread wherewithal they are tied, must be at least two or three inches long, betwixt each one, and you shall see diverse sorts of figures in the air, changing themselves into much variety.   

CHAP. XIV. How to make fiery boxes.  
THe fire-boxes are made of many rockets being put into a large cartoush the bottom whereof is covered with powder dust, and pierced in the middle to pass through a port fire or stoupell, to give fire to the rockets within the cartoush or coffin, which flying out, leave the cartoush most commonly whole and uncracked, those boxes are only to be cou●red with a piece of paper, that the rockets taking fire may fly out without resistance; and the reason why they are to be covered, is because if many be made one near to another, they should not take fire all at once.   

CHAP. XV. How to make fiery Lances.  
THe use of these lances is always required in all great fireworks, wherefore here shall be described the manner of their making. Their cartoushes are to be made as other cartoushes for rockets; only these may be made of pasteboard, and glued as they are a rolling, if it be to make great ones, but if for little ones, than paper shall suffice, the cartoushes being made, let them be filled with the dry composition prescribed for stars in the 8. chapter of this treatise, and prime them with wet gun powder, & the lower end of the cartoush is commonly stopped with a piece of wood, to the end that they may be nailed or stuck when they shall be needful, the wood being about two inches long out of the cartoush.   

CHAP. XVI. The manner how to make Rockets for the water.  
TAke fire & water being two elements of contrary qualities, the one to the other cause the rockets which work their effects in or upon the water to appear to the spectators more beautiful, and seem more rare and admirable, although that all sorts of rockets being fired, will work their effects upon or under the water, but this only is to be noted, that those which are made for the air, or for the ground, have such strength and force, that being once kindled and cast into the water, they will consume themselves there, and not rise up swimming as those which are made artificially for the water as followeth. To make good rockets for the water, the caliber of the mould ought to be one inch of diameter, and in length eight inches, the breech to enter one inch, having no broach; the rowler for the cartoush shall be three quarters of an inch of diameter, and the charger shall be somewhat lesser than the cartoush being ready to be filled as the others. Of two sorts of composition, you shall make one as followeth (if you desire to make a great fiery tail appear upon the water, Then take one pound of salt-peter, half a pound of powder-dust, half a pound of brimstone-dust, and two ounces of charcoale-dust: but if you desire to have it to burn clear like a candle upon the water, then take one pound of salt-peter, half a pound of brimstone-dust, and three ounces of powder-dust, all these must be well mingled together, as all other compositions) and with either of these compositions, fill your cartoush, and join to the upper end a saucisson, and then cover it all with melted pitch, rosine, grease, or painting, to hinder the water from spoiling the paper: and to make it float and swim upon the water, bind a road about two foot long to it as to the others for the air. Now if that you desire that this Rocket charge his actions upon the water, swimming now upon the water, and now under the water, put here and there powder dust, power to the quantity of half a spoonful as you charge it; also may you make it change colour, now red and then white fire, only changing the composition in filling it, and so the industrious may add many changes and diversities as they shall think fit: but if you are to make a great number for a great fireworke, than the plainest are the best, only filling therewith the first composition; but they must be filled two or three fingers breadth with fine powder dust, to make them fly far off, or else they must be made like rockets for the air, and afterward charge them four inches with composition made for water; but to be the more certain, it is best for those who have great works, to make to try one or two before they finish many, that the better they may attain their designs, which with eye they may accomplish, if the rules prescribed be observed; for as I think there is not one rule neglected appertaining to the fabrication of these fireworks prescribed; and now I will set down the manner how to make all sorts of ballouns and flying saucissons, which are absolutely the beautifullest parts of fireworks for pleasure, and after them I will describe a brave fireworke for pleasure with the manner how to place, dispose and range every piece and part thereof, to make their play in order.   

CHAP. XVII. How to make Girondells or fiery Wheels.  
A Girondell (or as some call it a fiery wheel) is often required in great or little fireworks for pleasure, and therefore I have thought fit and necessary to set down their description, as well as of all other parts of fireworks: for according to the application of the Girondels in a Fireworke, one may judge of the Enginiers industry, for being well applied, they adorn very much a fireworke, and being ill applied they spoil all. And to make them, you must make wheels of wood as big as you pretend to make Girondells, and unto those wheels bind fast rockets of a mean bigness, binding the mouth of one towards the tail of the other, and so continuing until you have filled your wheel quite round, which done, cover them with paper pasted very curiously about them, that the one taking fire, they take not fire altogether, but the mouth of one shall be left uncovered and ready primed, fit to take fire, the which ending shall give fire backward to the next, and so one to the other, every one his turn: there may be bound fiery Lances to these Girondells either upright, or near overthwart, which will show diversity of fire the wheel turning: also there may be added to these wheels boxes of fire, provided that they be artificially applied in such manner, that they balance the wheel equally, although the proper use of girondeles, is only to garnish the angles of a great Firework, without much trouble, for that confusion is too frequent in fireworks; and therefore it shall be expedient to avoid the multitude of fireworks too near compacted together; but placing them as much distant as conveniently may be admitted, by which means you shall be the better able to attain your designs pretended.   

CHAP. XVIII. The manner how to make Ballouns.  
THe ballouns being the perfectest part of all artificial fireworks for delight, I thought best to set their description here apart from the others, to the end that the learners might the better and more easier attain to the knowledge and perfection thereof; and being that I have already described the mortar, which is represented by the figure A, following, I will only refer the reader to the second chapter of the first Treatise for wars, where he he shall find sufficient instruction for the making of the instrument. But to make a balloon shall be required a wooden rowler, which is represented by the figure B following▪ this rowler must be of such bigness as you desire to make the inside of your balloune; upon which rowler, let there be rolled as many pastbords as you shall think sufficient for strength being well glued together and choke this cartoush at the on● end, leaving a little hole for a portfire as shall follow, and glue it in. This portfire shall be made just like unto ● rocket for the ground, but only tha● the composition may be somewhat slower; as for the air, and to know of what length this portfire ought to be, it shall not be amiss to try a balloon filled with earth; then your portfire being well fastened to the balloon; place all your serpents within it, putting nothing else amongst them, but only one or two saucissons, to break the balloon on high, when the serpents be all on fire, these saucissons must be made of the length of your serpents, and your serpents may be of the bigness of your rockets for the ground before prescribed, but not so long, and they may be filled with the composition either for the ground, or for the air: but it is to be noted, that they must be primed with the composition for the air, pressing very strongly the priming powder into the throat or gorge of the serpents, if it be put in dry: but if that we●te, then only with the end of a stick, or else with ones finger put in a little; as for other serpents and the two saucissons shall have their priming somewhat longer than the serpents, to the end that all the serpents may be a fire before that the balloon break. The serpents being not made too long, one may put two or three ranks in one balloon, as appears by the figure C, or one rank of serpents, and over them many Stars; all these things being thus disposed within the cartoush, let it be choked at the other end; and then prime the ballouns with cotton wieke steeped in gunpowder as the stoupel before spoken of; but this cotton must be well fastened to the end of the port-fire with strong packthread, which done, charge the balloon in your mortar, as is taught in the 6. chap. of the first Treatise; and you may shoot it when you please. This balloun is represented perfect by the figure D, but if you will fill your ballouns with stars only, another manner of proceeding shall be required; for as you put in your stars into the cartoush; let there be powder-dust mingled amongst them here and there to give fire to the stars, and to break the balloon being up in the air. There is also another sort of ballouns to be made, but the expense of making them is so excessive, that few or none will be at the cost, yet the description shall now follow. In stead of serpents made of paper, you may have them Iron about the bigness of your little finger, and one inch and a half of length, or two inches; and near to the bottom there must be a double bottom in the middle whereof there shall be a little hole, that one may fill the space between the two bottoms with corned powder, and the rest is to be filled with a composition somewhat slow, the difference between those of Iron, and the former of paper, is that many Iron ones may be laid in a lesser space; but in stead of a pasteboard cartoush, there must be had a wooden box, made by a Turner, and covered with canvas and glue; the portfire must be made also of Iron, like to a little hat, having in the bottom two or three little holes to give fire to the serpents; but for the expense of one of these, there may be made half a dozen of the others, nay near a dozen, wherefore I would wish no man to trouble themselves with this sort, unless it be for the pleasure of some great Prince.   

CHAP. XIX. How to make flying Saucissons.  
THe method which must be observed in making flying saucissons, is most easy, provided that one knoweth already the manner how to make the common sort before prescribed, for the cartoushes are to be made all alike; only excepted that those which are for flyers, be somewhat longer than the others, they are to be charged in like manner as the others, but the corned powder being put in, there must also be a little powder-dust put after it, and beaten with a mallet, as the rockets for the ground yet let not more than the thickness of a finger be of dust-powder, and then choke it almost quite together leaving only a hole as big as a small goose-quill, to which you shall put a little wet powder-dust for priming. They may be made otherwise, as followeth, which will make a more beautiful show; first of all, you must charge the cartoushesas the common saucissons, and then choke them as the common leaving only a little hole to take fire at, or if the hole chance to stop quite up, open it with a bodkin, but the saucissons being charged and choked, there must be part of the cartoush left over and above that which is charged, which part shall be filled either with powder dust, or else with composition of rockets for the air beaten wit a mallet, as before mentioned, I think it shall be needles to tell you that they ought to be covered with chord, being that it is sufficiently demonstrated before, it is to be noted that the saucissons which have the powder within the choking, doth turn wonderfully in the air, but the other sort which have the composition after the choking, fly up like to a rocket, almost carrying a great tail after them but the tail of the others showeth but very little, they are both sorts represented by the figure E, in the 18. chapter.   

CHAP. XX. How to make short Guns for the Saucissons.  
HAving already treated sufficintly and amply of the fabrication and making of the Mortar, in the imitation whereof most easily may be made Canons to shoot flying Saucissons very high in the air, making them of the one or the other matter or stuff prescribed for the Mortar in the second chapter of the first Treatise, but the powder sack needs not, and the touchhole ought to be in the middle of the bottom; also it shall be necessary that every Canon have a little breech to pass through a piece of wood, wherein they may be fastened, so that a port-fire may be laid from one to another, these are represented by the figure F, in the eighteenth chapter.   

CHAP. XXI. The manner how to dispose and build a great or little Firework.  
IT is a custom and usual thing among the brave and skilful Painters; 1 to teach their apprentices how to draw limbs, or members, as the eye, the nose, the mouth, the ear, the hand, the foot, and afterward the whole body; and in like manner, the most learned and famous Philosophers, beginning their schools, teach their disciples what is materia, forma, & primatio; and afterward, totum compositum; and to imitate them in this Treaty, I have taught the one after another, all the parties required are necessary for the composing & building of a beautiful fireworke for delight, having begun with the rockets for the air, afterward the rockets for the ground, the saucissons & rockets for the water, and following the ballouns and flying saucissons with all things belonging unto them; and to avoid falling into the old proverb (Ex omnibus aliquid, & in toto nihil: to know something of every thing, and nothing perfectly) I have here set down after all these parts the description and manner how to assemble, build and dispose any manner of artificial fireworke great or little: and to begin, you must first cause to be built a scaffold trianguler, square, round, or of such form and bigness as you shall desire to have your Firework, your scaffold being built, dispose thereupon your statues or figures which shall be prepared for it, let them be made of osiers, and covered with paper or canvas, and curiously painted as here is represented in this figure. Mountains, buildings and many statues, all which are to be supposed, only osiers and paper, or canvas painted, your figures or statues thus disposed upon your scaffold, round about the flore thereof, shall be laid your saucissons, being already fastened to pieces of timber, and to the pillars of your scaffold or rail, shall be fastened your fiery lances one right upward, and the other flat along, each distant, the one from the other about half a foot; and under the fiery lances, let there be nailed a rank of boxes of fire, which shall be ranged upon pieces of wood as the saucissons; all your fireworks being thus disposed, you may place in the side which is of least esteem, your partments of rockets made like unto square chests, or long boxes, of such bigness and length as shall be required to hold those rockets which you mean to place in each one: bu●1 these boxes must have a false bottom full of holes for to pass every rod of every rocket apart, and having filled let the box (which by proper name is called a partment) be covered with a leaf or two of paper pasted close; but to give fire to the rockets, let there be made a hole thorough the partment, through it shall pass a little piece of stoupel, or cotton wieke and giving fire to that, all the rockets within the partment, fly out, and in the same manner shall you give fire to all your fiery lances with a stoupel going from the one to the other; and as for your girondells, you shall give fire to them with a match, as you desire that they should play, and by this means all your fireworks shall begin to play at once, except those parts which shall be reserved without priming to be fired by the hand, as best shall seem to the artist or engineer, and so the industrious shall not fail to accomplish his designs, observing all the rules prescribed; which being at large laid down I will go forward to the next chapter, and there show how to make a most precious unguent for all manner of burnings, as well of common fire, as of artificial fire, if by hazard any mischance arise,   

CHAP. XXII. A most precious unguent for any burning.  
LEt no man wonder if (having ended this Treatise of fireworks) I take in hand to describe a little part of chirurgery, which I confess to have taken out of a Treatise written by Thybourel, a Chirurgeon of Lorraine, and having made experience of this unguent divers times as well for burnings, as that for other accidents, I may say with boldness & truth, that there was never the like secret of this kind left to posterity, specially against hurts come by fire, and which leaves less scars after the healing of the places wounded, and therefore I have set down the very words which Thybourel hath written in the last chapter of his 4th. book, entitled Recueil de plusieurs machine's militaires.  
Take fresh hog's grease or lard, as much as you please, and boil it, taking off the skim until there arise no more skim; then set the lard three or four nights in the air abroad, after which it must be washed in running water, to take away the saltish nature, and also to cleanse it white, then melt it, and keep it for your use. Bacon may serve in stead of lard.  
Otherwise.  
The white of an egg or fresh butter being mingled together and well beaten to an oil are excellent.  
Another sort most excellent.  
Take a stone of unslacked lime (or otherwise called quick lime) and let it dissolve in clear water, and when the water is settled, pour it gently out from the lime through a linen cloth, then put as much salad oil as you take water together, and beating it all to an oil, you shall have a most excellent unguent for all kind of burnings, neither of these unguents have any scar, but are precious remedies for the afflicted  
We have seen Impostures▪ cover sores with this water alone, but observing superstitious ceremonies, saying vain prayers, but we assure the posterity, that the water only is sufficient to heal wounds and s●res, only washing them with it, and covering them with a linen clothe wet in the same water, without any superstition, it doth modify and percute, by which it doth supply nature, and doth heal sores better than our ordinary unguents.  
Thus may you see how this brave Chirurgeon that set forth to the face of the world the perfection of this unguent (which cannot be sufficient-praised) confirming by his own confession, that the Surgeons do not use such good remedies in their shops and ordinary operations; wherefore I have set down his own words, that no man may esteem me inventor of Calumnies, against the practitioners of Chirurgery, nor any others, as many now adays seems to write with serpent's tongues, stinging virtue on every side, against whom viperous venom's patience is the only antedote, leaving them to sweat, vex, and torment themselves in their insatiable rage, and to end my discourse touching the perfection of this unguent, I will assure all these who shall have need and make use thereof, that they shall find in operation what I have here set down in description.    

A Treatise of practical Geometry.  
TO satisfy diverse of my friends and yield to their desires (whose treaties have been such powerful commands to me, that I have been constrained to lay aside and forsake my own proper will and follow theirs, I have set forth this little Treatise of practical Geometry to the view and censure of the world, which I acknowledge to be unworthy of so many singular and industrious wits, as yearly spring up in this Island, and the adiacents thereunto, notwithstanding I have given way to their requests for their private contentment, and to assure the world that I have nothing, no not my own will proper to myself, but that I will follow as near as possible I may the precepts and documents of that ancient, wise, and divine Philosopher Plato, who saith, Non nobis nati sumus, sed patriae & amicis; we are not borne for ourselves, but for the service of our country and friends: then for the satisfaction of my friends and service of those who will accept these my labours, I have clearly and in few words set down the manner how any man (who hath never so little studied Geometry) may take any distance, or height, depth, or breadth with two little sticks, yea e●en straws, being laid a cross, also I have set down a method how to take any kind of height, distance or depth, with the Sector without any arithmetic or rule thereof: and also by the sins tangents and secants, the whole being very portable, to refresh the memory, & to strengthen and augment the knowledge of those who (for want of practice) have not the perfect use of those instruments: then first of all I will set down the manner how to accommodate and dispose the sticks, twiggs, or straws for the measuring of any distance.  

The method how to make the Crosse.  
Having two sticks, the one long, and the other somewhat short as are represented in the figure following by CF, and DE, then mark upon the stick CF, points the one distant from the other precisely, half the length of DE, and let there be a hole made through the stick DE, so that it may slide upon CF, from end to end, and you shall know the two parts of the cross by these names, the longest part CF, shall be called the index, and the shortest DE, the cross. Now if you have any height to take, then fasten to one end of the cross, as to D, a perpendicular or plummet, and for the more easy and just operation, you should have a foot to support the cross; the instrument being thus prepared, you may measure with great facility either height breadth, or depth, as followeth.   

PROPOSITION I. How to take a height accessible.  
LEt it be proposed to take the height of the Tower AB, to the base or foot whereof one may easily approach▪ dispose the cross of your instrument in such sort that DCE be of equal distance there one from the other; as thus, settle your cross upon the first point of the index; then setting the instrument to your eye, go either nearer, or retire farther from the object you desire to measure until you see A the highest part thereof by the two extremities of the cross CD, (the inden being paralled to the earth) which will happen in the point C, and not elsewhere: that being measured, will be always the just height of the Tower required, only observe that you must always add to the distance between you and the Tower the length of the foot, which supporteth your instrument, and so you shall have precisely the height required: for scarce ever will it happen, that the instrument may be placed level with the base or foot o● the height required, but if that should chance to be, than the extremity of the index C would arrive in the point G, and so nothing to be added, but only measure the distance between G and B, and that would be the just height required; but if it chance that you may not plant the instrument in the point C, by reason of some inconveniences which may happen retire further back, and put forward the cross to the second point of the index, and then accord your visuele lines to see the point A, (the index always being parallel to the ground) which then will happen twice the height required, distant from the foot thereof: but if it happen that going back you ascend any little mountain, or descend into any little valley, you are then to observe some point in the wall of the Tower, which the index shall direct you to by the line visuel, & add only the height thereof unto half the distance 'twixt you and the Tower, and that shall be the height required most exactly: and thus you shall never fail in your operations.   

PROPOSITION. II. How to take a height inaccessible, or one height upon another height.  
SVppose the altitude BC, to be required, to the foot whereof one may come may come no nearer then to the point D, then at the point D, accord your visuell lines in B and C, the index being always level with the ground and the cross fixed upon the first marks or point of the index, than set a mark there in D, and going back towards F, as far as E, and put forward the cross to the second point of the index, and then direct your lines visuel again in B and in C which will arrive in E, and not elsewhere, then measure the distance between DE, which will be the altitude precisely required, but if you desire the height of the Tower B, and mountain level with the foot of your instrument, than you must add the length of the staff, which supporteth it as in former observations. And to have the altitude A B upon the top of BC, you must take two other observations in FG as before, setting your instrument in F, the cross being fixed upon the second point of the index, and direct your visuell lines in A and in C, and then set up a mark in F, and go back, putting forward the cross to the third point of the index, and direct your visuell lines again in A and in C, which will happen in G, and not elsewhere, then measure the distance between FG, which shall be equal to the altitude of the Tower AC, and ha●●ing taken already the altitude BC, you may very easily subtract the little altitude BC from the greater Altitude AC, and then will rest the altitude AB, which is required.   

PROP. III. How to take any distance upon a place accessible or inaccessible.  
IF it were proposed to take the distance AB, and that the place were accessible only in the middle upon the ●●ne CF, then dispose your instrument as before the cross placed upon the first point of the index, and going forward or backward upon ●he line CF, direct your visuel lines 〈◊〉 A and in B by the two extreami●es of the cross, the end of the index resting against your eye, then measure the distance between you and the point F, and that shall be just half the distance required between AB, but if the place be inaccessible, so that you may not approach nearer than the point D, put forward the cross to the second point of the index, & then direct your visuel lines to AB, and leave a mark at D, and put forward the cross one point more, and going back upon the line DE, until you may direct your visuel lines by the extremities of the cross again in A and B, then measure the distance between DE, and that will be half the distance between AB, and so may you operate going back, and putting forward the cross.   

PROPOSITION IU. Another manner how to take a distance inaccessible.  
SVppose the distance AB to be taken, and that B is the nearest place that may be required, then there must be set a mark, and withdrawing backward in a strait line towards C, and there again plant another mark at C, then going right towards one side, as towards F, counting your places equal to the number which you have already found B and C, and there direct your visuel lines in BGC, as in this figure at D, leaving there a mark, go strait along towards F, not moving the cross of your instrument, and going along make trial where your visuel lines may be directed again in A and in C, which will be in the point E, and not elsewhere, then leave there a mark, and measure the distance between DE, and that shall be the distance which you require, which is AB, the demonstration of this proposition is grounded upon the second and fourth propositions of 6 of Euclid.   

PROP. V. How to take a distance only upon a line parallel to it.  
LEt the distance AB be required, the which may neither be seen, nor come near unto, but only upon the line or bank CK, then upon that bank draw out a strait line with marks, as the line CK, parallel to the wall AB, and set a mark in K, then go back towards D, and direct your visuel lines in B and in K, by the extremities of the cross GH, then leaving the cross in the same state and a mark at D, withdraw yourself further back towards C, until you may direct your visuel lines to A and K, which will happen in C, and not elsewhere, and leave there another mark, then measure the distance between CD, and the same shall be the breach AB, equal to CD, and the demonstration of this proposition is grounded in the 29. and 33. propositions of the 1. of Euclid.   

PROP. VIII. How to take the depth of a valley.  
TO take the depth of a Valley, there is somewhat more difficult then in the former operations, because there are more observations to be made, and to begin you must from the point B observe in some place opposite a ●a●ke level with the horizon as A, the level whereof you may easily take at this instrument, as is taught in the second proposition by the help of the plumbet: then from the point B, take the distance BC, as in the fifth proposition, or else mecanikelie, which being done, direct your visuel lines from B to A, and to C, and leaving the instrument in same state, turning yourself about, frame the same angle upon the plain, which will be FBH, than plan● marks upon the lines BF and BH, and vpo● the line BH, count as many paces or fathoms as you shall have found between B and C, and at the end of your paces▪ 〈…〉 mark, which will be at E, thi● being done, dispose your instrument to make a right angle, placing the cross upon the ●irst point of the index, for the cross maketh always a right angle when the cross is upon the fifth point. The instrument being thus disposed, walk upon the line BF, until you may direct your visuell lines (by the extremities of the cross) to B and E, or else to FE, which will happen in D, and no where else, then measure the line DE, and it will be equal to the depth required G C, the demonstration of this proposition is grounded in the 26, of the 1. of Euclid, or upon the equality of the two triangles BG C, and BDE, which are both equal and equal angles. Although this operation be somewhat more obscure than the others, yet I think that it is sufficiently explained, and therefore I will go forward to the use of the sector.   

PROP. IX. The manner how to take either distance or altitude with the Sector.  
SVppose the altitude AD were to be taken to the foot whereof you may approach open the Sector 45 adding to it the sights then going forward or backward upon the line DI, until you may see the highest part of the altitude A, through the two upper sights, the inferior branch of the Sector, being parallel to the earth or horizon, then measure the distance between the centre of the Sector and the Tower, adding to that distance the length of the leg, which supporteth your Sector, and that shall be the altitude required, as appears in the figure following AB, and BC, are of equal distance, and adding the length of the foot of the sector, you will find, that it will be height of BD, which doth accomplish the altitude of the Tower. But to take a distance in any plain, as in the figure precedent, the same operation may be used, except only the two branches of the sector shall be turned parallel to the horizon or ground, having first made a right angle at the point G, or you may operate otherwise, first of all prolong a strait line as EGH of what length you please, then open your sector to a right angle, and set it the point G, so that you may see through two sights the point E, and where the line visuel of the other two sights strikes along, set up a merke as in I, then go towards it, and at your pleasure in the same line, set your sector opening it, so that you may see the point E and G, and keeping your instrument at the same width, & in the same place place, only turn it to the other side, so that you may see the point G thorough the sights which before saw E, and where the other visuell line shall cut the first line EGH, set a mark which will be in H precisely, and that shall be the distance required GE, the demonstration of this proposition is grounded upon the 4. and 26. of the 1. of Euclid.   

PROP. X. How to take any distance or altitude inaccessible with the Sector.  
TO obtain the altitude AB, you must first take the distance BC. as is taught by the 9 Proposition precedent, and knowing the distance BC, which I suppose to be 100 fathom, set in the point C your Sector, and direct your visuell line through the two upper sights to the top of the altitude A, and let the branch of the Sector be parallel with the ground, then leaving your instrument at the same width, let fall a perpendicular line upon the line of the Sector divided into equal parts, passing by the 100 number of the inferior branch noted D, and note what number the perpendicular doth cut upon the upper branch of the Sector noted H, which I suppose here to be● the 150 part or number, and then set the two points of a pair of compasses upon the two numbers, to wit, the one point upon 100, and the other point upon 150. of the equal parties, then transport the points of the compass all along one branch of the Sector upon the line of equal parties, and the two points shall denote as many parties upon the line of the Sector as the Tower doth contain fathoms in altitude, adding the length of the foot which supporteth the Sector, the demonstration of this Proposition is grounded upon the 4. Proposition of the 6. of Euclid.   

PROP. XI. Of Sines, Secants▪ and Tangents.  
BEcause the most noble, most artificial, and most certain way of taking of Altitudes, Distances, or other Dimensions, is by Sines, Secants, or Tangents, I have set down their operations in such Propositions as are usual in this subject at the end of this Treatise of Practical Geometry, and before I enter on the method of operation, it is necessary to define what the said Sins, Secants, & Tangents are.  
1. A righ● Sine is half the Subtence of of the double Ark.  
A Subtence, sometimes called a Cord, is a right line drawn from any part of the Circumference of a Circle, unto any other part of the same Circumference, so the right line DE is the Subtence or Cord of the Ark DGH half of which is DF, the Sine of the Ark DG; and so is MN the Subtence of the Circumference or Ark MGN, half of it is MI, which is the Sine of the Ark MG; now the Ark MG is half of the Ark MGN, and MI is half of the Subtence of that double Ark, viz. MGN; hence it is according to the definition aforesaid, that the Sine of any Ark is half the Subtence of the double Ark; by the same reason OT is the Sine of the Ark BOY and MS is the Sine of the Ark BOM and so of others.  
Note further that the total Sine, the Sine of 90, or the Radius, is nothing else but the Semidiameter of any Circle, viz. CB, or CG.  
If a line be drawn tou●h a Cir●l●, it is called a Tangent, (a tango) as ●he line AB toucheth the Circumference of the Circle in B, and so AB is called a Tangent line, and is situated on the Term of the Radius CB, and perpendicular thereto, by the 18 Proposition of the 3. of Euclid.  
2. The Secant of an Ark is a line contained between the Centre and the line Tangent, so that it pass by the given Ark, so if BOY be a given Ark, CQ is the Secant thereof, and if BM be a given Ark, then is CR the Secant thereof.  
3. The Tangent of an Ark is a line contained between the term of the Secant, and the touch point, so BQ is the Tangent of the Ark BOY, and BR is the Tangent of the Ark BM, etc.   

PROPOSITION. XII. How to take any altitude or distance by the Sins, Tangents and Secants.  
A of 51, degrees will have for his sins, 77715, parts.  
B of 90. degrees will have for his sins, 100,000. parts.  
C of 39 degrees will have for his sins, 62932. parts.  
Then working by the rule of three, you say if the sins of A 77715. give the side BC of 100 fathoms, how many fathoms will the sins of C, 6.932 for the side AB, and dispose your rule of three as followeth. If 77715— 100— 62932. And having multiplied the second number by the third or the third by the second, and divided the product by the first number, and the quotient of the division will be for the side AB 80. fathoms, 70. inches, 4. lines and 11/12 parts of a line And ●o find out the side AC, you must operate in the same manner by the rule of three, & say if the sine A of 79715 gives the side BC of 100 fathoms, how many fathoms will the sine B give, which is 100,000. for the side AC, and having multiplied the second number by the third, and divided the product by the first, the quotient will be 128. fathoms, 48. inches, 7. lines, & ●4875/77715. part of a line, which is almost one line for the side AC, and thus may you know any kind of distance with great facility, only knowing the angles and one side.   

PROP. XIII. How to take any altitude or distance inaccessible by the sins.  
LEt the altitude AC be proposed to be taken, and the nearest coming to it is B, then in that point B, take the angle ABC, as in the chapter before, which is of 61. degrees, then assemble 61. with 90. the right angle opposite and the whole is 151, which you must subtract from 180, and then will rest 29 for the angle A, and so the three angles of the triangle are known, but neither side; ●nd therefore to know one of the ●●des, that we may operate by the ●●es, tangents and secants, retire ●●rther back or aside to D, and mea●●re your paces or fathoms from B to ●, which I suppose here to be 300. ●aces, and in the point D, take the angle ADB, which is 37. degrees, and then may know that the angle exterior ABDELLA, is of a hundred and nineteen degrees, for it is equal to the two interior angles ABC and BAC by the 32. proposition of the 1. of Euclid, or otherwise it is the compliment of the half circle, of which the angle interior ABC of 61 degrees is taken out, now you have the knowledge of two angles ADB, and DAB, the which being joined, subtract them from 180. and then the third angle will remain DA B, then shall you know the three angles of the triangle A BD, and one of the sides by which you may come to the knowledge of the side A B of the first triangle A BC by the sine, and by the knowledge of the side A B, you may also know the side AC, by the sins, tangents and secants, as in the precedent chapter: and so in all manner of altitudes or distances, accessibles or inaccessibles, and it is to be noted, that one is not bound to wright backward, taking the second observation in D, but at the pleasure of the workman to make choice of the fairest place in the field, so I think no difficulty shall be found in these works   

The manner how to take the plane of a town, or any place out of musket-shot.  
SVppose you were to take the plane ABCDE; first, having considered that the situation of the plane hath a crooked ark BARNES; those two lines must be supposed strait as EQ and BP, making an obtuse angle at the point A, and being far off from this plane in the point P, consider that the lines PB and QE, make but one angle in the point A, then leaving a mark at P, withdraw from thence into some other place of field where your eye may rightly meet with the line A, which will happen in Q where you must set another mark now consider that if a line be drawn from Q to P, it will form a triangle APQ, whereof you may know two angles, P QA and AP Q which being done with your instrument, and you find the angle AP Q to be of 45. degrees, and PA Q of 34. degrees, then add 45. to 34. and cometh in the whole 79. which you must subtract from 180 degrees, and there will rest 101 for the third angle PAQ by the 32. proposition, lib. 1. of Euclid, and the angle EAB is equal to the angle PAQ by the 15. proposition 1. then let us conclude that we have the angle BAE of 101. degrees, which was unknown, then shall it be necessary in the place noted H, to know how the side CD is situated in respect of the parts of the world, and with a declinatorie we shall find it to be seated North, West & Southeast, which shall serve us for the knowledge of all the rest: and in the former manner may be taken all the other angles of the plane proposed, and to have the distance of each side of the plane, you must measure one only as before taught, which I suppose it shall be B C, which shall be found to be 100 fathoms, and having found aside homologe to it or like to it, and divided into an hundred parts equally, it shall serve for a scale to measure all the rest by.   

CHAP. I. A treaty of fortifications as well regularly as irregularly.  
IT is not my intention to treat here of all manner of fortifications, but only of those which by the general and common opinions are held and approved to be the best for the defence and conservation of deserving places; and to begin this Treatise, I will set down certain rules which are now held for maxims of the best fortifications.  
Let the flanked angle be open 90. degrees, or the nearest to 90. that may be it is the angle A in the next trianguler figure.  
Let the flanking angle be not above 150. degrees open, but the lesser, shall always be the best, this angle is BLC in the same figure.  
Let not the line of defence be longer than musket-shot, which is 100, or 110 fathom, it is CK or BY in the same figure.  
The largest neck or gorge of a bastion or bulwark is always the best and the longest flank lines, specially if ears be to be built upon them, CN or THERE are nose lines  
Now shall follow all the denominations of fortifications for the better intelligence of the better treating of them.   

CHAP. II. Denominations of the parties of Fortifications.  
A Flanked angle is the point of a bulwark in the figure following, it is the angle SBN.  
A flanking angle is BLC, which is made by two lines of defence meeting in the point L.  
A line of defence is the distance from the flank to the angle flanked, as here BY, or CK.  
The flank is K or L.  
The line of the flank is KN or IN.  
The curtain is the straight line drawn from one flank to the other, as here KING.  
The shoulder of a bulwark is N or M.  
The pane of a bulwark is BN.  
The ear is always built upon the line of the flank, as in the second figure following.  
The capital or chief line is BY or FC  
Bastion or bulwark is a piece contained within two lines, two panes, and two flanks.  
The neck or gorge of a bastion is the distance between the two flanks for the entry into the bastion.  
A rampard is the earth raised against the wall in the inside to strengthen it against Canon shot.  
The parapet is the elevation above the string of the wall.  
The counter escarp is the exterior brim of the town ditch.  
A half moon is a piece loose from the town raised of earth, and environed with a ditch to hinder the assault of the enemies, it is sometimes made in form trianguler, but sometimes oblong as occasion is, now shall follow the Treatise of the regular forts, and first of all the trianguler.   

CHAP. III. How to build a trianguler Fort.  
SVppose A BC to be the Triangle, where in a Fort three square is to be built.  
From the point C, as the centre, and from the distance CB, let the ark BTA for 60. degrees be described, which shall be divided into two equal parts at the point T.  
And again, the ark BT into four other equal parts, and O B shall be an ark of 7½ degrees, by which draw the line CO, and it shall form the diminished angle OCB of 7½ degrees.  
And to find the point for the flank divide the angle QCE in two equal parts by the strait lines CH, which teacheth us the point I, for the flank and then take the distance FI, and carry it to EKE, and the point K shall be another flank, then draw the lines of the flanks KN and IN perpendicularly to the courtine KING, which is drawn from the flank K to the flank I, and the same upon the two other sides, and you shall have the trianguler sort perfect.  
The flanked angle is of 45. degrees, and the flanking angle is 165. degrees open, which is somewhat far from the best maxims of fortifications.   

CHAP. IU. How to build a square Fort.  
LEt the square ABCD be in which you desire to build the Fort, draw the two diagonales AD. CB, which cut E the centre of the Fort from the point D as centre, and from the distance DC, let the ark CGF be drawn, which shall be divided in three equal parts, and from the point G draw the strait line GD, which formeth the diminished angle GDC of 15. degrees, and the intersection of the line GD in H doth show the line capitol CH, which shall be transported into IKL, and then draw the strait lines CI. DH and their fellows, and to find the point for the Flank, divide the angle FDG in two equal parts by the line DM, which cutting the line CI doth show us the point N for the Flank, then take the distance IN, and bear it to HO, and the point O shall be another Flank, then draw the courtine ON, and from the points O and N draw the two strait lines OPEN and N Q perpendicularly to the courtine ON. Do the same upon each other side, and you will have perfected the square fortres The Flanked angle is of 60. and the Flanking angle is open 150. degrees.   

CHAP. V. How to build the Pantagone Fort.  
THe five sides and five lines being drawn from the point A as centre, and from the distance AB. ●et the ●rke BD of 60. degrees be described, which shall be divided into ●oure equal parts, of which BE is one ●nd from the point E▪ draw the right ●ine A, which cutting the line BC●t ●t the point F, doth give the part BF●or ●or the capital line; which distance must be transported to the other lines AGNOSTUS, etc. then from the point B, draw the strait line BG, and so the others which shall be lines of defence.  
And to find the Flank, divide th● angle EAC in two equal parts by the right line HA and the intersection which it maketh upon the line BG at the point I shall be the Flank, then take the distance GI', and transport it into FK, and so to IG, and the point K shall be a Flank and the point I, then draw the line KING, which shall be a courtine, and draw the lines KL. and IN, which shallbe the Flank lines being drawn perpendicularly to the courtine, and doing the same to every other side, you shall have the desired Fort.   

CHAP. VI How to build the Fort Hexagone.  
Having divided your circle into six equal parts, and drawn the six strait diagonal lines and the six sides.  
From the point B as centre, & from the distance AB, let the ark ADC of 60. degrees be described, and it shall be divided into four equal parts, whereof AD is one.  
From the point D, draw the strait line DB, which cutting the line AC at the pont E, doth give the distance A for the capital line, then transport that distance to BF.  
And to find the point for the Flank divide the angle D BC in two equal parts by the strait line G B, which cutting the line OF at the point H doth show that point for the Flank, then take that distance HF, and transport it to EI, and to all the rest.  
From the point I, draw the strait lines IK perpendicularly to the curtin IH, and draw the other line HL perpendicularly also, and do the same upon each other side, and you shall have the Hexagone perfect, but if you will make ears, they must be built upon the two third parts of the lines of the Flank IK HL as appears at.   

CHAP. VII. How to build the Heptagone.  
LEt BKLMNO be the Heptagone given, and the centre thereof A from whence shall be drawn the seven lines AN. AK. AP, etc. Then from the point K as centre, and from the space KB, let the ark BCA of 60. degrees be described, which shall be divided into four equal parts, whereof BC shall be one of 15 degrees, from the point C draw the right line CK, the which cutting the line AN, at the point D, giveth that part for the capital line. Then take the distance ND, and carry it to KING, and to all the rest likewise, then from the point N draw the right line NI, which shall be the line of defence, and all the others like unto it.  
And to find the point for the flank, divide the angle AKC in two equal parts by the right line GK, which cutting the right line NI at the point F showeth that intersection F, for the point of the flank take the distance and carry it to D and E, and the point E shall be another Flank, then draw from E to F the line (which makes the courtine) than the two perpendicular lines for the Flank lines: whereupon shall be built the ears, and doing the same upon each side your fort shall be perfect, as required.   

CHAP. VIII. How to build the Octogone.  
FIrst let the square AB CD be made, & then the square EFGH.  
And to find the place for the Flank, divide the angle ADC by the straight line DIEGO, and the intersection K showeth the point for the Flank, as in the Forts precedent, and so transport to every side as in the other, and you shall have the Fort finished.   

CHAP. IX. The description of the height, depth, and thickness of every part of a complete Fortification.  
AB The largiour or breadth of the rampard 66. foot.  
AC Height of the rampard, 14. foot.  
GA' the interior heel of the rampard 14. foot.  
OF The breadth of the parapet 20. foot.  
EH Height of the parapet 6. foot.  
BY. The exterior heel of the parapet 7. foot.  
IK. The covered walk under the false bray 20 foot.  
KL The parapet of the false bray 20 foot.  
MN The breadth of the ditch 120 foot, & the depth thereof aught to be 10 foot.  
NO The heel of the ditch 10 foot.  
OPEN The covered walk of the counterescarpe.  
PQ The counter escarpe 6. foot of height.   

CHAP. X. The manner how to fortify places irregularly.  
Upon a line of 100 fathoms is to be made a tenayle; upon a line of 150. fathoms two half bastions. Upon a line of 200. fathoms a double tenayle. Upon a line of 250. fathoms or thereabout is to be made one bastion, and two half bastions.   

CHAP. XI. A Fortification irregular.  
SVppose the place QRSTV to be fortified without diminishing the place, and having taken it upon a fair sheet of paper and well considered, all the measures thereof then without this plane conduct the line AB of 250. fathom, and upon that line (according to the rules precedent) build one bastion, and two half bastions taken from the Octogon, and then shall you be at the point A, and from the point A, draw the line AP of 150. fathom, or thereabout, upon which you may build two half bastions; then draw the line PO, upon which you shall as before build one bastion and two half bastions, and so upon the line OX, and upon all the others according to the length that they shall be found. But if it should be required, that the Towne-walls should serve for the courtines, the figure following shall be an example either for a part or for all about a town.   

CHAP. XII. Another manner of fortifying irregularly.  
SVppose that one would fortify the plane BAPLMNO, but in such haste, and with so little cost that the town walls should serve round for the courtines, the plane then being taken upon a fair sheet of paper, and having drawn the line CDEF of 400. fathom, it shall be found necessary to build upon the same one bastion two half bastions and a tenaile, viz. the bastion E, the two half bastions D and F, and the tenaile DC, & having drawn the lines round about the town, leaving sufficient space between them and the walls to build bulwarks or bastions, you may take for the length of every line of defence, that is to say from the flank of one bastion to the point of another, 100 or 110 fathom and so every bastion shall be scoured with musket-shot, as the forts built regularly. This may serve for an example to those who by chance, may have occasion to repair or strengthen any part of a town, wall, or the whole, and for the fortifying of the wall, the earth must be raised in the inside as a rampard.    

A TREATISE OF Arithmetic.  

CHAP. I. Addition  
SInce that Arithmetic is absolutely necessary, and required in diverse and many Geometrical operations, I have added the examples following only to renew and refresh the memory of those who have already studied it; and not those who are quite ignorant therein, (commending them to large and ample Treatises together with masters of the science) but for such as by a weak memory have let slip the habitude which by practice they had in times past obtained, who may easily recover by this short Treaty sufficient knowledge to perform any ordinary operation beginning with Addition, which is a collection of many numbers in one, as he who would add together the numbers ABC, following to have the sum D, must begin with the first colon: And say 2 and 5, or 7 and 1 is 8, and set 8 under the line, as appears by the example above, then in the second colon, say 3 and 4 or 7, and pose them as before; afterward say for the 3. colon 6. are six, & posing or setting the numbers collected directly under the figures not collected as 8 under 2, and 7 under 3, and six under six, but if any should be above the number of 9, then to set 0 in the place, and set forward all the ten.  

The Proof.  
MAy be made in casting all the 9 away, though many times falls but by substraction is the way most certain, as thus, begin with the last collon and say 6, cut of 6, rest nothing, and give the 6 under the line a flash, than 4 and 3, or 7 and 7, out of 7 rest nothing, and as before give it a flash, than 1, 5 and 2, or 8, which as before shall be made nothing, the 8 underneath.    

CHAP. II. Substraction.  
SVbstraction is to take away a little number from, or out of a greater, as if one would from 8 6 4 2 take away 4 3 2 1, then must the numbers be disposed as followeth, the greater number uppermost, and the lesser undermost, then draw a line: Great number  8642. A Lesser number  4321. B  rest 4321. C And say he that of 2 in the rank payeth one, rests 1, the which set under the line, then of 4 pay 2, rests 2, which must also be set under the line, and who pays 3 out of 6. rests 3. setting them as before, then 4. out of 8 rests 4. which shall be also set as the others in the example above.  

The Proof.  
IS made in adding the lesser number B with the rest C as followeth.  
Little nummer.  4321  rest 4321 Great number.  8642 And the sum shall be the first number A, if the substraction have been well made.    

CHAP. III. Multiplication.  
MVltiplication is the assumption of one number, as many times as the other, containeth in itself unities. As if one would multiply 6. by 3, that is, to take as oft 6 as 3 contains unities, as in the example above, where 2 taken 4 times makes 8, and 5 taken 4 times makes 20. so that the number A multiplied by the number B multiplicator 4 times comprehended, make the product 2608.  

The Proof.  
IS made only dividing the product 2608 by the multiplication 4 and the quotient will be 652. if the multiplication were well made, as appears in the example following.    

CHAP. IV. Division.  
DIvision is the separation of a number into aliquot parts thereof, as to divide 5689 by 25. the figures must be disposed, as followeth, viz. the divisor under the first figures of the number, which is to be divided, as for example. And after the last figure of the number, shall be drawn a half circle to separate the quotient. The numbers being thus disposed, then say 2 in 5 how many times, and it shall be found twice; then set 2 behind the half circle as in the first example following; and then say twice 2. are 4 of 5 rests 1, dashing out the 2 and the 5. and set the 1 over 5, and say again twice 5 are 10. and strike out the 5 divisor under the 6. and also the 1. over the 5, then set forward your divisor one figure more as in the second example, and say 2 in 6. how how many times, and it shall be found 2, and say twice 2 are 4, and 4 out of 6. rests 2. which shall be set over the 6, then say twi●e 5 are 10. and 10. out of 18. rests 8 and 1 out of 2. rests 1. then dash out the 2. and set 1. over it; and also dash out the 5 and 2. divisors, and set more forward the divisors, as in the third example, and say 2 how many times 18. and it shall be found 7. times, and having set the 7, say 7 times 2 is 14. and 14 out of 18 rests 4, and set the 4 over the 8; and to conclude, say 7 times 5 are 35, and 35 out of 39 rests 4, and 3 out of 4 rests 1, and so your division is ended, as appears here following by three examples one of each operation.  

The Proof.  
Multiply the quotient 227. by the divisor 25, and to the product partial, add the numbers which rest, viz. 14. if any rest, and then add the whole together, and the product shall be the first number if the division hath been well made as in the example following.    

CHAP. V. Rules of Fractions.  
A Fraction is a number noting the parts aliquot of an entire or whole number, whereof it is said to be a fraction as a penny, the twelfth part of a shilling, one inch the four and fortieth part of an Ell, etc.   

CHAP. VI How to reduce intiers and Fractions all into Fractions.  
TO reduce 8¾ and 5⅔ all infractions you must multiply 8 by 4, saying 4. times 8 is 32, and the 3 being added, makes 35, which shall be set above a line just under the first figures, and under that line set 4, as in the example following, to show that the 45 are all fourth parts, and do in the like manner by the 5⅔, and say 3 times 5 are 15, and 2 makes 17. and so you shall have 17. thirds, as in the example following.  
To reduce 8¾ and 5⅔ all into fractions.   

CHAP. VI To reduce all fractions into one denomination.  
TO bring these two fractions to one deno. the 35 quarters numerators must be multiplied by the other numerators 17. thirds, and set the product over a line as here following, then multiply the 4 denominator by the other 3. denominator, and set the product of those under the line, and then will there be,  

Addition of Fractions.  
TO add ⅔ to ¼ the figures, dispose the figures as followeth, and say 3 times 1 is 3, and set the 3. over a line above the head of the others, and then say 4 times 2 are 8. and set them over a line above the head of the others also, then say 3 and 8 are 11, and set them over a line between the first numbers and the 11, shall be numerators, and say 4 times 3 are 12, & set them under the middle line, and those shall be denominators, and so you shall have eleven twelfth parts as followeth. But if the numerator be greater than the denominator, than it shall be divided by the denominator, and the product shall be an entire or entires, and what resteth (if any rest) shall be fraction, which ought to be abridged, as appeareth here following. Where the numerator 38 is greater than the denominator 24. then being divided by the denominator 24. doth yield 1 entire, and 14/14 the which being abridged come to 7/12, which is almost two thirds, and so of all others.    

CHAP. VII. Additions of intiers and fractions.  
BEing proposed to 243⅔ unto 462¼, the intiers must be added as in the first chapter of Addition and the fractions must be added as in the precedent chapter, and set product as appears following.   

CHAP. VIII. Substraction of Fractions.  
Whosoever would subtract ⅔ from ¾ must dispose the figures, as in the example following; and first multiply the numerators by the denominators a cross, as 3-times 3 are 9 and 4 times 2 are 8, and set 9 and 8 over the lines above the heads of the others, and then say take 8 out of 9, and there rests 1, which must be set over a line between both, and afterward say 3-times 4 is 12, multiplying denominators by denominators, and set them under the middle line, which shall be the denominators for the rest, as appears clearly in the example following.  
Out of 9 pay 8 rest 1   

CHAP. IX. Subtractions of intiers and fractions.  
TO subtract 1831 from 267 2/● the fractions must first be multiplied as in the example following; and say twice 2 are 4, and set the 4 over a line, than 3 times 1 are 3, and set them over a line, and then who pays 3 out of 4. rests 1, which must be set over the middle line, which done, multiply the one denominator by the other, and set the product under that which doth rest, as followeth.  
But if the fraction of the received should be less than the fraction of the sum paid, then must there be one borrowed from the whole or entire number, and count it according to the denomination of the fraction; for if the denominator be 4, then shall the intier be 4/4, if 5, then 5/5, if 6 then 6/6, etc.   

CHAP. X. Multiplication of Fractions.  
LEt there be proposed a superficies in the form of a paralellogram, vulgarly called square, the sides whereof the one is in length 7/8 of a fathom, and the other in breadth ¾, and these two fractions are to be multiplied together to find out how much the whole superficies doth contain the figures must be disposed as followeth. And multiply one numerator by the other numerator, and then one denominator by the other denominator, saying 3 times seven is 21, and set them over head, than four times 8 is 32, and set them underneath, and the whole will be 21/32 parts of a fathom, which certainly containeth the required superficies.   

CHAP. XI. Multiplication of entiers and Fractions.  
TO multiply ¼ by 2½, you must first of all reduce the whole into fractions, and then as here above multiply numerator by numerator, and denominator by denominator, and the product will be 45/●8, as plainly appears by the example following. But if it were proposed to multiply greater numbers, as 20 by 15 26/29, then multiply the 15. intiers by the deno. 29. of the fraction, & then add the numerator 26 of the same fraction; which done will mount to 461/29, than set the 461 over a line, and the 29. under it, and afterward multiply the 20 intiers by the 461. which done, divide the product of the whole by the denominator 29. and the numqer required shall be 317 27/19, as appears:   

CHAP. XII. The division of Fractions.  
TO divide ¾ by ⅓, each numerator is to be multiplied by each denominator opposite, and set the product over a line above them, and then divide the greatest product by the least as followeth.   

CHAP. XIII. To divide intiers and fractions by intiers and fractions.  
TO divide 12⅔ by 3⅙, they must first be reduced all into fractions as before, and then you must multiply the numerators by the denominators across as followeth, and then divide the greatest product by the least, as this example doth clearly demonstrate.   

CHAP. XIV. Evaluation of fractions which may not be abridged.  
SVppose you were to abridge 7/9 parts of a fathom, first you must consider what are the parts of the intier or whole, as 6. foot, or 72. inches: than you must multiply the numerator 7, by the denominator 72 parts, and let the product be divided by the denominator 9, and then you will find 56 inches for the evaluation of 7/9 parts of a fathom. By this means any fraction may be abridged as well in Geometry as as commerce, although they seem not to be abridged.   

CHAP. XV. For the evaluation of measuring lands.  
YOu must consider that the fathom of 6 foot in length, doth contain in superficies 36. and that the 72 inches in length doth contain in superficies 5184 inches, and of other measures then to valuate a fraction of 19/4● parts of a fathom, square in superficies you must multiply 5184 by 19 and divide the product by 47, and there will be 2095 inches for the square of 29/47 of a fathom square and so of other like measures.   

CHAP. XVI. Of the rule of three without fractions.  
MVltiply the second number 400 by the third 12. and product 4800 by the first number 4, and the quotient shall be the number required, and dispose your rule as followeth.  
months— pounds— months. If in 4— 400— 12  

The probation of this Rule.  
IS to multtplie the first number 4 by the fourth number 1200, and to multiply the second by the third, and the two products will be equal if the rule be well made.    

CHAP. XVII. Of the rule of three with intiers and fractions.  
FIrst all the intiers must be reduced into fractions as followeth. yard's pounds yards. If 2¼ 12½ 7½ 9/4 25/2 15/2 Which done, you must multiply the second number of fractions as by the third number of fractions 15, & then again multiply the product by 4 the denominator of the first number, and then say 2 times 2, or 4, and 4 times 9 is 36, which must be set under the line, by which you shall divide the first product 1500, and the quotient shall be the number required, as appears,  
here followeth two examples, differing the one from the other; whereof the manner of multiplying the one, is more easier than the other the first is multiplied as the precedent, but the last is multiplied first by all the intirres, viz. by 3, by 8 and by three, leaving the fraction ● by itself, and after all take the thir● of the intier, viz. of 50000, saying th● third part of 5 is one rest 2, for the 10 which is valuated at 20, then say the third part of 20 is 6, and so rests 2 for the second 0, and so to the end, and what shall rest at last, shall be set over a line, and your 3 4th or 5 under the line, than all being added together, you shall divide the product cutting off the figures to the quantity of the first number, saying, by ten, by a hundred, by a thousand, by ten thousand, by a hundred thousand, & the remainder is the number required, as appeareth, 191⅔.  

The first example.  
 
The second number being multiplied by the third, doth mount to 57500000, and being divided by the first multiplied by 3, as before is taught, the quotient will be 191 2/●0.   

The second example.  
   

CHAP. XVIII. Extraction of the square root.  
FIrst dispose your numbers as followeth out of which you mean to draw the root separating your figures by two and two beginning at the latter end; but first strike the half circle 73/21 21/01 (and then say the root of 73 is 8, and set 8 before the half circle, & rests 9, then double the quotient 8, and say 2 times 8 are 16, and set the 6 under the last figure of the second part of figures, and 1 under the first figure of the first part a● in this first example. Then say how many times is 1 in 9, and it shall be 5. which you shall also set under the 1 of the second separation, as apppeares in this second example. And than say 5 times 1 are 5, which taken out of 9 rests 4, and 5 times 6 are 30, and 30 out of 32 rests 2, and 3 out of 4 rest 1, and then again say 5 times 5 are 25, which out of 3● rest 6. and 3. out of 12. rest 9 and then double the quotient, and say twice 5 are 10 set 0 under 0 of the last separation, and keep ● in memory, and say twice 8 are 16, and 1 that I keep in mind makes 17, then set down 7 under the 5, and the 1 under the 6 of the middle separation, as appears in this example following. And than say how many times is 1 in 9 and it shall be 5 times, which shall be set down for quotient, and also under the last figure 1, and then say 5 times 1 are 5, out of 9 re●ts 4, and 5. times 7 are 35, which out of 36 rests 1, and 3 out of 4 rests 1, and 5 times 0 is 0, and 5. times 5 are 25. out of 31 rests 6, & 3 out of 10 rests 7, and 1 out of 1 rests 0, and so the rule is ended as appears following.   

CHAP. XIX. Another example of the square root.  
After you have separated your figures by two and two, and drawn 251 the square root, there doth yet rest 268, which must be reduced into fractions, and to begin set that rest 268 over a line at the end of your root, and that rest shall be numerator of the fraction; and to find the denominator only, double the root 251 if it be bigger than the rest, but if less as here, add one to the doubling of the first figure, saying twice 1 is 2, and 1 that I add makes 3, add only double the rest, and set it under the line, and that shall be the denominator of the fraction; and to have the root of this fraction here above, first take the root of the numerator, and set that root over a line, and it shall be numerator as appears following. Then draw root of the denominator, and set it under the line, and that shall be a denominator, and so you shall find 16/22, and what rests is unsensible. But because that this fraction 16/22 is not precisely perfect, and that there is a rest in each extraction, you may operate as followeth to have it move exactly, add as well to the numerator as to the denominator two 00, or four, or six, etc. and from each product or quotient out of one figure for every two 00 which you shall have added, and the more that you add 00, the more precisely you shall have the root as appears following. But if it were proposed to extract the square root of 16/25, there would nothing rest, nor would it be needful to add any 00, for the root of 16 is 4 and the root of 25 is 5, and so we should have 4/5, and the like in all such other accidents; and thus much for Arithmetic. Vale.   
FINIS.   

THE CONTENTS.  
A Treatise of Fireworks for War, 
page 1.  The manner how to make the Morterpeece. 
pag. 5.  The manner how to make Granades or mettle for the mortar or hand, 
pag. 10.  The manner how to make Granades of Canvas for the Mortar. 
page 16  How to make fiery arrows. 
pag. 20  How the Granads are to be charged into the Mortar. 
page 24  Than manner how to shoot the Granads, 
page 28  A most violent method to set a town on fire. 
page 33  How to make Granads to cast with men's hands. 
pag. 39  How to make fiery wheels to be cast with men's hands. 
pag 45  How to make a ship of wild fire. 
pag. 48  How to make a Petard. 
pag 55   
A Treatise of artificial fireworks for pleasure. 
page 61  A method to make moulds for rockets for the air. 
page 64  How to make flying rockets for the air. 
page 69  How to make moulds for rockets for the ground. 
pa. 74  How to make the composition for rockets upon the ground 
page 76  The manner how to make Serpents. 
pa 78  How to make golden rain. 
pag. 81  How to make stars, 
pag. 83  How to make Stars giving great reports. 
page 86  The manner how to make Saucissons, 
page 88  How to make Stoupell or preparing of your cotton-wieke. 
pag. 91  Thu manner how to assemble and set together the parts of a rocket. 
pag▪ 94  How to represent diverse sorts of figures in the air with rockets. 
page 97  How to make fiery boxes. 
pa. 101  How to make fiery lances. 
pag. 103  The manner how to make rockets for the water. 
page 105  How to make Girondels or fiery wheels 
pag. 110  The manner how to make Ballons. 
pag. 113  How to make flying Saucissons. 
pa. 120  How to make short Guns for the Saucissons. 
page 123  The manner how to dispose and build a great or little fireworke. 
pag. 125  A most precious unguent for any burning. 
pag. 131   
A Treatise of practical Geometry. 
page 136  The method how to make the Crosse. 
pag. 139  How to take a height accessible. 
pag. 141  How to take a height inaccessible, or one height upon another height. 
pag. 145  How to take any distance upon a place accessible or inaccessible. 
page 149  Another manner how to take a distance inaccessible. 
pa. 151  How to take a distance only upon a line parallel to it. 
pag. 165  How to take the depth of a Valley. 
p. 157  The manner how to take either distance or altitude with the Sector. 
pag. 161  How to take any distance or altitude inaccessible with the Sector. 
pag. 165  Definitions of Sines, Tangents, and Secants. 
pag. 168  How to take any altitude or distance by the Sins, Tangents, & Secants. 
pag. 171  How to take any altitude or distance inaccessible by the Sins. 
pag. 176  The manner how to take the Plane of a town or any place out of musket-shot. 
page 180   
A Treatise of Fortification, as well regularly as irregularly. 
pag. 184  Denominations of the parties of Fortification. 
pag. 186  How to build a trianguler Fort. 
p. 189  How to build a square Fort. 
pa 192  How to build the Pan●agone Fort▪ 
p. 195  How to build the Fort Hexagone. 
p. 198  How to build the Heptagone. 
pa. 201  How to build the Octogone. 
p. 204  The description of the height, de●th, and thickness of every part of a complete Fortication. 
p. 206  The manner how to fortify places irregularly. 
p. 209  A Fortification irregular. 
p. 210  Another manner of fortifying irregularly. 
p. 213   
A Treatise of Arithmetic. 

Addition. 
p. 216  Of Substraction. 
pag. 219  Of multiplication. 
p. 222  Of division. 
p. 224    How to reduce intiers and Fractions into Fractions. 
p. 229  To reduce all fractions into one denomination. 
pag. 230  Additions of intiers and fractions. 
p. 234  Substraction of Fractions. 
p. 235  Addition of Fractions. 
p. 231  Subtractions of intiers and fractions. 
p. 237  Multiplication of Fractions. 
p. 239  Multiplication of entiers, and Fractions. 
p 241  The division of Fractions. 
p. 243  To divide intiers and fractions by in●iers and fractions. 
p. 44  Eualuation of fractions which may not be abridged. 
p 245  For the evaluation of measuring lands. 
p 246  Of the Rule of three without fractions. 
p ●48  The probation of this Rule. 
p. 249  Of the Rule of three with intiers and fractions. 
p. 250  The first Example. 
p. 253  The second example. 
p. 25●  Extraction of the square root. 
p. 255  Another example of the square root. 
p. 258    

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