22 the quotient is the diameter or height of the circumference Likewise measure the greatest circumference of mettall at the mouth multiplying that measure by 7 deuide by 22 as before the quotient will shew the diameter of the mettall at the mouth substract that diameter last found from the diameter at the breech ½ the remaine is the true disparture Example A Culuering whose greatest circumference of mettall at the breech containeth 66 inches and at the mouth 44 inches I demaund how high is the diameter of the mettall both at the breech and mouth as also what is the true disparture of that peece Resolution Multiply 66 by 7 ariseth 462 deuide by 22 the quotient is 21 the height of the mettall at the breech likewise multiply 44 by 7 you haue 308 deuide by 22 the quotient is 14 the height of the mettall at the mouth which 14 abated from 21 rests 7 the which 7 deuided in 2 equall parts yeelds 3 inches ½ for a part the true disparture of that Culuering This is one of the principallest points belonging to a Gunner to know truely how to bring the concaue of the mettall of his peece euen diuers other waies there is to do the same As for chambred peeces there is no perfect or generall rule but is to be considered according to the chamber or concaue of the peece Euery reasonable Gunner may iudge in that case How by Arithmeticall skill you may mount any great peece of Ordinance by an inch rule vnto 10 degrees of the quadrant if you want a quadrant or other instrument First you must measure the iust length of the Cannon or bore of the peece reduce that measure into inches and double the same afterwards multiply the number of inches so doubled by 22 and deuide by 7 and note what the quotient number is which quotient deuided by 360 the degrees contained in the whole circumference of euery circle the last quotient number will shew you the number of inches and parts of an inch that will make a degree in the quadrant for that peece Example Admit there is a Saker or Fawcon whose concaue or bore containeth iust 7 foote in length and that you desire to know what parts of an inch rule will mount her to one degree of the quadrant you must reduce 7 foote into inches and you haue 84 inches that 84 doubled is 168 the which multiplied by 22 ariseth 3696 the which deuided by 7 the quotient will be 528 that quotient number being deuided againe by 360 wil yeeld 1 7 15 that is one inch and ½ wanting 1 15 part of an inch So I affirme that any peece of Ordinance whose chase or bore is but 7 foote long being mounted by an inch rule one inch and 7 15 parts that peece shall lye iust the height she wold haue done if you would haue mounted her one degree of the quadrant The like order is to be obserued in mounting any other peece of Ordinance by an inch rule of what length soeuer And note that in mounting any other peece of Ordinance to any degree of the quadrant by a Geometricall quadrant you must put the rule of the quadrant into the peece mouth lifting the peece vp or downe with a leauer or hand-spike towards the breech till the plummet cut iust vpon that degree of the quadrant you desire But to mount her by an inch you must place the rule vpon the highest part of the mettall at the breech of the peece coyning the peece vp or downe till through the sight or slit in your rule be lifted to that part or deuisiō in your rule that answereth the degrees you desire you espie the Carnoize or highest part of the mettall at the mouth of the peece and the marke all 3 in a streight line If you would mount the same peece to 2 degrees of the quadrant by an inch rule you must multiply the measure in your rule last found being 1 inch 7 15 parts by 2 in the order of fractions and you shall haue 44 15 the which 44 being the numerator of the fraction deuided by 15 the denominator the quotient being 2 inches 14 15 is your desire so may you affirme that 3 inches by the rule wanting 1 15 part of an inch will make 2 degrees by the quadrant And note that looke how much you would haue your peece mounted by an inch rule for to answer any number of degrees vnder 10 either multiply that number by the number of inches and parts of an inch that makes a degree of the quadrāt or else working as you did the first conclusion multiplying the first product by the number of inches desired and deuiding that product by the numbers afore mentioned your last quotient will resolue you of your desire Example I demaund how much the peece afore mentioned should be eleuated by an inch rule to answere to 8 degrees of the quadrant Resolution Reduce the length of the bore of the peece into inches as afore is shewed doubling that measure and it makes 168 as you see in the 1 conclusiō which 168 inches multiplied by 22 yeeldeth 3696 inches the which product afterwards multiplied by 8 ariseth 29568 which summe deuided by 7 the quotient is 4224 the same deuided by 360 yeelds in the quotient 11 inches 11 15 parts of an inch so many inches and partes of an inch must the same peece be eleuated to with an inch rule to answere to 8 degrees of the quadrant as by triall you may find How by Arithmeticke skill you may know the true thicknes of mettall in any part of any peece of Ordinance Take a paire of callapers and measure the height of the out side of the mettall in that place of the peece whereas you desire to know the thicknes of the mettall then with an inch rule or else a paire of streight compasses measure the diameter of the bore or concaue of the peece abating the height of the said diameter from the height of the whole thicknes of that part of the peece so measured And note the remainder the which deuide in 2 equall parts and the one of those parts is the iust measure of the thicknesse of the mettall in that part of the peece Example I prooued this conclusion with a Culuering whose bore or concauity at the mouth was 5 inches ½ height I found that the thicknes or height of the whole circūferēce of the sayd peece at the touch-hole was 16 inches ⅓ from the which I abated 5 inches ½ fraction wise rests 10 inches ⅚ parts of an inch that deuided in 2 equal parts the quotient is 5 inches and 5 12 or 5 inches ½ wanting the 1 12 part of halfe an inch so thicke was the mettall of that Culuering at the touch-hole Likewise I searched for the thicknesse of mettall in the same peece at the end of the trunions and I found that the thicknes or height of the superficies of all the mettall there contained 13 inches
onely to shake and beate the wall and the Ordinance on the two other side mounts or platformes shooting something slanting are to coyne or cut out that which the Ordinance from the middle platforme doth shake or loose The Baskets ramd full of earth being placed betweene each peece of Ordinance are to defend the Gunners and Laborers from hurt of them that are besieged as afore I haue said And further it is to be noted that to batter the coyne or cullion point of any wall two places is sufficient to plant your Ordinance in Also you may batter and beate downe the wall of a Towne or Castell as well by night as day so as the enemie shall haue no time to builde vp in the night that which was dung downe in the daie as thus Lay your peece or peeces to the marke in the day light and note well what degree of the quadrant she lieth at which is soone done in putting the rule of your quadrant into the peece mouth so laid against the marke letting a line and plummet fall to the ground from the said point of your quadrant and at the lighting of the plummet on the ground there driue in a stake or wooden pin and letting a plumbe line fall likewise from the midle part of the taile or breech of your peece to the ground driue therein another stake into the ground then stretch a line from the said 2 pinnes so as the ends of the said line may reach 2 or 3 yards further then the pinnes at each end And there make them fast in driuing a pin of wood or yron into the ground at each end then bringing your peece or peeces to lie streight aboue the said line or lines so drawne which is easily done hauing a lanterne with a close couer you may both charge and recharge and shoote aswell by night as day according to your desire How you may know the true weight of any number of shot for seuerall peeces of Ordinance how many soeuer they be and how many Tun weight they do all weigh Question Suppose a Ship is loaden with Bullets to be caried to the siege of a Towne c. in which ship is 500 shot for whole Cannons 800 demy Cannon shot 900 Culuering shot 1000 demy Culuering short 1100 Saker shot 1200 Minion shot and 1400 Fawcon shot the question is to know the true weight of all the shot and how many Tun they do all weigh Resolution In the beginning of this treatise I shewed how to find out the weight of any vnknowne bullet by the weight of a knowne bullet of the like mettall so that multiplying the number of euery seuerall sort by the weight that one of them weigheth and adding all the products into one summe and then deuiding that totall by 2240 pound which is the pounds in a Tun the quotient will shew you how many Tun all those bullets weigheth Example Admit the Cannon shot weigh 60 pound a peece by which I multiply 500 the number of that kind of bullet so ariseth 30000 pound weight and then there is 800 demie Cannon shot of 32 pound weight a peece which multiplied as before makes 25600 poūd weight And then there is 900 Culuering shot of 16 pound weight a peece which makes 14400 pound weight And then 1000 demie Culuering shot of 10 pound weight a peece which makes 10000 pound weight And then 1100 Saker shot of 5 pound weight a peece which makes 5500 pound weight And then 1200 Minnion shot of 3 pound weight a peece which makes 3600 pound And lastly 1400 Fawcon shot of 2 pound weight a peece which makes 2800 pound weight All these summes added together makes 91900 pound weight which deuided by 2240 yeelds in the quotient 41 Tun and 60 pound weight remaining In this order you may know how many Tun weight any number of shot weigheth so that knowing how many Tun any ship is of burthen you may easily know how many shot will loade the said ship How any Gunner or gunfounder may by Arethmiticke skill know whether the trunions of the peece be placed rightly on the peece or not Measure the length of the bore of the peece from the mouth to the breech deuide that measure by 7 and multiply the summe that commeth in the quotient by 3 the product will shew you how many inches or other measure the trunions ought to stand from the end of the lowest part of the concauity of the sayd peece at the breech And note that the trunions ought so to be placed as ⅔ parts of the circumference of the peece may be seene in that place whereas the trunions are set Example Admit the cilinder or concaue of a Cannon or other peece of Ordinance be 10 foote ½ long I demaund where the trunions of the sayd peece ought to stand Answere Reduce the length of the concaue of the peece into inches you haue 126 inches the which deuided by 7 the quotient is 18 that multiplied by 3 makes 54 inches or 4 foote ½ so farre ought the trunions to be placed from the breech or lowest part of the hollow concauity of the sayd peece Another way Or multiplying the length of the concaue of the peece by three and deuiding the product by 7 the quotient will shew the true place how farre the trunions ought to stand from the lowest part of the bore or concauity of the peece Example 126 inches the length of the concaue of the peece multiplied by 3 makes 378 inches which number deuided by 7 the quotient is 54 inches as before And note that the trunions of euery peece were inuented to hold the peece vp in her cariage to moue her vp and downe to make a perfect shot and to hold her fast in her cariage after she is discharged for if the trunions be placed too neare the mouth the peece will be too heauy towards the breech so as the Gunner appointed to serue with her shall haue much adoe to raise her to coyne her vp or downe or being placed too neare the breech the contrary will happen How you may know what empty caske is to be prouided to boy or carry ouer any peece of Ordinance ouer any riuer if botes or other prouision cannot be gotten It is thought sufficient that 5 Tun of empty caske will swimme and carry ouer a Cannon of 8 or 9000 pound weight 4 Tun will carry ouer a demy Cannon 3 Tun a Culuering and 2 Tun a Saker accounting all prouisions to be made fast thereto as plankes ropes c. so that knowing what number of Ordinance is to be ferried or caried ouer any riuer adding all their weights into one summe by framing the Golden rule you may presently know what empty caske is to be prouided to ferry ouer all the sayd Ordinance at one instant Example If a Cannon of 8000 weight require 5 Tun of empty caske how much emptie caske is to be prouided to carry ouer so many Ordinance as is
is 1331 4 and the cubike fraction of the greater is 9261 4 which knowne I set down vnder three pound the weight of the lesser bullet the vnite 1 and it will represent a fraction thus 3 1 and then multiplying and deuiding by the golden rule in fractions I find that the weight of the Culuering shot of 5 inches ¼ diameter will weigh 20 pound weight and almost ¾ pound as in the working you may find How by knowing the diameter and weight of an yron bullet to find the weight of a bullet of marble stone of the like diameter or how by knowing the weight and height of a bullet of marble to find out the weight of an iron bullet of like height Question Admit an iron bullet of 4 inches height weigh 9 pound I demaund what shall a bullet of marble stone weigh of like diameter Resolution In a theoreme afore mentioned I find that a bullet of yron to the like bullet of marble stone shall beare such proportion as 34 is to 15. And therefore I multiply the weight of the iron bullet knowne being 9 pound by 15 the proportion the stone bullet beareth thereto so ariseth 135 which deuided by 34 the quotient is 3 pound and 33 34 parts of a pound that is 4 pound wanting 1 34 part of a pound so much shall the bullet of marble stone weigh that is in Diameter and circumference equall to the like bullet of iron In like order reducing the weight of the stone bullet into his proper fractioÌ„ you shal haue 135 34 pound which deuided by 15 the proportioÌ„ the stone bullet beareth to the like bullet of iron your quotient is 9 the nuÌ„ber of pounds that the iron bullet weigheth How by knowing the height and weight of an iron bullet to find out the weight and height of the like bullet of lead or how to find the weight of an iron bullet by knowing the weight of a leaden bullet of like diameter Question There is a Cannon that shootes an iron bullet of 72 pound weight what shall a bullet of lead of the same diameter weigh Resolution To worke this I note that the theoreme before saith that a bullet of iron to the like bullet of lead shall beare such proportion as 28 is to 19 therefore I multiply 72 the pounds which the iron shot weigheth by 28 so ariseth 2016 which deuided by 19 the quotient is 106 pound 2 19 so much will a leaden bullet weigh that is proportionall to an iron bullet of 72 pound weight In this order by working as I haue shewed in the end of the last conclusion you may by knowing the weight of the leaden bullet find out the weight of the like bullet of iron How you may find out the weight of any stone bullet of marble by knovving the vveight of the like bullet of lead or hovv by knovving the vveight of the stone bullet to find out the vveight of a leaden bullet of like proportion Question If a bullet of lead weigh 106 pound what shall a bullet let of marble stone weigh of the selfe like proportion Resolution To answer this I find that a bullet of lead to the like bullet of marble beareth such proportion as 4 to 1. Therefore multiplying 106 by 1 and deuiding the product by 4 the quotient will be 26 pound ½ shewing the true weight of a stone bullet that is proportionall to the like bullet of lead And now to find out the weight of the leaden shot by knowing the weight of the stone shot reduce the stone bullet into his properfraction you shall haue 53 2 setting 1 vnder 4 fraction wise multiply the numerators together and likewise the denominators and deuiding the product arising of the numerators by the product of the denominators your quotient will be 106 pound shewing the true weight of the leaden bullet If you haue or do know the weight and true height of a bullet of stone or any other mettall and is desirous to know the weight and height of another bullet that is greater or lesser and of the same mettall in working as the first conclusion sheweth you shall haue your desire To find out the circumference of any circle or bullet Question I demaund how many inches is about the circumference of that bullet whose diameter is 9 inches Resolution To worke this or any such like there is a generall rule as thus that the proportion of the diameter to the circuÌ„ference is as 7 to 22 therefore multiplying the diameter 9 by â—2 ariseth 198 which summe deuided by 7 the quotient is 28 2 7 shewing the true number of inches about the circumference of a bullet of 9 inches diameter as the figure here demonstrated sheweth How you may by knowing the circumference of any bullet find out the height or diameter of the same Question The circumference of the bullet in the last conclusion contained 28 inches 2 7 as in the demonstration you may see I would know how I should worke to find how many inches the diameter of the same is Resolution To answer this and all such like I must worke contrarie to the former conclusion first reducing the whole number and broken being 28 inches 2 7 into his proper fraction and it will be 198 7 then multiplying by 7 according to Archimedes doctrine and deuiding by 22 the quotient will be 9. so many inches is the diameter of the same bullet In this order you may find out the diameter and circumference of all other bullets How to find out the solid content of any bullet c. Question There is a bullet of iron whose diameter containeth 9 inches how many square inches is in the solid content thereof Resolution To know this and all such like there is a generall rule as thus to multiply the diameter in his square I meane cubically and then multiply that product by 11 deuide the totall summe by 21 the quotient sheweth the number of square inches in that spherical globe or bullet for 9 multiplyed cubically ariseth 729 which augmented in 11 is 8019 that totall deuided by 21 yeeldeth 381 inches and 6 7 so many square inches of iron will be in a bullet of 9 inches diameter To find the true content of the superficies of any circle drawne vpon a flat as on a table or paper c. Question There is a circle whose diameter is 21 inches I demaund how many square inches is contained within the circumference of the same Resolution To resolue this ofr such like there is a generall rule in taking ½ the diameter and multiplying it in ½ the circumference or squaring the diameter and multiplying the product by 11 and deuiding the result by 14 the quotient sheweth the Area or content of all the superficies within the circumference thereof Example The square of 21 is 441 which multiplied by 11 is 4851 that deuided by 14 yeeldeth in the quotient 346 inches ½ Or other waies take the halfe of 21 inches that
from which I abated the diameter or concaue at the mouth being 5 ½ inches rested 7 ½ which deuided in 2 equall parts the quotient being 3 inches ¾ shewed the true thicknesse of the mettall at the trunions In this order you may find the true thicknesse of mettall in any part of any peece of Ordinance Another way to know the thicknesse of mettall in any part of any peece of Artillerie Take a letherne girdle and gird about that part of the peece you desire the thicknesse of mettall lay the same measure to an inch rule and note how many inches or other measure the same containeth then multiply that measure by 7 and deuiding the product by 22 your quotient is the true measure of the whole thicknesse of the peece in that place Thē substracting the diameter of the bore or concauity of the peece from that quotient note the remainder Deuide that remaine in two equall partes the one of those parts is the thicknesse of the mettall in that part of the peece so measured Example I prooued this conclusion with a demy Cannon of sixe inches diameter in girding the same about with a line hard behind the trunions and laying the same to an inch rule it cōtained 44 inches which summe multiplied by 7 amounted to 308 inches that summe deuided by 22 my quotient was iust 14. And so many inches was the height of the whole mettall in that part of the peece out of which quotient I did abate the diameter or bore of the peece being 6 inches and the remaine was 8 inches which deuided in 2 equall partes my quotient being 4 inches shewed the true thicknesse of mettall in that part of the peece being hard behind the trunions towards the breech And it is to be noted that euery peece of Ordinance if it be truly fortified with mettall ought to containe as much mettall in thicknesse round about so farre as the chamber where the powder and wad lyeth as the bullet is in height How to make a good shot in a peece that is not truly bored or to know how much any peece will shoote amisse that is thicker of mettall on the one side then on the other if you know the distance to the marke Question A certaine Gunner hauing shot diuers times in a Cannon at a marke supposed to be 500 paces from the peece findeth she shooteth still towards the right hand searching whether the fault were in him selfe or some impediment in the peece he findeth that the peece is 2 inches thicker of mettall on the right side then on the left And therefore requesteth how to lay the concaue of the peece being 9 foote in length equall with the marke so as he may make a straight shot Resolution To do this or the like there is a generall rule that looke how oftentimes the length of the cilinder or concaue of the peece is to the marke which is easily done by deuiding the distance to the marke by the length of the concaue of the said peece And knowing likewise how much the one side of the peece is thicker then the other the one halfe of that ouerplus being multiplied by the quotient first found the product will shew you how much the peece shooteth wide of the marke And this is a generall rule that looke which side of the peece is thickest of mettall towards that side shall the bullet fall for that the thinner side is more smart and the thicke side more dull in heating Example The Cannon in this conclusion is said to be 2 inches thicker of mettall more in thicknesse on the right side then on the left And the distance to the marke is supposed to be 500 paces that is 2500 feete the which deuided by 9 feete being the length of the hollow cilinder of the Cannon yeeldeth in the quotient 277 feete 7 9 the which multiplied by ½ the super fluitie of the mettall being one inch makes 272 feete 7 9 still and so much wide of the marke should the said peece haue shot at such a distance although she had beene laid full against the mids thereof How to remedie your peece being thicker of mettall in one part then another to make her shoote streight You must first search your peece with an instrument to know which is the thicker side then deuide the ouerplus of mettall in 2 parts setting the disparture of your peece one of those parts towards the thickest side of the peece mouth and bringing the midle part of mettall at the taile of your peece that disparture and the midle of the marke all in one streight line giue fire and you shall make a streight shot But beware of ouercharging of such peeces for they are dangerous If the thickest part of the mettall be aboue then you ought to make your disparture one inch more if vnder I meane towards the carriage an inch lesse To know the different force of any two like peeces of Ordinance planted against an obiect the one being further of from the said obiect then the other Question Admit there is a Castell or Fort to be battered being situate vpon a hill which hill is 50 paces in height and that 140 paces from the said Castell there is another hill of equall height to that hill whereon the Castell is built and Ordinance planted thereon to beat or batter the Castell wall and in the valley at the foote of the said hill 180 paces off from the Castell hill there is Ordiance planted and mounted at 20 degrees to shoot and beat downe the said castell I would know whether the Ordinance in the valley being 180 paces distance from the Castell and mounted at 20 degrees or the Ordinance on the height of the hill lying leuell to shoote a litle aboue the base of the wall being distant therefrom 140 paces shall worke the greatest effect in battering downe the said Castell wall the said peeces being of like length and height and hauing like charge in powder and bullet Resolution To resolue this or the like a man would thinke that the peece planted on the height of the hill lying leuell to shoote a litle aboue the ground-worke of the Castell would batter sorest because she is nearest yet by experience we find the contrary for the Castell being a great way within the reach of both the peeces that peece shall not onely shoote much further that is any thing eleuated but also pierce much sorer by so much as she is able to ouer shoot the other selfe like peece that lyeth leuell albeit the said peece so eleuated be planted furthest off from the said resisting obiect for euery Gunner knoweth and reason and experience doth teach euery reasonable man that no peece of Artillerie will shoote so far at point blanke as when the same is eleuated at any number of degrees because the bullet being ponderous flieth more heauily and sooner declineth being shot out of any peece lying leuell then out of any such like peece