THE ART OF GUNNERIE. Wherein is set forth a number of serviceable secrets, and practical conclusions, belonging to the Art of gunnery, by Arithmetic skill to be accomplished: both pretty, pleasant, and profitable for all such as are professors of the same faculty. Compiled by THOMAS SMITH of Berwick upon Tweed Soldier. LONDON, Printed for William Ponsonby. 1600. TO THE RIGHT HONOURABLE PERIGRIN BERTIE KNIGHT, LORD WILLOUGHBY Beak and Earsbie, Lord Governor of her majesties Town and Castle of Berwick upon Tweed, and Lord Wardon of the East marches of England, for and anempst Scotland, etc. IT is a common opinion Right Honourable amongst a great number, who may be termed more wayward than wise, that the Art of Soldiery may perfectly be attained in two or three months practice, and that any common man in a few weeks training, having seen two or three skirmishes may be called an expert soldier. Not considering that a Mariner may sail seven years, and yet be far from a Navigator. A number of Mechanical Artificers may labour diverse years, and yet be far from perfection; and a number of Soldiers may serve many years, and yet have but the bare name of a soldier. He may well be called a trained soldier, that knoweth by the sound of Drum, and Trumpet, without any voice, when to march, fight, retire, etc. that is able in marching, embattelling, encamping, and fight, and such like, to perform, execute, and obey the laws and orders of the field, that hath some sight in the Mathematicals, and in Geometrical instruments, for the conveying of Mines under the ground, to plant and manage great Ordinance, to batter or beat down the walls of any Town or Castle, that can measure Altitudes, Latitudes, and Longitudes, etc. such a one may be termed in my opinion an expert soldier, though he never buckled with the enemy in the field. Such perfections is well known to be in your Honour, that you are furnished with these and many morare qualities in the Art Military, and above all with wisdom and noble courage, to perform and execute any honourable enterprise whatsoever for the honour and service of God, your Prince and country, the which our proud enemies have felt to their pain and your everlasting fame. And although I myself be but one of the meanest soldiers in this Guarison now under your Lordship's government (whom we pray long to govern over us) being brought up from my childhood under a valiant Captain in Military profession, in which I have had a desire to practise and learn some secrets touching the orders of the field, and training of Soldiers: as also concerning the Art of Managing and shooting in great Artillery. I have thought it good (hearing of no other that hath done the like before) to frame together certain Arithmetical and Geometrical rules, to show in part how necessary Arithmetic and Geometry is for our profession, the which I have set down in two little books, the one entitled Arithmetical Military Conclusions, the other; The Art of gunnery: the first I wrote two or three years since, and bestowed on my Captain, Sir john Carie Knight, the which (God sparing life) I mean to correct & enlarge, & perhaps put to the Press: This other I have thought it my part, to offer to your Lordship's good consideration, to be shrouded under your Honourable buckler, to bear off the blows of envious tongues, which are ever ready to spit their spite against any virtuous exercise: which although it be unworthy to pass under so honourable a protection, I hope your Lordship will in indifferent balance weigh my willing mind, to do my country good, and your Honour any service my poor ability is able to perform, which if your Honour allow of, I shall think my pains well employed, and shall encourage me hereafter to bring this new found Art into some better perfection, so far as my poor ability is able to put in practice, or my simple skill in the Art will reach to. Thus loath to be tedious, I cease: beseeching God to preserve your Honour with much increase of honour, to God's glory and your hearts desire. Your honours dutifully at command, Thomas Smith Soldier. TO ALL GENTLEMEN, SOLDIERS, GUNNERS, AND ALL Favourers of Military Discipline, Thomas Smith wisheth increase of happiness. GENTLEMEN, there was never Author nor practised Gunner ever able (as I am persuaded) to describe at full, or could show perfectly the efficacy and force that Gunpowder is able to accomplish, it being a mixture of such a wonderful operation and effect, as by daily experience we find. And although diverse men in diverse ages, have invented diverse engines and Ordinance for offensive and defensive services by Gunpowder to be performed, yet none hath nor could ever attain to that full perfection, to know precisely what strange effects the said mixture is able to work. Also divers learned men have invented many excellent rules pertaining to the Art of Gunnerrie, and a great many of them have and do err in the principals of their inventions: and the cause is, for want of due practice therein. For the Art of Gunnery doth require great practice and experience, to declare the rare secrets thereof; which is not for mean men to attain to, for that the charges is great. And albeit, I am the least able of a great many to take any matter in hand, pertaining to the same Art, being but a sworn scholar thereto, and my ability far unable to put in practice that I would: yet because I have served a prentize-hood twice told since I took my oath, and never hearing of any that hath compiled any Arithmetical rules or secrets (which is the fountain head from whence all Arts or sciences do spring) into one volume, I thought it my part and duty (according to my skill) to do the best I could therein, for the benefit of others, and that in the plainest manner I could, that such as are not well seen in numbers Art, might the sooner understand the same. And albeit I have herein showed but a few Arithmetical conclusions belonging to the Art of gunnery, yet the experienced Gunner or skilful mathematician, by these few may devise a great many more, for service offensive and defensive, by Arithmetic and Geometry to be performed. All which conclusions (gentle Readers) I have thought best to frame in easy questions, showing the answers or resolutions thereof. And although they be but meanly framed, I hope you will accept the same in good part, the rather, for that they are a young Gunner's practices. And if there be aught herein that may profit you, yield me your friendly censure, I crave no more: or if in any place I have erred, either gently correct it, or pass it with silence, or in friendly sort admonish me thereof, I deserve no less. There is a great many that can spy a mote in another man's eye, that had need to have a beam pulled from their own: some will scan very curiously, and sooner find two faults then amend one. If you be of that mind (friendly readers) I mind not to make you my judges. The widows mite was aswell accepted as the gifts of the wealthy. A soldier in Alexander's camp, in the dry desert presented the king his helmet full of cold water, saying, if I could have gotten better drink, your Grace should have had part: the which the king gently accepted and liberally rewarded, answering, I weigh not thy gift, but thy willing mind. But I seek no reward for my travel, but only you will weigh my mind is willing to do my country good, and to profit the readers: and not to carp with Momus, nor disdain with Zoilus, nor sooth with Zantippus. In so doing you shall encourage me to set pen to paper, and to fly a higher pitch pertaining to this new found Arte. Otherwise, if you spit out your spite against me for my good will, I will as meanly account of your malice, and so as I find you, look to have of me. From my poor house in Berwick upon Tweed this 2. of May, 1600. Your friend and well-willer, Thomas Smith Soldier. PETER LUCAS CANNONNIER in commendation of the Author and his book. SHake silly pen to write of art, to him where art doth dwell, And say, the want of Eloquence doth so thy hand repel, That far thy Muse unable is to praise the Author's skill: Nor canst thou paint thy mind, nor finely tell thy will, But as there needs no sign at door, whereas the wine is pure, So need not I commend this work, it all men will allure, To love the Smith that forged this work, who hath such Art in store, That better is then Art which trieth gold from ore, As our proud foes have found, and felt by Ordinance might, And aid of the almighty jove, who doth defend our right. Therefore good Zeal go post-haste unto Fame, And bid her give this book an everliving name. Peter Lucas Gunner. Richard Hope Gentleman in commendation of the work. TO tell a tale without authority, Or feign a fable by invention, The one proceeds of quick capacity, The other shows but small discretion. Who writes conclusions how to use a piece, In my conceit deserves a golden fleece. Who takes in hand to write of worthy war, And never marched where any war was made, Nor never hopes to come in any jar, But tells the trial, knowing not the trade, To write of war, and note not what it is, May well be thought a work begun amiss. But he that by his study makes it known, What thing war is, and whereof it proceeds, Defensive and offensive reasons showing, To those that gape for honour by their deeds, A worthy work who doth not count the same, In my conceit he doth a Soldier shame. If so: Smith's travel cannot well offend, For so he meant before he set it forth, And if it chance to come where Soldiers wend. He it commands to seem of little worth: For what he writes, he writes to honour those, Which wade in wars to triumph over foes. Richard Hope Soldier. Richard Rotheruppe Gentleman in commendation. THat man whom Martial attempts May raise to honour high, Let him peruse with learned skill, Smith's work of gunnery. That fountain which such springs sends forth, Can never dry remain: I mean the Ground of Arts, from which All science we attain. As Grammar, Music, and Physic, With high Astronomy: And other Arts Mathematic, And brave Geometry. This Art of gunnery likewise, Amongst the rest let stand, Whose godfather this Author is, Which took the same in hand. Whose knowledge in this famous Art, Deserves eternal fame, For his conclusions excellent Doth well deserve the same. Richard Rotheruppe Soldier. THE ART OF GUNNERIE. A Table showing the deminite parts used for mensuration. FOrasmuch as some of these measures are to be used in the treatise following, it is requisite that I show what kind of measures are commonly used and now in force, beginning with a barley corn, from whence all these hereunder and a great many more do proceed, as An inch, containeth 3 barley corns laid end to end. a finger breadth, containeth 4 barley corns in thickness. a hand breadth, containeth 4 fingers. a foot, containeth 12 inches. a yard, containeth 3 feet. an ell, containeth 5 quarters of a yard. a span, containeth 3 handbredths. a foot, containeth 4 handbredths. a geometrical pace, containeth 5 feet. a fathom, containeth 6 feet, or 2 yards. 10 fathom, containeth a score, or 20 yards. a furlong, containeth 123 paces. our English furlong containeth 132 paces, or 660 feet. a perch or rood, containeth 5 yard's ½ or 16 feet ½. an acre, containeth 160 perches, 528 paces, or 2640 feet a league, containeth 1500 paces. an Italian or English mile, containeth 8 furlongs, or 1000 paces, or 5000 feet. a German mile, containeth 32 furlongs. a score, containeth 20 yards. an hundredth, containeth 600 feet, after 5 score to the 100 24 grains of wheat dry, containeth one penny of Troy's weight. 20 pence, containeth one ounce. 12 ounces, containeth one pound. 20 grains of barley, containeth one scruple of haberdepois weight 3 scruples, containeth one dram 8 drams, containeth one ounce. 16 ounces, containeth one pound. 112 pound, containeth 100 weight. a tun, containeth 20 hundredth A Table showing how to weigh a great deal with few weights. You may way any number of pounds from one to 40 with these 4 weights, 1. 3. 9 27. 1 to 121 with these 5 weights 1. 3. 9 27. 81. 1 to 364 with these 6 weights, 1. 3. 9 27. 81. 243. This rule of weighing many things with few weights proceedeth of Geometrical progression. The pounds to be weighed, are weighed with as many namelike weights, to be done either double or threefold, sometime by adding one weight to another, and sometimes by taking away and adding to the contrary balance. Example in a double respect: All terms to 15 are weighed with 4 weights of pounds: as, 1. 2. 4. 8. so in a triple respect, all pounds to 40 may be weighed with 4 weights, as 1. 3. 9 27. All pounds from 1 to 364 are to be weighed with these 6 weights, 1. 3. 9 27. 81. 243. and so infinitely. Measures. THe variety of measures are in a manner infinite, and yet are all comprehended under three general kinds, proceeding from a point in Geometry, as Arithmetic doth from an unite: that is to say, Lines, Superficies, Bodies. Lines having but only length without breadth of thickness, do measure only Altitudes, Latitudes, and Longitudes, etc. Superficies, being limited by lines, bearing length and breadth, without depth or thickness, in these are known the contents of Pavements, Glass, Board, Land, etc. Bodies, being bounden by Superficies, & containing length, breadth, and thickness, do make known the quantity of all solid or massive things, as timber, stone, etc. All which requires the aid of Arithmetic, to be truly measured. The definitions, terms, and orderly working of these and all other, the Elements of Geometry will teach you. Here I thought to have written briefly, or rather to have glanced at the wonderful strange effects that Arithmetic is able to work and attain to, but finding that that learned and famous man Master john Dee, in his Mathematical preface upon Euclids Elements, doth notably touch the same, showing the rare properties and incredible mysteries that numbers Art can reach to, affirming that the effects thereof, of man is notable fully to be declared, it soon struck me in the dumps, feeling myself far unable to soar so high. How to find the cubical radix or root of any number. AS in my book of the Art of war, entitled Arithmetical military conclusions, I began with the extraction of square roots, being a special rule to work diverse feats belonging to the said Art: So in this Treatise I have thought best to begin & show how to extract Cubic roots, for that diverse conclusions are to be done by the said rule, in the work following letting pass all former rules, as less necessary, the which are commonly known to every child, that hath any sight in the Art of numbering. To find the radix or root cubical of any number, you must note how many figures or numbers be in the total sum thereof, and then as is showed in the rule how to extract the square root of any number, you make a prick or point under every other number, beginning at the first number towards your right hand: even so in this rule, in searching for the cubical root of any number, you must put a prick under the first number towards your right hand, and so increase your number of pricks, under every third number, towards your left hand, and your quotient will contain so many figures as there be pricks. If your number propounded be cubical, multiply your quotient cubically, the product of that multiplication will be the number that was propounded. To multiply cubically, you must do as this example showeth. 5 multiplied in himself is 25, which 25 multiplied again by 5, makes 125, and is a cubick number. A cubical figure, is proportioned as these figures showeth, for a cube is a solid body of six equal squares or sides like a die. Example. It is requisite in learning to extract roots, to have in perfect memory all those cubic roots of digit numbers and the cubes they do make, the which will be a great help in working, the which I have here set down in a table after M. Records order. 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 Now to seek for the first figure or root, your table will show you what number shall stand in the quotient, being due to the last prick, towards your left hand, which figure so set in the quotient, multiplied cubickly, if it be equal to the number or numbers above that last prick, it doth show that the said number or numbers are cubic; but if it be more than a cube number, then abate the greatest cube number, that the quotient will make from the said numbers, and canceling the same, let the remain stand over the head of the said numbers, as is done in division of common numbers, and so have you done with the first prick. Secondly, triple your root, setting the said tripled number one place nearer from the last prick, towards your right hand. Thirdly, multiply the said triple, by the said quotient, the numbers arising thereof is your devisor, to set under your first tripled number. Fourthly, find out a number to be placed in your quotient, that may show how often times your devisor is contained in the devident, or numbers so remaining over it. Fiftly, you must multiply your divisor, by the number last placed in your quotient, first drawing a line under your devisor, and that which ariseth of the said multiplication must be placed under the said line. Sixtly, you must square the number last placed in your quotient, and multiply the said square by the triple of your first quotient number, & the sum arising of that multiplication set under the line, one place nearer towards your right hand. Seventhly, multiply the number last placed in your quotient cubickly, and set the same cube numbers under the line, beneath the other numbers, one place nearer towards your right hand: and then drawing a line under the same, add all those numbers together; the sum arising abate from the other prick that stands toward the right hand in your devident, and if nothing remain, the number propounded is a cubic number: but if any thing remain, the number propounded is no cubic number, but yet the quotient doth show the nearest cubic root in the proposition. In this order you must work by every prick, how many figures soever the numbers propounded containeth. To find a Denominator to the cubic remain. If the number propounded be not cubical, and that you desire to know the true denominator to the cubical remain, you must square your cubic root, and then triple the said square, and after triple the root, adding all those sums together, and to the total of the said addition, add one unity, so have you the true denominator cubical, the which you may abbreviate into lesser terms by Abbreviation, according to your desire. Or you may find the denominator cubical, by multiplying the root in the triple of another number that is more by one unite, nor the said root: and then adding one unite to the product of the said multiplication, you have your desire. An example how to work, to find the cubic root. Admit the sum or numbers, whose cubic root you desire to know be 32768. I set the prick under 8, and under the figure 2 standing in the fourth place, as in the work here you see, and I find that the greatest cubic number in 32 is 27, and 3 is his root, which 3 I place in the quotient, and his cube being 27, I subtract from 32, so resteth 5. And so I have done with the first prick towards my left hand, as here in the work you may see. Then I triple the quotient 3, & it is 9, which I set one place from the last prick nearer towards my right hand. And then I multiply the triple of the quotient being 9, by the said quotient 3, ariseth 27, the which I place under 57, drawing a line under my divisor 27, and then I seek how oft I can have 27 the divisor in 57, which is a part of the devident, the which I can have but 2 times, which 2, I place in the quotient, and by the said 2 I multiply the divisor 27, so ariseth 54, which I place under the line under the devisor, as here you see. And then I square the number last placed in the quotient being 2, and it is 4, which square I multiply by the triple of the first quotient number being 9, ariseth 36, which I place under 54, one place nearer towards the right hand, as here you may see. And then multiplying the digit 2. cubickly ariseth 8, to be set under the line one place nearer towards the right hand, & adding all these sums together, there ariseth 5768, the which substracted from the number belonging to the first prick there remaineth nothing, so I say that 32768 is a cubic number, and 32 is the true root thereof. You may prove it by multiplying the quotient cubickly, and abating the product from the number propounded, there will remain nothing. To find the nearest root of a number not cubic. Question. I demand the true cubic root of 117884. Resolution. The pricks placed in order as before, I find there will be but 2 figures in the quotient, & that the cubick number of 117 is 64, whose cubick root is 4, which 4 I place in the quotient, and his cube 64 being abated from 117, there remains 53 to be placed over the last prick: then tripling the quotient 4, ariseth 12 to be set down one place nearer towards my right hand, & then multiplying the quotient by the said triple, doth arise 48 for a divisor, which I set in his place, drawing a line under him as in the former work you see. And then I make search how oft I can have 48 in 538, which I can have many times, but more than 9 times I must not take; and therefore I set down 9 in the quotient, and multiplying the same by the divisor 48, ariseth 432, to be placed under the line under the divisor, than I do multiply the said 9 squarely, ariseth 81, the which multiplied by 12 being the triple of the first quotient, ariseth 972, the which I set down one place nearer towards my right hand; and then I multiply 9 cubickly, ariseth 729 to be set down yet one place nearer towards my right hand: and adding all those sums together, the total is 53649, which abated from 53884, rests 235. And thus I affirm, that 49 is the nearest cubic root in whole numbers of 117884, as here by the work you may see. Now to find a denominator for the 235 remaining, I square the root 49, so ariseth 2401. Then I triple the said squared number and there ariseth 7203, and then I triple the root 49, ariseth 147, to which I add one, and it makes 148. All which sums joined together, makes 7351, and so the true cubic root of 117884 is 49 and 235/7351 parts of an unite. Theormes showing the true proportion that a bullet of one metal beareth to the like bullet of a contrary metal, as also the proportion that the circumference of any buller or globe etc. beareth to the diameter, and of the superficial content thereof to the diametral square thereof, the which according to Archimedes are thus proved. All circles are equal to that right angled triangle, whose containing sides, the one is equal to the semidiameter, the other to the circumference thereof. The proportion of all circles to the square of their Diameter, is as 11 to 14. All globes bear together triple that proportion that their Diameters do. The circumference of any circle, is more nor the triple of his diameter, by such proportion as is less than 1/7 and more nor 10/27. A bullet of iron, to the like bullet of marble stone is in proportion as 15. to 34. A bullet of lead to the like bullet of iron, is in proportion as 28 is to 19 A bullet of lead to the like bullet of marble stone is in proportion as 4 to 1. The Diameter of any bullet etc. is in proportion to the circumference as 7 to 22. How by knowing the true weight of any one bullet, and the diameter of the piece due for the said bullet, to find out the weight of any other bullet belonging to a contrary piece of Ordinance. Question. Admit a Demy Cannon of 7 inches Diameter shoot an iron bullet of 32 pound weight, I demand what weight shall that bullet be of, that serves a Cannon of 9 inches diameter? Resolution. To answer this and such like, there is a general rule; for Ewclid in his sixth book of geometrical elements, hath demonstrated and proved that all globes are in triple proportion to their Diameters, therefore I multiply the proportion of each bullet cubically, and I find the cube of 7 is 343, and the cube of 9 is 729. Then by the rule of proportion I say, if 343 yield 32 pound weight, what shall that bullet weigh whose cube is 729? So multiplying 729 by 32 pound, the weight of the lesser bullet, ariseth 23328. which divided by the 343, being the cube of the lesser bullet, yields in the quotient 68 pound & 4/343 parts of a pound, so much shall that bullet, weigh, that serves a Cannon of 9 inches diameter, as by working the rule you shall find. Another easy conclusion, how by the weight of a small bullet known, to find out the weight of a greater. Question. A bullet of 3 inches diameter weighing four pound weight, what shall a bullet of the same metal weigh whose diameter is twice the height of the former (that is 6 inches high?) Resolution. I work in the order of the former conclusion, multiplying the diameter of each bullet cubically, and dividing as afore is showed, the quotient is 32 pound weight, so much shall the greater bullet weigh. Example. In the last conclusion the weight of the greater bullet weighed 32 pound, being 6 inches diameter, how shall I find the weight of a bullet of the same metal that is but half that height. Resolution. I find the cube of 6 is 216, and the cube of 3 is 27, so framing the converse rule of 3, I say: if 216 yield 32 pound weight, what will 27? And multiplying 27 by 32, and dividing the product by 216, the quotient yields 4 pound, the true weight of the lesser bullet. And note that if you know the diameter and weight of any bullet, and would know the weight of one that is but ½ the height of the first, the lesser shall be in weight but the ⅛ part of the greater. Or knowing the weight of any bullet, if you would know the weight of another of the same metal, being twice the height of the former, the greater shall weigh 8 times as much as the lesser, as in a figure demonstratively hereafter drawn you may see. How by knowing the weight of any bullet whose diameter containeth both whole inches and parts of whole, how you should work to find out the true weight of another whose diameter ends with a fraction. Question. If a Sakeret shoot a bullet of 2 inches ¾ diameter, of 3 pound weight, what shall a Culvering shot weigh of 5 inches ¼ diameter? Resolution. To answer this or such like, I reduce each bullet into his proper fraction, and I find that the bullet of 2 inches ¾ diameter will be in a fraction 11/4 or 11 quarters, and the Culvering bullet of 5 inches ¼ height, will be 21/4 than I multiply each of these 2 fractions cubically, and I find that the cube fraction of the lesser bullet is 1331/4 and the cubike fraction of the greater is 9261/4 which known, I set down under three pound (the weight of the lesser bullet) the unite 1, and it will represent a fraction thus 3/1, and then multiplying and dividing by the golden rule in fractions, I find that the weight of the Culvering 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 iron 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 demand what shall a bullet of marble stone weigh of like diameter. Resolution. In a theorem afore mentioned, I find that a bullet of iron to the like bullet of marble stone, shall bear such proportion as 34 is to 15. And therefore I multiply the weight of the iron bullet known being 9 pound by 15, (the proportion the stone bullet beareth thereto) so ariseth 135, which divided 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, equal to the like bullet of iron. In like order reducing the weight of the stone bullet into his proper fraction, you shall have 135/34 pound, which divided by 15, the proportion the stone bullet beareth to the like bullet of iron, your quotient is 9, the number 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 shoots an iron bullet of 72 pound weight, what shall a bullet of lead of the same diameter weigh? Resolution. To work this, I note that the theorem before saith, that a bullet of iron to the like bullet of lead, shall bear such proportion as 28 is to 19, therefore I multiply 72 (the pounds which the iron shot weigheth) by 28, so ariseth 2016, which divided by 19, the quotient is 106 pound 2/19, so much will a leaden bullet weigh that is proportional to an iron bullet of 72 pound weight. In this order by working as I have showed 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 knowing the weight of the like bullet of lead, or how by knowing the weight of the stone bullet to find out the weight 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 self 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 dividing the product by 4, the quotient will be 26 pound & ½ showing the true weight of a stone bullet, that is proportional 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 have 53/2, & setting 1 under 4 fraction wise, multiply the numerators together, and likewise the denominators, and dividing the product arising of the numerators by the product of the denominators, your quotient will be 106 pound, showing the true weight of the leaden bullet. If you have or do know the weight and true height of a bullet of stone, or any other metal, and is desirous to know the weight and height of another bullet that is greater or lesser, and of the same metal, in working as the first conclusion showeth, you shall have your desire. To find out the circumference of any circle or bullet. Question. I demand how many inches is about the circumference of that bullet whose diameter is 9 inches. Resolution. To work this or any such like, there is a general rule, as thus, that the proportion of the diameter to the circumference is as 7 to 22, therefore multiplying the diameter 9 by ●2 ariseth 198, which sum divided by 7, the quotient is 28, 2/7 showing the true number of inches about the circumference of a bullet of 9 inches diameter, as the figure here demonstrated showeth. 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 work to find how many inches the diameter of the same is. Resolution. To answer this and all such like, I must work contrary 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 dividing 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, etc. 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 general rule, as thus, to multiply the diameter in his square, I mean cubically, and then multiply that product by 11, divide the total sum by 21, the quotient showeth the number of square inches in that spherical globe or bullet, for 9 multiplied cubically ariseth 729, which augmented in 11 is 8019, that total divided by 21, yieldeth 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 drawn upon a flat, as on a table or paper, etc. Question. There is a circle whose diameter is 21 inches, I demand how many square inches is contained within the circumference of the same? Resolution. To resolve this ofr such like, there is a general rule, in taking ½ the diameter, and multiplying it in ½ the circumference, or squaring the diameter, and multiplying the product by 11, and dividing the result by 14, the quotient showeth 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 divided by 14, yieldeth in the quotient 346 inches ½. Or other ways, take the half of 21 inches, that is, 10 inches ½, and take ½ of the circumference, which is 33 inches, reduce them into fractions according to the rule, you have 21/2 for the diameter and 33/1 for the circumference, then multiplying the one by the other, the product is 1386/4, which divided by the denominator 2, yieldeth in the quotient 346 ½ as before. In this order you may find out the content of the plain of any circle. To find out the circumference of any bullet or globe diverse and sundry ways. Question. How many inches is about the circumference of that bullet or globe, whose diameter is supposed to be 21 inches high? Resolution. After you have with your callaper compasses, found out the height of the diameter, multiply the same by 22, so there will arise 462, the which divided by 7, the quotient will be 66 inches, the true measure of the circumference Another way. Triple your diameter, and thereto add the 1/7 part of the same, your product is the circumference. Example: The triple of 21 is 63, and the 1/7 part of 21 is 3, which added to 63 is 66 inches, as before. Another way to work the same. Look how many times you can have 7 in the diameter, so many times must you have 22 in the circumference. Example. The diameter being 21 inches, divided by 7, yields in the quotient 3, by which if you multiply 22, your product will be 66 inches, for the circumference, as before. In this order you may find out the circumference of any bullet, or spherical body, etc. To find out the superficies of any round body, as bullet, globe, etc. diverse and sundry ways. Question. I have a demi Cannon bullet of 7 inches diameter, I demand how many inches the superficial content thereof is? Resolution. To answer this and all such like, I must in the order before showed, find out the circumference of the bullet, and I find that a bullet of 7 inches diameter, shall contain 22 inches in circumference, which circumference being multiplied in the diameter, ariseth 154 inches, the true number of inches contained upon the superficies of a bullet of 7 inches diameter. Another way. Multiply the square of the diameter of any bullet or globe by 22/7 the product is your desire. Example: The bullet whose diameter was 7 inches being squared, the square thereof is 49, which multiplied by 22, yields 1078 which sum divided by 7, the quotient is 154 inches as before. Another way. Divide the square of the circumference of any bullet by 22/7 your quotient numbers will show you the superficial measure of the same. Example: The circumference of the bullet aforenamed of 7 inches diameter containeth 22 inches, the square thereof is 484 inches, that number divided by 22/7 as you do in fractions, in setting an unite under the square number thus, 484/1 and multiplying the said square number by the denominator of the other fraction being 7, ariseth 3388, which divided by the numerator 22, the quotient is 154 inches, the superficial content thereof, as before. How you may find out the solid content or crassitude of any round bullet or globe, etc. diverse ways. Question. In the question before propounded of the bullet, whose diameter was 21 inches, I would know how many inches is in all the solid or massive content thereof? Resolution. I multiply the diameter cubickly, and after multiplieth that cubic number by 11, so ariseth 101871, the which divided by 21, my quotient is 4851, showing there is so many inches in the solid content of a bullet or globe of 21 inches diameter. Another way. Multiply the cube of ½ the circumference by 49, and divide the product arising thereof, by 363, your quotient will show your desire. Example: The circumference of a bullet whose diameter is 21 inches, containeth 66 inches, the ½ thereof is 33 inches, the cube whereof is 35937, that sum multiplied by 49 is 1760913, which divided by 363, the quotient is 4851 inches as before. How you may by knowing the diameter and weight of any bullet, or other round body, find out the diameter of any bullet or globe that weigheth twice the weight of the former. Question. There is a demi Culvering bullet of 4 inches diameter weighing 9 pound, I demand the true height of that bullet which weigheth 18 pound weight. Resolution. To work this and all such like demands, this rule is general in multiplying the height of the lesser bullet whose weight is known cubically, then doubling that sum, and extracting the cubic root thereof, the quotient will answer your question. Example. The bullet afore named of 4 inches diameter being multiplied cubically is 64, that sum doubled is 128, the cubic root thereof is 5 inches and a fraction remaining scarce the 1/30 part of an inch, showing the true height of a bullet that weigheth 18 pound. In this order if you have a bullet that is 3 times as heavy as another of like metal, whose weight is known, and that you desire to know the diameter of the greater bullet: in tripling the cubic number of the lesser bullet whose diameter is known, & extracting the cubic root thereof, you shall find out the true height of the greater bullet. Or if you would find out the height of any bullet of like metal, that weigheth 4 times as much as an other bullet whose weight is known, quatriple the cubic number of the diameter of the lesser bullet, and extract the cubic root thereof, your quotient will satisfy you. Or if 5 or 6 times etc. in working as I have showed you may find your request. How you may Geometrically find out the diameter of any bullet, that weigheth twice as much as another known bullet. Take the true height or diameter of the lesser bullet whose weight you know, and square the same as you see in the figure following. Then draw a line that may divide the said square in 2 equal parts, in the 2 opposite angles, and that line shall be the diameter of a bullet twice the weight of the other: then divide that diametral line in 2 equal parts, setting one foot of your compass in the centre or mids thereof, and with the other foot draw a circle, and that circumference will represent the proportion of a bullet, twice as much in weight as the lesser. How you may Arithmetically prove this conclusion. The diameter of the lesser bullet is 5 inches, the square of it is 25. that some double is 50. the square root of 50, is 7. 1/7 and so much is the diameter of the greater bullet, as in the figure here drawn you may see. Another way Geometrically, to find the diameter of any unknown bullet that is double the weight of a known bullet. Draw a strait line of what length you think good, as you see the line A. B. then draw another cross line perpendicular to the ground line as you see the line C. D. note the meeting or crossing of the lines, as is the point E. This done, open your compass the just length of the diameter of the lesser bullet whose weight you would double, setting one foot of the compass in E. and the other in D. and measure towards B. twice that diameter, as is done in the points F. G. Then divide the line E. F. in 2. equal parts in the point H. and after divide the line E. H. in 2 equal halves, as in the point I. And lastly divide the line I H. in 2 equal parts in the point K. Which done, open your compass, placing one foot in K. and the other in G. draw ½ a circle, as you see, I do the semi circle L. C. G. After divide the line C. D. in 2 equal parts in the point M. and opening your compass the just wideness of one of those parts, set one foot in M. and with the other foot draw the line C. N. L. Which done, the bullet whose diameter is the line L. E. will weigh twice as much as the bullet whose diameter is the line E. D. as Ewclid in his 6. book of Geometrical Elements doth demonstrate and prove. The greater circle O. doth show the proportion of a bullet that weigheth twice as much as the lesser circle N. both the said bullets being cast of one like metal. Another demonstration to prove the former conclusion by numbers. In a conclusion before set down, where the bullet of a demi culverin of 4 inches diameter weighed 9 pound, I proved that a bullet whose weight was 18 pound should be more than 5 inches diameter. Even so I have hereunder divided the line E. D. of the former conclusion, being supposed to be the diameter of a bullet whose weight is known, into 4 equal parts or inches. And likewise dividing the Diameter F. E. into the like divisions it containeth 5 of those parts, and almost the 1/30 part of an inch more, showing the true height of a bullet that is twice as much in weight as the lesser bullet of 4 inches diameter, as this figure showeth. As the upper face or side of any square being doubled, the square arising of that doubled side shall be in proportion just 4 times as much as the first square was, whereas a great many would think it would be but twice as much. Even so the diameter of any circle being doubled, the Area or superficial content of the flat of the same circle so doubled, shall be four times as much as the other. Also any cube, globe or bullet, whose diameter is in double proportion to another, the solid content of that bullet whose diameter is so doubled, shall be in weight 8 times as much as the lesser, as these two examples in the conclusions following figuratively drawn showeth. How by knowing the superficial content of the plain of any circle, to find out the superficial content of another that is twice the diameter of the first. Question. There is two circles drawn, the one 7 inches diameter, the other 14 inches: how much is the content of the greater circle more than the lesser? Resolution. To answer this or the like, by the theorem afore mentioned, I square the diameter of the lesser circle being seven inches, so ariseth 49 inches, that square multiplied by 11, yields 539, the which divided by 14, the quotient is 38 inches ½ showing the superficial content of the circle of 7 inches diameter. Also working in the same order, I find the content of the greater circle of 14 inches diameter to contain 154 inches, which divided by 38. ½ the quotient is 4, showing that the superficial content of the greater circle is just 4 times as much as the lesser. By knowing the weight and height of any one bullet to find out the weight of another of twice the height of the former. Question. If a bullet of 4 inches diameter weigh 9 pound, how much shall a bullet of 8 inches height weigh. Resolution. To know this or the like, multiply the diameter of each bullet cubically, and I find the cube of 4 is 64, & the cube of 8 is 512, which known, I frame the rule of proportion saying, if 64 yield 9 pound, what will 512? and in multiplying and dividing according to the rule, my quotient is 72 pound, the weight of the greater bullet, (that is just 8 times the weight of the lesser bullet.) For further proof behold these 2 figures in cubick form, where you may see that the greater figure whose side is in double proportion to the lesser, doth contain 8 times the quantity of the lesser. An easy rule to find out the diameter of any bullet, and how to know how much one bullet is higher than another by Arithmetic skill, without any callaper compasses. If you want a pair of callaper compasses, take a line or a garter etc. and gird the bullet or bullets whose height you desire just in the mids, laying that measure to an inch rule, noting how many inches or other measure the same containeth, then multiplying the said measures by 7, and dividing by 22, the quotient will show you your request. And then abating the lesser diameter from the greater, the remain will show you how much the one is higher than the other. Example. Suppose the circumference of the one bullet be 16 inches, and the circumference of the other 26 inches, in working as above is taught, I find the diameter of the lesser bullet is 5 inches 1/11 and the diameter of the greater bullet 8 inches 4/11, so abating the lesser from the greater, the remain is 3 inches and 3/11 parts of an inch, showing the greater bullet is so much in height more than the lesser. The like is to be observed with any other. By this rule you may know how much the circumference or any part of your piece is higher than another. A table showing the weight of all iron bullets from the Fawconet to the Cannon in Habberdepoiz weight. Height of the shot in inches and parts of inches. Weight of the shot in pounds and parts of pounds. Height. Weight. 2. 1. 2/ 7 2. ¼ 1. ¾ 2. ½ 2. ⅓ 2. ¾ 3. 3/ 7 3. 4. ½ 3. ¼ 5. 3. ½ 6. 2/ 9 3. ¾ 7. 6/ 7 4. 9 4. ¼ 10. ¾ 4. ½ 12. ⅔ 4. ¾ 14. 5/ 8 5. 16. ¼ 5. ¼ 19 ⅔ 5. ½ 22. 1/ 7 5. ¾ 25. ⅚ 6. 29. ½ 6. ¼ 32. ⅛ 6. ½ 36. ⅝ 6. ¾ 40. ¾ 7. 46. 7. ¼ 52. 6/ 7 7. ½ 56. ⅝ 7. ¾ 64. ½ 8. 70. 8. ½ 76. ⅔ How you may Arithmetically know the true breadth of the plate of the ladle that is due for any piece of Ordinance▪ by knowing the height of the bullet fit for the said piece. Take a line and compass the bullet in the mids, laying the same measure to an inch rule, divide the same measure into 5 equal parts, 3 of those parts is the just breadth you ought to make your plate of, which being orderly placed on the staff, and bend circularly, serves to hold the powder in: the other ⅖ parts being cut and taken away, and so left open, serves to turn the powder into the piece, the which to do Gunner like, as soon as you have filled the ladle so full that you may strike the same with a rule, and put the same into the mouth of the piece, fix your thumb upon the upper part of the staff, towards the end next the tampion or head thereof, and so thrusting the ladle gently home to the breech of the piece, turn the rammer staff, so as your thumb fall directly under the staff, and so shall you empty your ladle orderly. Now to know the ⅗ parts of the bullets circumference, that you may make the plate of your ladle orderly, and of that just breadth, lay the measure of the whole circumference to an inch rule, and then multiply the same by 3, and divide the product by the denominator 5, your quotient will tell you truly the breadth that the plate of your ladle ought to be of. Example. A Cannon whose bullet is 7 inches high, will be 22 inches in the circumference, that multiplied by 3 is 66, which divided by 5 the quotient is 13 inches ⅕, the true breadth that the plate for a cannon ladle of 7 inches diameter ought to be of. The length of the ladle is to be made according to the length, height, and weight of the piece for which it is made, which in a table in the end of the book you may find set down for all sorts of pieces. How to make a ladle for a chamber-bored piece. Open your compass the just diameter of the chamber, within ⅛ part of an inch thereof. Divide that measure in 2 equal parts, than set your compass to one of those parts, and with the one foot fixed on a paper or smooth board, draw with the other foot a circle, the diameter thereof will be a just quarter of an inch shorter than the diameter of the chamber-bore, by the circumference whereof, you may find out the true breadth of the plate of a ladle that is fit for such a chamber-bored Cannon, by the rule afore set down how to find the true breadth of the plate of any ladle, for any other piece of Ordinance, in taking the ⅗ parts of the circumference thereof, the length ought to be twice the diameter, and ⅔ parts, to hold at 2 times the just quantity of corn powder that is due to charge such a chamber-bored cannon with Example. The diameter of the circle drawn for any cannon whose chamber-bore is 7 inches containeth 6 inches ¾9 the circumference whereof is 21 inches 6/7, the ⅗ parts thereof is 12 inches ¾, and so much aught that ladle to be in breadth, and in length it ought to be 18 inches ⅔. In this order you may work to make a ladle in length and breadth for any belbored Cannon: and to find out the thickness of the metal at the touchhole, or the height of the bore thereof, the conclusion following will show you. How to find out the height or diameter of the chamber, in any chamber-bored Cannon, or other piece of Ordinance, and how to find out the thickness of the metal round about the chamber thereof. Take your priming iron, or else a strait piece of wire, and bow the end thereof in manner of a hook, and then put the same into the touchhole, down to the lowest part of the concavity of the piece, and then with your knife or else with a piece of chalk, make a stroke upon the wire hard by the upper part of the metal, without the piece at the touchhole, then measure by your inch rule, how long the wire is from that stroke to the end. After put in the same wire again, and pull it up, so as the bowed end may restor stay within the cylinder or concave of the piece: and make an other mark or stroke on the said wire, hard by the upper part of the metal, the distance between those 2 strokes, is the just thickness of the metal, round about the chamber, the which abated from the length of the wire (I mean from the first stroke to the lowest end) the remain is the true diameter of the chamber-bore in that piece. Example. Admit the length of the wire from the end of the concavity to the first stroke containeth 15 inches, and the distance between the 2 strokes is 8 inches: then those 8 inches is the just thickness of the metal about the chamber; which abated from 15 inches, rests 7 inches, the just diameter of the chamber in such a piece. By Arithmetic skill, how to know whether the carriage for your piece be truly made or no: or how the carriage for any other piece of Ordinance ought to be made. Measure the just length of the cylinder or bore of your piece, the planks of your carriage ought to be once and a half that length. Also measure the diameter of the piece, and the said planks at the fore end should be in depth 4 times the diameter, and in the midst 3 times and ½ the diameter, and at the end next the ground, two times and ½ the diameter, and in thickness once the diameter. Example. Admit a Culvering of six inches diameter is in length in the bore thereof 20 times that measure (that is 10 foot long,) than I say that the planks of her carriage ought to be 15 foot in length; and at the fore end next the piece 2 foot in breadth, and in the midst one foot three quarters, and at the lowest end next the ground one foot and a quarter: and in thickness half a foot. Also every carriage ought to have four transoms, and aught to be strengthened with strong iron bolts. The holes or centres wherein the trunions ought to lie, aught to be three times and ½ the diameter from the fore end of the carriage, and in depth ⅔ parts of the thickness of the trunions, which depth you may easily find out, as thus: take the height or diameter of the trunions, and multiply the same measure by 2, and dividing by the denominator 3, the quotient will show your desire. How by knowing the weight of any one piece of Ordinance, to find out the weight of any other. Question. If a Saker of four inches diameter weigh 1600 pound weight, what will a Culvering weigh that is six inches diameter? Resolution. Some would think that the rule of proportion plainly wrought, would answer this question: but in that they are deceived, for the content of solid bodies being massy, are Spherical or cubical inproportion, therefore you must multiply the diameters of every piece cubically, & set down the weight of the piece known in the middle number, and so working according to the rule of proportion, you shall find out the true weight of the greater piece. Example. 4 inches the diameter of the lesser piece, multiplied cubically, ariseth 64 inches. Likewise the cubic number of the diameter of a Culvering of 6 inches high, is 216 inches: then framing the rule of proportion, I say, if 64 being the cube of 4 yield 1600 pound weight, (being the weight of a Saker of 4 inches bore) what will 216 being the cubic number of 6 inches, so multiplying 216 by 1600, ariseth 345600. which divided by 64 yields in the quotient 5400 pound weight, so much weigheth the Culvering of 6 inches diameter. In working by the converse rule of proportion, you may not only prove this conclusion, but also may find out the weight of any lesser piece of ordinance, by knowing the weight of a greater. Example. If 216 being the cube of 6 inches, yield 5400 pound in weight, what will 64 being the cube of 4 inches? so multiplying 5400 by 64 there ariseth 345600. which divided by 216, the quotient is 1600 pound weight, showing the true weight of the Saker of 4 inches diameter, as before. Or if the diameters of the pieces whose weight you would know, contain both whole numbers and broken, in reducing each diameter into his proper fraction, and multiplying the same cubically, setting down the weight of the piece known, in the middle place, for the second number, and multiplying and dividing as afore is taught, the quotient will show you your request, as the conclusion following will teach you. Question. If a demi culverin of 5 inches ¼ diameter weigh 2600 pound weight, what will a Cannon of 7 inches ¾ diameter? Resolution. I reduce the diameter of each piece into his proper fraction, and I find that the broken number of 5 inches ¼ diameter containeth 21/4, which multiplied cubically ariseth 9261/4. Likewise I reduce the diameter of the Cannon, being 7 inches ¾ into his fraction, and it is 31/4;, whose cube is 29791/4;: then 1 set an unite I under 2600, and it doth represent a fraction thus 2600/1. Now to find out the weight of the greater piece, I set down these 3 new made fractions in the order of whole numbers, and working by the rule of proportion, I find the greater piece weigheth 8363 pound, and almost ¾ of a pound: for in multiplying 29791 by 2600, there ariseth 77456600, the which augmented by the denominator 4 maketh 309826400 for the devident or number to be divided. Likewise the fraction of the lesser piece being 9261, multiplied by his denominator 4, makes 37044 for a divisor, which devident being divided by the divisor, yieldeth in the quoent 8363 pound, and certain parts of a pound, so much will a Cannon of 7 inches ¾ weigh being proportional in metal to the other piece. How you may be Arithmetical skill, know how much of every kind of metal any brass piece of Ordinance containeth. Question. Every Gunfounder doth commonly use for every 100 pound weight of copper, to put in 10 pound weight of latin, and 8 pound weight of pure Tin: I demand how many pound weight of every of those metals is in a Culvering of 5600 pound weight? Resolution. To answer this or all such like, I join all the several mixtures together, and they make 118 pound, which I reserve for my divisor. Then I multiply the weight of the piece by every mixture severally, and there ariseth of the 100 weight of copper being the greatest mixture, 560000, the which sum is to be divided by the divisor common (to wit, 118 pound) and the quotient is 4745 pound and 90/118 parts of a pound: so much copper is in the said piece. Now to know how much latin is in the same, I multiply the whole weight of the piece by 10 the second mixture, and the product is 56000, which number divided by the divisor common, the quotient is 474 pound 68/118: so much latin is in the same piece. And lastly to know how much Tin was in the same piece, I multiply the weight of the piece by 8, ariseth 44800, which divided by the divisor 118, the quotient is 379 78/118: and so much Tin was put into the said piece. Now to prove the work if it be truly wrought or not, I add all the 3 quotients together, and because they do all make the true sum of the whole weight of the piece according to the proposition, I affirm the same to be truly wrought. The Gun-founders do hold and affirm, Note. that the latin doth incorporate, and causeth the piece to be of a good colour, and the Tin doth strengthen and bind the other mixtures. How you may know how far any piece of Artillery will convey her bullet at the best of the random, by knowing the utmost range and point blank of another piece, and how to prove the same: by which rule, you may know how far any piece will reach at point blank and utmost range. Question. If a Saker at point blank convey her bullet 200 paces, and at the best of the random shoot 900 paces, what will that Cannon do which at point blank shoots 360 paces? Resolution. To resolve this or the like, I set down the numbers proportional according to the rule, multiplying 900 paces (the utmost random of the Saker) by 360 paces, (the point blank of the Cannon,) so ariseth 324000, which divided by 200 the number of paces the Saker shoots at point blank, the quotient is 1620. And so many paces will a Cannon shoot at the best of the random, that at point blank rangeth 360 paces, as by working you may find, and by experience better understand. You may prove this conclusion by the converse rule of proportion, multiplying 900 the number of paces the Saker shoots at the best of the random, by 360, the paces that the Cannon shoots at point blank; and dividing that product 1620 the number of paces the Cannon shoots at the best of the random, the quotient is 200. showing the number of paces that the Saker shall shoot at point blank. In this order you may work the like conclusion by any other piece of Artillery, and find out the point blank and utmost range thereof. To know how much a bullet of iron will out fly a bullet of lead of the like diameter, being both discharged out of one piece, with one like quantity in powder. Question. If a bullet of lead of 24 pound weight, being shot out of a piece with ⅔ parts of the said bullets weight in powder, range at point blank 240 paces, how far will a bullet of iron of like height range, being discharged out of the said piece at point blank with the like quantity of powder? Resolution. The proportion between a bullet of iron and a bullet of lead of the same height, I have showed by the theorems and conclusions afore set down: by which I find that a bullet of iron being of equal diameter to a leaden bullet of 24 pound weight, the said iron bullet shall weigh 16 pound 2/7 parts. And for as much as the leaden bullet is shot with ⅔ parts in powder of his weight, that is, with 16 pound of powder, which is very near the full weight of the iron bullet, I find that the said bullet of iron shall out fly the leaden bullet ⅓ part of the level range (that is) the iron bullet shall fly being shot as afore at point blank 320 paces, that is, 80 paces further than the leaden bullet rangeth at point blank. But if the piece out of which the said bullets were shot, had been mounted at any number of degrees of random, the range of the iron bullet would shorten somewhat of the ⅓ of the overplus of the said range: so that if the piece were mounted to the best of the random, the said bullet of iron would not out fly the leaden bullet, not the ⅕ part of the said range. By knowing how much powder is sufficient to charge any one piece of Ordinance, to know how much of the same powder will charge any other piece of Ordinance. Question. If a Saker of 4 inches diameter▪ require 5 pound of corn powder for her due loading, how much of the same like powder will charge a Cannon of 7 inches diameter? Resolution. The plain rule of proportion cannot resolve this conclusion, except you multiply every number cubically, and then the quotient will show you your desire. Example. The cube of 4 is 64, and the cube of 7 is 343. which multiplied by the weight of the charge of powder due to load the lesser piece, ariseth 1615, which divided by the cubic number of the diameter of the lesser piece, yields in the quotient 25 pound and almost ¼ part of a pound: so much corn powder must a Cannon of 7 inches diameter have to charge her with. And note, that for as much as now the shooting with Serpentine powder is not used, being of no great force, and the making of corn powder never better known, nor of more force than now it is made & daily used in shooting in great Ordinance; as also the great Ordinance now cast, not so fully fortified with metal as they ought to be, being made more nimble and lighter than in times past, therefore the experienced Gunners do observe as a general rule to abate ¼ part of the ordinary charge of corn powder in all pieces above 6 inches bore. How by knowing how much Serpentine powder will charge any piece of Ordinance, to know how much corn powder will do the like, or chose by knowing how much corn powder will charge any piece of Ordinance, to know how much Serpentine powder will serve. Question. I demand how much corn powder will charge that culverin that shoots 24 pound of Serpentine powder at a shoot? Resolution. You must note for a general rule, that 2 parts of corn powder will do as much as 3 parts of Serpentine powder: so that the proportion between the quantities or charges of these powders, is as 2 to 3, therefore I multiply 24 by 2, ariseth 48, which divided by 3, my quotient is 16 pound: so much corn powder will charge the said culverin. Or if you know how much corn powder will charge her, you may know how much Serpentine powder will serve, in multiplying 16 pound the due charge of corn powder, by 3, and dividing the product by 2, your quotient is 24, as before. In this order you may do the like by any other piece. And note that her due charge of corn powder, will less hurt the piece, then of Serpentine powder, for if Serpentine powder be ramd any thing hard, it is long a firing. And a little heat long continued (which the Serpentine powder will do) dangereth the piece more than a great heat presently gone, which effect corn powder works. How by knowing how far your piece will shoot with her due charge in powder and shot, how to give a near estimate how far she will shoot with a charge more or less than her common charge. Question. Admit a Culvering shoot a bullet of 18 pound weight 900 paces, being charged with ⅔ parts in powder of the bullets weight, I demand how far should the said piece shoot that bullet if she had been charged with as much powder as the bullet weighteth? Resolution. By the rule of proportion I find she should shoot ⅓ part further than she did at the first shot, being charged with ⅓ part of more powder, that is 1200 paces: yet it is known she will not drive the bullet full out the ⅓ part of this range further, although she will come very near it, and the reason is, because the bullet flieth in a circular proportion more or less, a part of the range, after the insensible straight line or motion of the bullet be past, according to the degrees of random the piece is elevated at. Also the concave of the piece being filled up with the powder, wadd and bullet, further than it ought to be, is a hindrance to the range of the bullet in proportion according to that little quantity of the concave which the overplus of powder and wad filleth up; which though it be but little in comparison of the whole concavity to the range, yet it is a great hindrance in the bullets range, for that the bullet being so much nearer to the mouth, is driven into the air before the powder be all fired, and have effected his force thereon: so that giving the piece her bullet's weight in corn powder, she will shoot much further nor with an ordinary charge, but it will both put the piece in danger of breaking, and those that are near thereto in danger of their lives, if the piece be not all the better fortified with metal. How by knowing how much powder a few pieces of Ordinance have spent, being but a few times discharged, to know how much powder a greater number of the same pieces will spend to be often discharged. Question. If 4 Cannons being twice discharged at any service, shoot 240 pound of powder, how much powder will charge 5 Cannons to shoot every one 6 shots? Resolution. Work by the double rule of proportion, saying; If 4 Cannons shoot 240 pound of powder, what will five Cannons? your quotient will be 300 pound: then say again, if 2 times discharging yield 300 pound of powder, what 6 times? and your quotient being 900 pound weight, showeth that so much powder is due to 5 Cannons, to shoot every one 6 shots. To know how much powder every Cannon spent in the former conclusion at one shoot. Question. If 5 Cannons burn 900 pound weight of powder, being but 6 times discharged, how much powder did every one shoot at one shoot? Resolution. Multiply 4 the number of pieces first propounded by 2, the times they were discharged, ariseth 8, by which divide 240 the number of pounds in powder spent, the quotient is 30 pound, and so much powder did every Cannon fire at one shoot. Or else you may multiply the other 5 Cannons by the times they were discharged, and dividing that product by the powder spent, you shall have 30 pound weight of powder in your quotient also. How to know how much powder every little cask or firken aught to contain, and how many of those casks makes a Last of powder, and how many shots any quantity of powder will make for any piece of Ordinance. Every little cask or firken being empty, aught to weigh 12 pound, and being filled aught to hold an hundredth pound weight of powder: so that the full cask ought to contain 100 of Habberdepoize weight, and 24 of those casks or firkens filled makes a Last of power. Question. How many shots will one of those casks filled with powder make to a Culvering that shoots 15 pound weight of corn powder at one shot? Resolution. Divide the 100 pounds of powder in each firken by 15, the quotient will show you that 100 weight of powder will be 6 shots to a Culvering that burns 15 pound of powder at a shoot, and 10 pound to spare. How by knowing how many shots a firken of powder will make for a Culvering, to know how many shots a Last of powder well make for a Canon. Question. If a firken of powder of one hundredth weight charge a culverin 5 times, shooting 20 pound of powder at every shot, how many of those shots will be in a Last of powder (containing 24 hundredth weight) to a Cannon that shoots 30 pound of powder at every shot? Resolution. Reduce 24 hundredth weight into pounds, you have 2400 pound; then say by the rule of 3 direct; If 100 pound weight of powder be but 5 shot, what will 2400? and you shall have in the quotient 120 shot, for the said culverin that shoots 20 pound weight at one shoot. And whereas the question says the Cannon shooteth 30 pound of powder at a shot, you must frame the backer rule of 3, and say if 20 bear proportion to 120. what will 30? so multiplying 20 by 120, and dividing by 30, the quotient is just 80: so many shots of powder will be in a Last for any Cannon that shoots but 30 pound weight at a shot. The like is to be done with any other. To know how many shots of powder will be in a Grand barrel for any piece of Artillery. Question. If an ordinary Culvering shoot 15 pound weight of good corn powder at one shoot, how many times will a Grand barrel full of powder serve to charge her, the said holding 300 weight? Resolution. Divide 300 by 15, the quotient is 20, your desire: the proof is easy; for multiplying 20 by 15, you have 300 the number first propounded: the like is to be done if you would know how many shoots will be in a grand barrel, for any other piece of Ordinance, in dividing the pounds of powder contained in the said Barrel by the number of pounds of powder due to charge the said piece. To know Arithmetically what proportion of every receipt is to be taken to make perfect good powder, what quantity so ever you would make at a time. Question. The best ordinary corn powder made in these days, containeth 12 parts of Mr. 3 parts of coal, and 2 parts of Sulphur. The order how to compound and make the same is not peculiar to this treatise, being mere Arithmetical; I demand how many pound weight of every sort is to be taken to make 1000 pound weight of powder? Resolution. Add all the parts or pounds of the receipts together, ariseth 17 pound for your divisor. Then frame the golden rule saying, if a mixture of 17 pound weight of powder, require 12 pound of the saltpeter, what will 1000 pound weight? In multiplying and dividing according to the rule, the quotient will be 705 pound, and 15/17 parts of a pound: so many pound of the Mt. is to be taken to make 1000 pound weight. Again, say by the same rule, if that a mixture of 17 pound weight, do require 2 pound of Sulphur, what will 1000? your quotient will show you, that 117 pound and 11/17 is to be taken. And lastly by the said rule say, if a mixture of 17 pound take 3 pound of Cole, what will 1000 pound take? and your quotient will tell you that 176 pound and 8/17 parts is to be taken. The which 3 quotient numbers being all added together, will be just 1000 pound weight, and so proves the work to be truly done. And note that the goodness or badness of powder may be known divers ways, as by the colour, the taste of the tongue, the quick burning, etc. Also the brimstone is that material substance that is most apt to kindle with any spark, the coal most fit to continue or maintain the flame, and the Mr. being resolved into a windy exhalation worketh the effect, as chief and principal of the three. Before I frame these conclusions following, of the random or range of the bullet, and the diversity thereof, it is requisite to make known to the Reader, how that divers have written, and some will vaunt that by the range or flight of the bullet out of any one piece of Ordinance known, they will or can tell the utmost range of all other, thinking that the range of the bullet out of any one piece, should be proportional to the bullet and charge of powder out of any other piece. Also some do affirm, that out of any one piece of Ordinance discharged with sundry quantities of powder, they can tell the utmost range of the bullets discharged; and their reason is, that the range of those bullets shall be proportional to the weight of powder wherewith they were charged. And hereupon some have given out rules which are false and full of errors: for the diversity of proportions cannot by the plain rule of proportion be resolved, as they affirm: but this may they do; Out of any one piece of Ordinance charged with one and the same like charge in powder and bullet, find by the rule of proportion, the near difference or ranges of the bullets, the piece being mounted or dismounted at any degree of random; or by knowing how many paces, yards, feet, or other measure any piece will reach at point blank, by knowing the point blank and utmost range of another piece of Ordinance, they may find the furthest range of the first. Or chose, by knowing the utmost range and point blank of one piece, and the utmost random of another piece, they may find out the point blank of that other piece, as by the rules following shallbe proved. And it is to be noted that any piece of Ordinance being mounted to the best of the random or highest degree of the quadrant, the mouth and hollow cylinder of the said piece, must be erected to 45 degrees, that is at the 6 point of the scale in the quadrant (as the most part of quadrantes now are made:) but some pieces will shoot as far at the 5 point, or at 41, 42, or 43 degrees according as the wind is of calmness, for if any piece be mounted higher than 45 degrees, she shall shoot shorter in every degree about the 1/45 part of her utmost range. And therefore to know how to work these conclusions, you must buy an instrument Geometrical, or by some line of measure truly divided, measure the distance from the piece to the place where the shot first fell or grazed, noting how many perches, paces, yards, or other measure that distance is; which known, divide that distance by the degrees in the best of the random, being 45, your quotient will tell you how many paces, yards, feet, or other measure your piece will shoot further or shorter in mounting or dismounting a degree: the which known as I have said, by one truly measured, you may before you shoot, know very near how far or short your piece will shoot, at the raising or dismounting of any degree, allowing one and the self like proportions in charging, both with powder, bullet and wad. How by Arithmetic skill you may know how with one and the self same like charge in powder and shot, how much far or short, any piece of Ordinance will shoot, in mounting or dismounting of any degree: whereby you may know how far your piece will shoot at any degree of the random, by knowing the distance she shoots at the utmost grave. Question. If a Cannon at her utmost random (that is, at 45 degrees) carry the bullet 1440 paces from the piece, how far shall the same piece shoot being dismounted but one degree? Resolution. To answer this or all such like, I set down the numbers according to the rule of proportion, and multiplying and dividing accordingly, I find she shall shoot short in dismounting a degree, 32 paces, or 53 yards, or 160 foot, which substracted from 1440, rests 1408 paces; so far shall the Cannon shoot in dismounting her one degree of her furthest range. Or you may do the like in framing the golden rule, saying: If 45 degrees range 1440 paces, what will one? and you shall have 32 paces in your quotient as before. How by knowing the distance to the mark, by the conclusion or rule before, you may know whether your piece will shoot short, or over the mark, or you may know how far it is from your platform to any mark, within the reach of your piece, only by knowing the utmost range of your piece, and the degrees she is elevated at. Question. Admit the same Cannon in the former conclusion, which ranged at the best of the random 1440 paces, having the like charge in powder, shot and wad, is laid to shoot at a mark being mounted at 30 degrees, I demand how far it is from the piece to the said mark, or how far the said piece doth carry so mounted? Resolution. To answer this, I multiply the paces my piece reacheth at the best of the random, by those degrees in the proposition (to wit) 30 degrees, and there ariseth 43200, which divided by 45, my quotient is 960 paces, (that is 40 paces less than a mile) so far will that piece shoot being mounted at 30 degrees. And if you would know how much this is short of the utmost range, abate the same from the said range, the remain is your desire. As 960 paces abated from 1440, rests 480 paces, so much doth she shoot short of her best random. In this order by 2 shoots known, you may know what any piece of Ordinance will do being mounted above 10 degrees to the best of the random, but under 10 degrees you should err something in this practice, because the range of the bullet flieth a great part of the way in an insensible straight line, and the piece mouth elevated above 10 degrees, shoots or drives the bullet in a more circular proportion. The range or flight of the bullet by the draft in the next leaf may be understood. And note that in service there is no piece of Ordinance lightly mounted above 15 or 20 degrees, except mortar pieces, and such like. The direct strait range at 90 degrees This draft here drawn doth show you the range or motion of the bullet through the air, shot out of any piece of Ordinance at any degree of the random. How to make a table of randoms, or go very near to know the true range of the bullet out of all sorts of pieces, being mounted from degree to degree. Many Authors have taught how to make a table of randoms, whereas some of them never shot in any piece of Ordinance in their lives. And for as much as I find their writing and reasons differing, I think it will be a very hard matter to make a perfect table of randoms, except the same be tried and experimented with some piece of Ordinance in some convenient ground. I never heard nor read of any that hath as yet fully put the same in practice, the which would be much available to every Gunner, to know what every piece would do at the mount of every degree or point in the quadrant, the motion or range of the bullet being something variable at the mount of every degree. You shall very near find out the true range or random of the bullet shot out of any piece of Ordinance, the piece mounted at any degree of random, as thus: Charge your piece with her due loading, in powder, shot and wad, laying the piece at point blank, which you may easily try, by putting the rule of the quadrant into the piece mouth, & coining the piece at the breech, so as the plummet may cut the quadrant in the line of level, as you see in the first figure hereafter drawn, that piece lieth point blank: which done give fire, & mark where the bullet first grazeth, after bring your piece to the same platform, so as the wheels and carriage stand neither higher nor lower than they did the first shoot: and being charged with one & the self like quantity in powder, bullet & wad as before, the piece being of like temper raise her mouth one degree, as the second figure showeth: discharge her, and mark where the pellet falleth or grazeth first; then measure how far the first graze of the second bullet is beyond the graze of the first bullet, so much will the piece convey the bullet further at the mount of every degree, or very near thereto. But being mounted above 20 degrees, she will shoot shorter & shorter, a little at the mount of every degree to the best of the random, according to the height & circular motion of the bullet. If the piece be mounted to the best of the random, the plummet will cut the 45 degree of the quadrant, as the 3 figure showeth. Or you may make a table of randoms like the other, as thus: Measure the distance the piece conveyeth the bullet at the best of the random, from which abate the distance the piece conveyeth her bullet at point plank, divide the remain by 45, the quotient will show you how far the shoot is carried at the mount of every degree: or dividing the said remain by so many degrees as you would elevate your piece at, the quotient will likewise show you how far the bullet doth range beyond point blank. Example. If a Cannon at point blank range 300 paces, and at the best of the random shoot 1500 paces, how far shall she shoot at the mount of one degree? Resolution. Abate 300 from 1500, rests 1200, which divided by 45, the quotient is 26 2/3, so many paces shall she shoot at the mount of every degree. This conclusion or rule, I do not affirm to be clean without error, for that I never tried the same, yet it will come very near to this proportion, being tried on a plain ground that is water level, for the piece being mounted from 1 to 10 degrees, conveyeth the bullet with little bending at the fall thereof, and from 10 degrees to 20, as the motion of the bullet decreaseth: so it falleth more bowing then in the first 10 degrees. And mounted from 20 grades, to the best of the random, conveyeth the bullet in a more circular course. And it is to be noted, that any piece of Ordinance having her due charge, will drive the bullet more ground mounted at 20 degrees, then from 20 grades to the best of the random. And being truly loaden and discharged at the best of the random, will drive the bullet 5 times the distance of her level range, or rather better. How you may Arithmetically know how much wide, over, or short, any piece of Ordinance will shoot from the mark, by knowing the distance to the mark, and how your piece is laid to shoot at the said mark. Question. If a Culvering or Cannon of 10 foot long, be shot at a mark 700 yards from the piece, the mouth of the said piece planted an inch wide, how far shall the bullet light wide of the mark? Resolution. Reduce the measure of the length of the piece into inches, because the denomination of wideness is by inches, and the piece of 10 foot length, will yield 120 inches. Likewise reduce the length from the piece to the mark into inches, you have 25200 inches. Then by the rule of propotion: say, if 120 inches shoot wide one inch, what will 25200 inches? And in multiplying and dividing according to the rule, you shall find in your quotient 210 inches, that is 17 foot ½: so much shall the bullet light wide of the mark. For this is a general rule, that look how many times the length of the cylinder or concave of the piece is to the mark, so many inches shall the piece shoot amiss, being laid over one inch, or under, or wide of the mark, if the wind do not alter it. The like is to be done of any other. A remedy to lay your piece strait, if she lie either over, under, or wide of the mark. Let a plumb line fall perpendicularly over the middle part of the breech of the piece, and with a hand-spike or lever, wind the carriage of the piece too and fro till you espy the middle part of the metal at the mouth of the piece, and the said line divide the mark in 2 equal parts: so shall you make a straight shot, giving the piece her true disparture and length. Another way. Or you may take the true diameter of the concave at the mouth of the piece, laying an inch rule to the same, divide the said diameter in 2 equal parts; to the point of which division being the centre of the cylinder of the piece, let a thread and plummet fall, or else erect a squire, so as the containing angle touch the centre or middle point of the diameter, by the edge of which rule or squire draw a line with the point of your knife, from the height of the metal at the mouth: that line would cross in the centre if it were continued, and it is a perpendicular or plumb line to the other, by which line or strike so drawn, with a little piece of soft wax, set up a strait straw, to reach a little above the metal. And knowing likewise the middle metal at the breech of the piece, it is an easy matter to make a strait shot, if the 2 sights (to wit) the sight at the breech and mouth be laid so as they divide the said mark in 2 parts: for this is general, that any three things that the eye can comprehend at once, being equal with the eye, are in a straight line from the eye, whether the same be at ascent or descent. The line or strike thus drawn at the mouth of the piece, will show you presently where and how to set up your disparture of your piece at any occasion. In shooting without disparting your piece at any mark within point blank, to know how far the bullet will fly over the mark by knowing the distance to the mark. Question. A Cannon or culverin of 12 foot in length, having three inches more metal at the breech on each side then at the mouth, shooting at a mark supposed to be within the level range, and 600 yards from the mouth of the piece, being shot without her disparture, how much shall the shot fly over the mark? Resolution. It is a general rule, that look how much the piece is thicker of metal, in any one side at the breech, then at the thickest part at the mouth, as also look how many times the length of the piece is to the mark, so many times that overplus of thickness shall the bullet fly over the mark, being no higher than the piece, and the said piece discharged without her disparture. Example. Divide 600 yards (being the distance from the piece to the mark) by 4, (the length of the piece) your quotient is 150, which multiplied by 3 inches the overplus of metal, ariseth 450 inches: so much shall the bullet fly over that mark, the mark being placed on the side of a hill or bearing bank, and within the level range of the piece. In like manner shooting at any mark within ½ the utmost range of the piece, and not disparting your piece, you shall over shoot something, giving the piece her due length and due loading. How you may lay your piece point blank without instrument. If you bring the height of the metal at the mouth of the piece, and the height of the metal at the breech, equal with the horizon, the hollow cylinder of the piece will lie point blank. How you may Arithmetically dispart any piece of Ordinance truly divers ways. If you measure with a pair of Callapers the greatest height of metal at the mouth of the piece, and likewise at the breech, abating the less out of the greater, ½ the remainder is the just disparture. Example. A Culvering that is 19 inches high at the greatest part of metal in the breech, will be 13 inches high at the greatest part of metal at the mouth: which 13 inches abated from 19, rests 6, which divided in 2 equal parts, the quotient being 3 inches showeth the true disparture of that Culvering. Another way to dispart any piece without Callapers. Take a line and measure the greatest circumference of metal in the breech, then multiply that measure by 7, dividing the product by 22, the quotient is the diameter, or height of the circumference. Likewise measure the greatest circumference of metal at the mouth, multiplying that measure by 7, divide by 22 as before, the quotient will show the diameter of the metal at the mouth: subtract that diameter last found, from the diameter at the breech ½, the remain is the true disparture. Example. A Culvering whose greatest circumference of metal at the breech containeth 66 inches, and at the mouth 44 inches, I demand how high is the diameter of the metal both at the breech and mouth, as also what is the true disparture of that piece? Resolution. Multiply 66 by 7, ariseth 462, divide by 22, the quotient is 21, the height of the metal at the breech: likewise multiply 44 by 7, you have 308, divide by 22, the quotient is 14, the height of the metal at the mouth, which 14 abated from 21 rests 7, the which 7 divided in 2 equal parts, yields 3 inches ½ for a part, the true disparture of that Culvering. This is one of the principallest points belonging to a Gunner, to know truly how to bring the concave of the metal of his piece even: divers other ways there is to do the same. As for chambered pieces, there is no perfect or general rule, but is to be considered according to the chamber or concave of the piece. Every reasonable Gunner may judge in that case. How by Arithmetical skill you may mount any great piece of Ordinance by an inch rule unto 10 degrees of the quadrant, if you want a quadrant or other instrument. First you must measure the just length of the Cannon or bore of the piece: reduce that measure into inches, and double the same: afterwards multiply the number of inches so doubled by 22, and divide by 7, and note what the quotient number is, which quotient divided by 360 the degrees contained in the whole circumference of every circle, the last quotient number will show you the number of inches, and parts of an inch, that will make a degree in the quadrant for that piece. Example. Admit there is a Saker or Falcon, whose concave or boar containeth just 7 foot 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 foot into inches, and you have 84 inches, that 84 doubled is 168, the which multiplied by 22 ariseth 3696, the which divided by 7, the quotient will be 528; that quotient number being divided again by 360, will yield 1 7/15 (that is) one inch and ½, wanting 1/15 part of an inch. So I affirm that any piece of Ordinance whose chase or boar is but 7 foot long, being mounted by an inch rule one inch and 7/15 parts, that piece shall lie just the height she would have done if you would have mounted her one degree of the quadrant. The like order is to be observed in mounting any other piece of Ordinance by an inch rule, of what length soever. And note that in mounting any other piece of Ordinance, to any degree of the quadrant, by a Geometrical quadrant, you must put the rule of the quadrant into the piece mouth, lifting the piece up or down with a leaver or hand-spike towards the breech, till the plummet cut just upon that degree of the quadrant you desire. But to mount her by an inch, you must place the rule upon the highest part of the metal at the breech of the piece, coining the piece up or down, till through the sight or slit in your rule (be lifted to that part or division in your rule that answereth the degrees you desire) you espy the Carnoize or highest part of the metal at the mouth of the piece, and the mark, all 3 in a straight line. If you would mount the same piece 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 have 44/15, the which 44 being the numerator of the fraction divided by 15 the denominator, the quotient being 2 inches 14/15 is your desire; so may you affirm that 3 inches by the rule wanting 1/15 part of an inch, will make 2 degrees by the quadrant. And note, that look how much you would have your piece mounted by an inch rule for to answer any number of degrees under 10, either multiply that number by the number of inches and parts of an inch, that makes a degree of the quadrant, or else working as you did the first conclusion, multiplying the first product by the number of inches desired, and dividing that product by the numbers afore mentioned, your last quotient will resolve you of your desire. Example. I demand how much the piece afore mentioned should be elevated by an inch rule, to answer to 8 degrees of the quadrant? Resolution. Reduce the length of the bore of the piece into inches, as afore is showed, doubling that measure, and it makes 168, as you see in the 1 conclusion: which 168 inches multiplied by 22, yieldeth 3696 inches, the which product afterwards multiplied by 8, ariseth 29568, which sum divided by 7, the quotient is 4224: the same divided by 360, yields in the quotient 11 inches 11/15 parts of an inch, so many inches and parts of an inch must the same piece be elevated to with an inch rule, to answer to 8 degrees of the quadrant, as by trial you may find. How by Arithmetic skill you may know the true thickness of metal in any part of any piece of Ordinance. Take a pair of callapers, and measure the height of the out side of the metal in that place of the piece whereas you desire to know the thickness of the metal, then with an inch rule, or else a pair of straight compasses, measure the diameter of the boar, or concave of the piece, abating the height of the said diameter from the height of the whole thickness of that part of the piece so measured. And note the remainder, the which divide in 2 equal parts, and the one of those parts is the just measure of the thickness of the metal in that part of the piece. Example. I proved this conclusion with a Culvering, whose boar or concavity at the mouth was 5 inches ½ height, & I found that the thickness or height of the whole circumference of the said piece at the touchhole, was 16 inches ⅓, from the which I abated 5 inches ½ (fraction wise) rests 10 inches ⅚ parts of an inch: that divided in 2 equal parts, the quotient is 5 inches, and 5/12 or 5 inches ½ wanting the 1/12 part of half an inch, so thick was the metal of that Culvering at the touchhole. Likewise I searched for the thickness of metal in the same piece at the end of the trunions, and I found that the thickness or height of the superficies of all the metal there contained 13 inches, from which I abated the diameter or concave at the mouth, being 5 ½ inches, rested 7 ½, which divided in 2 equal parts, the quotient being 3 inches ¾ showed the true thickness of the metal at the trunions. In this order you may find the true thickness of metal in any part of any piece of Ordinance. Another way to know the thickness of metal in any part of any piece of Artillery. Take a leathern girdle, and gird about that part of the piece you desire the thickness of metal, 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 dividing the product by 22, your quotient is the true measure of the whole thickness of the piece in that place. Then substracting the diameter of the boar or concavity of the piece from that quotient, note the remainder. Divide that remain in two equal parts, the one of those parts is the thickness of the metal in that part of the piece so measured. Example. I proved this conclusion with a demi Cannon of six inches diameter, in girding the same about with a line hard behind the trunions, and laying the same to an inch rule, it contained 44 inches, which sum multiplied by 7, amounted to 308 inches: that sum divided by 22, my quotient was just 14. And so many inches was the height of the whole metal in that part of the piece, out of which quotient I did abate the diameter or bore of the piece being 6 inches, and the remain was 8 inches, which divided in 2 equal parts, my quotient being 4 inches, showed the true thickness of metal in that part of the piece, being hard behind the trunions towards the breech. And it is to be noted, that every piece of Ordinance if it be truly fortified with metal, aught to contain as much metal in thickness round about, so far as the chamber where the powder and wad lieth, as the bullet is in height. How to make a good shot in a piece that is not truly bored: or to know how much any piece will shoot amiss, that is thicker of metal on the one side then on the other, if you know the distance to the mark. Question. A certain Gunner having shot divers times in a Cannon at a mark supposed to be 500 paces from the piece, findeth she shooteth still towards the right hand, & searching whether the fault were in himself, or some impediment in the piece, he findeth that the piece is 2 inches thicker of metal on the right side then on the left. And therefore requesteth how to lay the concave of the piece (being 9 foot in length) equal with the mark, so as he may make a strait shot. Resolution. To do this or the like, there is a general rule, that look how oftentimes the length of the cylinder or concave of the piece is to the mark, which is easily done by dividing the distance to the mark, by the length of the concave of the said piece. And knowing likewise how much the one side of the piece is thicker than the other, the one half of that overplus being multiplied by the quotient first found, the product will show you how much the piece shooteth wide of the mark. And this is a general rule: that look which side of the piece is thickest of metal, towards that side shall the bullet fall, for that the thinner side is more smart, and the thick side more dull in heating. Example. The Cannon in this conclusion, is said to be 2 inches thicker of metal more in thickness on the right side then on the left. And the distance to the mark is supposed to be 500 paces, (that is, 2500 feet) the which divided by 9 feet, being the length of the hollow cylinder of the Cannon, yieldeth in the quotient 277 feet 7/9, the which multiplied by ½ the supper fluity of the metal being one inch, makes 272 feet 7/9 still, and so much wide of the mark should the said piece have shot at such a distance, although she had been laid full against the mids thereof. How to remedy your piece being thicker of metal in one part then another to make her shoot straight. You must first search your piece with an instrument, to know which is the thicker side, then divide the overplus of metal in 2 parts, setting the disparture of your piece one of those parts towards the thickest side of the piece mouth, and bringing the middle part of metal at the tail of your piece, that disparture and the middle of the mark, all in one straight line, give fire and you shall make a straight shot But beware of overcharging of such pieces, for they are dangerous. If the thickest part of the metal be above, than you ought to make your disparture one inch more: if under (I mean towards the carriage) an inch less. To know the different force of any two like pieces of Ordinance planted against an object, the one being further of from the said object then the other. Question. Admit there is a Castle or Fort to be battered, being situate upon a hill, which hill is 50 paces in height, and that 140 paces from the said Castle there is another hill, of equal height to that hill whereon the Castle is built, and Ordinance planted thereon to beat or batter the Castle wall, and in the valley at the foot of the said hill 180 paces off from the Castle hill, there is Ordiance planted, and mounted at 20 degrees, to shoot and beat down the said castle: I would know whether the Ordinance in the valley being 180 paces distance from the Castle, and mounted at 20 degrees, or the Ordinance on the height of the hill, lying level to shoot a little above the base of the wall, being distant therefrom 140 paces, shall work the greatest effect in battering down the said Castle wall, the said pieces being of like length and height, and having like charge in powder and bullet? Resolution. To resolve this or the like, a man would think that the piece planted on the height of the hill, lying level to shoot a little above the groundwork of the Castle, would batter sorest, because she is nearest: yet by experience we find the contrary, for the Castle being a great way within the reach of both the pieces, that piece shall not only shoot much further, that is any thing elevated, but also pierce much sorer, by so much as she is able to over shoot the other self like piece that lieth level: albeit the said piece so elevated, be planted furthest off from the said resisting object: for every Gunner knoweth, and reason and experience doth teach every reasonable man, that no piece of Artillery will shoot so far at point blank, as when the same is elevated at any number of degrees; because the bullet being ponderous, flieth more heavily and sooner declineth, being shot out of any piece lying level, then out of any such like piece mounted at any degree of the random. So that of force it must needs follow, that the piece planted in the valley 180 paces off from the Castle, shall pierce and batter a great deal sorer than the like piece planted on the height of the hill being but 140 paces from it. Example. Example. Suppose a Cannon or culverin at point blank shoot 240 paces, and being mounted at one degree outshoote the same 30 paces, what will the said piece do being mounted 20 degrees? By proportion I find, that if at the mount of one degree, any bullet range 30 paces beyond the level range, that at 20 degrees if shall outfly the same 600 paces: albeit the said bullet range not in every degree a just like number of paces, yet the proportion will be very near thereto. And because the piece at the foot of the hill is said to be 40 paces further from the Castle, than the like piece planted on the height of the hill, I abate 40 out of 600, rests 560 paces: so far would the piece in the valley out shoot the other like piece on the hill; so that it must needs follow, her bullet shall pierce sorest, for that it hath most strength to fly furthest. Another exmaple or trial of the former conclusion. The piece planted upon the hill, is said to be 140 paces from the Castle, and the like piece at the soot of the hill 180 paces. Now suppose each of those pieces being laid at point blank, would not range above 240 paces, abate 140 paces (the length to the mark of the piece on the hill) from 240 paces her level range, and the remain is 100 paces; and so many paces shall that piece strike the mark before the end of her level range. Now to find the like in the piece planted in the valley 180 paces from the Castle, mounted at 20 degrees, I find by the conclusion afore set down, that she shall out shoot the other 600 paces: so that abating the distance from the piece to the Castle, being 180 paces from 840 paces, her whole range mounted at those degrees, there remains 660 paces. And forasmuch as the said piece elevated at 20 grades, doth strike the mark 660 paces before the full end of the range of her bullet, it must of force pierce or batter sorer than the other piece whose bullet beats the mark but 140 paces before the full end of his range. How you may having diverse kinds of Ordinance to batter the walls of any Town or Castle, etc. tell presently how much powder will load all those Ordinance one or many times. Question. There is a Castle besieged, and to batter the same there is appointed 4 Cannons, 6 demi Cannons, 6 culverins, 8 demi culverins, and 5 Sakers: these pieces are charged every time with corn powder, the whole Cannons shoots at every shot 32 pound of powder a piece, the demi Cannons 18 pound, the whole culverin 16 pound, the demi culverin 12 pound, and the Sakers 6 pound a piece. All which pieces being 10 times discharged, did make a breach sufficient for 9 or 10 men to enter in by rank (a breach of such a wideness is thought sufficient to be assaultable,) I demand how much powder was spent before the breach was made? Resolution. To answer to this demand, I multiply the number of every sort of pieces, by the weight in powder that one of them shoots, and the product showeth me how much powder every sort of the said pieces did spend at one bout: then I add every number together, and the total of that addition showeth me how much powder will load all those pieces one time, which addition multiplied by 10, being the times they were supposed to be discharged, the product showeth the just quantity of corn powder occupied at the said siege by the great Ordinance. Example. I multiply 32 pound the weight of powder due to load every Cannon by 4 the number of Cannons, ariseth 128. Likewise 18 pound of powder being the duty of every demi Cannon multiplied by 6 the number of the same pieces, ariseth 108, and 16 pound of corn powder being the duty of every culverin multiplied by 6 the number of those pieces, is 96. And 12 pound of powder being the due loading of every demi culverin multiplied by 8. the number of the same is 96. And lastly 6 pound of powder the duty of every Saker, multiplied by 5 the number of that sort of pieces, is 30. These sums or additions put together makes 458 pound weight of powder: and so much will discharge all those pieces one time; the which sum multiplied by 10, is 4580 pound of powder, that is, two Last of powder wanting 220 pound. In this order if you have 20 Last of powder, by knowing the number of every sort of several Ordinance, you may presently know how many shots, or how many times the said powder will load all the said Ordinance, as this table showeth. Names of the Pieces. Number of each sort of Pieces. Powder due to load each sort of Pieces one time. Cannons. 4. 128. Demy Can. 6. 108. culverins. 6. 96. Demy Culuer. 8. 96. Saker. 5. 30. Sum 458 pound of powder, which multiplied by 10, makes 4580 pound weight. And it is worthy the noting, that in planting of Ordinance to batter or beat down any curtain, wall, or Cullion point, you must plant the same in 3 or 2 several places at the least, from the thing to be beaten down; so as the said Ordinance be a pretty distance from other, upon convenient platforms, having Gabbions or Baskets, about 8 foot high, ramd full of earth conveniently placed between each piece, to save the Gunners and labourers from the danger of the enemy's shot: which Ordinance would be planted within 200 or 240 paces of the object to be overthrown, if it be possible to have convenient platforms and to bring them so nigh the said object. The which Ordinance (if so you have made 3 mounts or platforms, the Ordinance from the 2 side mounts doth coin or cut out that which the Ordinance from the middle mount doth batter or pierce, or shake, as this draft here drawn showeth. The best shooting to batter down the broad side or curtain of any wall, is to level something under the middle part of the wall, and after to shoot 2 or 3 foot higher: for the lower part being beaten down, the height or upper part of the said wall must fall of necessity. And a special regard must be had to give fire from each platform or mount at one instant, for that the bullets beating all together, do more shake and batter the said wall, then lighting now one and then another. In the figure or draft which I have drawn showing how Ordinance may be planted to ding down or batter the broad side or curtain of any wall, castle or Fort, the middle Ordinance placed on the middle mount or platform, directly against the object to be beaten down, are called the peircers, and are only to shake and beat the wall, and the Ordinance on the two other side mounts, or platforms shooting something slanting, are to coin or cut out that which the Ordinance from the middle platform doth shake or loose. The Baskets ramd full of earth being placed between each piece of Ordinance are to defend the Gunners and labourers from hurt of them that are besieged, as afore I have said. And further it is to be noted, that to batter the coin or cullion point of any wall, two places is sufficient to plant your Ordinance in. Also you may batter and beat down the wall of a Town or Castle as well by night as day, so as the enemy shall have no time to build up in the night that which was dung down in the day, as thus: Lay your piece or pieces, to the mark in the day light, and note well what degree of the quadrant she lieth at, which is soon done in putting the rule of your quadrant into the piece mouth, so laid against the mark, 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 drive in a stake or wooden pin; and letting a plumb line fall likewise from the middle part of the tail or breech of your piece to the ground, drive therein another stake into the ground, then stretch a line from the said 2 pings, so as the ends of the said line may reach 2 or 3 yards further than the pings at each end. And there make them fast in driving a pin of wood or iron into the ground at each end, then bringing your piece or pieces to lie straight above the said line or lines so drawn (which is easily done having a lantern with a close cover) you may both charge and recharge, and shoot aswell by night as day, according to your desire. How you may know the true weight of any number of shot, for several pieces of Ordinance, how many soever they be, and how many Tun weight they do all weigh. Question. Suppose a Ship is loaden with Bullets to be carried to the siege of a Town, etc. in which ship is 500 shot for whole Cannons, 800 demi Cannon shot, 900 Culvering shot, 1000 demi Culvering short, 1100 Saker shot 1200 Minion shot, and 1400 Falcon 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 showed how to find out the weight of any unknown bullet, by the weight of a known bullet of the like metal, so that multiplying the number of every several sort by the weight that one of them weigheth, and adding all the products into one sum; and then dividing that total by 2240 pound, which is the pounds in a Tun, the quotient will show you how many Tun all those bullets weigheth. Example. Admit the Cannon shot weigh 60 pound a piece, 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 piece, which multiplied as before, makes 25600 pound weight. And then there is 900 Culvering shot of 16 pound weight a piece, which makes 14400 pound weight. And then 1000 demie culverin shot of 10 pound weight a piece, which makes 10000 pound weight. And then 1100 Saker shot of 5 pound weight a piece, which makes 5500 pound weight. And then 1200 Minion shot of 3 pound weight a piece, which makes 3600 pound. And lastly, 1400 Falcon shot of 2 pound weight a piece, which makes 2800 pound weight. All these sums added together makes 91900 pound weight, which divided by 2240, yields 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 burden, you may easily know how many shot will load the said ship. How any Gunner or gunfounder may by Arethmiticke skill, know whether the trunions of the piece be placed rightly on the piece or not. Measure the length of the bore of the piece, from the mouth to the breech, divide that measure by 7, and multiply the sum that cometh in the quotient by 3, the product will show you how many inches or other measure the trunions ought to stand from the end of the lowest part of the concavity of the said piece at the breech. And note that the trunions ought so to be placed, as ⅔ parts of the circumference of the piece may be seen in that place whereas the trunions are set. Example. Admit the cylinder or concave of a Cannon, or other piece of Ordinance be 10 foot ½ long, I demand where the trunions of the said piece ought to stand? Answer. Reduce the length of the concave of the piece into inches, you have 126 inches, the which divided by 7, the quotient is 18, that multiplied by 3, makes 54 inches, or 4 foot ½, so far ought the trunions to be placed from the breech or lowest part of the hollow concavity of the said piece. Another way. Or multiplying the length of the concave of the piece by three, and dividing the product by 7, the quotient will show the true place, how far the trunions ought to stand from the lowest part of the boar or concavity of the piece. Example. 126 inches the length of the concave of the piece, multiplied by 3, makes 378 inches, which number divided by 7, the quotient is 54 inches as before. And note that the trunions of every piece were invented to hold the piece up in her carriage, to move her up and down to make a perfect shot, and to hold her fast in her carriage, after she is discharged: for if the trunions be placed too near the mouth, the piece will be too heavy towards the breech, so as the Gunner appointed to serve with her, shall have much ado to raise her, to coin her up or down, or being placed too near the breech, the contrary will happen. How you may know what empty cask is to be provided to boy or carry over any piece of Ordinance over any river, if boats or other provision cannot be gotten. It is thought sufficient that 5 Tun of empty cask will swim and carry over a Cannon of 8 or 9000 pound weight, 4 Tun will carry over a demi Cannon, 3 Tun a Culvering, and 2 Tun a Saker, accounting all provisions to be made fast thereto, as planks, ropes, etc. so that knowing what number of Ordinance is to be ferried or carried over any river, adding all their weights into one sum, by framing the Golden rule, you may presently know what empty cask is to be provided to ferry over all the said Ordinance at one instant. Example. If a Cannon of 8000 weight require 5 Tun of empty cask, how much empty cask is to be provided to carry over so many Ordinance as is supposed to be 100000 weight? Resolution. I multiply 100000 by 5, so ariseth 500000, the which being divided by 8000, the quotient is 62 ½, so many Tun of empty cask is to be provided to carry over so many Ordinance as weigeth 100000 pound weight. The which empty cask made fast head to head a row on each side, by such as have skill in such services, and planked above, would serve for a bridge to carry over a whole Army with all provisions thereto belonging. All which necessaries in time of service, and many more, belongeth to the Master of the Ordinance his office, to have in readiness, as also to be provided of Trunks, Arrows, Balls, and all kind of fireworks, wet or dry, and the receipts for making thereof. As also engines for mounting or dismounting of Ordinance, wheels, Axeltrees, Bullets, Powder, Ladles, Sponges, Ropes, Shovels, Anckors, etc. Also it is the duty of the Master of the Ordinance, the Master Gunner, and every chief officer or quarter Master under them, to be expert in the Art of Gunnery, the better to teach and instruct their inferiors, the which without some practice in Arethmiticke and Geometry they cannot well accomplish. They ought to have some sight in the mathematicals, the better to teach and instruct such as would shoot at all randoms, to know what Ordinance is convenient for an Army, or to batter or beat down the walls of any Town or Castle, to know what powder and shot is to be provided for that or such like purpose, what carriage horses, labourers and other necessaries is to be allowed for the same. They ought to practise all Geometrical instruments, for the measuring of heights, lengths, breadthes, depths, etc. To practise how to convey mines under the ground, and how the same should be truly wrought, to blow up any Tower, Castle, etc. To know what length the mine will contain with all his windings to and fro to the place appointed. To have skill, in the handling of all engines and inventions belonging to the Ordinance. To appoint to every piece of Ordinance in time of service ', Gunners that know perfectly how to manage their pieces, to charge, shoot, cleanse, scour, wad and ram the same, and what labourers are to attend thereon. To know in every platform appointed, how to place the baskets or gabbions, and what proportion of wideness, height, or thickness they ought to contain: and that the loops have their due proportion of wideness. To see that every Gunner be able to discharge his duty, and not for favour or affection to prefer such as can say most, and do least: but that every man be preferred to place of credit, and esteemed according to his honest behaviour and skill in this singular Arte. That none be permitted to the profession of a Gunner, but that he be first truly instructed in the principals of the Art, by such as have skill therein. And not to make or suffer every tag and rag to be a Gunner, as is too much used in these days in Towns of garrison, who was never practised in the Art, nor bathe discretion nor desire to practise therein: a great number of such have but only the bare name of a Gunner, although their standing hath been of long time: for as a great many of Mariners have sailed 7 or 8 years and yet far from a Navigator, so a great many such have continued in pay a large prentise-hood, and yet far from a good Gunner. Such in time of service would work as the builders did at the Tower of Babel, when one called for one thing, he had delivered a contrary thing. In service the Prince by such is not truly served, the Art less esteemed, and themselves discredited. The Art is like to a circle without end, or like to a Labyrinth, wherein a man being well entered in, knoweth not how to get out again, and therefore it must be exercise and industry that must make a perfect Gunner. Many things here could I write pertaining to the duty of a Gunner, and every officer pertaining to the Ordinance, but for as much as the same is not peculiar to this arithmetical treatise, and sufficiently handled by other Authors, I omit. How to know the true time that any quantity of Gunmatch being fired, shall burn, to do an exploit at any time desired. Take common match, and rub or beat the same a little against some post or stool to soften it, and then either dip the same in saltpeter water and dry it again in the sun, or else rub it in a little powder and brimstone beaten very small and made liquid with a little Aqua vitae, and dried afterwards. Now when you would occupy the same, try how long one yard will burn, which suppose to be ¼ part of an hour, than 4 yards will be a just hour in burning. Now suppose you have laid some powder or balls of wild fire to burn some house, ship, mine, corne-stacks, etc. or that you have placed the said powder or balls in some secret place to burn some thing you are desirous to spoil, and that you would be going from the place 3 hours before it effect, then binding the one end fast to the balls, laying loose powder under & about the same, or wrapping the one end like a wreath amongst the powder loosely, draw out the other end, or lay it crookedly, or wrap it softly about something, so as one part do not touch another, and fire it at the other end: which match so drawn or rolled, being just 12 yards in length, shall kindle the thing you would burn at the end of 3 hours, according to your desire: for the rule of proportion showeth, that if one yard require a quarter or ¼ of an hour, that 12 yards of match will burn out in 3 hours. The like order you may observe, to answer to any time appointed. How by Arithmetical skill you may know what number of men, horses, or oxen, is sufficient to draw any piece of Artillery, and how much every one draweth a piece, so as they all draw together. Question. If 90 men be able sufficiently to draw a Cannon of 9000 pound weight, accounting carriage and all, I demand how many men is able to draw a culverin of 2500 pound weight, and how much every man drew for his part? Resolution. I answer: If a Cannon of 9000 pound weight, require 90 men, the quotient showeth me that a culverin of 2500 weight requireth 25 men to draw the same: and dividing the weight of the piece to be drawn by the number of men appointed to draw the same, the quotient will show you how much every man drew to his part (to wit) 100 weight. To know how many horses is to be provided to draw any Piece of Ordinance, and how much every one draweth. Question. If three horses draw a Falcon of 900 weight, how many horses will draw a culverin of 3000 weight? Resolution. I say as before, if a piece of 900 weight require 3 horses, what will a piece of 3000 weight? and in working according to the rule, the quotient is 10, showing that 10 horses must be provided to draw a culverin of 3000 weight. Also dividing 3000, the weight of the said piece, by 10 (being the number of horses) there will stand in the quotient 300, showing the draft of each horse. To know how many Oxen is to be provided to draw any piece of Artillery. It is to be noted that 3 yoke of oxen is thought to draw as much as three horses, and that 3 yoke of oxen is sufficient to draw a Saker of 1400 weight. Question. How many oxen must be provided for a Cannon of 8000 weight? Resolution. In working as before, I find that 34 oxen, or 17 yoke of oxen, will serve to draw a Cannon of 8000 pound weight. And note that whereas there doth remain 2/7 parts of a whole number, neither men, horses, nor other cattle, can in any such millitare questions be brought into a fraction, but yet the rule it showeth that 17 yoke of oxen is sufficient for the draft of a Cannon of 8000 pound weight, when 3 yoke of oxen serve for to draw a Saker of 1400 pound weight. If you divide the weight of the whole Cannon being 8000 pound weight by 34, the oxen appointed to draw the same, the quotient is 235 pound 5/17: so much did every ox draw. How you may wanting both oxen and horses to draw any piece of Ordinance, know presently how many men is able sufficiently to draw the same, either on plain or marish ground. Question. I showed in a conclusion before, that 3 yoke of oxen would draw a piece of 1400 pound weight, and that 90 men would draw a Cannon of 9000 pound weight; now if there want both horses and oxen, or that you are occasioned to draw the said piece through some marish ground, whereas horses and oxen cannot pass, I demand how many men is sufficient to hale a Saker of 1400 pound weight through the said marish ground? Resolution. If a Cannon of 9000 weight require 90 men to draw the same, I find that a Saker, weighing 1400 pound weight must have 14 men to draw the same, and every one shall draw 100 weight for his part. In drawing Artillery through any soft marish ground it is requisite to have in readiness, in the Master of the Ordinance his carts, which carrieth the provisions for the Ordinance certain hurdles of boards, or rather flat bottomed boats, in which any piece of Ordinance may be placed carriage and all, and by force or strength of men may be drawn as easily, as to draw the said piece on the firm land, for that the said boat is apt to slide or swim on the soft owish, the ropes being made fast to the forestearne or sides of the said boats, which boats do serve also for carriage of the Ordinance, and all things thereto belonging, over any river or soft owish ground, etc. How you may by the rule afore, know how many oxen will draw any piece of Ordinance, if you want men and horses. I showed that 90 men is able to draw a Cannon of 9000 pound weight, and that three yoke of oxen will serve to draw a Cannon of 1400 pound weight: now wanting men and horses, I say if a Saker of 1400 pound weight require 6 oxen, what will a Cannon of 9000? and in multiplying the weight of the Cannon by 6, the number of oxen appointed to draw the Saker, and dividing that product by the weight of the lesser piece, the quotient is 38 oxen or 19 yoke, so many must be provided to draw a Cannon of 9000 pound weight, which weight divided by the 38 oxen appointed to draw the same, the quotient showeth that every ox drew 236 pound weight. How you may wanting men and oxen to draw any piece of Ordinance, know how many horses is requisite to draw the same. Also I noted before, that 3 horses would serve to draw a Falcon of 900 pound weight: I demand how many horses will serve to draw a Cannon of 9000 pound weight? In working as before, the quotient is 30, so mamy horses is requisite for that purpose: which piece, her weight divided by the number of horses appointed to draw the same, the quotient showeth that every horse drew 300 pound weight. In this order you may know what number of men, horses, or oxen, is able to draw any piece of Ordinance, and what every one severally doth draw. How to know how many 100 of Haberdepoize weight any piece of Ordinance, or other gross weight containeth. In the conclusions afore set down, thou must note gentle reader, that every 100 weight of most things, is accounted after five score to the hundredth: but if thou be desirous to know how many hundredth of Haber depoize weight any piece of Ordinance or other gross weight containeth, thou mayst by Arithmetic soon be resolved, for every 100 of Haberdepoize weight containeth 112 pound, the half hundredth 56 pound, the quarter 28 pound, and the pound 16 ounces: so that dividing the weight of any great piece by 112, thou mayst easily know how many hundredth of Haberdepoize weight the same containeth. I would know how many hundredth of Haberdepoize weight is in a Cannon of 9000 pound weight, I divide the same by 112 as aforesaid, and the quotient being 80 40/112, showeth that a Cannon of 9000 pound weight contains 80 hundredth of Haberdepoize weight, one quarter and 12 pound. A Tun containeth 2000 of Haberdepoize weight. How you may proportionally prove all sorts of pieces of Artillery for service whether they will hold or no. All pieces that shoot a bullet under 10 pound weight, and duly fortified with metal, being shot 3 times, first with the whole weight of the iron bullet. Secondly with 5/4 parts thereof, and lastly with 3/2 parts of the same, will hold for any service, being charged with her ordinary charge, albeit the said piece were discharged 100 times in one day. How you may find out the proportional charge afore named as thus. Suppose a piece shoot a bullet of 6 pound weight, and that you desire to know what 5/4 parts in powder of the weight of the bullet is: multiply the weight of the said bullet by the numerator 5, and divide by the denominator 4, the quotient is your desire. Example. 6 multiplied by 5, is 30: the same divided by 4, the quotient is 7 ½. The like order you must use in giving her 3/2 parts in powder to the weight of the shot, and your quotient is 9 pound. How to prove any piece that shooteth a bullet under 50 pound weight, and above 10 pound weight. Any piece that shooteth a bullet above 10 pound in weight, and under 50 pound, would for the first shot be charged with ⅔ parts in powder of the pellets weight: for the second shot with ⅚ parts, and lastly with the whole weight of the bullet. Example. Admit a piece shoot a bullet of 40 pound weight, the ⅔ parts thereof is 26 pound ⅔, and ⅚ parts thereof is 33 pound ⅔ parts. And note that in proving any piece of Ordinance, whether she be serviceable or not, her mouth would be mounted to 20 or 30 degrees of the quadrant, or thereabout. To know how much one coyler rope, for the draft of any piece of Ordinance is bigger than another, and how thick any of them is. Take the compass of the lesser, and likewise the circumference of the greater, abating the lesser out of the greater, the remain is your desire, which known by the rule of proportion you may find out the height or thickness of the lesser. Example. Suppose you have a coyler rope of 6 inches compass, and another of 4 inches compass, abating 4 inches from 6 inches the compass of the greater, rests 2 inches, the diameter or height of the greater: which known, frame the rule of proportion saying: If 6 yield 2, what 4? the quotient is one inch ⅔ parts, showing the true thickness or height of the lesser. To know how much one coyler rope is more than another. Take the compass of your rope, and multiply it in itself, and look how much you would have the other greater, augment your product by the same proportion, extract the square root, you have your desire. Example. A coyler rope of 6 inches compass squared, makes 36 inches. Now if you would have one 3 times as much, then multiply 36 by 3, the product is 108, the square root thereof is 10 inches and something better, and so thick ought a rope to be that is 3 times the compass of the other. How by knowing the weight of a fathom of one rope, to know the weight of a fathom of any other. A cable or coyler rope of 10 inches compass weighing 16 pound every fathom, how much will a fathom of that rope weigh, that is 15 inches compass, and made of the same stuff? I multiply the greater in itself, ariseth 225, and that multiplied by 16 pound the weight of a fathom of the lesser rope, ariseth 3600, the which divided by 100, being the square of the lesser rope, the quotient is 36 pound, and so much will every fathom of the greater rope weigh. In this order by knowing what a fathom of the greater rope weigheth, you may soon find what a fathom of the lesser rope weigheth. How by knowing the quantity or compass of any small rope, to find out the same in another that is many times that bigness. Admit I have a small rope of 3 inches compass, and that it is required to know the height of another that is 5 times that compass. I square the number 3. ariseth 9, which multiplied by 5 makes 45, the square root thereof is 6 inches ¼ so high is the greater. The like is to be done of all such like demands. To know the weight of a whole coyler rope for the draft of any piece of Ordinance. Question. There is a coyler rope of 8 inches compass weighing 12 pound every fathom, I demand the whole weight of that rope being 20 fathom long? Resolution. Multiply the number of faddoms in the rope (being 20) by the weight of one fathom, the product is 240 pound weight, your desire. The length of a coyler rope for a whole Cannon ought to be 70 fathom or thereabouts. For an ordinary Cannon 64 or 66 fathom, and for a demi Cannon 60 fathom or thereabouts. For a culverin 40 fathom, a demi culverin 36 fathom, and a Saker 30 fathom, etc. To find out the superficial content of the hollow concavity of any piece If you multiply the length of the cylinder or bore of the piece, by the circumference of the hollow concave about the mouth, the product will show you the superficial content of the cylinder of the said piece. Example. A Cannon of 7 inches diameter having her concave or hollow cylinder 12 foot in length, how much is the superficial content thereof? Resolution. Reduce the length of the hollow concave of the pecce into inches, ariseth 144 inches, which multiplied by 22 inches, the circumference of the concave at the mouth of the piece, ariseth 3168 inches, the superficial content of the metal compassing the concave of the piece. To find out the crassitude or solid content of the cylinder or concave of any piece. First you must by the rules taught in the beginning of the book, find out the content of the base or plain of the concavity at the mouth of the piece, in multiplying ½, the diameter in half the circumference or else squaring the diameter and multiplying that product by 11, and dividing the result by 14, the quotient will also show you the content, the which multiplied in the length of the cylinder of the piece, the product is your desire. Example. The Cannon above named of 7 inches diameter, wrought as is showed, yieldeth 38 inches ½ at the base or circular content of her mouth, which multiplied by 144 inches, the length of the cylinder, yieldeth 8280 inches, the solid content of the concave of the said piece. If you desire to know how many foot in square measure the solid content of the empty or hollow concavity of the piece aforenamed or any other doth contain, you must work thus; divide the number of inches in the solid content thereof by the number of inches in a foot square being 1728, the quotient is your desire. Example. The solid content of the piece of 7 inches diameter above named, containeth 8280 inches, which divided by 1728, the quotient is 4 57/72, that is 5 feet in square measure wanting 15 inches. The like is to be done in any other piece, or in measuring the cylinder or Cone in any other solid body. How you may Arithmetically know how much any piece of Ordinances is taper-bored, or whether the same be taper-bored or not. Put upon your rammer staff a tampion of wood, that is just the height of the hollow concave of your piece, and thrust the same home into the piece; if it go not home to the breech, than the piece is taper-bored, if it go home the piece is not taper bored: if she be taper-bored, then put on such a tampion of wood upon your rammer staff, as may fill the concave of the piece in the narrowest part where she is taper-bored, and be sure that it go home to the breech of the piece, and afterwards with your compasses, measure the diameter of either tampion, abating the lesser measure out of the greater, the remain is your desire. And note that the tampion at the end of every rammer staff, is to thrust home the wad and bullet close to the chamber or place where the powder lieth, and every rammer staff ought to have a sponge at the one end, to cleanse the piece with, and a tampion of wood at the other end, to put home the bullet and wad with, in the centre of which ought to be a hollow screw wherein the Gunner may screw in a wad hook to unload any piece at his pleasure. How to shoot in any mortar piece. mortar pieces were invented only to annoy the enemy, when other Ordinance cannot be used against them, as being charged with stone to beat down the houses of the enemy, or to fall amongst men being assembled together, or charged with balls of wildfire to burn the enemy's ships, houses, or corn. To make a perfect shot in one of these pieces, it is requisite you know 2 things belonging to the same (that is to say) how far your mortar piece will carry a bullet, or a ball of firework, as she is to shoot at the best of the random: and likewise how far it is from your piece to the mark you intent to shoot at, which known you may make a perfect shot, as thus. Example. If a mortar piece shoot a bullet or firework 700 paces, and that the mark which you intent to shoot at is but 500 paces; I demand at what degree of the quadrant, shall the piece be laid at, to make a good shot? Resolution. To answer this and all such like, reason and experience teacheth, that the lesser ground you intent to shoot, you must raise the mouth of your mortar piece so many degrees above the best of the random, as is sufficient to reach the mark desired: and therefore I say if 700 paces require 45 degrees of the quadrant, what will 500? and the quotient tells me, that at 63 degrees of the quadrant the mouth of the said piece must be elevated at, to cause the bullet or fireball to light accordingly. If you abate 45 degrees (being the best of the random) from 63 degrees, that the piece was elevated at, the remain is 18 degrees, & so many degrees of the quadrant was the mouth of the mortar piece elevated at to reach the mark. To know how far or short any mortar piece will shoot further or shorter, at the mount or dismount of one or many degrees. Question. A mortar piece that shoots 450 paces at the best of the random, I would know how much shorter shall she shoot, being elevated one degree above the utmost range? Resolution. Divide the distance of the utmost range being 450 paces, by 45 the degrees in the best of the random, the quotient is 10, so many paces will the said piece shoot shorter, her mouth elevated one degree. How you may know very near how far from your piece the bullet shall light, the said mortar piece mouth being raised to what degree you think good. Question Suppose there is a Castle etc. besieged, and that the Gunners had brought their Ordinance as near as they would wish, so that having discharged the mortar piece in the former conclusion, at the mount of 60 degrees, they find that the bullet falls in or about the mids of the said Castle or Fort. The question is how far it is between the piece and the fall of the said bullet? Resolution. You must first seek what difference of degrees is between 60 and 45, and you shall find 15, then by the rule of proportion say, if one degree abate 10 paces, what will 45? and you shall find 150 paces in your quotient. And in this order by the help of Arithmetic you may find how far it is from the piece to the mark. Also it is possible to shoot so directly upright in a quiet, fair, and calm day, that the bullet shot out of your mortar piece, shall fall into the piece mouth again or hard besides the same, if you raise the piece mouth just to 90 degrees of the quadrant, which albeit it be not serviceable, yet it is possible to be done: For this is a general rule, that no piece of Ordinance whatsoever can shoot a bullet to continue still in a straight line, during the motion of the said bullet, except you elevate or raise the concave of the said piece directly towards the zeneth of the sky, or else plumb down towards the centre of the earth. The diameter of the chamber mouth in every mortar piece, aught to be equal to the semi-diameter in the mouth of the said mortar. The length of every chamber in a mortar piece, aught to be once and a half the diameter of the chamber. The metal at the breech of every mortar piece, aught to be fortified equal in thickness to the diameter of the mouth of the chamber within, and at the trunions to the semi-diameter, and at the forepart or neck of the piece, to the ⅓ part of the diameter of the chamber mouth. To mount a mortar piece by the quadrant, some use to put the rule of the quadrant into the piece mouth, close to the metal, or inside of the piece, noting at what degree the plummet hangs; but for as much as there be many mortar pieces a little taper-bored at the mouth, (I mean the diameter at the mouth is something wider than it is within) therefore it is the best to have a rule made for the purpose, which among the experienced Gunners is common, the said rule being about 18 inches length, at the middle point or prick whereof is another shorter rule, framed artificially about a foot long, joined close, and falling perpendicularly on the longer rule, whose containing angle lighteth justly on the middle point or mids of the longer rule, from which point is drawn by Art the ⅛ part of a circle, and divided into 45 equal divisions or degrees, so as the 90 degree stands just on the centre or middle point of the longer rule: so that laying the longer rule cross the mouth of the piece, you shall presently know at what degree the said mortar piece is elevated at by the plummet, the piece being mounted at any grave above 45. And thus may you mount your mortar piece, to shoot at what degree you think good. The pattern of the rule this figure showeth, plainly drawn. The orderly flight or motion of the bullet or fireball shot out of any mortar piece, by the figure or draft hereunder may be perceived. Having planted Ordinance upon any mount or platform, to besiege any Town, etc. and that you desire to make some little trench or ditch about the same for the defence thereof, how you may know how much the earth and turf that is cast out of the said ditch, shall raise a wall in height, being laid orderly at the brim of the said ditch, on the inside thereof, making the same wall to any proportion assigned. Question. Suppose the General command the captain of the Pioneers, that a ditch be made about the mounts or platforms where the Ordinance plays, making the same 18 foot in breadth at the brim, 12 foot in breadth at the bottom, and 8 foot in depth, and that the earth and turf digged out of the said trench be laid orderly in the inside thereof at the brim of the said ditch, so as a wall may be made in breadth at the bottom 12 foot, and at the top 8 foot, I demand how high shall that wall be when it is finished? Resolution. To work this, there is a general rule, (as thus.) Add the wideness or breadth of the brim, and the breadth or wideness at the bottom together, the ½ of that addition multiplied by depth of the ditch the product of that multiplication shall be your devident, or number to be divided. Now to find the height of the wall, add the thickness of the bottom of the wall which you mean to make, to the thickness or breadth that you intent to make it at the head; the ½ of that addition shall be your devisore, which devident divided by the divisor, the quotient will show you the height of the wall. Example. The trench in this conclusion is said to be 18 foot broad at the mouth or brim thereof, and 12 foot at the bottom, which 2 numbers being added, makes 30, the half whereof is 15 feet, which 15 feet multiplied by 8 feet being the depth, ariseth 120 feet for my devident. Likewise, add twelve foot (the thickness of the wall at the bottom) to 8 foot the breadth you mean to make it at the head, so ariseth 20 feet, the ½ thereof is 10 feet for my devisor, (and so thick the said wall will be in the mids: the which devident being 120, being divided by the divisor 10, the quotient is 12, and so many foot in height shall the earth and turf casten out of the trench aforesaid, make a wall being 12 foot broad at the bottom, 8 foot at the head, and 10 foot in breadth at the mids: the said trench being 18 foot broad at the brim, 12 foot broad at the bottom, and 8 foot deep. In this order you may find out the height, breadth, or depth of any such like wall or ditch, in making the same after any proportion assigned. Brief observations of certain principals in the Art of Gunnery, for every Gunner to consider of, to practise and learn, viz. To know what disparture every piece of Ordinance ought to have in shooting either at or within point blank, or with an inch rule at any advantage. To use a mediocrity in ramming and wadding, and in giving every piece her due loading in powder and bullet. To know the goodness and badness of powder, and how to mix and make perfect good powder, and how to fine the peter, etc. To consider the wind, whether it blow with you or against you, or on any side of the piece, and how to weather your piece to make a good shot. To consider the platform, whether it be flat, or else declining for the recoil of your piece, and whether the mark be higher or lower than your platform, as also to know the distance thereto. To know whether your piece be truly bored or not, and how to make a perfect shot in a piece that is not truly bored. To consider whether the one wheel be more glad or reverse faster upon the axle-tree than the other, or whether the one wheel stand higher than the other, lest you do shoot wide. To know whether a short piece will outshoote a long piece or not, keeping the length of the mark by the like degrees of the quadrant. To know that leveling with the quadrant towards a hill (the mark standing higher than your platform) you shall shoot short: and shooting into a valley, you do overshoot the mark, but shooting on a level ground you keep the length with the quadrant, and how you ought to lay your piece to make a perfect shot with ●he quadrant at every mark. To know that giving level with an inch rule (which some call the rule of flat) it is erroneous in shooting in ●eeces of contrary length, as also at several marks: observing one method. To learn to know the distance to the mark, and what distance your piece will shoot at point blank, or mounted from degree to degree (which is the best rule to snoote by. To know whether the carriage or stock of your piece have her due length or not, and whether the piece be truly placed therein or not. To consider that in shooting diverse pieces from one platform, to discharge that piece which stands to the ley wards first, and to set your match or fire ever on the ley side, and your powder on the wind hand. To know the true order in mixing and making all kind of fireworks, wet and dry. To know the height and weight of all pieces of Ordinance, and whether the same lie straight in the carriage or not. To know the height and weight of all bullets of like metal, and the circumference thereof: and what proportion a bullet of one metal beareth to the like or unlike bullet of a contrary metal. To know how much Serpentine or corn powder is requisite to charge any piece of Artillery. To know what necessaries belongeth to any piece of Ordinance, being in service by land or sea; as ladles, sponges, handspikes, ropes, coins, etc. and what labourers should attend the same. To know likewise what men, horses, or oxen, is able to draw any piece of Ordinance in service, or on the sudden. To be circumspect of lighted matches and candles etc. for fear of powder, being in sea-service: and to keep a perfect register of every thing pertaining to your Ordinance, both what you have present, and what you have spent, to keep your Ordinance dry within, and to have in readiness all kind of serviceable fireworks, which fireworks ought to be made either in the boat or on land, but not in the ship for fear of had I wist. To know the use of all Geometrical instruments belonging to the profession of a Gunner, as also to have some sight in Arithmetic and Geometry, thereby to shoot at all randoms, and how to manage and handle all engines, for the mounting or dismounting of any piece of ordinance, in or out her carriage, etc. To know that every piece ought to be as thick of metal in every part from the lowest part of the concave at the breech, to that part of the chamber that holds the powder, as the bullet due to that piece is in height. A breviary of certain secrets in the Art of Gunnery. A bullet violently driven out of any piece of Ordinance by the force of the powder, flieth swiftest and straightest from the mouth, till it be past ½ the distance of the level range. The great noise or roar that the piece makes in delivering the bullet (or discharged without bullet) ariseth between the air within the piece, violently driven out into the open air by the force of the fire (the Petre or Master being resolved into a windy exhalation.) And according to the quantity of the fire and air, bursting out of the piece, so is the crack more or less. Any bullet shot out of a piece lying level, doth fly more heavily, and worketh less effect in piercing an object, then when the piece is elevated at any degree or degrees of the random. A heavy bullet violently moving pierceth sorer than a lighter bullet, having the like motion. A bullet of lead shall work as great effect against an object, as the like bullet of iron, having the like motion, by reason of his overplus of weight. A bullet shot out of any piece of Artillery, will pierce more against any thing standing firm, then against a movable object, and shot at an object a reasonable distance from the piece, will pierce more effectually, then shot at the same nearer hand. Every bullet doth make a long or short range, according to the elevation of the piece out of which it is shot. A bullet flieth ever furthest in his straight motion (or in an insensible straight line) the higher that the piece is elevated at the mouth. Any piece discharged twice with one and the self like quantity of powder, wad, and bullet, having one and the self like proportion in ramming and wadding, and shot at one like degree of random, the piece of like temper at either shot shall make like ranges, but the said piece discharged as afore, but not of like temper, shall make several grazes. Two pieces in all respects equal, save only that the one is something longer than the other, discharged with one like quantity in powder and bullet, shall make several grazes, according to the length of the cylinder of the piece, the longer shall outshoote the shorter. Two pieces in all respects equal, save only in length, discharged at a mark of equal distance from each piece, and being within the range of both pieces, the bullet shot out of the shorter piece, shall graze or beat the mark, before the bullet shot out of the longer piece. Two pieces proportional in all respects, being discharged with one like quantity and kind of powder, but differing in bullet, as the one iron: the other lead, and both bullets of like height, shall make several ranges, the iron bullet shall outfly the leaden bullet, but discharged with a bullet of metal, and afterwards with a like bullet made of wood, observing one and the like quantity in powder at every shot, the bullet of wood shall not fly so far as the like bullet of metal. A piece any whit elevated at the mouth, will shoot further in an insensible state line, then lying level: and by how much more any bullet is driven more swifter through the air, by so much it is made the more lighter in the moving or drift thereof. Two pieces a like in every respect, shot with one like bullet, but different quantity of powder, shall make several ranges. Also the said pieces and bullets equal in all respects, and the powder also in quantity equal, saving that the mixtures of the said powder is not alike, shall make several ranges. One piece discharged diverse times with one like bullet, first with the quarter of the weight of the bullet in powder, after with half the weight, thirdly with ⅔ parts of the weight, and lastly with the whole weight of the bullet in corn powder, and the ranges differing at point blank noted, the ranges at the utmost random differing, shall be proportional, one method in charging, etc. being observed. To every piece of Ordinance, according to the proportion of the diameter, length of the cylinder, and weight of the bullet belonging thereto, there is a due quantity of powder to be allowed, so that charging the piece with more or less than the said due proportion, shall rather hinder then further the bullet in his furthest range. By how much the metal of any piece is made hotter by often shooting, than it was before you made the first shot, by so much is the concave or bore of the piece made more attractive, the metal more dulled and the piece worketh less effect than in the beginning. All pieces in whose metal is mingled most tin, lead, or copper, is more attractive a great deal then those pieces in whom is put most bel-mettall. A brass piece made hot with often shooting, is more apt to break then when it is cold; and any piece of Artillery is more apt to break at the first or second shot in a hard frost being cold, then made hot with often shooting. Any piece of Ordinance discharged, having her full charge in powder, will range and pierce further, then wanting any part thereof; and having a little quantity more than her due charge in powder, will overshoot the other, but it will danger the piece; but doubling the weight of the bullet in powder, shall shoot less ground than having a mean proportional charge in powder (to wit between ⅔ parts and the whole weight of the bullet) for that the cylinder of the piece is too much choked, and the bullet driven out into the open air before the powder be all fired. Every piece of Artillery ought to have her convenient length and weight of metal, according to the proportion of the diameter or bore of the same, and being made longer or shorter than her said due length, will rather hinder then further her utmost range. Any piece of Ordinance made hot through much shooting, will neither range so far, nor pierce so deep, as being temperately cold. No piece of Artillery can shoot a bullet to range still in a perfect straight line, except you shoot the same either directly upright towards the zeneth of the sky, or else directly plum down towards the centre of the earth. The right line of the utmost random in all pieces, is more than the right line of the level range; and the right line of the utmost range, is not so much as the right line of 90 degrees. The utmost range in all sorts of pieces, is not at just 45 degrees of random, as Tartallia and diverse others do affirm, but shooting with the wind in a quiet or calm day, is at or about 45 degrees, but the wind against, or on any side, or rough, or the air thick, etc. will range as far at or about 40 degrees. Two pieces in all respects equal save only in length, discharged with a like quantity in powder, wad, & bullet, and shot at a mark within the reach of both pieces, mounted at like degrees of random with the quadrant, the shorter piece shall outshoote the longer. The right lines made by any 2 pieces at one degree of random discharged, are proportional to the ranges of their bullets at the same degrees of random, and the right lines made by any 2 pieces at any random, are proportional to their utmost ranges. Any piece of Ordinance first discharged with the whole weight of the bullet in Serpentine powder, & after discharged with ½ the weight of her bullet, in such corn powder as shall cause the piece to range the same ground: and lastly discharged with half the quantity of either sort of powder, the second ranges shall not be equal, although the manner of charging and temper of the piece be all alike. Three pieces in all respects equal, save every one exceeds other in like proportion in length, the utmost ranges of their bullets shall not be alike proportional, although the form of charging be uniform and alike. A piece twice charged, first with an iron bullet fit for the same piece, and after with a leaden bullet of the like weight, but differing in height, and with one and the like quantity in powder and wad, at either time the iron bullet shall outfly the leaden bullet. A piece discharged first with an iron bullet, and after with a leaden bullet of like height, and at either time discharged with the weight of the bullet in Serpentine powder, shall make unequal ranges. A piece twice discharged at like degree of random, first with an iron, and then with a leaden bullet, and after discharged with any other quantity of powder, the ranges of the bullets shall not retain the same proportion. If 2 pieces of one length but differing in boar, the one discharged with an iron, the other with a leaden bullet at one like random, having the weight of either bullet in course powder, do range both alike ground, and the said pieces after discharged with half the weight of their bullets, of the same or any other powder, shall not range one like distance of ground. Two pieces of one metal and length, but of different bullets equally mounted, discharged with any like quantity of one powder, shall not range justly one distance of ground. The proportion of the different ranges, that iron and leaden bullets make, being found by experience in any one piece of Ordinance, the same proportion will not hold in all other pieces of Ordinance of contrary length, that shoots the same like bullet. Any piece of Ordinance being thicker of metal on the one side then on the other, discharged at a mark, will cast the bullet towards that side, that is thickest of metal. Two pieces of contrary length, but of like diameter, having both one like charge, being shot off at a mark within the reach of both pieces giving level with an inch rule, at one like height of the rule, shall make several grazes, the shorter piece shall outshoote the longer. Any piece of Ordinance will convey the bullet more ground, her mouth elevated at 18 or 20 degrees, then from the said grades to the best of the random, although there be 7 degrees vantage in the latter. Any piece of Ordinance having her due loading will convey the bullet more than five times the distance of her level range. A Table showing the contents of this book. A Table of the deminite parts used in mensurations. 1. A table showing how to weigh any great quantity with few weights. 2. How to extract the cubic radix or root of any number, and how to find a true denominator to the cubic remain, and how to prove if you work right or not. 4.5 Theorems, showing the proportion between a bullet of one metal, to a bullet of contrary metal, and between the diameter and circumference thereof, etc. 8. How by knowing the true weight of any bullet, and diameter of the piece due for the same, to find the weight of any other bullet of like metal belonging to a contrary piece of Ordinance. 8. How by the known weight of any small bullet, you may find out the weight of a greater, and how to prove if you work right or not. 9 By knowing the weight of any bullet, whose diameter containeth both whole numbers and broken, how to find the weight of any other of like metal. 10 By knowing the diameter height and weight of an iron bullet, to find the height and weight of a bullet of marble stone: or chose, by knowing the height and weight of a bullet of marble stone, to find the weight of the like bullet of iron. 11. By knowing the weight and diameter of an iron bullet, to find the height and weight of a leaden bullet of the same proportion: or chose, by knowing the weight of a leaden bullet, to find the weight of an iron bullet of like height. 12. To find out the weight of any bullet made of marble stone, by knowing the weight of the like bullet of lead, or else by knowing the weight of any leaden bullet, to find out the weight of a bullet of marble of like diameter. 12. To find out the circumference of any bullet or round body, etc. 13. By knowing the circumference of any bullet, how to find out the diameter thereof. 14. To find the solid content of any bullet or globe. 15. To find the superficial content of any bullet, etc. 15. To find out the circumference of any circular body diverse ways. 16. How to find the superficial content of any round body, as bullet or globe diverse ways. 17. How to find the crassitude or solid content of any bullet, etc. diverse ways. 18. By knowing the diameter and weight of any bullet, etc. to find the diameter of another of like metal, that is twice the weight of the first. 18. How you may diverse ways Geometrically find out the weight of any unknown bullet, that is double the weight of a known bullet, and how to prove the same conclusions by numbers. 19 20. By knowing the superficial content of the flat or plain of any circle, to find out the superficial content of another, that is twice the diameter of the first. 23. By knowing the weight and height of any one bullet, to find out the true weight of another that is twice the height of the former. 24. How you may Arithmetically find the diameter or height of any bullet, and to know how much any one bullet is higher than another, without any callapers. 25. A table showing the weight of all iron bullets, from the Fawconet to the Cannon, in Haberdepoize weight. 26. How you may Arithmetically know the true breadth of the plate of any ladle due to any piece of Ordinance, by knowing the diameter of the bullet fit for the piece. 27. How to make a ladle for a chamber-bored piece. 28. To find out the height of the diameter of the chamber in any chamber-bored Cannon, or other piece: and how to find out the thickness of metal, round about the chamber thereof. 29. How you may Arithmetically know whether the carriage for your piece be truly made or not, and how the carriage for any piece of Ordinance ought to be made. 30. By knowing the weight of any one piece of Ordinance, to find the weight of any other. 31. How by Arithmetic skill you may know how much of every kind of metal is in any brass piece of Ordinance. 33. How to know how far any piece of great Artillery will convey her bullet at the best of the random, by knowing the utmost range and point blank of another piece, and by the same rule how you may know how far any great piece will range at point blank and utmost random. 35. To know how much a bullet of iron will fly further than the like bullet of lead, being discharged the one after the other out of any great piece, with one like quantity in powder. 36. By knowing how much powder is sufficient to charge any one piece of Ordinance, to know how much of the same powder will charge any other piece of Ordinance. 37. By knowing how much Serpentine powder will charge any piece of Ordinance, to know how much corn powder will do the like: or chose, by knowing how much corn powder will charge any piece, to know how much Serpentine powder will serve. 38. By knowing how far any piece shoots with her due charge of powder, to give a near estimate how far the said piece will shoot, with a charge more or less in powder then the other. 39 How by knowing how much powder a few pieces of Ordinance hath spent, being but a few times discharged, to know how much powder a great number of the like pieces will spend to be often discharged. 40. How to know how much powder every little cask or firken aught to contain, and how many of those casks doth make a Last of powder, and how many shoots any quantity of powder will be for any great piece of Artillery. 41. By knowing how many shoots a firken of powder will make for a Culvering, to know how many shoots a Last of powder will make for a Cannon. 41. To know how many shoots of powder will be in a grand barrel, for any piece of Ordinance. 42. How you may Arithmetically know what proportion of every receipt is to be taken to make perfect good powder: what quantity soever you would make at a time. 43. How by Arithmetic skill you may know how with one and the self like charge in powder and bullet, how much far or short any piece of Ordinance will shoot, in mounting or dismounting her any degree, whereby you may know how far your piece will shoot at any degree of the random, by knowing how far she will reach at the utmost random. 46. By knowing the distance to the mark by the conclusion above, you may know whether your piece will shoot short or over the mark, or you may know how far any mark is from your platform, being within the reach of your piece, only by knowing the distance of the utmost range of your piece, and the degrees she is elevated at. 46. 47. How to make a table of randoms, or go very near to know the true range of the bullet, out of all sorts of great pieces of Artillery, being mounted from degree to degree. 48. How you may Arithmetically know how much wide, over, or short any piece will shoot from the mark, by knowing the distance to the mark, and how your piece is laid to shoot at the said mark. 51. How to lay your piece to make a straight shot at any mark. 51. In shooting at any mark within point blank, not disparting your piece, to know how far the bullet will fly over the said mark, only by knowing the distance to the mark. 53. How to lay your piece point blank without iustrument. 54. How you may Arithmetically dispart any great piece of Artillery diverse ways. 54. How by Arithmetical skill you may mount any great piece by an inch rule to 10 degrees of the quadrant, if you want a quadrant or other instrument. 55. How you may know the true thickness of metal in any part of any great piece of Ordinance diverse ways. 58. 59 How to make a good shot in a piece that is not truly bored, or to know how much any piece will shoot amiss, that is thicker of metal on the one side then on the other, if you know the distance to the mark: & how to remedy your piece, being thicker of metal in one part then another, to make her shoot straight. 60. To know the different force of any 2 like pieces of Ordinance planted against an object, the one being further off from the said object then the other. 62. How you may having diverse kinds of Ordinance to batter the walls of any Town or Castle, etc. tell presently how much powder will load all those Ordinance, one or many times. 65. How you may know the true weight of any number of shot for several pieces of Ordinance, how many soever they be, and how many Tun weight they do all weigh. 70. How any Gunner or gunfounder may by Arithmetic skill know whether the trunions of any piece be rightly placed on the piece or not. 72. How you may know what empty cask is to be provided to boy or carry over any piece of Ordinance over any river, if boats or other provision cannot be gotten. 73. How to know the true time that any quantity of gun-match, being fired shall burn to do an exploit, at any time desired. 76. How by Arithmetic skill you may know what number of men, horses, or oxen, is sufficient to draw any great piece of Artillery, and how much every one draweth, so as they all do their endeavour. 77. To know how many hundredth of Haberdepoize weight any piece of Ordinance or other gross weight containeth. 81. How you may proportionally prove all sorts of pieces of Artillery for service, whether they will hold or not. 82. To know how much one coyler rope is more than another, for to draw any great piece of Ordinance. 83. By knowing the weight of a fathom of one coyler rope, to know the weight of a fathom of any other. 84. By knowing the quantity or compass of any small rope, to find out the same in another that is many times that bigness, and how to find out the weight of a whole coyler rope, for the draft of any piece of Ordinance. 84. To find out the superficial content of the hollow concavity of any piece. 85. To find out the crassitude or solid content of the cylinder or concavity of any piece, and how much the same containeth in square measure. 86. How you may know how much any piece of Ordinance is taper-bored by Arithmetic skill, or whether any great piece of Ordinance be taper-bored or not. 87. A table wherein you may know the names of all pieces of Artillery, their height and weight, and thickness of metal in any part of them, and what men, horses, or oxen, is sufficient to draw the same, and the height, weight, and compass of the bullet belonging to every piece: and how much powder will charge every of the said pieces, and the length and breadth of the ladle fit for any piece, and how thick, broad, long, or deep, the carriage of every piece should be, and how long every coyler rope should be, for the draft of any great piece of Ordinance. 87. Conclusions for shooting in mortar pieces. 87. To know how much further or shorter any mortar piece will shoot at the mount or dismount of one or many degrees. 88 To know very near how far from your piece the bullet shall light, the mortar piece raised at what degree you think good. 89. Notes to be learned concerning mortar pieces. 89. To know how much the earth and turf that is digged & thrown out of any ditch, shall make a defensible rampart or brickwall at the brim of the said ditch, making the same to any proportion assigned for the better defence of the Ordinance in time of service. 93. Certain brief observations of certain principals of the Art of Gunnery, to be known of every gunner: with a breviary of certain secrets of the same Art, very necessary for all professors of the Art of Gunnery. 94. FINIS.