OF THE ART OF GREAT ARTILLERY, Viz. THE EXPLANATION of the most excellent and necessary Definitions, and Questions, pronounced and propounded, by that rare Soldier and Mathematician, Thomas Digges Esquire; and by him published, in his Stratiaticos, and Pantometria, concerning great Ordinance, and his Theorems thereupon. Together, With certain Expositions, and answers thereunto adjoined: Written by Robert Norton Gunner. And by him Dedicated, to the Worshipful john Reinolds Esquire, Master Gunner of England. LONDON, Printed by Edw: Allde, for john Tap, and are to be sold at his Shop, at the corner of Saint Magnus' Church. 1624. ¶ To the Worshipful John Reinolds Esquire, Master Gunner of England: Robert Norton Gunner, wisheth all health and happiness. GOod Sir, (as your sometimes Scholar, & your now substitute) give me leave, to show some part of my dutiful thankfulness, for your many loving favours received; your kind instructions, your free helping hand to the beginning of my encouragement, by your loving Certificate (to the right Honourable the Lord Carew Mr. of his Majesty's Ordinance) of my sufficiency to perform the place and Office of a Gunner, for his Majesty's service: I therefore have presumed to demonstrate a part of my duty herein, the rather for the loving respect you have ever showed unto me, and to all others under your Command, that endeavour to become serviceable for our King, and Country, by the diligent practice in the excellent Art of great Artillery: It being my chance of late Sir, to turn over Mr. Digges his Books, entitled Stratiaticos, and Pantometria, to light upon certain dificill Definition, obscure Theorems, and some subtle questions, concerning the use of great Ordinance, which as Mr. Diggs there saith, though he long since publicly propounded them, that none hath yet undertaken to answer any one of them: wherefore in his last edition with short Marginal notes, yes and no, himself hath darkly resolved some of them, as hereafter in the Margin appeareth, whereof I have undertaken to make a more plain Exposition of them & the rest, aswell for mine own, as for others furtherance. And whereas he hath there coated certain published Errors in this Science, I have thought fit to join them together, with some other Errors that I have also espied in other Authors writing of this Art: All which I have presumed to publish under your judicious Patronage, hoping that my willing pains shall neither incur disgrace, nor displease, but rather that these my first Fruits will be accepted with such love as I truly offer them, whereby I shall be encouraged (with your good leave) hereafter to proceed further with my Treatise of the Art of Artillery, wherein I persuade myself, that the most necessary particulars belonging to the Gunners Art, are more acutely showed, then in any other Treatise in any Language yet extant: And so I shall endeavour to deserve the continuance of your love and favour, and will rest at his Lordships, and your Worship's command, both whom, I pray God ever to bless. Robert Norton. The Preface. TO commend the Art and Practice of Artillery, it were utterly needless, for that it is apparent, that all other Sciences are therein used, as in their convenient and proper Medium, and grounded on supreme virtue, seeing that without it no Kingdom or Commonweal, can either be or continue in Peace, or defend itself, nor offend their Enemies; it being the powerful Regent in our Modern Millitia: for that it destroyeth Enemies, depresseth Tyrants, chasteneth Rebels, increaseth Dominions, and is the common make-peace, and Conseruor of tranquillity in Kingdoms and Commonweals. It consisteth of many ingenious Theorical and Practical parts, wherein Knowledge must be the Pilot of Action, or else the action will prove but sillily simple. The Theoric in this (as in all other Sciences) being the fundamental groundwork of the practice part thereof; Therefore Knowing from Doing, must be no more separated than Letters from Arms: this made the Hebrew, Egyptian, and Persian Soldiers, to be aswell Priests as Gentlemen; and the Grecians to be both Philosophers and Captains, and the Romans' to be both Soldiers and Scholars; whereby each of them got for their Nation a world of honour. M. Diggs said very well, that it fares in this Art, as in Soldiary, and in Navigation; for as many a private Soldier, whose brains will only reach how to stand Sentinel, and to March to his Guard, yet takes himself to be a perfect Soldier, but if once his wit be capable to become Corporal of the field, or Sargeant Maior, than he comes to see his former ignorance and wants. Even so the Common Sailor, if he can but say his Compass, furl a Sail, and take turn at Helm and Lead, doth less know his ignorance, than such a Master or Pilot at hath sailed a ship by his Chart, Compass and Art, round the world; And so likewise in this Art, many silly Gunners that never sounded the deep Channel of this Art, will not stick to say, they know enough, and scorn to learn more, when they God knows understand not the first principles of good Art or practice; but if by chance, or mischance, they made a good shot once, though without understanding of the true cause of it, it must uphold their reputation for ever, and be sufficient to make Fools proud. This may chance to be returned me home, yet I care not, for I hope to escape the name of a Coward (though my purse be cudgeled) for that I have taken up those weapons, that with a challenge were laid down so long ago; But let them that are envious commendor come mend it, and so I end. Theirs that love Art and Practice, Robert Norton. Certain Definitions taken out of Mr. Digges his Pyrotechnie, and published in his Pantometria concerning Great Ordinance. Explained by R. N. Gunner. Mr. Digges. FOR as much as by the hollow Cylinder or Trunk of the Piece, the violence of all shot of Great Artillery is not only directed, but also increased, I call that hollow Cylinder of the Piece, her Soul. Mr. Norton. This Soul is usually termed by the most experienced Gunners, the Concave Cylinder, or Bore of the Piece: And when she is loaded, so much thereof as containeth her Charge, is called her Chamber, or Charged Cylinder, whether she be Equally Bored, or Camber Bored; and the rest that is unfilled, is called the vacant Cylinder of her Boar. M. D. The Mettaline substance of the Piece, of what shape, kind, or proportion soever, I call The Body of the Piece. M. N. The several parts of the Mettallyne substance of each Piece, are distinctly known to Gunners, by diverse and several Names: As the thickest of her Mettle at the most eminent ring of her Breech is called her Basering or Carnooze, the whole length of her shaft, is called her Chase; and those parts of Mettle that M. Digges elsewhere calls her Ears, are by Gunners, called her Trunnions; and before her Trunnions that part of her chase towards her Neck (which is near the Mouth) is called her Coronice: Her foremost extreme of her Chase, is called her Mouth or Muzzle ring, etc. M. D. The Soul in all principal Pieces of Battery is ever a perfect uniform Cylinder comprehended in a Circular Column, and two equal Circles, whereof the one I call the Head the other the Base. M. N. The Soul or Concave aught indeed to be a perfect uniform Cylinder, except in Chambered Cambered Taper and Belbored Pieces, but by reason of ill and careless Founding, few Pieces come to that perfection; neither do scant any of their Souls lie exactly in the midst of their Metals, which (unless for them the Dispart and Dispartline be arteficially, and accordingly varied) causeth them to shoot awry, or over or under, or traverse. M. D. The direct line, which by Mathematical imagination, doth conjoin the Centres of the two Circles, is the perfect and true direction of all Shott made out of great Ordinance, I term the Axis of that pieces Soul. M. N. Which may very properly be so called, but it is usually termed the Axis of her Boar, or the middle line of her Concave Cylinder, those two Circles at the extremes of that line doth with their planes make right and Orthogonall Angles therewith. M.D. A mark is said to lie within Point blank, when the Piece being directed with her convenient Bullet and Charge, is able to strike and reach that mark. M. N. True it is, it hath improperly been called pointblank, as fare as any Piece conveyeth her Shott in a right line, nevertheless the Piece be mounted to the third or fourth point; But then if you ask how much that Piece shooteth at poynt-blanke, unless you also assign the Mount, it cannot be certainly answered, And so consequently there is no certainty of that pointblank for any Piece, because then every Piece may have a thousand such several Point-blanks. Wherefore I suppose it were more proper to call that only distance Poynt-blanke, which the Piece conveyeth her Shott in a right or insenceable crooked line; the Axis of her Boar lying level with the Horizon, that is, she being neither mounted nor embased to any point, or minute of a point, above or under the Level, that being the only Blank point, that is without numeration, as being the beginning, both of elevation and depression. M. Diggs. A mark is said to lie within the Mettle of the Piece, being directed not by the Axis of the Soul, but by the Coronice and uppermost ring of her Head, and coile, is able to reach the Mark. M.N. This is by Gunners termed shooting by the Mettle of the Piece, (or Mira commune) whereby it is meant, that the Large line (which is that line which passeth upon the uppermost of the Pieces Mettle, from the Breech to the Mouth, vertically over the Axis of the Soul of the Piece) be directed to the mark, and is able to reach the same. M. D. The difference of these two Ranges, I call the Difference of the level Range of the Soul and Body of the Piece. M. N. Which by Gunners is called the difference between the shooting a Piece by her due dispart, and by the uppermost of her Mettle (or most eminent, the Base and Muzle-rings, at her Breech and mouth) and which (upon the level) is near about twice so much ground, as with a dispart for her Range it is: But this holds not so above the level, in or near double, for it only elevates the piece about five or six degrees (in some pieces more, in some less) to shoot by the Mettle more than by the dispart increasing under the mount of the best random, but less being mounted above the same, so much as 4. or 5. or 6. deg. would increase or decrease the Range for the Elevations assigned. M. D. The Axis of the Body of any Piece I term that straight line, which passeth between the Centres of the two outermost Circles at the coile and Head of the Piece, which in all Pieces, truly founded, is also the very same with the Axis of the Soul. M. N. Gunners and Gun-founders call the greatest ring at the Breech (the Cornooze or Basering of the Piece (which Mr. Diggs terms the outermost Circle at the coile, the other which he calleth the outermost Circle at the Head, they term the greatest ring at the Mouth or the Muzle-ring: But that Axis cannot be the very same with the Axis of the Concave Cylinder, in pieces that are not truly Founded. M. D. If the two Axis differ, the Piece is false founded, and then they are either parallels, or make an Angle: if they be parallels, I term it the Distance of the Axes of the Body and Soul. M. N. The definition thereof is very significant and proper, for if in the Casting or Founding of the Piece, the Axis of the Nowell that maketh the hollow Cylinder swerve paralelly aside, it will make the Piece thicker of Mettle all along upon one side, than it is upon the opposite side thereof. M.D. If they be not parallel, their Angles of variation are considered two ways; That is to say, in Altitude and Latitude; and those Angles accordingly named the Anomaly: Angles of Altitude or Latitude of those Pieces. M. N. Those Angles of variation may be three ways considered, that is, in Altitude only, in Latitude only, and in Both jointly, each of which will make the Piece shoot amiss, either over and aside, under and aside, or directly over or under the mark; and so for remedy thereof, the Piece must be differently disparted, and accordingly alter her large line upon the top of the Mettle, otherwise than it should have been, if her Concave Cylinder, had lain directrectly in the middle of her Mettle. M. D. The first part of the violent course (of a shot thrown out of any piece of Ordinance) of Gunners termed Point blank reach, I call the direct line of the Bullets circuit. M.N. Most understanding Gunners, leave that improper phrase or Name of Poynt-blanke, although it be very ancient (because as I said of the uncertainty therein, for there may be for one Piece 1000 several poynt-blanks, if that distance which the Piece conveyeth her shot in a straight line, should be Point-blank, it carrying so at every several degree or point of mounting, a several distance; namely four or five times so fare upon the best Random, as upon the Level: Besides the level right Range, might most properly be called the poynt-blanke only, because it is the Blank point, and the beginning of Elevation and depression: each other point in either, hath its proper denomination and Numeration, as the first, second, third, etc. to the twelfth point, which lastly is perpendicular, either above or under the Horizon: And in place of Pointblank therefore, they call that distance the straight Line or right Range, proper for the mounting or embasing assigned. M. D. The second part being a Curve Circuit, beginning at the foresaid declination from the Axis, ascending to the highest Altitude above the Horizon, and ending at a like Altitude to his beginning, I term for distinction sake his Middle Helicall or conical Arck. M. N. Gunners call it the Circuit of the Crooked range or Circular motion of the Shott, or the Mixed or Compound motion thereof; It perticipating of the violent, and the Natural motions mixed together, beginning at the farther end of the straight Line or Right Range, and ending at the first graze of the Shott, and being peculiarly differing in every mount or Random from all the rest; And therefore in two or more several Pieces, each like Random in the one, is proportional to the others like Randon: That is, having both the Randons' of one Piece, and one like Randon of another Piece given. The rule of proportion will produce the Range for the other like Randon for the mounting sought. M. D. The altitudes of any Bullets Circuit, I call that Line perpendicular, which by imagination Mathematical, falleth from the Bullet, at his very highest of his Mount, perpendicularly down to the plain horizontal: Which line of Altitude, coupled together with the right lines from the Top and Foot, Concurring at the Centre of the Pieces Circular Base, doth make a right angled Triangle. M.N. This and the two next Definitions, although they need no expositions, yet they have good uses in the Theorems following. M.D. The horizontal line of that Triangle I call the Base. Master Diggs. The other slope Line is the hypothenusal. M. D. The Pieces direct line of that Circuit which is always above the hypothenusal, (for distinction sake) I call the Line diagonal, For that there are several of these diagonal lines to all Angles of Randon, and together with the Horizantall line do comprehend the Angle of the Mount. M. N. In my book called the Art of Artillery, I term that hypothenusal Line, the Secant Range of the Piece, because the same properly representeth as in the Doctrine of right lined Triangles the Secant of the Mount: As the aforesaid perpendicular doth the Tangent, and the horizontal the Radius thereof; which I so do, the rather for the more easy and certain Calculations therein requisite. M. D. The Pieces horizontal level Range, I term the distance between the Piece and the first graze of the Bullet, when the Piece upon her discharge, lieth level upon her Carriage, not mounted upon any lofty platform, but such as lieth, even with the true horizontal plane, whereon the Bullet must play. M. N. This is amongst Gunners most usually termed the Level Range, which I call the Level-dead range, and so look how fare the shot goeth directly, (without any sensible declining) is usually called the Level right range. M. D. All other Ranges made on any horizontal plane, by the Bullet, when the Piece is mounted, at any several Randons', I term the horizontal Range. M. N. They are by Gunners called Randons' or Ranges, and by me in the said Book, termed Dead-ranges. M. D. And because every Piece hath some certain grave, of the Quadrant whereunto mounted, she maketh her uttermost horizontal Range, in such sort, that if ye mount the Piece higher, the Bullet shall fly a shorter distance; and the horizontal Range return less and less again: That point of the uttermost Randon horizontal, I term the Tropic point or grave. M. N Which Range is called by Gunners the best of the Random, and by me the greatest dead Range being the furthest distance that the same Piece can possibly convey her shot: and that hath been many years supposed to be at the mount of 45. degrees (as the mean or middle between the Level, and 90. degrees) But now it is found to be rather at the mean or middle degree between the Level and the degree of mounting, that in decreasing, conveyeth the shot just the distance of the Level Range, which is about 82. degrees, so that above 40. degrees above the Horizon is the best of the Random if accidents be excepted. M. D. And that which serves for the discovery of the different violence of all Pieces right Lines or Right Ranges howsoever Mounted, (By me called the lines Direct of the Bullets Circuit, and Lynes diagonal, I call for distinction sake The Theoric of Lynes Dyagonall. M. N. For the Theoric, all Dyagonals (or more properly termed Right Ranges) for each several degree of mount of the Quadrant from 1. degree to 10. and so for each 10. degrees to 90. I have thought good to adjoin an abreviation of one of my Tables A Table of Right Ranges. Gr. paces Gr. paces. Level 0 192 20 524 1 209 30 695 2 227 40 855 3 244 45 930 4 261 50 1005 5 278 60 1140 6 285 70 1220 7 302 80 1300 8 320 90 1350 9 337 10 354 out of my Book called the Art of Artillery, (which hereafter I purpose to publish the use of the said Table of Right Ranges, is thus, First, if you know the right range of the Piece upon the Level, or upon any degree mounted, you may thereby (with this Table) know upon every other degree of mounting the same Piece, how far she will convey her shot in a straight Line, being like loaded, and having like accidents. As suppose for example, your Piece upon the Level shoots 250. paces upon the straight Line; and you desire to know how fare she will shoot in a straight Line being mounted unto 40. degrees above the horizontal plane; say by the Rule of three: if 192. (the number of the Table for the Level) give 250. your Pieces level right Range, what shall 855. (the number of this Table for 40. degrees) give? and you shall find 1113. (fere) the fourth proportional, which is the Number of paces she will convey her shot, being so mounted to 40. degrees elevation in the straight Line or right Range sought. M. D. The other that discovereth the variety of Ranges of all Pieces, at all degrees of Randon, I call the Theoric or Scale of Randons'. M. N. For the Theoric of Randons' I have also here epitomised another Table, out of my said Book of the Art of Artillery, whereby, for the six first points of the Gunners Quadrant, you having the dead or horizontal Range of one shot made out of any Piece, whether it be of the nature of a Culvering (which is between 30. and 40. Dyamitres of her Boar in length by the Range of numbers set against the Letter S.) or whether it be of the quality of a Cannon (that is between 18. and 24 Diametres of her Boar in length, by the Rank of Numbers set against the Letter C.) you may having one shot by the same, and the rule of Proportion, or rule of 3. know her dead or Horizantall Range for any other point of mounting sought. As for example: suppose for a Culvering that shooteth level 260. paces: I would know how fare she will convey her shot upon a horizontal plane, she being mounted to the third point (that is to 22. gr. 4.) Say as 192. (the number against S.) for the level range, in the Table giveth 2032. the number under the third point, what will 260. give? work by the Rule of three, and you shall find 2752. for the number of paces that she will convey her shot upon a horizontal plane being so mounted. A Table of Dead or horizontal ranges. 0 1 2 3 4 5 6 points. S. 192 985 1623. 2032 2185 2281 2300 paces. C. 201 958 1600 1983 2135 2232 2250 paces. M.D. The other Composed of All, and by Conference of all their parts together, framing a Theoric of perfection, differing in all planes horizontal, or varying for all kind of Pieces, and Bullets whatsoever; Their Ranges and Randons': The Altitude of their Circuits, together with their Lines diagonal and hypothenusal shall be named, The Theoric of Artillery general. M. N. I have not here run through all those curious varieties, because they require many experiments, much Art, and large discourse & grounded upon Mathematical and practical demonstration: But, to know how much of the horizontal Line, lieth under the right Range or straight Line of the shot (being found as before) for the Elevation given, it is thus to be gotten: As the Radius is to the right sign of the Compliment of the Elevation given, So is the right Range to the Level distance under it. And for the Level under the crooked Range, that is found by substracting the distance under the right Range, out of the whole distance of the horizontal or dead Range sound as aforesaid, and the remainder shall be the level Distance sought. Lastly, for the Circuits of the crooked Ranges, Mr. Nicholas Tartaglia in the fourth, fifth, and sixth Propositions of his second Book of his Sciencia Nova (which I purpose one day to Translate, if God give life, and no other man prevent me) doth demonstrate, that the crooked Range of a shot made above the Level, maketh more than a Quadrant, and upon the Level a just Quadrant, and under the Level, less than a Quadrant. But for all his subtle Demonstrations, it is not exactly so found by experience. For it is neither perfectly Circular elipsicall parabolical, Hyperbolical, nor conical, but merely Helicall, or Helisphericall, according to the proper Levelling, mounting, or embasing of the Piece assigned, which let now suffice. Mr. Digges his Theorems, concerning the new Science of great Ordinance, resolving the most part of his Artillery Questions in Stratiaticos proponed. The first Theorem. THere are three chief most material and efficient causes of the greater violence of any shot made out of great Ordinance, viz. the Powder, the Piece, and the weight of the Bullet. The 2. Theorem. Powder is compounded of three Principles or Elements, Saltpetre, Sulphur, and Cole, whereof Saltpetre is it that gives the chiefest violence. The 3. Theorem. Albeit Saltpetre be indeed the only or most material cause of the violence, And that Powder commonly found most forcible, that is richest of Petre, yet is there a certain proportion of Perfection of these three Components. And that in such sort, as if you add more or less Petre, the violence shall abate. The 4. Theorem. Although Powder be also the most efficient cause of the force and violence of any shot, yet is there such a proportional charge of Powder to be found for every several Piece, in regard of the proportion of her charged and vacant Cylinders, as giving more or less, than the same proportional charges, it shall diminish, and not increase the violence of the shot. The 5. Theorem. If any two Bullets of equal quantity, but unequal weight, be let fall from any lofty place to the Horizon, the more weighty shall ever fall the more swiftly: albeit not proportionally to their weight; which Axiom is indeed erroneous, albeit a great Philosopher hath averred the same. The 6. Theorem. If two equal Bullets of different weight, be shot out of one and the same Piece directly to the Zenith, both Bullets being of massy mettle, and charged with one quantity and kind of Powder, the lighter shall always outfly the heavier. But such kind of Bullets they may be charged with all, as the Heavier shall outfly the Lighter, although they be both discharged with the same Piece, and quantity of the same Powder. The 7. Theorem. There is such a convenient weight to be found of the Bullet, in respect of the Powder and Piece, as the Bullets mettle being either heavier or lighter than that weight, shall rather hinder then further the violence or fare range of the shot. The 8. Theorem. There is such a convenient Proportion to be found of the length of every Piece to his Boar or Bullets Diameter in respect of the Powder, and weight of the Ball, as either increasing or diminishing that proportion it shall abate also, and hinder the violence of the shot. The 9 Theorem. This proportion exactly found in any one Piece, doth not hold in all other, and yet the difference and alteration such, as may be reduced to Rules certain. The 10. Theorem. Besides these three most material causes of violence, the Randons' also and different Mounts of Pieces, cause a great alteration, not only of the fare shooting of all Pieces, but also of their violent Battery. And albeit the different alterations are very intricate and strange, yet have they a Theoric certain. The 11. Theorem. There are also many other Accidental alterations happening by reason of the wind, the thickness or thinness of the Air, the heating or cooling of the Piece, the different manner of charging by Ramming fast or lose the Powder, by close or lose rolling or lying of the Bullet, by the unequal recoil of the Piece, either by reason of the unequality of the Platform or Wheels, or by the uneven lying of the Piece in his Carriage or deformity of the Axtree, with diverse other such like, whereof no rules certain can be prescribed, to reduce these uncertain differences to any certain proportions: but all these are by Practice, Discretion, and judgement to be considered, and uniformly guided and performed in their best perfection. The 12. Theorem. Any Piece mounted 90. grades above the Horizon, throweth his Bullet most violently immediately after the discharge, and then the motion groweth slower, till the Bullet be come to his utmost Altitude, and then by Perpendicular falling, increaseth by little and little, his swiftness again, even till it come to the Horizon. But at all other Randons', it falleth not so out. The 13. Theorem. Albeit in the subtlety of Geometrical Demonstration, no part of the Bullets violent motion, can be truly averred a right or direct line, save only the Perpendicular: yet in these experiments Mechanical, That first part of the violent motion (I mean so fare as the Piece is said to carry Pointblank) being so near the direct, is, and may well be termed the direct line. As all water levels are accounted in all Mechanical operations, the Perfectest levels and directest lines. Albeit the subtlety of Geometrical Demonstration, doth find them not right or direct, but Curve or Circular. The 14. Theorem. When any Piece is mounted directly to the Zenith. Then doth his Motion violent (being in that situation directly opposite to the natural) carry the Bullet in a perfect right line, directly upward, till the force of the violence be spent, and the Natural motion have gotten the victory. And then doth the Natural return the Bullet downward again, by the very same Perpendicular line. And so is the whole motion of the Bullet in this case a very direct Perpendicular to the Horizon. The 15. Theorem. But if any Piece be discharged upon any Angle of Randon, albeit the violent motion contend to carry the Bullet directly by the line diagonal; Yet the Perpendicular motion being not directly opposite, doth though unsensibly, even from the beginning by little and little draw it from that direct and diagonal course. And as the violent doth decay, so doth the natural increase: and of these two right lined motions, is made that mixed Curve Helicall Circuit of the Bullet. The 16. Theorem. Any Piece therefore discharged at any Mount or Randon, first throweth forth her Bullet directly a certain distance, called of Gunners their Pointblank Range, and then it maketh a Curve declining Ark, and after finisheth either in a direct line, or nigh inclining towards it. The 17. Theorem. The further that any Piece shooteth in her direct line, commonly called Pointblank, the deeper also she pierceth in her Battery, if the Bullet be not of substance brickle or frangeable. The 18. Theorem. The more ponderous a Bullet is, the more it shaketh in Battery, albeit it pierce not always so deep as the lighter or letter shot conveniently charged. The 19 Theorem. Any two Pieces of Battery Ordinance, charged with one kind of Bullet, and shot into one Rampire of massy uniform kind of Substance, shall ever make their Profundities of piercing Proportional to their level Ranges horizontal, if they be discharged either level or at one grave of Randon, and at like distance. The 20. Theorem. Any two Pieces of Battery discharged into any Rampire of uniform massy substance, shall ever make their Piercing depths proportional to their lines diagonal, albeit these Pieces be discharged from different Randons', so as they batter at like distance. The 21. Theorem. As Archimedes' line Helicall or spiral, is made by the direct motion of a point carried in a right line, while that right line is Circularly turned as Semidiameter upon his Circles Centre; So is this Artillery Helicall line of the Bullets Circuit created only by two right lined motions becoming more or less Curve according to the difference of their Angles, occasioned by the several Angles of Randon. Whereupon by demonstration Geometrical, a Theoric may be framed, that shall deliver a true and perfect description of those Helicall lines at all Angles made between the Horizon and the Pieces lines diagonal. The 22. Theorem. These direct or diagonal lines, are always longest when the Pieces Axis is directed to the Zenith. And always as the Pieces Axis declineth more and more to the Horizon. So do the diagonal lines grow shorter, and at the level horizontal, shortest of all. The 23. Theorem. These direct lines diagonal, albeit they increase in length at every grave of Random from the Horizon to the Zenith, yet is not their increase uniform or proportional, either to their degrees of Randon or Horizontall Ranges, nor yet to their Circuits or Altitudes, and yet such as may be reduced to a theoric certain. The 24. Theorem. The middle Curve Arkes of the Bullets Circuits, compound of the violent and natural motions of the Bullet, albeit they be indeed mere Helicall, yet have they a very great resemblance of the Arks conical. And in Randons' above 45. they do much resemble the Hyperbole, and in all under the Ellepsis: But exactly they never accord, being indeed spiral mixed and Helicall. The 25. Theorem. Any Piece discharged at any one Random with like Bullets, and several charges of Powder, shall make both their lines diagonal and Curve Circuits of different longitude, but the Curve Arkes shall always be as Parallels, and their Longitudes Proportional to their lines diagonal. The 26. Theorem. The last declining line of the Bullets Circuit, albeit it seems to approach somewhat to the nature of a direct line again, yet is it indeed still Helicall and mixed, so long as there remaineth any part of the motion violent. But after that is clean spent, the rest of his course to the Horizon is direct and Perpendicular, and a perfect right line indeed, which is best discerned in those Grades of Randon, which are between the Zenith and the Mount or Randon Aequorizontall. The 27. Theorem. This declining line doth always make a greater and greater Angle with the Horizon, as you raise the Piece to a greater Mount, till you come to the Mount Equorizantall, about which Point the same declining line becometh Perpendicular before the Bullet fall to the Horizon. The 28. Theorem. The horizontal Ranges in all Pieces mounted from the Horizon toward the Zenith, doth not still increase, but at every grave of Randon are longer, till you come to the Point or mount Tropical, commonly called the utmost Random, which hath been generally thought to be the grave 45. but is not so. And from that Tropical grave upward, the Ranges decrease again till you come to the grave Aequorizontall, so called because the Bullet than falleth a like distance to the level Ranges. The 29. Theorem. This Aequorizontall Grade is as far distant from the Zenith, as that Grade is from the Horizon, which shall cause the Piece to shoot in the horizontal plain a distance equal to his highest Altitude or longest line diagonal. The 30. Theorem. The Mounting of any Piece above his Aequorizontall grave doth still decrease her horizontal Ranges even till it come to the Zenith. But in a Proportion different from any of the former, her Bullet ending every of those Circuits in a direct line Perpendicular. The 31. Theorem. The Gradual increase and decrease of these Ranges horizontal, albeit they are equal in the Quadrant, yet are they neither equal nor proportional in the Horizon, neither the Ranges nor their intervals. Neither compared between themselves, nor yet conferred with the Chords or Sins of their Arks. And yet is there such a kind of Proportional increase and decrease of the proportion of their intervals, as may be reduced to a theoric certain. The 32. Theorem. The Tropical grave commonly called the utmost Random, is not as hath been generally supposed the Medium or Middle between the Horizon and the Zenith, Viz. 45. but rather between the Horizon and the grave Aequorizontall, which will fall out much nigher 50. from the Zenith, and 40. from the Horizon. The 33. Theorem. The highest Altitude of any Bullets Circuit is farthest distant from the Piece, when she is discharged at her utmost Random or point Tropical, and at all other Randons' either above or beneath that Tropical Point: That highest Altitude is ever least distant, and the bases of these Triangles do ever increase to the Random Tropical, and decrease after, even as the horizontal Ranges; but in Proportion more different every one from other. The 34. Theorem. The Altitudes of the Circuits of Randons' do not increase and decrease as their Ranges Reciprocally, but from the Horizon in every grave to the Zenith, do still increase, but yet neither equally nor Proportionally, neither conferred between themselves, neither yet with sins or Chords of their Arkes of Randon. And yet the increase and decrease of their intervals Proportions, such as may be reduced to a theoric certain. The 35. Theorem. The Hypothenusall lines of all these different Circuits carry a mixed proportion of the composition of the Proportions of these Altitudes and bases by addition of their Squares, But are not proportional to the lines diagonal of their corresponding Angles of Randon. The 36. Theorem. Any two Pieces of Ordinance being mounted to any one Grade of Randon shall make their Horizontall Ranges of their Bullets Proportional to the Altitudes of their Circuits. The 37. Theorem. The Ranges horizontal of any two Pieces discharged at one Randon, will be always proportional to their lines diagonal of the same Pieces Circuits. The 38. Theorem. The horizontal level Ranges of any two Pieces of Artillery are ever proportional to the utmost Ranges horizontal of the same Pieces. The 39 Theorem. Any two Pieces whatsoever discharged at one Randon, do ever make their lines diagonal, and lines of Altitude Proportianall howsoever the Proportions of their charges vary. The 40. Theorem. Any two Pieces whatsoever discharged at one grave of Random upon any inclining or declining Plain shall nevertheless make their Ranges proportional to their lines diagonal and Altitudes of those their different Ranges. Albeit the Pieces be charged with a different kind of proportion of Powder and Bullet, so as the shot be made in a fair Calm day, as is in these cases always presupposed, because for such uncertain Accidents there cannot certaine Rules Artificial be prescribed. The 41. Theorem. One Piece discharged at several Randons' under the utmost Random, being a like charged and discharged, and the Piece also of one temper, at both times, shall ever make several Ranges. But if she be discharged at several Randons', the one above the Tropic point, the other under, Then may their Ranges be equal notwithstanding their Randons', Lines diagonal, Altitudes, Bases, and Lines hypothenusal, be all different. The 42. Theorem. When any Piece (being twice discharged at several Randons', the one above, the other beneath the Tropic point) shall make the same or equal Ranges in a horizontal plain. The middle grave between those several Mounts is very nigh the grave of utmost Random: and the Piece Mounted to that middle grave, shall then make very nigh his utmost horizontal Range. The 43. Theorem. The grave of utmost Random or point Tropical of any Piece in a Plain horizontal, shall not be the Tropical grave of that Piece, in a plain declining or inclining, but an other Peculiar to that Angle of Inclination or Declination. The 44. Theorem. Any Piece discharged at his grave of utmost advantage horizontal upon a Plain inclining, shall not make so great a Range as on his plain horizontal: But contrariwise on a Plain descending shall make a farther Range. The 45. Theorem. A Piece discharged first at his due level, and again at his Aequorizontall grave, albeit in the plain horizontal they make equal Ranges, yet in Plains declining shall they not so do, but always the Level Ranges shall ever out-shoot in all declining Plains the Range of that grave Aequorizontall. The 46. Theorem. A Piece discharged at any grave from the Zenith to the grave Aequorizontall, shall always make a greater Range in any Plain inclining or declining, then on the Plain horizontal. The 47. Theorem. In all plains inclining at all Randons' between the horizontal Level and point Tropical, all Pieces shoot farther in their Plains horizontal, then on any Plains inclining, and contrariwise in Plains Declining: But above the Tropic grave not always so, but sometimes, and not always contrary. The 48. Theorem. In any Plain whether it be inclining or declining, if any Piece of Ordinance be discharged being Parallel or Equidistant to that plain, and the first graze or bond noted. If the same Piece be with like charge uniformly charged and discharged at such an high grave of Random as may cause the Bullet Range the former Distance: That middle grave of the Quadrant, which lieth between these two Mounts, shall be very nigh the grave of utmost advantage, for that inclining or declining plain. The which in all plains inclining, will be above the utmost Range horizontal, and in all declining under. The 49. Theorem. In all inclining or declining plains, as the grave Tropike of greatest advantage doth ; So doth also the proportions of their Ranges, at every grave of Randon differ, whether they be accounted from the Zenith, Horizon or Plains, inclining or declining. But yet in such an assured and certain manner as may be reduced to a Theoric perfect. The 50. Theorem. In all Grades of Randons', and in all manner of Pieces, whether the plains be horizontal or vary by Inclination or Declination, the diagonal Lines are still proportional to those of the plains horizontal respectively taken by Graduation from the Zenith, in all Pieces whatsoever. But the Lines of Altitudes, their Bases and Lines Hipothenusall are ever different in every several Angle, both of Inclination and declination, and vary by such a different Proportion from the horizontal, as they are to be discovered by a several Method of Calculation. The 51. Theorem. Such Theorikes, Scales, and Instruments, may be framed for the Invention of these strange Proportions of Altitudes, Lines diagonal, and Ranges horizontal, as thereby with the aid of Calculations Arithmetical, and some Rules Geometrical, a man may exactly and readily discover the true Circuits and Ranges of the Bullets of all Pieces of Ordinance whatsoever, mounted howsoever; and upon all grounds or plains inclining or declining, that can be Imagined, as shortly to the world by God's grace shall be made apparent. Mr. Digges his Questions, in the Art of Artillery with Mr. Nortons' Answers to them as followeth. Of Powder. 1. Mr. Dig:. WHether there be not for any Piece proponed such a certain quantity of Powder to be found, as duly to the charge of the same Piece agreeth, And that in such sort, that charging the Piece with more or less than that quantity, it shall hinder the fare ranging of the Bullet. Mr. N. By the fourth Theorem aforesaid, There is such a certain proportional charge of powder to be found for every Piece, in regard of her Charged and vacant Cylinder: But there must further be a consideration had concerning weight of the Shott, the Fortification of the Piece, & the different force of the sorts of powder, each to be proportional to other, and so three Dyametres of the Boar, or ⅓. of the weight of her Shott in Corn powder for Cannons. Or ⅔. of the Diametres, or ⅘. of the weight of the Iron Shot of Corned (Artillery) powder, for the Culuerings. And four Dyametres, or the whole weight of the Shott of such powder for Sakers, Falcons, and lesser Ordinance, is usually accounted as their due proportional Charges, which charge (if it could be readily found) would be just so much powder, as being all fired within the Cylinder, will at that instant have carried the shot just to the mouth of the Piece. 2. M. D. Whether one and the selfsame Piece twice charged with one and the selfsame quantity of Powder and Bullet, discharged also at the same Random, shall make the same Ranges? M. N. No, for at the Second time it will shoot further than at the first: As well because the Air that before was still quiet and unbroken, will be then moved that way the shot goeth, and by the course of the shot become broken. And also for that the charged Cylinder will then be drier and warmer than at the first, and cause the Powder to fire quicker, and better together, which will add more force thereunto. 3. M. D. If a Piece be discharged with the weight of his Bullet in Serpentine powder, and afterwards discharged with half the weight of his Bullet in such Corn powder as shall cause the Piece to cast the same ground; No. I demand if the same Piece be again Charged with half the quantity of either sort, whether these Ranges shall also be equal? The form of Charging being uniform, and temper of the Piece alike. M. N. No, for the last halves will one of them be then more farther off from due proportional Charge of the said Piece, than the other was from his Mate at the first; And therefore it is certain, that the last half of the Corn Powder, will shoot much further than the half of the Sorpentine Powder can do, because there is more Petre in the quantity of the last half, then in so much of the first half of the Corn Powder, and less in the other. 4. M. D. If two Pieces of the same Length and Bullet be charged with one kind of Powder, but several weights; I demand whether the Ranges shall be proportional to the said weights, No. Or to the Square, Cube, or Squared square roots of the said weights: Or whether the Proportion of the Ranges, be not to be found, without any further respect, either to the length of the Piece, or Ponderosity of the Bullet? Considering (by Hypothesis) all those are equal. M.N. Their Ranges will not be proportional to those Roots, (besides, whether they should be alike or differently mounted, being not here expressed. Neither the different weights of the Powder) therefore without them, those nor any other proportion certain can hold; yet they may he found in a Compounded proportion, having respect unto the proportions between the force of the Powder, weight of the Shott, and length of the Piece. 5. M. D. Whether the Proportion found in one kind of Powder, No. hold not in all other kinds, of what mixture soever it be, the Pieces and Bullets (being as is before supposed) equal? M.N. No, for the proportions of all different kinds of Powder, equal in weight or Measure, do differ in Force accordingly more or less, as there is more or less Petre, and working therein; Nevertheless the equality of the rest, for that a lesser weight or measure of stronger Powder will equal the proportional Force of a greater weight or measure of weaker Powder, and these are for Action in Geometrical proportion augmented, but for Resistance they are augmented in Arithmetical proportion, as an excellent Philosopher elsewhere doth largely Demonstrate. 6. M. D. Whether the proportion of such Ranges, He referreth this to a Book never yet extant. be not a Mean proportional resulting of the Commixion of the equality of the Pieces lengths and Bullets; and the inequality of the Powders Quantity? The Quality being supposed alike. M.N. I say it comes so near to such a mean proportionallity, as that in practise it might be accepted for the same indeed; But it not being exactly so, and also not being easily found, by reason it lurketh under so many compounded varieties and contrarieties, we must content ourselves with such a necessary Nearness, as in my Exposition of M. Digges his Definitions are exemplified, or to be showed elsewhere more largely. Of the length of the Piece and Powder. 7. Mr. Digges. Whether two Pieces being in all respects equal, saving only in length, being charged with one Bullet, Yes. and one quantity of Powder shall not make equal Grazes? M.N. No, for both by Master Digges his answer and mine to the first Question, there is a Charge certain; which is, that two Pieces of several lengths, and otherwise equal, cannot with one Quantity of like Powder, have both of them such a Charge, as can possibly be equally or proportionally nearest to their Charge certain; And therefore their several Grazes must needs be also different. 8. M. D. Not always. Whether the longer Cannon shall make the greater Range, whatsoever Quantity, or kind of Powder they be discharged withal, the Quantity of Powder being equal? M. N. I imagine that Master Digges meant the Quality or Kind of Powder, which may cause proportional difference, according to their several Forces: Otherwise I may answer yes, unless the longer Pieces charge be as much over, as the shorter is under, the Charges duly certain. 9 M.D. Yes, but not without respect of the Powder. Whether there be not a certain convenient Length of the Piece, in respect of his Boar or Bullet, to make the utmost Range, in such sort, that making the Cannon longer, shall rather hinder then further. M. N. Surely Master Digges meant herein as in the last, the length of the Cylinder or Chace, by name of making the Cannon longer, & then the question is by him truly answered yes. If the proportional Force and Quantity of the Powder be therein also considered and excepted; as I have under his former Definitions exemplified. 10. M.D. Whether this Length and proportion being found in one Piece, No. hold in all other (the proportion I mean for the Length) must of necessity alter? M. N. One proportion cannot hold for Cannons and Minions of (not above 24. Dyametres) and also for Culverings and Sakers (not less than 30. Dyametres of their Boars in Length) and the greater weight of the greater Shott, resisting the powders Force so much the more, by how much the more the Piece is mounted, and doth likewise in his Course more (by the greater gravity thereof) affect to descend out of the straight Line of her Course, than the lighter doth, though both beginning with equal swiftness, each of these and all, or some of them together do hinder general proportions, besides the less proportion of fortifying of the Cannon, (which Mettle) then of the Culverin, and yet the proportion of the shot more increasing in the Cube, causeth that the Cannon can neither burn within her, nor endure ⅘. in Corn powder of the weight of her shot, as the Culverin can do, much less as the Falcon, Saker, etc. which can endure to burn within them, their shotts whole weight in such powder, therefore they must needs convey the shot a greater proportional distance, than the Cannon or Minion can possibly do, and consequently break the proportion above in his question mentioned. 11. M. D. The proportion being by Experience found with Serpentine Powder according to the ordinary Charge; Unanswered. Whether giving like quantity of Corn Powder the same Proportion of Perfection shall hold, Unanswered. or a new be sought, in respect of the alteration of the Powder? M. N. No, that perfect proportion will be altered and anew to be sought, for there will a different quantity of Petre be found in the like quantity of those several powders; wherefore their Forces, and consequently the Ranges must needs also differ. And further you may understand, that although Serpentyne powder be grown out of use (because Corn powder is found better for Ordinance) and that the Force thereof was commonly accepted in compared proportion to Corn Powder, as ●. to 1. Yet for that there is also in several sorts or Receipts of Corn powder greater differences of Force found between them: Therefore also the said proportion cannot hold, for if in one pound of Corn powder of the receipt of 6. 1. and 1. there be 12. ounces of Petre, whereas in one pound of Powder of the receipt of 4. 1. and 1. there will be but 10. ounces and ⅔. of petre found; You may thereby also know what quantity of any one sort or receipt of Powder will be of equal Force, with any assigned quantity of any other sort of powder, whose receipt or mixture you already know, or can find out, which you may easily do many ways; As in my Book of the Art of Artillery at large is showed. 12. M. D. If two Pieces being in all respects equal, save only in Length, be discharged with one Bullet, and Quantity of one sort of Powder, make several Ranges (at Point blank discharged) I demand whether the same proportion of Ranges shall still continue, with whatsoever Quantity or kind of Powder the same Pieces be Charged? The Quantity being always equal, No. and all other Circumstances in Charging and Discharging in either of them alike? M. N. The proportions cannot continue the same; For if in the former two shotts the assigned Charge be as much over for the shortest Piece, as it is under for the Longest, or in any other proportion, it cannot be in like proportion in the Latter two shotts. 13. M. D. If two Pieces in all respects alike, save only in length, be charged with one Bullet, in Ordinary Serpentyne-Powder; I demand whether their Ranges shall bear the proportion of the length of their Cannons, or of the Vacant-hollow-Cannons, from the Charge to the Mouth? Or if it bear not the same proportion, whether they carry not the proportion of the Square, Cube, No. or Squared-square Roots? Considering all other Circumstances, all other things being equal saving only the Cannons? it is apparent, that from their Proportion, as the Original cause, the proportion of Ranges must in this case be derived. M. N. We must conceive that which Mr. Digges here and elsewhere termeth the Pieces Cannons, is the Vacant and Charged or Concave Cylinder of the Piece, in her whole length within, or the Chase without. And to the first part of this double Question, I Answer, that the assigned Charge must needs be nearer to her most due Charge certain for one of the Pieces then for the other. And for the latter part I answer No, as unto the fourth Question: But to the Conclusion I answer, as I did to the first Question, that from proportionallity of the quantity and force of the Charge, with the length of the Piece, and height and weight of the shot, the Proportion of Ranges for each several Mounting is derived. 14. M. D. If there be three Pieces in all respects equal, saving in length of their Cannons or Cylinders, and yet those three longitudes proportional: No. I demand whether the Ranges of their Bullets shall not be proportional? All other Circumstances save only this of Lengths being one, equal, and alike. M. N. The longest will outrange the two shorter, if that Piece be so well fortified, that she can endure her due charge of powder, and so by the same reason the middlemost may outrange the shortest; And yet their lengths may shorten in such sort, that their Ranges upon any like Mounting, or upon the Level, may be proportionals, but then the several lengths of their Chases will not be proportional thereto: For that one, or those two which are nearest the due charge of Powder, shall carry the shot with more advantage, than the more or most remote from their due charge, in regard of that which their lengths and weight of the shot can do. 15. M. D. If three Pieces as afore, having the Longitude of their hollow or vacant Cylinders proportional, whether, Unanswered. then (all the rest being equal and uniform,) the Ranges shall not be proportionals? M.N. I say no: If their Boar be equal, their Charge equal, and yet their Lengthes shortening or lengthening by Proportion, because the Charge will only in one of them come nearest unto the due charge, which advantage will break the proportion of it, with the other two; But if the middlemost for length were the nearest unto the certain length due to the assigned charge of Powder, than the longest must needs be too long, and the shortest too short, which disadvantages would come nearer to proportionallity, but not be exactly so; for diverse causes elsewhere herein showed. 16. M. D. Or if in one of these Cases the proportion of the Ranges be not a Mean Proportional, Unanswered. resulting of the commixion of the Equality of the Bullets weights, the Equality of the Powder, and the Inequality of the Longitudes of the Cylinders, either whole or vacant? M. N. Yes, it is doubtless a mean resulting proportionally of the Comixion of the equalities, with the inequality, but always with the former Cautions mentioned in the two last precedent Answers. Of the Powder & length of the Piece considered with the Bullet. 17. M. D. If a Piece twice charge d with one quantity of the same Powder, but the Bullets in weight different, Note there is a mean convenient. I demand whether the lighter shall always outsly the heavier, or that some convenient weight may be found? M.N. Doubtless there is a mean convenient weight may be found, which must be proportional to the Force that moveth it; For a man's hand can throw a weight of 4. pound of Lead further than a weight of 20. pound of Iron, or then 2. pound of Feathers. And so shooting severally in a Saker three shotts, one of Lead, another of Iron, and the third of Stone with 12. pound of powder (as the Leaden shot would weigh) then will the Shot of Lead outrange the other two, and that of Iron will out-range that of Stone; But severally shooting those three with 8. pound of powder, as the Iron shot weigheth: it will outrange the Lead shot, and the Led the Stone shot: But shooting them again severally with 4. pound of powder, then will the Stone shot outrange the Iron, and the Iron the Lead. Besides, the more a Piece is mounted, the more doth the heavier weight of the shot resist the Mover, which if it be too much or too little for the Force, it than impaireth the Motion; Therefore will the convenient mean weight be proportional. 18. M. D. Yes. Whether this convenient weight of the Bullet altar not, according to the Quantity or Validity of the Powder? M. N. Yes, for the Leaden shot will require the whole weight of ordinary powder, or ⅘. of the Cornepowder; And the Iron shot ⅘. of ordinary, or ⅔. of the best; And the stone ⅔. of the ordinary, or ½. of the best powder, As hath been ordinarily observed to do generally his best execution; But this holds not always for the force of powder and other proportions and accidents are continually variable. 19 M. D. Whether to find the said convenient Ponderosity of the Bullet, it be sufficient to consider the Powder, Both. or that the Longitude of the Piece also cause therein a diversity? M. N. Yes, the longer Piece will require the more powder to be fired within her, before the shot can arrive to the just mouth of the Piece to be then discharged out, than the shorter will, For if all the powder be fired before the shot arrive at the mouth, the after-running thereof within the rest of the Cylinder will hinder the swiftness thereof, by the Cylinders being too long. And likewise if the shot be discharged out of the Pieces mouth before all the powder be fired, and that it have received all the force of the Powder for want of sufficient length in the Cylinder of the Piece, it will be also hindered. Therefore there is a convenient length for the convenient weight to accompany with it continually proportional. 20. M. D. If two Pieces be twice charged, first with an Iron, then with a Led shot; The Quantities of Powder, Unanswered. at both times equal: Whether the differences of the Ranges be not derived only from the proportions of the Weights of these Bullets? (All other circumstances by proportion differing not) And what relation to the Ponderosity of the Bullets these Ranges have? M. N. To this I have already sufficiently Answered, especially in the three last precedent Answers, For as much as herein the proper weight of the Shot, & due length of the Piece, proportional to the Force of the Powder and Height of the Boar, do all domineer in altering their Shotts Ranges. And beside we seldom use Lead shot, but often Stone shot in great Ordinance, wherefore the Questions and Answers may be better applied to lion and stone, then to Iron and Lead. 21. M. D. Unanswered. Two Pieces being in all respects equal, and charged with one kind of Powder, but different Bullets, the one Iron, the other Led; And either having the weight of his Bullet: I demand whether the ranges be equal? M. N. No, but proportional according as the length of the Piece is nearest proportion as aforesaid. 22. M. D. No. If one Piece be charged three several times, first with a Stone Bullet, then with Iron, and finally with Lead: And the Iron of such temper, that it be an exact mean proportional in weight between the other two, being all discharged with one quantity of Powder, I demand whether the Ranges shall be in continual proportion? M. N. No, for the length of the Piece; Height of the Boar, and force of the Powder; will be nearer the convenient mean, for one of the said shotts, then for any of the other two: Therefore the continued proportion will not hold herein. 23. M. D. No. Whether a Piece being twice charged, first with Iron, then with Lead, having one quantity of Powder, and the Range noted; I demand whether, being charged with any other quantity of Powder, the Ranges of the same Bullets, shall not always retain the same proportion? M. N. I also deny that the Ranges shall retain the same Proportion, for that the weight of one shot will nearer approach then the other towards the convenient length of the Piece and force of the other quantity of Powder, And therefore that advantage will alter the proportion therein. 24. M. D. If in a Falcon, for example, Unanswered. by experience I find two such quantities of Powder, as discharging the Falcon with the first Quantity of Powder, with an Iron, shot; and again discharging her with the second quantity and a Led shot, they Range both duly one ground: I demand whether in a Saker of the same Length with the Falcon, Charging her first with an Iron Bullet, then with a Lead Bullet, using the same quantities of Powder, whether their Ranges shall be proportionals? And whether doubling either quantity of Powder, it shall alter the proportion of the Ranges? M. N I say unto the first part of this double Question, that the Saker shall not make proportional Ranges of ground. First, for that the Saker is of a higher Boar; And secondly, for that the convenient proportions of Powder cannot agree with those quantities (But I never heard of a Saker so short as a Falcon by a foot at the least.) And to the second part I say, if at the first they had been proportionals, yet they would alter their Ranges by doubling convenient proportions of either quantity of Powder. 25 M. D. If two Pieces of one Length be of such different Quantity of Bullet, No. that the one being discharged with a Lead Bullet, the other with an Iron Bullet, either having Powder the weight of their Bullet, and so make equal Ranges: I demand whether either of them discharged with half the weight of their Bullets in Powder shall Range alike also? M. N. No certainly, for the halves will be further of, (than the whole weights were) from the Convenient mean: Therefore their Ranges will not be alike; for then the Iron shot will outrange the Leaden. 26. M. D. Not always. If two Pieces be of one Length, but of several quantity of Bullet, and yet of one kind of Mettle or substance, and discharged with the weight of the Bullet, in one kind of Powder: I demand whether they shall not Range one ground, being equally Mounted? M. N. Not at any time, if the Piece of the lower Boar have her due length: For than she will over-range the other of the greater Boar. 27. M. D. Unanswered. If there be once found by experience in some one Piece such a perfection of a Cannon, as whether ye make him longer or shorter, he shall Shoot less ground, having always the weight of his Bullet, of one kind of Powder to his Charge: I demand whether if another Piece whose Cannon or Hollow Cylinder is in proportion like to the same, although greater in Quantity, shall not be of the same perfection? M. N. No, unless there be a due convenient Fortification and length of her Cylinder proportional unto the height of her Boar weight of her shot and Force of the Powder of her convenient Charge. But with those conditions I say, yes, it shall be of like perfection. 28. M. D. If two Pieces having their Hollow Cylinders similes or proportional be discharged with the weight of their Bullets in Powder at like Randon, Unanswered. I demand the proportion of their Ranges; the quantity of their Cylinders known? M. N. If the range of one shot be truly known of either piece, made (with like loadings and accidents) either upon the level or any Mount assigned, then by my Expositions before set down under M. Digges his three last Definitions (preceding h s Theorems) both the said Ranges may be found, as is well demonstrated by Nicholas Tartaglia in his seaventh Proposition of his second Book entitled his Nova Scientia. Where he saith, That every Range or violent motion of a body equally heavy (as round shot is) be it great or small, equally mounted above the Horizon, or equally obliqne or parallel to the plane of the Horizon, they will make their Ranges like, and consequently be proportional in their distances. 29. M. D. Of any two Pieces presented, to know which shall shoot furthest, being both Charged with the weight of their Bullet in Powder, The force of the Powder being first in some one approved? M. N. If the Length of the Cylinders, be in like proportions convenient to their Boar's Height and Charge, than the longer the Piece the farther she shooteth. Otherwise the Cylinder (may by being too long) make the longest shoot shortest. 30. M. D. Unanswered. Any two Pieces proponed, how to charge them with such quantity of Powder as they may both at like Randons', Range like ground. M. N. There in the Force of the said quantities of Powder, conpared with the length of the Concave Cylinder, and weight of the Shott, and all other circumstances must be considered: and so applied that the Piece that would shoot farthest must be proportionally abated of her allowance of Powder, as that which would shoot shortest is the more weak. 31. M. D. Unanswered. Having proved any Piece at his utmost Random with any one kind of Powder, to know how to diminish the Proportion of Powder from time to time in such proportion, as the Piece keeping that Random, shall shoot any part you will assign? M. N. Having by my answer to Master Digges his 11. Question, or otherwise found how much Petre is in that quantity of powder proved, and deminished it according to the force you desire to lessen; and having by my Expositions upon M. Digges three last Definitions (preceding his Theorems) found the Ranges, you may find Leading marks: But proportional proofs manually acted, exceedeth Art herein, by reason of unseen accidents. 32. M. D. Not equal but proportional. Whether the right Line of the utmost Randon be equal to the right Line of the Level Range, Or whether in all Pieces they retain proportion? M. N. The Level straight Line or course of a short explained in the 11. Definition contained in the 5. page, is at the best Random, as 1. to 5. as Tartaglia and others have demonstrated agreeing nearly with experience; Therefore they cannot be equals, but they are always proportionals in all Pieces. 33. M. D. Whether the right Line of the utmost Range be less than the right Line of 90. Grades of Randon? Yes. M. N. No, for the right Range or straight Line of the Bullets course at the mounting of any Piece to 90. degrees, is there longest, and is directly perpendicular to the Horizon. 34. M. D. Whether the right Line of the utmost Range be a Meane-proportionall between the right line of the Level Range, and the right Line of the vertical Range, Yes. viz. mounted to 90. Grades? M. N. No, by the next answer following, but every straight Line of any Piece, for each Mount is proportional to the straight Line of any other Piece like loaded and mounted, compared with any Ranges of equal Mounts for those Pieces given. 35. M. D. Whether the right Line of the utmost Random, Yes. be not rather a mean proportional, between the Level right Range, and that grave of Randon that Rangeth the ground of the Level Range? M. N. Yes, especially in a Calm, and that mean proportional will be found to be somewhat above 40. degrees by the 19 Definition and the Exposition thereof. 36. M.D. Yes in a Calm. Whether the Right lines made by any two Pieces discharged, be not proportional to the Ranges of their Bullets at the same Random? M. N. Yes, by the 20. and 21. Definitions and Expositions hereof: And as it is well demonstrated by Tartaglia in his Nova Scientia, and his Coloquys. 37. M. D. Not answered. Whether the right Lines made by any two Pieces at any Random, be not proportionals with their utmost Ranges? M. N. Yes doubtless, nevertheless the 23. of M. Digges his said Theorems concerning great Ordinance herein inserted; And as it will appear by my two Explanations and Tables following his 20. and 21. Definitions aforesaid. 38. M. D. No. Whether the Utmost Random, (I mean to make the utmost Range) be always one, whether the ground be level or ascending? M. N. No, it cannot be so, for that the ascending ground doth sooner meet with the way of the shot, than the Level, and the Level sooner than the descending plane. 39 M. D. No. Whether the Bullet end his Range with a Line, not sensibly different from a straight Line; As it doth begin his Circuit? M. N. Yes, under 60. degrees of Mount, but about 70. 80. and unto 90. it endeth in a perpendicular right Line, which is a right and straight Line. 40. M. D. Whether all Pieces at one Random discharged, Not answered. as they make one Angle at the beginning of their course, do make one Angle at the end of their Race? M. N. Yes, they do as in Tartagliaes Nova Scientia is demonstrated, upon all like planes, and meeting with like Accidents. 41. M. D. Whether the Angle at the end of the Circuit made with the Bullet be equal with the Angle of Randon? No. M. N. No, it doth not; for it is always greater, except at the Mount of 90. degrees. 42. M. D. Whether the upper part of the Circuit made with the Bullet be a proportion of a Circle as Tartaglia supposeth? No. M.N. No, for it is a Mixed, Curved or Helisphericall Line or Circuit, proper to the degree of mounting. 43. M. D. If a Falcon of three Inches Bullet weigh 700. pound, I demand how much a Cannon of eight inches will weigh, that is able to receive his proportional Charge to that Cannon? M.N. This cannot be answered by the simple Rule of proportion, because weight is not appertaining to Lines nor to Superficies, but unto Solid Bodies. And also you may note that although M. Digges here compareth the Falcon which is at least 30. of the Dyametres of her Boar in Length, and fortified with Mettle to endure the weight of her shot in powder, with the Cannon that is but 18. or 24. Dyametres of her Boar in length, and no better fortified then to endure ⅔. of her shotts' weight in Powder: But comparing the Falcon with a Demi-Culvering that is of the same kind and Fortification, whose shot weigheth 10. pound, the Question is Answered thus: for Example. The Cube of three (the inches of the height of the Falcon's Boar) is 27. & the Cube of 4½. (the inches of the Demi-culverings Bore) is 91. (fere) Now say by the Rule of three, Unanswered. if 27. give 91. what shall 700. give? And so having accordingly Multiplied the third number by the second, and divided the product by the first, you shall have in the Quotient 2359. pound 7/27. for the weight of the said Demi-Culvering sought. 44. M. D. If a Falcon of 3. inches Bullet require 3. pound of Powder for his charge, I demand how much Powder will charge a Cannon of 8. inches Bullet? M. N. Because this Question is of several kinds of Ordinance, as is said in the last precedent Answer, I have also applied this Answer, and the Example to the said Demi-Culuering thus: for Example. Multiply 91. the Cube of 4 1/●. by 3. l. the charge of the Falcon, and the product will be 273. which I divide by 27. the Cube of 3. (the Diametre of the Falcon) And the Quotient I find to be 10. pound and ●/●. for the Powder of the due charge for the said Demi Culvering sought. 45. M. D. Not answered. If the Falcon that carrieth Point Blank 150. paces, and at the utmost Random 1300. paces: I demand how fare a Culverin at his utmost Random will reach, that at Poynt-blanke, or Level rangeth 250. paces? M. N. Say by the Rule of 3 if 150. give 1300. what shall 250. give? answer, 2166. paces, and ⅔. for the utmost range of the Culverin sought. M. D. And thus by observations used in one Piece, by this Art of Proportion, a man may discover the Force of all Pieces. M. N. The Gunner may know by her level right Range how fare his Piece will convey her shot at any Elevation mounted: If he multiply the number of Paces she carrieth in a straight line, (she lying level) by 11. and divide the product by 50. the Quotient will be the greatest digression, which it maketh at the first degree more than at the Level. But all other degrees proceed always diminishing unto the utmost Random: and to know how they diminish, take the number of degrees from one to 41. the best of the Random, and that will be 40. by which dividing the said number of the former Quotient; This Quotient will be the number of Paces, which shall decrease from degree to degree, from the first unto the utmost Random. As for Example: For the Culverin that shot 250. paces level in a straight Line; I do multiply that, by 11. and it produceth 2750. which I divide by 50. and the Quotient will be 50. paces; which it shooteth at the first degree more than at the level; which 50. paces, divided again by 40. (the degrees between the first degree and the utmost Randon) & this Quotient will be 1. pace and ¼. which I take from 350. namely from the 250. & the 50. increased at the first, and also the 50. to be increased for the second digression, it leaveth 348. paces ¾ for the Range of the second degree, unto which add 48. paces ¾. abating the one pace; ¼. and there will be 195. ¼. for the paces of the Range of the third degree, and so proceed. Certain Erroneous Positions and Grounds published by professors of this Art of Great Artillery, noted by Mr. Digges. 1. THat in all Pieces of Ordinance mounted above the utmost Random, the Bullet is violently carried in a right Line to his utmost distance from the Earth, and then falleth perpendicularly down to the Horizon. 2. That all Pieces of one Bullet being charged with one quantity of the same Powder, and discharged at one Randon, shall make their Ranges proportional to the Length of their Pieces. 3. That if you Charge any one Piece, with several quantities of one kind of Powder (the Piece being discharged two several times, at one Randon, and with the same Bullet, shall make different Ranges proportional to the weight or quantity of the powder. 4. The fourth and chief of all the rest, is the Grade of utmost Random; For most Writers that ever I read, agree in this, that the mount of 45. grades above the horizontal plane, should make there the utmost Range. 5. That every degree of Random doth equally increase the Range in any one Piece, from the Zenith to 45. their Tropic grave, and so in decreasing likewise, and that proportionally in all pieces more or less according to their Force. 6. That in all sorts of Pieces, the difference of their utmost Ranges should be in proportion answerable to the weight of their Bullets and Charges of Powder. Other Erroneous Positions published concerning Ordinance; Noted by Robert Norton. 1. THat when a Piece is to shoot at a mark above the Level, being within distance of the right Range, than it by the virtue of the Fire that ascendeth overshooteth the Mark. And that therefore to remedy it, the Piece must be so much embased, Sior. Collado. fol. 61. until a Plum line at the Cornice let fall, will not enter but touch the lower side of the mouth of the Piece. The same Author there affirmeth Folio 60. that if any Piece be to shoot from above downwards to a mark under the Level, and the Pieces Mettle directed to the mark, that then the Piece must be so much Elevated more, as may equal the Angle, that the Pieces flat of her mouth maketh with a Plum line applied thereunto to equal the Natural defect caused by the ponderous descending of the shot in her mixed motion or crooked Range. Mr. Smith hath published in his Art of Gunnery, page 35. that if a Saker will convey her Bullet at Point blank 200. paces, and at the best of the Random 900. paces, that then that Cannon will shoot 1620. paces at the best Random that carrieth at Pointblank 360. paces, which is very erroneous, for that at the best of the Random the shot rangeth rather 10. times as much as at the Level Range. And also page 39 he affirmeth, that if a Culverin be shot off with 2/●. of the weight thereof, and then with the whole weight of her shot in Powder, that then the second shot shall be near ⅓. further than the first. And page 46. that a Cannon that shooteth 1440. paces at 45. degrees, will for every degree of less Mount, abate 32. paces in her Range. And page 47. he saith that a Cannon that shooteth at her best Random 1440. paces, will shoot at 30. degrees, but 960. paces, whereas in that case she will convey her shot being like laden and mounted about 1269. paces. And page 49. he saith, that if you abate the Level Range of a Piece from the utmost Range thereof, and divide the remainder by 45. degrees, than the Quotient will show you how fare the shot is carried at every degree. Or by deuiding the same number by so many degrees, as you would elevate your Piece at, The Quotient he saith doth show how fare the Bullet doth range beyood Poynt-blanke, and that thereby you may make a Table of Randons'. But he either forgetting, or I think rather not knowing that the best Random is but little above 40. degrees Elevation, or that the Piece shooteth thereat about 10. times as much as the Level Range, Or that the Ranges diminish from 1. to the best Range, and increase from the best to 90. degrees Elevation; It hath made him publish these and many other Erroneous positions. FINIS.