falls to be under 2 therefore seek in the Table above in the ranck of Squares for this number 2 or the nearest number less which here we find to be 1 and over it we find the Root to be 1 which must be placed in the quotient and likewise under the first prick to the left hand then having 1 for a Divisor and 1 for the quotient say but the common Rule of Division 1 times 1 is one 1 from 2 and there remains 1 which sent over the 2 then double the quotient and it makes 2 which place between the two first pricks to the left hand that is under 7 then say how many times 2 in 17 here you must be very cautious not to take too many which here may be six times place the 6 in the quotient as before and under the second prick that is under 3 and divide as before then double the quotient which is now 16 and it makes 32 place the 2 between the second and third prick viz. under 5 and the 3 before it under the 6 so the 32 will stand under the 175 which is above then say how many times 3 in 17 which you will find to be 5 place it in the quotient and under the third prick and divide as before always setting the Remainder over the head of its proper figures then double the quotient again which is now 165 and it makes 330 place the oâ— between the two pricks as before and place the figures before it to the left hand as you see above and the first figure to the left will be 3 which stands under 13 then say how many times 3 in 13 which will be 4 which place in the quotient and under the fourth or last prick and divide as before so you will finâ— no Remainder which assures the number given to be a square number The proof of these is known by multiplying the square Root found in it self taking in the remains if any be and it must produce that given number otherwise it is false Note how many pricks you have and so many numbers must the quotient consist of If the number given be not a true Square then a fraction will remain which fraction you may find out the value thereof to a tenth hundredth or a thousandth part c. Doing thus set next to the right hand after the Sum proposed two four or six cyphers or more for the more cyphe 〈…〉 you put the less is your Error and every two cyphers will produce a fractional figure more than the Integers belonging to the proper quotient which are tenths hundredths or thousand parts of a Unite according to the number of cyphers added that is if you add two cyphers then you find the tenths of a Unite c. But the Square Root being not of so much use in Gunnery as the Cube Root we shall proceed no farther to Exemplifie the same supposing it to be done already in the Treâ—tise of Military Discipline The Extraction of the Cube Root Begin at your right hand as you did in Extracting the Square Root and set pricks under every fourth figure that is leave two figures unprickt or between the pricks and so proceed to the left until you have done as here you see 7 5 6 7 8 7 3 2 the number of pricks shew the number of figures that will be in the quotient Then see by the Table before in this Chapter the nearest Cube to the numbers standing over the first prick to the right hand which is 75 I search in the Table of Cubes and find the nearest number to it in the Table of Cubes to be 64 and its Root 4 which must be set down in the quotient and likewise its Cube 64 under the prick and if that number doth not amount to so much as the number standing over the prick then substract it from the same and set the Remainder over head Then triple the quâ—tient and that triple you must set under the next number to the right hand before that prick where you did last end Multiply that tripled number by the quotient and set it dâ—â—â— under the first triple and that number let be your Divisor Then as in common Division must you look how many times the Divisor in the figures is standing over them and place that in the quotient This done Multiply your quotient by your Divisor and set it under your Divisor with a Line between Then multiply the last figure in the quotient by it self and then in the triple and set that figure under the former one figure more to the right hand Lastly Multiply the last figure cubically and set that Sum also one figure to the Right hand then add all these three multiplications together and substract it out of figures standing over the first and second prick and the Remainder set over them This done again triple the quotient and proceed exactly as before c. If your number be not an exact Cube but some numbers remain whereof you desire to find the exact fraction that is as near as possible may be viz. to a tenth hundredth or a thousandth part c. To find the tenths add three cyphers the hundreds 6 cyphers the thousands nine cyphers at the Right hand of your figures according to the directions given in finding the fractional of a square But these Rules being somthing tedious to many men we will for their encouragement and ease add a Table of Squares and Cubes whereby any man may find by inspection only the Square and Cube of any number of Inches and parts of an Inch to a tenth part provided your number exceed not 100 inches which will be found very necessary and save much labour as will appear by the following Examples But first we will present you with the Table it self A Table of Squares and Cubes very useful for the speedy Extracting of Square and Cube Roots for the Resolution of Questions in Military Affairs Whether for the Ordering of Battalions or Gunnery c. R Aq Ac 1 1 1 2 4 8 3 9 27 4 16 64 5 25 125 6 36 216 7 49 343 8 64 512 9 81 729 10 100 1000 11 121 1331 12 144 1728 13 169 2197 14 196 2744 15 225 3375 16 256 4096 17 289 4913 18 324 5832 19 361 6859 20 400 8000 21 441 9261 22 484 10648 23 529 12167 24 576 13824 25 625 15625 26 676 17576 27 729 19683 28 784 21952 29 841 24389 30 900 27000 31 961 29791 32 1024 32768 33 1089 35937 34 1156 39304 35 1225 42875 36 1296 46656 37 1369 50653 38 1444 54872 39 1521 59319 40 1600 64000 41 1681 68921 42 1764 74088 43 1849 79507 44 1936 85184 45 2025 91125 46 2116 97336 47 2â—09 103823 48 2304 110592 49 2401 117649 50 2500 125000 51 2601 132651 52 2704 140608 53 2809 148877 54 2916 157464 55 3025 166375 56 3136 175616 57 3249 185193 58 3364 195112 59 3481 205379 60 3600 216000 61
proportionably must be allowed for Pieces of greater or lesser weight The 12 Pounders fortified of Brass of 3200 l. for Guns of this weight and nature is usually allowed 3 ounces and a half for every hundred weight of Metal Demy-Culverin Brass of 3300 l. there is allowed by the Tower for Pieces of Ordnance of this nature 3 ounces and a half and somthing more to every hundred weight of Metal the which is approved a very sufficient Allowance Demy-Culverin Drakes of 2900 l. is allowed by most two ounces three quarters to each hundred weight of metal which will be durable in time of Service Saker fortified Brass of 2000 l. is allowed 3 ounces and somthing more for every hundred weight of Metal but there may be a small abatement in time of Service CHAP. XXVII To know whether a Piece of Ordnante be truly bored or no. YOu must provide a Pike-staff about a foot longer than the bore of the Piece and at the end thereof fasten a Rammer head that will just fill all the bore to the touch hole and at the other end of the staff you must bore a hole big enough to put through a Rod of Iron which must hand from the same and at the other end of the Rod must be made a weight about the bigness of a Saker Shot this is done to make the Pike-staff and Rammer head to lie with the same side upward when they are taken out of the Piece as they did when they were within the Piece then you must put your Instrument thus prepared into the Piece letting the Iron Ball that is at the end of the Rod which is put through the hole bored a cross the Pike-staff hand perpendicular then take your priming Iron or some other bodkin and put it down the touch hole to the Rammer head making a mark therewith this done draw out your Instrument and lay the same on a long Table with the Iron Ball hanging off the end perpendicular as it did when the Instrument was in the Piece then observe whether the mark you made upon the Rammer head when it was in the Piece be just upon the uppermost part of the same if it be the bore of the Piece lies neither to the right hand nor to the left but if you find it any thing to the right or left hand so much lyeth the bore either to the right or left and the Piece in Shooting must be ordered and charged accordingly But if you would know whether the bore lie more upwards or downwards then bend a Wire at the very end so that it being put in at the very touch hole may ketch at the metal when it is drawn out then put the Wire down the touch hole till it touch the bottom of the metal in the Chamber then holding it in that place make a mark upon the wire just even with the touch hole after draw up the wire until it ketch at the metal on the top of the Chamber and holding it there make a mark as before the difference between the two marks is the just wideness of the Chamber and the distance between the first mark and the end of the Wire having half the Diameter of the Chamber of the Piece substracted from it will leave half the Diameter of the Piece if the Piece be true bored but if the Piece's number be more than half the Diameter of the Piece the bore lieth too far from the touch hole and the upper part of the metal is thickest but if lesser the lower part of the metal is thickest or hath most metal CHAP. XXVIII Of the necessary Instruments for a Gunner with several other necessary things A Master Gunner intending upon service ought most chiefly to be prepared with these Instruments as Calabers Compasses height board Sight Rule Gunners Scale and a Gunners quadrant to divide as well into 12 as 90 equal parts with a Geometrical Square to make Montures Levels heights Breadths Distances and Profundities of which you shall read more in the Second Part also with a little brass Level Scales Weights Priming-Irons Moulds to make Cross-bar Shot for Musquets a Book of Accompts and an Iron wire or Spring and a Transome to dispart a Piece of Ordnance that the Transome may go up and down according to the Diameter and thickness of the Piece let the Transom be long enough to reach the base Ring from the touch hole In the next place he ought to be very expert in the knowledg of cutting out making up and finishing all sorts of Ladles Spunges Rammers Cartredges c. For which purpose you may have Recourse to the foregoing Table And because it may somtimes happen by reason of the steepness badness and unevenness of the way you may be driven to dismount and remount your Piece e're you get up to the top of a Hill therefore you must carry with you a Gynn and a Wynch with all the appurtenances thereunto belonging as wind Ropes an Iron Goats-foot with a Crow Pins Truckles Pullies to help you at a dead lift CHAP. XXIX The making of Rammers Spunges Ladles and Cartredges Formers Carriages Wheels Trucks c. with the Height of Shot fit for any Piece FOr the better expedition of this work we have in the former Table shewed the length and breadth of each Ladle always remembring that you cut each Ladle somewhat longer that is allowing so much more as must be fastned to the staff or so much as the staff goes within the Plate The Buttons or heads of the Ladles must be near the height of the shot For Spunges the bottoms and heads must be of soft wood as Birch and Willow and to be one Diameter and three quarters in length and three quarters or very little less of the height covered with Sheeps skin and nayled with Copper nayles so that together they may fill the hollow of the Piece Let the bottoms and heads of the Rammers be made of good hard wood and the height one Diameter of the Shot and the length one third of the Diameter of the Shot To make Ladles for Chamber bor'd Pieces open your Compasses to the just Diameter of the Chamber within one eighth part of an Inch Divide that measure in two equal parts then set the measure to one of them and by that distance upon a flat or paper draw a Circle the Diameter of that Circle is one fourth part shorter than the Diameter of the Chamber Take three fifths of that Circle for the breadth of the Plate of the Ladle But for Cannon the length ought to be twice and two third parts to hold at twice the just Diameter of the Powder As for Example The Diameter of a Circle drawn for a Cannon whose Chamber bore is 7 Inches containeth six and three quarters the circumference whereof is 21 Inches 6 7 and three fourth parts thereof is 12 ¾ and so much ought the Ladle to be in breadth and in length 18 ⅔ parts By this Rule you may
for example if the range of 45 degrees be double the height of the Perpendicular or vertical shot in a space not hindring so is the range of 45 in the air to the vertical in the air and so of the rest which observations only will teach which yet are most difficult in the greater sort of Guns or Bows especially the perpendicular whose height we can scarce certainly know unless some rock might be found high enough to whose top or some certain place the Bullet or Arrow may come the height of which top or place we may afterwards measure No Towers surely are high enough and by the time of the descent or fall of the Bullet to conclude a place may be found to which bullets darts or other things that are cast upright or vertically ascending do come doth therefore fail because they do not observe the same rule of swiftness in descending as is evident from darts to which seeing it happens in their ascent or rise of 50 fathoms to be slackned in their descent or fall something like this may be also thought to happen to Bullets to wit when they descend from the height of a thousand fathoms But you may avoid these difficulties for if from that rock in the Dolphinate whose height 't is said is 600 fathoms or more a stone or bullet of Iron or any other matter be let fâ—ll * the time of its falling being noted as for example if in the space of 18 seconds it fall from the height of 648 fathoms as Experiments of this kind the Reader will doubtless find as also others about Pendulums in the Opuscula Posthuma of 1. Bap. Batiani truly it should fall if the spaces be in Duplicate ratio or as the square of the times in the whole descent then we have rightly judged before of the vertical altitude or perpendicular shot which the Bullet of an indifferent Gun reaches that is the height of 288 fathoms which yet I cannot credit otherwise the middle shot or range of 45 degrees of that Gun at least would be double to that perpendicular or vertical one that is to say it would be 576 fathoms whenas I found it not 400 fathoms Besides these observations I shall add those which the industrious Galeus an Engineer to divers Dâ—kes in whose presence he made them writ with his own hand and gave me which that you may more easily understand let the greater Gun K which we commonly call a Cannon be parallel to the Horizon and let the eye be taking aim by the points I and O the horizontal shot being supposed O P or in the figure beneath S X or T V he sayes that the remainder of the rang which bends till it touches the horizon in the point Λ is almost equal to the horizontal shot that is that there is almost as much space made by the bullet from that point from which it begins to bend towards the horizon un til it touch it as it had made before the bending of it But now let us suppose that horizontal shot O P or T V removed to the lower figure in which let A B be the horizontal plain and let the the aforsaid horizontal shot be A I Galeus contends that the middle range of 45 degrees which is the longest of all is eleven fold the length of the horizontal shot O P or A I And in those Guns which are half the weight of the foregoing greater Guns to be in respect of A I as 10 and ½ is to 1 and in lesser Guns as 10 to 1 that is our figure as A B in respect of A I in which the middle range is A G E B for that is the middle range which passes through the middle of the quadrant Φ 5 which they call the sixth point because it is the middle part of the half circumference A 1 2 3 4 5 6 7 8 9 10 11 12 divided into 12 equal parts which joynd to the quadrant Φ 5 may be useful for levelling the gun at any elevation above the horizon if it shall be divided not only into 12 parts but also into an 180 degrees And hence he concludes that the dead or exact Horizontal Shot or range that is in the figure R Λ I to be in Proportion to the middle range as 1 to 6 or in lesser Guns as one to five which dead Horizontal ranges is to the range of an Elevationof one degree as five to six or more exactly as 55 to 67 or as 14 to 17. But when the recoyling of the greater Gun is hindred the dead Horizontal range will be greater by a seventh eighth ninth or tenth part than that range which is made with recoyling in lesser Guns it will be a twelfth or a fifteenth part less Moreover he asserts that the middle range A 6 E does proceed righton without arching by the line A G which may be almost equal to A 5 that is almost 5 fold or 4 ½ the Horizontal shot then not only that it does ascend to the point D so that the greatest height of the middle range may be fourfold the Horizontal and be over the line A C sixfold the same which some affirm but he by observation saies is false against Tartaglia affirming that the greatest altitude must be F E answering the point 7 that is seven distances of Horizontal range from the gun A that F E may be almost five fold the Horizontal-range Galeus did likewise well conjecture that the Curve for the middle range does come near to the Curve of an hyperbolical or parabolical line and that not by force of reason but only from observations Moreover the greatest range at forty five degrees elevation he makes 16200 foot that is 2700 of the French fathoms who because he used feet less than ours you might account it for 2500 fathoms that the said range may answer to our league and that the bullet might pass through the air in near half a minute or 30 seconds of time and because the dead Horizontal range may be 1 â— of the utmost it will be 2700 foot or 450 fathoms which being supposed the Horizontal range will scarce exceed 200 fathoms Mersennus in this place hath published a table of ranges made by the said Galeus but it being apprehended that the same is not so near the truth as that of Torricellio or another here published in English by the antient well known Teacher of the Mathematicks Mr. Henry Bond that the same may be preserved and become more common in use we have inserted the same Two Tables of RANGES According to Degrees of MOVNTVRE By H. Bond. The first Table deg  1 8758 2 7813 3 7077 4 6482 5 5991 6 5581 7 5234 8 4932 9 4669 10 4440 11 4237 12 4055 13 3889 14 3741 15 3606 16 3483 17 3370 18 3266 19 3279 20 3080 21 2996 22 2978 23 2845 24 2776 25 2712 26 2651 27 2593 28 2538 29 2486 30 2437 31 2391 32 2344 33 2300