The arte of gunnerie. Wherein is set foorth a number of seruiceable secrets, and practical conclusions, belonging to the art of gunnerie, by arithmeticke skill to be accomplished: both pretie, pleasant, and profitable for all such as are professors of the same facultie. / Compiled by Thomas Smith of Barwicke vpon Tweed souldier.

The arte of gunnerie. Wherein is set foorth a number of seruiceable secrets, and practical conclusions, belonging to the art of gunnerie, by arithmeticke skill to be accomplished: both pretie, pleasant, and profitable for all such as are professors of the same facultie. / Compiled by Thomas Smith of Barwicke vpon Tweed souldier. Smith, Thomas, fl. 1600-1627. This text is an enriched version of the TCP digital transcription A12531 of text S116363 in the English Short Title Catalog (STC 22855). Textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. The text has been tokenized and linguistically annotated with MorphAdorner. The annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). Textual changes aim at restoring the text the author or stationer meant to publish. This text has not been fully proofread Approx. 192 KB of XML-encoded text transcribed from 64 1-bit group-IV TIFF page images. EarlyPrint Project Evanston,IL, Notre Dame, IN, St. Louis, MO 2018 A12531 STC 22855 ESTC S116363 99851580 99851580 16860 2264183950

This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.

Early English books online. (EEBO-TCP ; phase 1, no. A12531) Transcribed from: (Early English Books Online ; image set 16860) Images scanned from microfilm: (Early English books, 1475-1640 ; 939:16a; 939:16b) The arte of gunnerie. Wherein is set foorth a number of seruiceable secrets, and practical conclusions, belonging to the art of gunnerie, by arithmeticke skill to be accomplished: both pretie, pleasant, and profitable for all such as are professors of the same facultie. / Compiled by Thomas Smith of Barwicke vpon Tweed souldier. Smith, Thomas, fl. 1600-1627. [11], 103, [6], 76 p. : ill. (woodcuts). Printed [by Richard Field] for VVilliam Ponsonby, London : 1600 [-1601]. Printer's name from STC. Printer's device on title page, McK. 192; head- and tail-piece, initial. The "plates" are woodcuts. Includes second part, with separate t.p. and register: Certaine additions to the booke of gunnery, with a supplie of fire-workes. All done by the former author Thomas Smith souldior of Barwicke vpon Tweede: both pleasant and profitable. [device McK. 192] London, Printed for William Ponsonby. 1601. Reproduction of the original in the Henry E. Huntington Library and Art Gallery. eng Gunnery -- Early works to 1800. Artillery -- Early works to 1800. Ordnance -- Early works to 1800. Military fireworks -- Early works to 1800. eebo The arte of gunnerie· Wherein is set foorth a number of seruiceable secrets, and practical conclusions, belonging to the art of gunnerie, by arithmeti Smith, Thomas, souldier of Barwick upon Tweed 1601 126 35024 4 0 4 0 0 0 2.28 B The rate of 2.28 defects per 10,000 words puts this text in the B category of texts with fewer than 10 defects per 10,000 words. 2005-04 TCP Assigned for keying and markup 2005-05 Aptara Keyed and coded from ProQuest page images 2005-07 Emma (Leeson) Huber Sampled and proofread 2005-07 Emma (Leeson) Huber Text and markup reviewed and edited 2005-10 pfs Batch review (QC) and XML conversion

THE ARTE OF GVNNERIE . Wherein is set foorth a number of seruiceable secrets , and practical conclusions , belonging to the Art of Gunnerie , by Arithmeticke skill to be accomplished : both pretie , pleasant , and profitable for all such as are professors of the same facultie . Compiled by THOMAS SMITH of Barwicke vpon Tweed Souldier .

LONDON , Printed for William Ponsonby . 1600.

TO THE RIGHT HONORABLE PERIGRIN BERTIE KNIGHT , LORD WILLOVGHBIE Beake and Earsby , Lord Gouernour of her Maiesties Towne and Castle of Barwicke vpon Tweed , and Lord Wardon of the East marches of England , for and anempst Scotland , &c.

IT is a common opinion Right Honourable amongst a great number , who may be tearmed more wayward then wise , that the Art of Soldiery may perfectly be attained in two or three moneths practise , and that any common man in a few weekes trayning , hauing seene two or three skirmishes may be called an expert soldier . Not considering that a Mariner may saile seuen yeares , and yet be far from a Nauigator . A number of Mechanicall Artificers may labour diuerse yeares , and yet be far from perfection ; and a number of Souldiers may serue many yeares , and yet haue but the bare name of a souldier . He may well be called a trained souldier , that knoweth by the sound of Drum , and Trumpet , without any voice , when to march , fight , retire , &c. that is able in marching , embattelling , encamping , and fighting , and such like , to performe , execute , and obey the lawes and orders of the field , that hath some sight in the Mathematicals , and in Geometricall instruments , for the conueying of Mines vnder the ground , to plant and mannage great Ordinance , to batter or beat down the wals of any Towne or Castle , that can measure Altitudes , Latitudes , and Longitudes , &c. such a one may be tearmed in my opinion an expert souldier , though he neuer buckled with the enemie in the field .

Such perfections is well knowne to be in your Honour , that you are furnished with these and many morare qualities in the Art Militarie , and aboue all with wisedome and noble courage , to performe and execute any honorable enterprise whatsoeuer for the honour and seruice of God , your Prince and countrie , the which our proud enemies haue felt to their paine and your euerlasting fame .

And although I my selfe be but one of the meanest souldiers in this Guarison now vnder your Lordships gouernement ( whom we pray long to gouern ouer vs ) being brought vp from my childhood vnder a valiant Captaine in Militarie profession , in which I haue had a desire to practise and learne some secrets touching the orders of the field , and trayning of Souldiers : as also concerning the Art of Managing and shooting in great Artillerie . I haue thought it good ( hearing of no other that hath done the like before ) to frame together certaine Arithmeticall and Geometricall rules , to shew in part how necessarie Arithmeticke and Geometrie is for our profession , the which I haue set downe in two litle bookes , the one intituled Arithmeticall Militarie Conclusions , the other ; The Art of Gunnerie : the first I wrote two or three yeares since , and bestowed on my Captaine , Sir Iohn Carie Knight , the which ( God sparing life ) I meane to correct & enlarge , & perhaps put to the Presse : This other I haue thought it my part , to offer to your Lordships good consideration , to be shrouded vnder your Honourable buckler , to beare off the blowes of enuious tongues , which are euer ready to spit their spite against any vertuous exercise : which although it be vnworthie to passe vnder so honourable a protection , I hope your Lordship will in indifferent ballance weigh my willing mind , to do my countrie good , and your Honour any seruice my poore abilitie is able to performe , which if your Honour allow of , I shall thinke my paines well employed , and shall encourage me hereafter to bring this new found Art into some better perfection , so farre as my poore abilitie is able to put in practise , or my simple skill in the Art will reach to . Thus loth to be tedious , I cease : beseeching God to preserue your Honour with much increase of honour , to Gods glorie and your hearts desire .

Your honours dutifully at command , Thomas Smith Souldier .

TO ALL GENTLEMEN , SOVLDIERS , GVNNERS , AND ALL FAuourers of Militarie Discipline , Thomas Smith wisheth increase of happinesse .

GENTLEMEN , there was neuer Author nor practised Gunner euer able ( as I am perswaded ) to describe at full , or could shew perfectly the efficacie and force that Gunpowder is able to accomplish , it being a mixture of such a wonderfull operation and effect , as by dayly experience we find . And although diuerse men in diuerse ages , haue inuented diuerse engines and Ordinance for offensiue and defensiue seruices by Gunpowder to be perfourmed , yet none hath nor could euer attaine to that full perfection , to know precisely what straunge effects the said mixture is able to worke . Also diuers learned men haue inuented many excellent rules pertaining to the Art of Gunnerrie , and a great many of them haue and do erre in the principals of their inuentions : and the cause is , for want of due practise therein . For the Art of Gunnery doth require great practise and experience , to declare the rare secrets thereof ; which is not for meane men to attaine to , for that the charges is great .

And albeit , I am the least able of a great many to take any matter in hand , pertaining to the same Art , being but a sworne scholler thereto , and my abilitie far vnable to put in practise that I would : yet because I haue serued a prentize-hood twise told since I tooke my oth , and neuer hearing of any that hath compiled any Arithmeticall rules or secrets ( which is the fountaine head from whence all Arts or sciences do spring ) into one volume , I thought it my part and duty ( according to my skill ) to do the best I could therein , for the benefit of others , and that in the plainest maner I could , that such as are not well seene in numbers Art , might the sooner vnderstand the same . And albeit I haue herein shewed but a few Arithmeticall conclusions belonging to the Art of Gunnerie , yet the experienced Gunner or skilfull Mathematitian , by these few may deuise a great many moe , for seruice offensiue and defensiue , by Arithmeticke and Geometrie to be performed . All which conclusions ( gentle Readers ) I haue thought best to frame in easie questions , shewing the answers or resolutions thereof . And although they be but meanly framed , I hope you will accept the same in good part , the rather , for that they are a yong Gunners practises . And if there be ought herein that may profit you , yeeld me your friendly censure , I craue no more : or if in any place I haue erred , either gently correct it , or passe it with silence , or in friendly sort admonish me thereof , I deserue no lesse .

There is a great many that can spie a mote in another mans eye , that had neede to haue a beame pulled from their owne : some wil scan verie curiously , and sooner find two faults then amend one . If you be of that mind ( friendly readers ) I mind not to make you my iudges .

The widowes mite was aswell accepted as the gifts of the wealthie . A souldier in Alexanders campe , in the dry desert presented the king his helmet full of cold water , saying , if I could haue gotten better drinke , your Grace should haue had part : the which the king gently accepted and liberally rewarded , answering , I wey not thy gift , but thy willing mind . But I seeke no reward for my trauell , but onely you will wey my mind is willing to do my countrey good , and to profite the readers : and not to carpe with Momus , nor disdaine with Zoilus , nor sooth with Zantippus . In so doing you shall encourage me to set penne to paper , and to flie a higher pitch pertaining to this new found Arte. Otherwise , if you spit out your spite against me for my good will , I will as meanely account of your malice , and so as I find you , looke to haue of me .

From my poore house in Barwicke vpon Tweed this 2. of May , 1600. Your friend and welwiller , Thomas Smith Souldier .

PETER LVCAS CANNONNIER in commendation of the Authour and his booke . SHake silly pen to write of arte , to him where arte doth dwel , And say , the want of Eloquence doth so thy hand repell , That farre thy Muse vnable is to praise the Authors skill : Nor canst thou paint thy mind , nor finely tell thy will , But as there needs no signe at dore , whereas the wine is pure , So need not I commend this worke , it all men will allure , To loue the Smith that forg'd this worke , who hath such Art in store , That better is then Arte which trieth gold from ore , As our proud foes haue found , and felt by Ordinance might , And ayde of the almightie Ioue , who doth defend our right . Therefore good Zeale go post-hast vnto Fame , And bid her giue this booke an euer-liuing name . Peter Lucas Gunner .

Richard Hope Gentleman in commendation of the worke . TO tell a tale without authoritie , Or faine a fable by inuention , The one proceeds of quicke capacitie , The other shewes but small discretion . Who writes conclusions how to vse a peece , In my conceipt deserues a golden fleece . Who takes in hand to write of worthie warre , And neuer marched where any warre was made , Nor neuer hopes to come in any iarre , But tels the triall , knowing not the trade , To write of warre , and note not what it is , May well be thought a worke begun amisse . But he that by his studie makes it knowne , What thing warre is , and whereof it proceeds , Defensiue and offensiue reasons shewing , To those that gape for honor by their deeds , A worthie worke who doth not count the same , In my conceit he doth a Souldier shame . If so : Smiths trauell cannot well offend , For so he meant before he set it forth , And if it chaunce to come where Souldiers wend. He it commaunds to seeme of litle worth : For what he writes , he writes to honor those , Which wade in warres to triumph ouer foes . Richard Hope Souldier .

Richard Rotheruppe Gentleman in commendation . THat man whom Martiall attempts May raise to honor hie , Let him peruse with learned skill , Smiths worke of Gunnerie . That fountaine which such springs sends forth , Can neuer drie remaine : I meane the Ground of Arts , from which All science we attaine . As Grammer , Musicke , and Phisicke , With high Astronomie : And other Artes Mathematicke , And braue Geometrie . This Art of Gunnerie likewise , Amongst the rest let stand , Whose god-father this Author is , Which tooke the same in hand . Whose knowledge in this famous Arte , Deserues eternall fame , For his conclusions excellent Doth well deserue the same . Richard Rotheruppe Souldier .

THE ART OF GVNNERIE .

A Table shewing the deminite parts vsed for mensuration .

FOrasmuch as some of these measures are to be vsed in the treatise following , it is requisite that I shew what kinde of measures are commonly vsed and now in force , beginning with a barly corne , from whence all these hereunder and a great many moe do proceed , as An inch , containeth 3 barley cornes layed end to end . a finger bredth , containeth 4 barley cornes in thicknesse . a hand bredth , containeth 4 fingers . a foote , containeth 12 inches . a yard , containeth 3 feete . an ell , containeth 5 quarters of a yard . a span , containeth 3 handbredths . a foote , containeth 4 handbredths . a geometricall pace , containeth 5 feete . a fadome , containeth 6 feete , or 2 yards . 10 fadome , containeth a score , or 20 yards . a furlong , containeth 123 paces . our English furlong containeth 132 paces , or 660 feete . a pearch or rood , containeth 5 yards ½ or 16 feet ½ . an aker , containeth 160 perches , 528 paces , or 2640 feet a league , containeth 1500 paces . an Italian or English mile , containeth 8 furlongs , or 1000 paces , or 5000 feet . a Germane mile , containeth 32 furlongs . a score , containeth 20 yards . an hundreth , containeth 600 feet , after 5 score to the 100. 24 grains of wheate dry , containeth one penie of Troyes weight . 20 pence , containeth one ounce . 12 ounces , containeth one pound . 20 graines of barley , containeth one scruple of haberdepois weight 3 scruples , containeth one dramme 8 drammes , containeth one ounce . 16 ounces , containeth one pound . 112 pound , containeth 100 weight . a tunne , containeth 20 hundreth

A Table shewing how to weigh a great deale with few weights .

You may way any number of pounds from one to 40 with these 4 weights , 1. 3. 9. 27. 1 to 121 with these 5 weights 1. 3. 9. 27. 81. 1 to 364 with these 6 weights , 1. 3. 9. 27. 81. 243.

This rule of weighing many things with few weights proceedeth of Geometricall progression . The pounds to be weighed , are wayed with as many namelike weights , to be done either double or three-fold , sometime by adding one weight to another , and sometimes by taking away and adding to the contrary ballance . Example in a double respect : All termes to 15 are weighed with 4 weights of pounds : as , 1. 2. 4. 8. so in a triple respect , all pounds to 40 may be weighed with 4 weights , as 1. 3. 9. 27. All pounds from 1 to 364 are to be weighed with these 6 weights , 1. 3. 9. 27. 81. 243. and so infinitely .

Measures .

THe varietie of measures are in a maner infinite , and yet are all comprehended vnder three general kinds , proceeding from a point in Geometrie , as Arithmeticke doth from an vnite : that is to say , Lines , Superficies , Bodies .

Lines hauing but only length without bredth of thicknesse , do measure onely Altitudes , Latitudes , and Longitudes , &c.

Superficies , being limited by lines , bearing length and bredth , without depth or thicknesse , in these are knowne the contents of Pauements , Glasse , Boord , Land , &c.

Bodies , being bounden by Superficies , & containing length , bredth , and thicknes , do make knowne the quantitie of all solide or massiue things , as timber , stone , &c. All which requires the aide of Arithmeticke , to be truly measured . The definitions , termes , and orderly working of these and all other , the Elements of Geometrie will teach you .

Here I thought to haue written briefly , or rather to haue glaunced at the wonderfull strange effects that Arithmeticke is able to worke and attaine to , but finding that that learned and famous man Master Iohn Dee , in his Mathematical preface vpon Euclids Elements , doth notably touch the same , shewing the rare properties and incredible mysteries that numbers Art can reach to , affirming that the effects thereof , of man is notable fully to be declared , it soone strake me in the dumps , feeling my selfe farre vnable to soare so high .

How to finde the cubicall radix or roote of any number .

AS in my booke of the Art of warre , entituled Arithmeticall militarie conclusions , I began with the extraction of square rootes , being a speciall rule to worke diuerse feats belonging to the sayd Art : So in this Treatise I haue thought best to begin & shew how to extract Cubicke rootes , for that diuerse conclusions are to be done by the sayd rule , in the worke following letting passe all former rules , as lesse necessarie , the which are commonly knowne to euery child , that hath any sight in the Art of numbring .

To finde the radix or roote cubicall of any number , you must note how many figures or numbers be in the totall summe thereof , and then as is shewed in the rule how to extract the square root of any number , you make a pricke or point vnder euery other number , beginning at the first number towards your right hand : euen so in this rule , in searching for the cubicall roote of any number , you must put a prick vnder the first number towards your right hand , and so increase your number of prickes , vnder euery third number , towards your left hand , and your quotient will containe so many figures as there be prickes .

If your number propounded be cubicall , multiplie your quotient cubically , the product of that multiplication will be the number that was propounded .

To multiply cubically , you must do as this example sheweth . 5 multiplied in himselfe is 25 , which 25 multiplied againe by 5 , makes 125 , and is a cubick number .

A cubicall figure , is proportioned as these figures sheweth , for a cube is a solide body of sixe equal squares or sides like a die .

Example .

It is requisite in learning to extract rootes , to haue in perfect memorie all those cubicke rootes of digit numbers and the cubes they do make , the which will be a great helpe in working , the which I haue here set downe in a table after M. Records order . 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729

Now to seeke for the first figure or roote , your table will shew you what number shal stand in the quotient , being due to the last prick , towards your left hand , which figure so set in the quotient , multiplied cubickly , if it be equall to the number or numbers aboue that last pricke , it doth shew that the said number or numbers are cubicke ; but if it bee more then a cube number , then abate the greatest cube number , that the quotient will make from the sayd numbers , and cancelling the same , let the remaine stand ouer the head of the said numbers , as is done in deuision of common numbers , and so haue you done with the first pricke .

Secondly , triple your roote , setting the said tripled number one place nearer from the last pricke , towards your right hand .

Thirdly , multiply the said triple , by the said quotient , the numbers arising thereof is your deuisor , to set vnder your first tripled number .

Fourthly , find out a number to be placed in your quotient , that may shew how often times your deuisor is contained in the deuident , or numbers so remaining ouer it .

Fiftly , you must multiply your deuisor , by the number last placed in your quotient , first drawing a line vnder your deuisor , and that which ariseth of the said multiplication must be placed vnder the said line .

Sixtly , you must square the number last placed in your quotient , and multiplie the said square by the triple of your first quotient number , & the summe arising of that multiplication set vnder the line , one place nearer tow●rds your right hand .

Seuenthly , multiply the number last placed in your quotient cubickly , and set the same cube numbers vnder the line , beneath the other numbers , one place nearer towards your right hand : and then drawing a line vnder the same , adde all those numbers together ; the summe arising abate from the other pricke that standes toward the right hand in your deuident , and if nothing remaine , the number propounded is a cubicke number : but if any thing remaine , the number propounded is no cubicke number , but yet the quotient doth shew the nearest cubicke roote in the proposition .

In this order you must worke by euery pricke , how many figures soeuer the numbers propounded containeth .

To find a Denominator to the cubicke remaine .

If the number propounded be not cubicall , and that you desire to know the true denominator to the cubicall remaine , you must square your cubicke roote , and then triple the said square , and after triple the roote , adding all those summes together , and to the totall of the sayd addition , adde one vnitie , so haue you the true denominator cubicall , the which you may abbreuiate into lesser termes by Abbreuiation , according to your desire .

Or you may find the denominator cubicall , by multiplying the roote in the triple of another number that is more by one vnite , nor the said roote : and then adding one vnite to the product of the said multiplication , you haue your desire .

An example how to worke , to find the cubicke roote .

Admit the summe or numbers , whose cubicke roote you desire to know be 32768. I set the pricke vnder 8 , and vnder the figure 2 standing in the fourth place , as in the worke here you see , and I finde that the greatest cubicke number in 32 is 27 , and 3 is his roote , which 3 I place in the quotient , and his cube being 27 , I substract from 32 , so resteth 5. And so I haue done with the first prick towards my left hand , as here in the work you may see .

Then I triple the quotient 3 , & it is 9 , which I set one place from the last pricke nearer towards my right hand . And then I multiply the triple of the quotient being 9 , by the said quotient 3 , ariseth 27 , the which I place vnder 57 , drawing a line vnder my deuisor 27 , and then I seeke how oft I can haue 27 the deuisor in 57 , which is a part of the deuident , the which I can haue but 2 times , which 2 , I place in the quotient , and by the sayd 2 I multiplie the deuisor 27 , so ariseth 54 , which I place vnder the line vnder the deuisor , as here you see .

And then I square the number last placed in the quotient being 2 , and it is 4 , which square I multiply by the triple of the first quotient number being 9 , ariseth 36 , which I place vnder 54 , one place nearer towards the right hand , as here you may see . And then multiplying the digit 2. cubickly ariseth 8 , to be set vnder the line one place nearer towards the right hand , & adding all these sums together , there ariseth 5768 , the which substracted from the number belonging to the first prick there remaineth nothing , so I say that 32768 is a cubicke number , and 32 is the true root thereof . You may proue it by multiplying the quotient cubickly , and abating the product from the number propounded , there will remaine nothing .

To find the nearest root of a number not cubicke .

Question .

I demaund the true cubicke root of 117884.

Resolution .

The pricks placed in order as before , I find there will be but 2 figures in the quotient , & that the cubick number of 117 is 64 , whose cubick root is 4 , which 4 I place in the quotient , and his cube 64 being abated from 117 , there remaines 53 to be placed ouer the last prick : then tripling the quotient 4 , ariseth 12 to be set down one place nearer towards my right hand , & then multiplying the quotient by the said triple , doth arise 48 for a deuisor , which I set in his place , drawing a line vnder him as in the former worke you see . And then I make search how oft I can haue 48 in 538 , which I can haue many times , but more then 9 times I must not take ; and therefore I set downe 9 in the quotient , and multiplying the same by the deuisor 48 , ariseth 432 , to be placed vnder the line vnder the deuisor , then I do multiply the said 9 squarely , ariseth 81 , the which multiplied by 12 being the triple of the first quotient , ariseth 972 , the which I set down one place nearer towards my right hand ; and then I multiply 9 cubickly , ariseth 729 to be set downe yet one place nearer towards my right hand : and adding all those sums together , the totall is 53649 , which abated from 53884 , rests 235. And thus I affirme , that 49 is the nearest cubicke root in whole numbers of 117884 , as here by the worke you may see .

Now to find a denominator for the 235 remaining , I square the roote 49 , so ariseth 2401. Then I triple the sayd squared number and there ariseth 7203 , and then I triple the roote 49 , ariseth 147 , to which I adde one , and it makes 148. Al which summes ioyned together , makes 7351 , and so the true cubicke roote of 117884 is 49 and 235/7351 partes of an vnite .

Theormes shewing the true proportion that a bullet of one mettal beareth to the like bullet of a contrary mettall , as also the proportion that the circumference of any buller or globe &c. beareth to the diameter , and of the superficiall content thereof to the diametrall square thereof , the which according to Archimedes are thus proued .

All circles are equall to that right angled triangle , whose containing sides , the one is equall to the semidiameter , the other to the circumference thereof .

The proportion of all circles to the square of their Diameter , is as 11 to 14.

All globes beare together triple that proportion that their Diameters do .

The circumference of any circle , is more nor the triple of his Dyameter , by such proportion as is lesse then 1/7 and more nor 10/27.

A bullet of iron , to the like bullet of marble stone is in proportion as 15. to 34.

A bullet of lead to the like bullet of iron , is in proportion as 28 is to 19.

A bullet of lead to the like bullet of marble stone is in proportion as 4 to 1.

The Diameter of any bullet &c. is in proportion to the circumference as 7 to 22.

How by knowing the true weight of any one bullet , and the diameter of the peece due for the said bullet , to find out the weight of any other bullet belonging to a contrarie peece of Ordinance .

Question .

Admit a Demy Cannon of 7 inches Diameter shoot an yron bullet of 32 pound weight , I demaund what weight shall that bullet be of , that serues a Cannon of 9 inches diameter ?

Resolution .

To answer this and such like , there is a generall rule ; for Ewclid in his sixt booke of geometricall elements , hath demonstrated and proued that all globes are in triple proportion to their Diameters , therefore I multiply the proportion of each bullet cubically , and I find the cube of 7 is 343 , and the cube of 9 is 729. Then by the rule of proportion I say , if 343 yeeld 32 pound weight , what shall that bullet weigh whose cube is 729 ? So multiplying 729 by 32 pound , the weight of the lesser bullet , ariseth 23328. which deuided by the 343 , being the cube of the lesser bullet , yeelds in the quotient 68 pound & 4/343 parts of a pound , so much shall that bullet , weigh , that serues a Cannon of 9 inches diameter , as by working the rule you shall find .

Another easie conclusion , how by the weight of a small bullet knowne , to find out the weight of a greater .

Question .

A bullet of 3 inches diameter weighing foure pound weight , what shall a bullet of the same mettall weigh whose diameter is twise the height of the former ( that is 6 inches high ? )

Resolution .

I worke in the order of the former conclusion , multiplying the diameter of each bullet cubically , and deuiding as afore is shewed , the quotient is 32 pound weight , so much shall the greater bullet weigh .

Example .

In the last conclusion the weight of the greater bullet weighed 32 pound , being 6 inches diameter , how shall I find the weight of a bullet of the same mettall that is but halfe that height .

Resolution .

I find the cube of 6 is 216 , and the cube of 3 is 27 , so framing the conuerse rule of 3 , I say : if 216 yeeld 32 pound weight , what will 27 ? And multiplying 27 by 32 , and deuiding the product by 216 , the quotient yeelds 4 pound , the true weight of the lesser bullet . And note that if you know the diameter and weight of any bullet , and would know the weight of one that is but ½ the height of the first , the lesser shall be in weight but the ⅛ part of the greater . Or knowing the weight of any bullet , if you would know the weight of another of the same mettall , being twise the height of the former , the greater shall weigh 8 times as much as the lesser , as in a figure demonstratiuely hereafter drawne you may see .

How by knowing the weight of any bullet whose diameter containeth both whole inches and partes of whole , how you should worke to find out the true weight of another whose diameter ends with a fraction .

Question .

If a Sakeret shoote a bullet of 2 inches ¾ diameter , of 3 pound weight , what shall a Culuering shot weigh of 5 inches ¼ diameter ?

Resolution .

To answer this or such like , I reduce each bullet into his proper fraction , and I find that the bullet of 2 inches ¾ diameter will be in a fraction 11/4 or 11 quarters , and the Culuering bullet of 5 inches ¼ height , will be 21/4 then I multiply each of these 2 fractions cubically , and I find that the cube fraction of the lesser bullet is 1331/4 and the cubike fraction of the greater is 9261/4 which knowne , I set down vnder three pound ( the weight of the lesser bullet ) the vnite 1 , and it will represent a fraction thus 3/1 , and then multiplying and deuiding by the golden rule in fractions , I find that the weight of the Culuering shot of 5 inches ¼ diameter will weigh 20 pound weight and almost ¾ pound , as in the working you may find .

How by knowing the diameter and weight of an yron bullet , to find the weight of a bullet of marble stone of the like diameter : or how by knowing the weight and height of a bullet of marble , to find out the weight of an iron bullet of like height .

Question .

Admit an iron bullet of 4 inches height weigh 9 pound , I demaund what shall a bullet of marble stone weigh of like diameter .

Resolution .

In a theoreme afore mentioned , I find that a bullet of yron to the like bullet of marble stone , shall beare such proportion as 34 is to 15. And therefore I multiply the weight of the iron bullet knowne being 9 pound by 15 , ( the proportion the stone bullet beareth thereto ) so ariseth 135 , which deuided by 34 , the quotient is 3 pound , and 33/34 parts of a pound : that is , 4 pound wanting 1/34 part of a pound , so much shall the bullet of marble stone weigh that is in Diameter and circumference , equall to the like bullet of iron . In like order reducing the weight of the stone bullet into his proper fraction , you shal haue 135/34 pound , which deuided by 15 , the proportion the stone bullet beareth to the like bullet of iron , your quotient is 9 , the number of pounds that the iron bullet weigheth .

How by knowing the height and weight of an iron bullet , to find out the weight and height of the like bullet of lead , or how to find the weight of an iron bullet , by knowing the weight of a leaden bullet of like diameter .

Question .

There is a Cannon that shootes an iron bullet of 72 pound weight , what shall a bullet of lead of the same diameter weigh ?

Resolution .

To worke this , I note that the theoreme before saith , that a bullet of iron to the like bullet of lead , shall beare such proportion as 28 is to 19 , therefore I multiply 72 ( the pounds which the iron shot weigheth ) by 28 , so ariseth 2016 , which deuided by 19 , the quotient is 106 pound 2/19 , so much will a leaden bullet weigh that is proportionall to an iron bullet of 72 pound weight .

In this order by working as I haue shewed in the end of the last conclusion , you may by knowing the weight of the leaden bullet , find out the weight of the like bullet of iron .

How you may find out the weight of any stone bullet of marble , by knowing the weight of the like bullet of lead , or how by knowing the weight of the stone bullet to find out the weight of a leaden bullet of like proportion .

Question .

If a bullet of lead weigh 106 pound , what shall a bullet let of marble stone weigh of the selfe like proportion ?

Resolution .

To answer this , I find that a bullet of lead to the like bullet of marble , beareth such proportion as 4 to 1. Therefore multiplying 106 by 1 , and deuiding the product by 4 , the quotient will be 26 pound & ½ shewing the true weight of a stone bullet , that is proportionall to the like bullet of lead .

And now to find out the weight of the leaden shot , by knowing the weight of the stone shot , reduce the stone bullet into his properfraction , you shall haue 53/2 , & setting 1 vnder 4 fraction wise , multiply the numerators together , and likewise the denominators , and deuiding the product arising of the numerators by the product of the denominators , your quotient will be 106 pound , shewing the true weight of the leaden bullet .

If you haue or do know the weight and true height of a bullet of stone , or any other mettall , and is desirous to know the weight and height of another bullet that is greater or lesser , and of the same mettall , in working as the first conclusion sheweth , you shall haue your desire .

To find out the circumference of any circle or bullet .

Question .

I demaund how many inches is about the circumference of that bullet whose diameter is 9 inches .

Resolution .

To worke this or any such like , there is a generall rule , as thus , that the proportion of the diameter to the circumference is as 7 to 22 , therefore multiplying the diameter 9 by ●2 ariseth 198 , which summe deuided by 7 , the quotient is 28 , 2/7 shewing the true number of inches about the circumference of a bullet of 9 inches diameter , as the figure here demonstrated sheweth .

How you may by knowing the circumference of any bullet , find out the height or diameter of the same .

Question .

The circumference of the bullet in the last conclusion , contained 28 inches 2/7 as in the demonstration you may see , I would know how I should worke to find how many inches the diameter of the same is .

Resolution .

To answer this and all such like , I must worke contrarie to the former conclusion , first reducing the whole number and broken being 28 inches 2/7 into his proper fraction , and it will be 198/7 then multiplying by 7 according to Archimedes doctrine , and deuiding by 22 , the quotient will be 9. so many inches is the diameter of the same bullet .

In this order you may find out the diameter and circumference of all other bullets .

How to find out the solid content of any bullet , &c.

Question .

There is a bullet of iron whose diameter containeth 9 inches , how many square inches is in the solid content thereof ?

Resolution .

To know this and all such like , there is a generall rule , as thus , to multiply the diameter in his square , I meane cubically , and then multiply that product by 11 , deuide the totall summe by 21 , the quotient sheweth the number of square inches in that spherical globe or bullet , for 9 multiplyed cubically ariseth 729 , which augmented in 11 is 8019 , that totall deuided by 21 , yeeldeth 381 inches , and 6/7 so many square inches of iron will be in a bullet of 9 inches diameter .

To find the true content of the superficies of any circle drawne vpon a flat , as on a table or paper , &c.

Question .

There is a circle whose diameter is 21 inches , I demaund how many square inches is contained within the circumference of the same ?

Resolution .

To resolue this ofr such like , there is a generall rule , in taking ½ the diameter , and multiplying it in ½ the circumference , or squaring the diameter , and multiplying the product by 11 , and deuiding the result by 14 , the quotient sheweth the Area or content of all the superficies within the circumference thereof . Example :

The square of 21 is 441 , which multiplied by 11 is 4851 , that deuided by 14 , yeeldeth in the quotient 346 inches ½ . Or other waies , take the halfe of 21 inches , that is , 10 inches ½ , and take ½ of the circumference , which is 33 inches , reduce them into fractions according to the rule , you haue 21/2 for the diameter and 33/1 for the circumference , then multiplying the one by the other , the product is 1386/4 , which deuided by the denominator 2 , yeeldeth in the quotient 346 ½ as before . In this order you may find out the content of the plaine of any circle .

To find out the circumference of any bullet or globe diuerse and sundrie waies .

Question .

How many inches is about the circumference of that bullet or globe , whose diameter is supposed to be 21 inches high ?

Resolution .

After you haue with your callaper compasses , found out the height of the diameter , multiply the same by 22 , so there will arise 462 , the which deuided by 7 , the quotient wil be 66 inches , the true measure of the circumference

Another way .

Triple your diameter , and thereto adde the 1/7 part of the same , your product is the circumference . Example :

The triple of 21 is 63 , and the 1/7 part of 21 is 3 , which added to 63 is 66 inches , as before .

Another way to worke the same .

Looke how many times you can haue 7 in the diameter , so many times must you haue 22 in the circumference . Example . The diameter being 21 inches , deuided by 7 , yeelds in the quotient 3 , by which if you multiply 22 , your product will be 66 inches , for the circumference , as before . In this order you may find out the circumference of any bullet , or sphericall body , &c.

To find out the superficies of any round body , as bullet , globe , &c. diuerse and sundry waies .

Question .

I haue a demy Cannon bullet of 7 inches diameter , I demaund how many inches the superficiall content therof is ?

Resolution .

To answer this and all such like , I must in the order before shewed , find out the circumference of the bullet , and I find that a bullet of 7 inches diameter , shall containe 22 inches in circumference , which circumference being multiplied in the diameter , ariseth 154 inches , the true number of inches contained vpon the superficies of a bullet of 7 inches diameter .

Another way .

Multiply the square of the diameter of any bullet or globe by 22/7 the product is your desire . Example : The bullet whose diameter was 7 inches being squared , the square thereof is 49 , which multiplied by 22 , yeelds 1078 which sum deuided by 7 , the quotient is 154 inches as before .

Another way .

Deuide the square of the circumference of any bullet by 22/7 your quotient numbers will shew you the superficiall measure of the same .

Example :

The circumference of the bullet aforenamed of 7 inches diameter containeth 22 inches , the square thereof is 484 inches , that number deuided by 22/7 as you do in fractions , in setting an vnite vnder the square number thus , 484/1 and multiplying the said square number by the denominator of the other fraction being 7 , ariseth 3388 , which deuided by the numerator 22 , the quotient is 154 inches , the superficiall content thereof , as before .

How you may find out the solid content or crassitude of any round bullet or globe , &c. diuerse wayes .

Question .

In the question before propounded of the bullet , whose diameter was 21 inches , I would know how many inches is in all the solid or massiue content thereof ?

Resolution .

I multiply the diameter cubickly , and after multiplieth that cubicke number by 11 , so ariseth 101871 , the which deuided by 21 , my quotient is 4851 , shewing there is so many inches in the solid content of a bullet or globe of 21 inches diameter .

Another way .

Multiply the cube of ½ the circumference by 49 , and deuide the product arising thereof , by 363 , your quotient will shew your desire . Example : The circumference of a bullet whose diameter is 21 inches , containeth 66 inches , the ½ thereof is 33 inches , the cube whereof is 35937 , that summe multiplied by 49 is 1760913 , which deuided by 363 , the quotient is 4851 inches as before .

How you may by knowing the diameter and weight of any bullet , or other round bodie , find out the diameter of any bullet or globe that weigheth twise the weight of the former .

Question .

There is a demy Culuering bullet of 4 inches diameter weighing 9 pound , I demaund the true height of that bullet which weigheth 18 pound weight .

Resolution .

To worke this and all such like demaunds , this rule is generall in multiplying the height of the lesser bullet whose weight is knowne cubically , then doubling that summe , and extracting the cubicke roote thereof , the quotient will answer your question . Example . The bullet afore named of 4 inches diameter being multiplied cubically is 64 , that summe doubled is 128 , the cubicke roote thereof is 5 inches and a fraction remaining scarse the 1/30 part of an inch , shewing the true height of a bullet that weigheth 18 pound . In this order if you haue a bullet that is 3 times as heauie as another of like mettall , whose weight is knowne , and that you desire to know the diameter of the greater bullet : in tripling the cubicke number of the lesser bullet whose diameter is knowne , & extracting the cubicke roote thereof , you shall find out the true height of the greater bullet . Or if you would find out the height of any bullet of like mettal , that weigheth 4 times as much as an other bullet whose weight is knowne , quatriple the cubicke number of the diameter of the lesser bullet , and extract the cubicke root thereof , your quotient will satisfie you . Or if 5 or 6 times &c. in working as I haue shewed you may find your request .

How you may Geometrically find out the diameter of any bullet , that weigheth twise as much as another knowne bullet .

Take the true height or diameter of the lesser bullet whose weight you know , and square the same as you see in the figure following . Then draw a line that may deuide the said square in 2 equall partes , in the 2 opposite angles , and that line shall be the diameter of a bullet twise the weight of the other : then deuide that diametrall line in 2 equall parts , setting one foote of your compasse in the center or mids thereof , and with the other foote draw a circle , and that circumference wil represent the proportion of a bullet , twise as much in weight as the lesser .

How you may Arithmetically prooue this conclusion .

The dyameter of the lesser bullet is 5 inches , the square of it is 25. that some dubble is 50. the square roote of 50 , is 7. 1/7 and so much is the diameter of the greater bullet , as in the figure here drawne you may see .

Another way Geometrically , to find the diameter of any vnknowne bullet that is double the weight of a knowne bullet .

Draw a straight line of what length you thinke good , as you see the line A. B. then draw another crosse line perpendicular to the ground line as you see the line C. D. note the meeting or crossing of the lines , as is the pointe E. This done , open your compasse the iust length of the diameter of the lesser bullet whose weight you would double , setting one foote of the compasse in E. and the other in D. and measure towards B. twise that diameter , as is done in the points F. G. Then deuide the line E. F. in 2. equall parts in the point H. and after deuide the line E. H. in 2 equall halfes , as in the point I. And lastly deuide the line I. H. in 2 equall partes in the point K. Which done , open your compas , placing one foote in K. and the other in G. draw ½ a circle , as you see , I do the semi circle L. C. G. After deuide the line C. D. in 2 equall partes in the point M. and opening your compasse the iust widenes of one of those parts , set one foote in M. and with the other foote draw the line C. N. L. Which done , the bullet whose diameter is the line L. E. wil weigh twise as much as the bullet whose diameter is the line E. D. as Ewclid in his 6. booke of Geometricall Elements doth demonstrate and proue .

The greater circle O. doth shew the proportion of a bullet that weigheth twise as much as the lesser circle N. both the said bullets being cast of one like mettall .

Another demonstration to proue the former conclusion by numbers .

In a conclusion before set downe , where the bullet of a demy Culuering of 4 inches diameter weighed 9 pound , I proued that a bullet whose weight was 18 pound should be more then 5 inches diameter . Euen so I haue hereunder deuided the line E. D. of the former conclusion , being supposed to be the diameter of a bullet whose weight is knowne , into 4 equall parts or inches . And likewise deuiding the Diameter F. E. into the like diuisions it containeth 5 of those parts , and almost the 1/30 part of an inch more , shewing the true height of a bullet that is twise as much in weight as the lesser bullet of 4 inches diameter , as this figure sheweth .

As the vpper face or side of any square being doubled , the square arising of that doubled side shall be in proportion iust 4 times as much as the first square was , whereas a great many would thinke it wold be but twise as much . Euen so the diameter of any circle being doubled , the Area or superficiall content of the flat of the same circle so doubled , shall be foure times as much as the other . Also any cube , globe or bullet , whose diameter is in double proportion to another , the solide content of that bullet whose diameter is so doubled , shall be in weight 8 times as much as the lesser , as these two examples in the conclusions following figuratiuely drawne sheweth .

How by knowing the superficiall content of the plaine of any circle , to finde out the superficiall content of another that is twise the diameter of the first .

Question .

There is two circles drawne , the one 7 inches diameter , the other 14 inches : how much is the content of the greater circle more then the lesser ?

Resolution .

To answer this or the like , by the theoreme afore mentioned , I square the diameter of the lesser circle being

seuen inches , so ariseth 49 inches , that square multiplied by 11 , yeelds 539 , the which deuided by 14 , the quotient is 38 inches ½ shewing the superficiall content of the circle of 7 inches diameter . Also working in the same order , I find the content of the greater circle of 14 inches diameter to containe 154 inches , which deuided by 38. ½ the quotient is 4 , shewing that the superficiall content of the greater circle is iust 4 times as much as the lesser .

By knowing the weight and height of any one bullet to find out the weight of another of twise the height of the former .

Question .

If a bullet of 4 inches diameter weye 9 pound , how much shall a bullet of 8 inches height weye .

Resolution .

To know this or the like , multiply the diameter of ech bullet cubically , and I find the cube of 4 is 64 , & the cube of 8 is 512 , which knowne , I frame the rule of proportion saying , if 64 yeeld 9 pound , what will 512 ? and in multiplying and deuiding according to the rule , my quotient is 72 pound , the weight of the greater bullet , ( that is iust 8 times the weight of the lesser bullet . ) For further proofe behold these 2 figures in cubick forme , where you may see that the greater figure whose side is in double proportion to the lesser , doth containe 8 times the quantitie of the lesser .

An easie rule to find out the diameter of any bullet , and how to know how much one bullet is higher then another by Arithmeticke skill , without any callaper compasses .

If you want a paire of callaper compasses , take a line or a garter &c. and gird the bullet or bullets whose height you desire iust in the mids , laying that measure to an inch rule , noting how many inches or other measure the same containeth , then multiplying the said measures by 7 , and deuiding by 22 , the quotient will shew you your request . And then abating the lesser diameter from the greater , the remaine will shew you how much the one is higher then the other .

Example .

Suppose the circumference of the one bullet be 16 inches , and the circumference of the other 26 inches , in working as aboue is taught , I find the diameter of the lesser bullet is 5 inches 1/11 and the diameter of the greater bullet 8 inches 4/11 , so abating the lesser from the greater , the remaine is 3 inches and 3/11 partes of an inch , shewing the greater bullet is so much in height more then the lesser . The like is to be obserued with any other .

By this rule you may know how much the circumference or any part of your peece is higher then another .

A table shewing the weight of all yron bullets from the Fawconet to the Cannon in Habberdepoiz weight .

Height of the shot in inches and parts of inches . Weight of the shot in pounds and partes of poundes . Height . Weight . 2. 1. 2 / 7 2. ¼ 1. ¾ 2. ½ 2. ⅓ 2. ¾ 3. 3 / 7 3. 4. ½ 3. ¼ 5. 3. ½ 6. 2 / 9 3. ¾ 7. 6 / 7 4. 9. 4. ¼ 10. ¾ 4. ½ 12. ⅔ 4. ¾ 14. 5 / 8 5. 16. ¼ 5. ¼ 19. ⅔ 5. ½ 22. 1 / 7 5. ¾ 25. ⅚ 6. 29. ½ 6. ¼ 32. ⅛ 6. ½ 36. ⅝ 6. ¾ 40. ¾ 7. 46. 7. ¼ 52. 6 / 7 7. ½ 56. ⅝ 7. ¾ 64. ½ 8. 70. 8. ½ 76. ⅔

How you may Arithmetically know the true breadth of the plate of the ladle that is due for any peece of Ordinance ▪ by knowing the height of the bullet fit for the said peece .

Take a line and compasse the bullet in the mids , laying the same measure to an inch rule , deuide the same measure into 5 equall parts , 3 of those parts is the iust bredth you ought to make your plate of , which being orderly placed on the staffe , and bent circularly , serues to hold the powder in : the other ⅖ partes being cut and taken away , and so left open , serues to turne the powder into the peece , the which to do Gunner like , as soone as you haue filled the ladle so full that you may strike the same with a rule , and put the same into the mouth of the peece , fixe your thombe vpon the vpper part of the staffe , towards the ende next the tampion or head thereof , and so thrusting the ladle gentlie home to the breech of the peece , turne the rammer staffe , so as your thombe fall directly vnder the staffe , and so shall you empty your ladle orderly .

Now to know the ⅗ parts of the bullets circumference , that you may make the plate of your ladle orderly , and of that iust breadth , lay the measure of the whole circumference to an inch rule , and then multiplie the same by 3 , and deuide the product by the denominator 5 , your quotient will tell you truely the breadth that the plate of your ladle ought to be of .

Example .

A Cannon whose bullet is 7 inches high , will be 22 inches in the circumference , that multiplied by 3 is 66 , which deuided by 5 the quotient is 13 inches ⅕ , the true breadth that the plate for a cannon ladle of 7 inches diameter ought to be of .

The length of the ladle is to be made according to the length , height , and weight of the peece for which it is made , which in a table in the ende of the booke you may find set downe for all sorts of peeces .

How to make a ladle for a chamber-bored peece .

Open your compasse the iust diameter of the chamber , within ⅛ part of an inch thereof . Deuide that measure in 2 equall partes , then set your compasse to one of those parts , and with the one foote fixed on a paper or smooth boord , draw with the other foote a circle , the diameter thereof will be a iust quarter of an inch shorter then the diameter of the chamber-bore , by the circumference whereof , you may find out the true breadth of the plate of a ladle that is fit for such a chamber-bored Cannon , by the rule afore set downe how to find the true breadth of the plate of any ladle , for any other peece of Ordinance , in taking the ⅗ partes of the circumference thereof , the length ought to be twise the diameter , and ⅔ partes , to hold at 2 times the iust quantitie of corne powder that is due to charge such a chamber-bored cannon with

Example .

The diameter of the circle drawne for any cannon whose chamber-bore is 7 inches containeth 6 inches ¾9 the circumference whereof is 21 inches 6/7 , the ⅗ partes thereof is 12 inches ¾ , and so much ought that ladle to be in breadth , and in length it ought to be 18 inches ⅔ . In this order you may worke to make a ladle in length and breadth for any bel-bored Cannon : and to find out the thicknes of the mettall at the touch-hole , or the height of the bore thereof , the conclusion following will shew you .

How to find out the height or diameter of the chamber , in any chamber-bored Cannon , or other peece of Ordinance , and how to find out the thicknes of the mettall round about the chamber thereof .

Take your priming yron , or else a straight peece of wyer , and bow the end thereof in manner of a hooke , and then put the same into the touch-hole , downe to the lowest part of the concauity of the peece , and then with your knife or else with a peece of chalke , make a stroke vpon the wyer hard by the vpper part of the mettal , without the peece at the touch-hole , then measure by your inch rule , how long the wyer is from that stroke to the end . After put in the same wyer againe , and pull it vp , so as the bowed end may restor stay within the cilinder or concaue of the peece : and make an other marke or stroke on the said wyer , hard by the vpper part of the mettal , the distance betweene those 2 strokes , is the iust thicknes of the mettall , round about the chamber , the which abated from the length of the wyer ( I meane from the first stroke to the lowest end ) the remaine is the true diameter of the chamber-bore in that peece .

Example .

Admit the length of the wyer from the end of the concauity to the first stroke containeth 15 inches , and the distance betweene the 2 strokes is 8 inches : then those 8 inches is the iust thicknesse of the mettall about the chamber ; which abated from 15 inches , restes 7 inches , the iust diameter of the chamber in such a peece .

By Arithmeticke skill , how to know whether the caryage for your peece be truly made or no : or how the caryage for any other peece of Ordinance ought to be made .

Measure the iust length of the Cilinder or bore of your peece , the plankes of your caryage ought to be once and a halfe that length . Also measure the diameter of the peece , and the sayd plankes at the fore end should be in deapth 4 times the diameter , and in the midst 3 times and ½ the diameter , and at the ende next the ground , two times and ½ the diameter , and in thicknesse once the diameter .

Example .

Admitte a Culuering of sixe inches diameter is in length in the bore thereof 20 times that measure ( that is 10 foote long , ) then I say that the plankes of her caryage ought to be 15 foote in length ; and at the fore end next the peece 2 foote in breadth , and in the midst one foote three quarters , and at the lowest end next the ground one foote and a quarter : and in thicknesse halfe a foote . Also euery caryage ought to haue foure transomes , and ought to be strengthened with strong yron boltes .

The holes or centers wherein the trunions ought to lye , ought to be three times and ½ the diameter from the fore end of the caryage , and in depth ⅔ parts of the thicknesse of the trunions , which depth you may easily find out , as thus : take the height or diameter of the trunions , and multiply the same measure by 2 , and deuiding by the denominator 3 , the quotient will shew your desire .

How by knowing the weight of any one peece of Ordinance , to find out the weight of any other .

Question .

If a Saker of foure inches diameter weigh 1600 pound weight , what will a Culuering weigh that is sixe inches diameter ?

Resolution .

Some would thinke that the rule of proportion plainly wrought , would answer this question : but in that they are deceiued , for the content of solide bodies being massie , are Sphericall or Cubicall inproportion , therefore you must multiply the diameters of euery peece cubically , & set downe the weight of the peece knowne in the middle number , and so working according to the rule of proportion , you shall find out the true weight of the greater peece .

Example .

4 inches the diameter of the lesser peece , multiplied cubically , ariseth 64 inches . Likewise the cubicke number of the diameter of a Culuering of 6 inches high , is 216 inches : then framing the rule of proportion , I say , if 64 being the cube of 4 yeeld 1600 pound weight , ( being the weight of a Saker of 4 inches bore ) what will 216 being the cubicke number of 6 inches , so multiplying 216 by 1600 , ariseth 345600. which deuided by 64 yeeldes in the quotient 5400 pound weight , so much weigheth the Culuering of 6 inches diameter .

In working by the conuerse rule of proportion , you may not onely prooue this conclusion , but also may find out the weight of any lesser peece of ordinance , by knowing the weight of a greater .

Example .

If 216 being the cube of 6 inches , yeeld 5400 pound in weight , what will 64 being the cube of 4 inches ? so multiplying 5400 by 64 there ariseth 345600. which deuided by 216 , the quotient is 1600 pound weight , shewing the true weight of the Saker of 4 inches diameter , as before .

Or if the diameters of the peeces whose weight you would know , containe both whole numbers and broken , in reducing each diameter into his proper fraction , and multiplying the same cubically , setting down the weight of the peece knowne , in the middle place , for the second number , and multiplying and deuiding as afore is taught , the quotient will shew you your request , as the conclusion following will teach you .

Question .

If a demy Culuering of 5 inches ¼ diameter weigh 2600 pound weight , what will a Cannon of 7 inches ¾ diameter ?

Resolution .

I reduce the diameter of each peece into his proper fraction , and I find that the broken number of 5 inches ¼ diameter containeth 21/4 , which multiplied cubically ariseth 9261/4. Likewise I reduce the diameter of the Cannon , being 7 inches ¾ into his fraction , and it is 31/4 ; , whose cube is 29791/4 ; : then 1 set an vnite I vnder 2600 , and it doth represent a fraction thus 2600/1. Now to find out the weight of the greater peece , I set down these 3 new made fractions in the order of whole numbers , and working by the rule of proportion , I finde the greater peece weigheth 8363 pound , and almost ¾ of a pound : for in multiplying 29791 by 2600 , there ariseth 77456600 , the which augmented by the denominator 4 maketh 309826400 for the deuident or number to be deuided . Likewise the fraction of the lesser peece being 9261 , multiplyed by his denominator 4 , makes 37044 for a deuisor , which deuident being deuided by the deuisor , yeeldeth in the quoent 8363 pound , and certaine partes of a pound , so much will a Cannon of 7 inches ¾ weigh being proportionall in mettall to the other peece .

How you may be Arithmeticall skill , know how much of euery kind of mettall any brasse peece of Ordinance containeth .

Question .

Euery Gun-founder doth commonly vse for euery 100 pound weight of copper , to put in 10 pound weight of lattine , and 8 pound weight of pure Tinne : I demand how many pound weight of euery of those mettals is in a Culuering of 5600 pound weight ?

Resolution .

To answere this or all such like , I ioyne all the seuerall mixtures together , and they make 118 pound , which I reserue for my deuisor . Then I multiply the weight of the peece by euery mixture seuerally , and there ariseth of the 100 weight of copper being the greatest mixture , 560000 , the which sum is to be deuided by the deuisor common ( to wit , 118 pound ) and the quotient is 4745 pound and 90/118 partes of a pound : so much copper is in the said peece . Now to know how much latin is in the same , I multiply the whole weight of the peece by 10 the second mixture , and the product is 56000 , which number deuided by the deuisor common , the quotient is 474 pound 68/118 : so much latin is in the same peece . And lastly to know how much Tin was in the same peece , I multiply the weight of the peece by 8 , ariseth 44800 , which deuided by the deuisor 118 , the quotient is 379 78/118 : and so much Tin was put into the said peece .

Now to proue the worke if it be truely wrought or not , I adde all the 3 quotients together , and because they doe all make the true sum of the whole weight of the peece according to the proposition , I affirme the same to be truely wrought .

The Gun-founders do hold and affirme , Note . that the lattin doth incorporate , and causeth the peece to be of a good colour , and the Tin doth strengthen and bind the other mixtures .

How you may know how far any peece of Artillery wil conuey her bullet at the best of the randon , by knowing the vtmost range and point blancke of another peece , and how to proue the same : by which rule , you may know how far any peece will reach at point blancke and vtmost range .

Question .

If a Saker at point blanke conuey her bullet 200 paces , and at the best of the randon shoot 900 paces , what will that Cannon do which at point blancke shoots 360 paces ?

Resolution .

To resolue this or the like , I set downe the numbers proportionall according to the rule , multiplying 900 paces ( the vtmost randon of the Saker ) by 360 paces , ( the point blanke of the Cannon , ) so ariseth 324000 , which deuided by 200 the number of paces the Saker shoots at point blanke , the quotient is 1620. And so many paces will a Cannon shoot at the best of the randon , that at point blanke rangeth 360 paces , as by working you may find , and by experience better vnderstand .

You may proue this conclusion by the conuerse rule of proportion , multiplying 900 the number of paces the Saker shoots at the best of the randon , by 360 , the paces that the Cannon shoots at point blanke ; and deuiding that product 1620 the number of paces the Cannon shoots at the best of the randon , the quotient is 200. shewing the number of paces that the Saker shall shoot at point blanke . In this order you may worke the like conclusion by any other peece of Artillery , and finde out the point blanke and vtmost range thereof .

To know how much a bullet of yron will out flie a bullet of lead of the like diameter , being both discharged out of one peece , with one like quantitie in powder .

Question .

If a bullet of lead of 24 pound weight , being shot out of a peece with ⅔ partes of the said bullets weight in powder , range at pointe blanke 240 paces , how far will a bullet of yron of like height range , being discharged out of the said peece at point blanke with the like quantitie of powder ?

Resolution .

The proportion betweene a bullet of yron and a bullet of lead of the same height , I haue shewed by the theoremes and conclusions afore set downe : by which I finde that a bullet of yron being of equall diameter to a leaden bullet of 24 pound weight , the said yron bullet shall weigh 16 pound 2/7 partes . And for as much as the leaden bullet is shot with ⅔ parts in powder of his weight , that is , with 16 pound of powder , which is very neare the full weight of the yron bullet , I find that the said bullet of yron shall out flie the leaden bullet ⅓ part of the leuell range ( that is ) the yron bullet shall flie being shot as afore at point blanke 320 paces , that is , 80 paces further then the leaden bullet rangeth at point blanke . But if the peece out of which the said bullets were shot , had beene mounted at any number of degrees of randon , the range of the yron bullet would shorten somewhat of the ⅓ of the ouerplus of the said range : so that if the peece were mounted to the best of the randon , the said bullet of yron would not out flie the leaden bullet , not the ⅕ part of the said range .

By knowing how much powder is sufficient to charge any one peece of Ordinance , to know how much of the same powder will charge any other peece of Ordinance .

Question .

If a Saker of 4 inches diameter ▪ require 5 pound of corne powder for her due loading , how much of the same like powder will charge a Cannon of 7 inches diameter ?

Resolution .

The plaine rule of proportion cannot resolue this conclusion , except you multiply euery number cubically , and then the quotient will shew you your desire .

Example .

The cube of 4 is 64 , and the cube of 7 is 343. which multiplied by the weight of the charge of powder due to load the lesser peece , ariseth 1615 , which deuided by the cubicke number of the diameter of the lesser peece , yeelds in the quotient 25 pound and almost ¼ part of a pound : so much corne powder must a Cannon of 7 inches diameter haue to charge her with . And note , that for as much as now the shooting with Serpentine powder is not vsed , being of no great force , and the making of corne powder neuer better knowne , nor of more force then now it is made & daily vsed in shooting in great Ordinance ; as also the great Ordinance now cast , not so fully fortified with mettall as they ought to be , being made more nimble and lighter then in times past , therefore the experienced Gunners do obserue as a generall rule to abate ¼ part of the ordinarie charge of corne powder in all peeces aboue 6 inches bore .

How by knowing how much Serpentine powder will charge any peece of Ordinance , to know how much corne powder will do the like , or contrariwise by knowing how much corne powder will charge any peece of Ordinance , to know how much Serpentine powder will serue .

Question .

I demaund how much corne powder will charge that Culuering that shoots 24 pound of Serpentine powder at a shoot ?

Resolution .

You must note for a generall rule , that 2 partes of corne powder will doe as much as 3 partes of Serpentine powder : so that the proportion betweene the quantities or charges of these powders , is as 2 to 3 , therefore I multiply 24 by 2 , ariseth 48 , which deuided by 3 , my quotient is 16 pound : so much corne powder will charge the said Culuering . Or if you know how much corne powder will charge her , you may know how much Serpentine powder will serue , in multiplying 16 pound the due charge of corne powder , by 3 , and deuiding the product by 2 , your quotient is 24 , as before . In this order you may doe the like by any other peece . And note that her due charge of corne powder , will lesse hurt the peece , then of Serpentine powder , for if Serpentine powder be ramd any thing hard , it is long a fiering . And a little heate long continued ( which the Serpentine powder will doe ) dangereth the peece more then a great heat presently gone , which effect corne powder works .

How by knowing how far your peece will shoot with her due charge in powder and shot , how to giue a neare estimate how far she will shoot with a charge more or lesse then her common charge .

Question .

Admit a Culuering shoote a bullet of 18 pound weight 900 paces , being charged with ⅔ parts in powder of the bullets weight , I demaund how far should the said peece shoot that bullet if she had been charged with as much powder as the bullet weighteth ?

Resolution .

By the rule of proportion I find she should shoot ⅓ part further then she did at the first shot , being charged with ⅓ part of more powder , that is 1200 paces : yet it is knowne she will not driue the bullet full out the ⅓ part of this range further , although she will come very neare it , and the reason is , because the bullet flieth in a circular proportion more or lesse , a part of the range , after the insensible streight line or motion of the bullet be past , according to the degrees of randon the peece is eleuated at . Also the concaue of the peece being filled vp with the powder , wadd and bullet , further then it ought to be , is a hinderance to the range of the bullet in proportion according to that litle quantitie of the concaue which the ouerplus of powder and wad filleth vp ; which though it be but little in comparison of the whole concauity to the range , yet it is a great hinderance in the bullets range , for that the bullet being so much nearer to the mouth , is driuen into the ayre before the powder be all fiered , and haue effected his force thereon : so that giuing the peece her bullets weight in corne powder , she will shoote much further nor with an ordinary charge , but it will both put the peece in danger of breaking , and those that are neare thereto in danger of their liues , if the peece be not all the better fortified with mettall .

How by knowing how much powder a few peeces of Ordinance haue spent , being but a few times discharged , to know how much powder a greater number of the same peeces will spend to be often discharged .

Question .

If 4 Cannons being twise discharged at any seruice , shoote 240 pound of powder , how much powder will charge 5 Cannons to shoote euery one 6 shots ?

Resolution .

Worke by the double rule of proportion , saying ; If 4 Cannons shoote 240 pound of powder , what will fiue Cannons ? your quotient will be 300 pound : then say againe , if 2 times discharging yeeld 300 pound of powder , what 6 times ? and your quotient being 900 pound weight , sheweth that so much powder is due to 5 Cannons , to shoote euery one 6 shots .

To know how much powder euery Cannon spent in the former conclusion at one shoote .

Question .

If 5 Cannons burne 900 pound weight of powder , being but 6 times discharged , how much powder did euery one shoote at one shoote ?

Resolution .

Multiply 4 the number of peeces first propounded by 2 , the times they were discharged , ariseth 8 , by which deuide 240 the number of pounds in powder spent , the quotient is 30 pound , and so much powder did euery Cannon fire at one shoote . Or else you may multiply the other 5 Cannons by the times they were discharged , and deuiding that product by the powder spent , you shall haue 30 pound weight of powder in your quotient also .

How to know how much powder euery little caske or firken ought to containe , and how many of those caskes makes a Last of powder , and how many shots any quantity of powder will make for any peece of Ordinance .

Euery little caske or firken being empty , ought to weigh 12 pound , and being filled ought to hold an hundreth pound weight of powder : so that the full caske ought to containe 100 of Habberdepoize weight , and 24 of those caskes or firkens filled makes a Last of power .

Question .

How many shots will one of those caskes filled with powder make to a Culuering that shootes 15 pound weight of corne powder at one shot ?

Resolution .

Deuide the 100 pounds of powder in each firken by 15 , the quotient will shew you that 100 weight of powder will be 6 shots to a Culuering that burnes 15 pound of powder at a shoote , and 10 pound to spare .

How by knowing how many shots a firken of powder will make for a Culuering , to know how many shots a Last of powder well make for a Canon .

Question .

If a firken of powder of one hundreth weight charge a Culuering 5 times , shooting 20 pound of powder at euery shot , how many of those shots will be in a Last of powder ( containing 24 hundreth weight ) to a Cannon that shoots 30 pound of powder at euery shot ?

Resolution .

Reduce 24 hundreth weight into pounds , you haue 2400 pound ; then say by the rule of 3 direct ; If 100 pound weight of powder be but 5 shot , what will 2400 ? and you shall haue in the quotient 120 shot , for the said Culuering that shoots 20 pound weight at one shoote . And whereas the question sayes the Cannon shooteth 30 pound of powder at a shot , you must frame the backer rule of 3 , and say if 20 beare proportion to 120. what will 30 ? so multiplying 20 by 120 , and deuiding by 30 , the quotient is iust 80 : so many shots of powder will be in a Last for any Cannon that shootes but 30 pound weight at a shot . The like is to be done with any other .

To know how many shots of powder will be in a Graund barrell for any peece of Artillerie .

Question .

If an ordinarie Culuering shoot 15 pound weight of good corne powder at one shoote , how many times will a Graund barrell full of powder serue to charge her , the said holding 300 weight ?

Resolution .

Deuide 300 by 15 , the quotient is 20 , your desire : the proofe is easie ; for multiplying 20 by 15 , you haue 300 the number first propounded : the like is to be done if you would know how many shoots will be in a graund barrell , for any other peece of Ordinance , in deuiding the poundes of powder contained in the said Barrell by the number of poundes of powder due to charge the said peece .

To know Arithmetically what proportion of euery receit is to be taken to make perfect good powder , what quantitie so euer you would make at a time .

Question .

The best ordinary corne powder made in these daies , containeth 12 partes of Mr. 3 partes of cole , and 2 partes of Sulphur . The order how to compound and make the same is not peculiar to this treatise , being meer Arithmeticall ; I demaund how many pound weight of euery sort is to be taken to make 1000 pound weight of powder ?

Resolution .

Adde all the parts or pounds of the receits together , ariseth 17 pound for your deuisor . Then frame the golden rule saying , if a mixture of 17 pound weight of powder , require 12 pound of the Salt-peter , what will 1000 pound weight ? In multiplying and deuiding according to the rule , the quotient will be 705 pound , and 15/17 parts of a pound : so many pound of the Mt. is to bee taken to make 1000 pound weight . Againe , say by the same rule , if that a mixture of 17 pound weight , do require 2 pound of Sulphur , what will 1000 ? your quotient will shew you , that 117 pound and 11/17 is to be taken . And lastly by the said rule say , if a mixture of 17 pound take 3 pound of Cole , what will 1000 pound take ? and your quotient will tell you that 176 pound and 8/17 parts is to be taken . The which 3 quotient numbers being all added together , will be iust 1000 pound weight , and so proues the worke to be truly done .

And note that the goodnes or badnes of powder may be knowne diuers waies , as by the colour , the tast of the toung , the quicke burning , &c. Also the brimstone is that materiall substance that is most apt to kindle with any sparke , the cole most fit to continue or maintaine the flame , and the Mr. being resolued into a windie exhalation worketh the effect , as cheife and principall of the three .

Before I frame these conclusions following , of the randon or range of the bullet , and the diuersitie thereof , it is requisite to make knowne to the Reader , how that diuers haue written , and some will vaunt that by the range or flight of the bullet out of any one peece of Ordinance knowne , they will or can tell the vtmost range of all other , thinking that the range of the bullet out of any one peece , should be proportionall to the bullet and charge of powder out of any other peece . Also some do affirme , that out of any one peece of Ordinance discharged with sundry quantities of powder , they can tell the vtmost range of the bullets discharged ; and their reason is , that the range of those bullets shall be proportionall to the weight of powder wherewith they were charged . And hereupon some haue giuen out rules which are false and full of errors : for the diuersity of proportions cannot by the plaine rule of proportion be resolued , as they affirme : but this may they do ; Out of any one peece of Ordinance charged with one and the same like charge in powder and bullet , find by the rule of proportion , the neare difference or ranges of the bullets , the peece being mounted or dismounted at any degree of randon ; or by knowing how many paces , yards , feete , or other measure any peece will reach at point blanke , by knowing the point blanke and vtmost range of another peece of Ordinance , they may find the furthest range of the first . Or contrariwise , by knowing the vtmost range and point blanke of one peece , and the vtmost randon of another peece , they may find out the point blanke of that other peece , as by the rules following shal be proued .

And it is to be noted that any peece of Ordinance being mounted to the best of the randon or highest degree of the quadrant , the mouth and hollow cilinder of the said peece , must be erected to 45 degrees , that is at the 6 point of the skale in the quadrant ( as the most part of quadrantes now are made : ) but some peeces will shoote as far at the 5 point , or at 41 , 42 , or 43 degrees according as the winde is of calmenesse , for if any peece be mounted higher then 45 degrees , she shall shoote shorter in euery degree about the 1/45 part of her vtmost range . And therefore to know how to worke these conclusions , you must buy an instrument Geometricall , or by some line of measure truely deuided , measure the distance from the peece to the place where the shot first fell or grazed , noting how many pearches , paces , yardes , or other measure that distance is ; which knowne , deuide that distance by the degrees in the best of the randon , being 45 , your quotient will tell you how many paces , yardes , feete , or other measure your peece will shoote further or shorter in mounting or dismounting a degree : the which knowne as I haue said , by one truely measured , you may before you shoote , know very neare how far or short your peece will shoote , at the raising or dismounting of any degree , allowing one and the selfe like proportions in charging , both with powder , bullet and wad .

How by Arithmeticke skill you may know how with one and the selfe same like charge in powder and shot , how much far or short , any peece of Ordinance will shoote , in mounting or dismounting of any degree : whereby you may know how far your peece will shoote at any degree of the randon , by knowing the distance she shoots at the utmost grade .

Question .

If a Cannon at her vtmost randon ( that is , at 45 degrees ) carry the bullet 1440 paces from the peece , how far shall the same peece shoote being dismounted but one degree ?

Resolution .

To answere this or all such like , I set downe the numbers according to the rule of proportion , and multiplying and deuiding accordingly , I find she shall shoot short in dismounting a degree , 32 paces , or 53 yeards , or 160 foote , which substracted from 1440 , rests 1408 paces ; so far shall the Cannon shoote in dismounting her one degree of her furthest range . Or you may do the like in framing the golden rule , saying : If 45 degrees range 1440 paces , what will one ? and you shall haue 32 paces in your quotient as before .

How by knowing the distance to the marke , by the conclusion or rule before , you may know whether your peece will shoote short , or ouer the marke , or you may know how far it is from your platforme to any marke , within the reach of your peece , onely by knowing the vtmost range of your peece , and the degrees she is eleuated at .

Question .

Admit the same Cannon in the former conclusion , which ranged at the best of the randon 1440 paces , hauing the like charge in powder , shot and wad , is laid to shoot at a marke being mounted at 30 degrees , I demaund how far it is from the peece to the said marke , or how far the said peece doth carry so mounted ?

Resolution .

To answere this , I multiply the paces my peece reacheth at the best of the randon , by those degrees in the proposition ( to wit ) 30 degrees , and there ariseth 43200 , which deuided by 45 , my quotient is 960 paces , ( that is 40 paces lesse then a mile ) so far will that peece shoot being mounted at 30 degrees . And if you would know how much this is short of the vtmost range , abate the same from the said range , the remaine is your desire . As 960 paces abated from 1440 , rests 480 paces , so much doth she shoote short of her best randon . In this order by 2 shoots knowne , you may know what any peece of Ordinance will do being mounted aboue 10 degrees to the best of the randon , but vnder 10 degrees you should erre something in this practise , because the range of the bullet flieth a great part of the way in an insensible streight line , and the peece mouth eleuated aboue 10 degrees , shootes or driues the bullet in a more circular proportion .

The range or flight of the bullet by the draught in the next leafe may be vnderstood . And note that in seruice there is no peece of Ordinance lightlie mounted aboue 15 or 20 degrees , except morter peeces , and such like .

The direct straight range at 90 degrees

This draught here drawne doth shew you the range or motion of the bullet through the ayre , shot out of any peece of Ordinance at any degree of the randon .

How to make a table of randons , or go very neare to know the true range of the bullet out of all sorts of peeces , being mounted from degree to degree .

Many Authors haue taught how to make a table of randons , whereas some of them neuer shot in any peece of Ordinance in their liues . And for asmuch as I find their writing and reasons differing , I thinke it will be a very hard matter to make a perfect table of randons , except the same be tried and experimented with some peece of Ordinance in some conuenient ground . I neuer heard nor reade of any that hath as yet fully put the same in practise , the which would be much auailable to euery Gunner , to know what euery peece would do at the mount of euery degree or point in the quadrant , the motion or range of the bullet being something variable at the mount of euery degree . You shall very neare find out the true range or randon of the bullet shot out of any peece of Ordinance , the peece mounted at any degree of randon , as thus :

Charge your peece with her due loading , in powder , shot and wad , laying the peece at point blanke , which you may easily try , by putting the rule of the quadrant into the peece mouth , & coyning the peece at the breech , so as the plummet may cut the quadrant in the line of leuell , as you see in the first figure hereafter drawne , that peece lyeth point blanke : which done giue fire , & marke where the bullet first grazeth , after bring your peece to the same platforme , so as the wheeles and cariage stand neither higher nor lower then they did the first shoote : and being charged with one & the selfe like quantity in powder , bullet & wad as before , the peece being of like temper raise her mouth one degree , as the second figure showeth : discharge her , and marke where the pellet falleth or grazeth first ; then measure how farre the first graze of the second bullet is beyond the graze of the first bullet , so much will the peece conuey the bullet further at the mount of euery degree , or very neare thereto . But being mounted aboue 20 degrees , she will shoote shorter & shorter , a litle at the mount of euery degree to the best of the randon , according to the height & circular motion of the bullet . If the peece be mounted to the best of the randon , the plummet will cut the 45 degree of the quadrant , as the 3 figure sheweth . Or you may make a table of randons like the other , as thus : Measure the distance the peece conueyeth the bullet at the best of the random , from which abate the distance the peece conueyeth her bullet at point planke , deuide the remaine by 45 , the quotient will shew you how far the shoote is caried at the mount of euery degree : or deuiding the sayd remaine by so many degrees as you would eleuate your peece at , the quotient wil likewise shew you how far the bullet doth range beyond point blanke .

Example .

If a Cannon at point blanke range 300 paces , and at the best of the randon shoote 1500 paces , how farre shall she shoote at the mount of one degree ?

Resolution .

Abate 300 from 1500 , rests 1200 , which deuided by 45 , the quotient is 26 2/3 , so many paces shall she shoot at the mount of euery degree .

This conclusion or rule , I do not affirme to be cleane without error , for that I neuer tried the same , yet it will come very neare to this proportion , being tried on a plain ground that is water leuel , for the peece being mounted from 1 to 10 degrees , conueyeth the bullet with litle bending at the fall thereof , and from 10 degrees to 20 , as the motion of the bullet decreaseth : so it falleth more bowing then in the first 10 degrees . And mounted from 20 grades , to the best of the randon , conueyeth the bullet in a more circular course . And it is to be noted , that any peece of Ordinance hauing her due charge , will driue the bullet more ground mounted at 20 degrees , then from 20 grades to the best of the randon . And being truely loaden and discharged at the best of the randon , will driue the bullet 5 times the distance of her leuell range , or rather better .

How you may Arithmetically know how much wide , ouer , or short , any peece of Ordinance will shoote from the marke , by knowing the distance to the marke , and how your peece is laid to shoote at the said marke .

Question .

If a Culuering or Cannon of 10 foote long , be shot at a marke 700 yardes from the peece , the mouth of the said peece planted an inch wide , how far shall the bullet light wide of the marke ?

Resolution .

Reduce the measure of the length of the peece into inches , because the denomination of widenesse is by inches , and the peece of 10 foote length , will yeeld 120 inches . Likewise reduce the length from the peece to the marke into inches , you haue 25200 inches . Then by the rule of propotion : say , if 120 inches shoot wide one inch , what will 25200 inches ? And in multiplying and deuiding according to the rule , you shall find in your quotient 210 inches , that is 17 foote ½ : so much shall the bullet light wide of the marke . For this is a generall rule , that looke how many times the length of the cilinder or concaue of the peece is to the marke , so many inches shall the peece shoote amisse , being laid ouer one inch , or vnder , or wide of the marke , if the winde doe not alter it . The like is to be done of any other .

A remedie to lay your peece straight , if she lie either ouer , vnder , or wide of the marke .

Let a plumbe line fall perpendicularly ouer the middle part of the breech of the peece , and with a hand-spike or leuer , winde the carriage of the peece too and fro till you espie the middle part of the mettall at the mouth of the peece , and the said line deuide the marke in 2 equall partes : so shall you make a streight shot , giuing the peece her true disparture and length .

Another way .

Or you may take the true diameter of the concaue at the mouth of the peece , laying an inch rule to the same , deuide the said diameter in 2 equall partes ; to the point of which deuision being the center of the cilinder of the peece , let a threed and plummet fall , or else erect a squire , so as the containing angle touch the center or middle point of the diameter , by the edge of which rule or squire draw a line with the point of your knife , from the height of the mettall at the mouth : that line would crosse in the center if it were continued , and it is a perpendicular or plumbe line to the other , by which line or strike so drawne , with a litle peece of soft waxe , set vp a straight straw , to reach a litle aboue the mettall . And knowing likewise the midle mettall at the breech of the peece , it is an easie matter to make a straight shot , if the 2 sights ( to wit ) the sight at the breech and mouth be laid so as they deuide the said marke in 2 partes : for this is generall , that any three thinges that the eye can comprehend at once , being equall with the eye , are in a streight line from the eye , whether the same be at ascent or descent .

The line or strike thus drawne at the mouth of the peece , will shew you presently where and how to set vp your disparture of your peece at any occasion .

In shooting without disparting your peece at any marke within point blanke , to know how far the bullet will flie ouer the marke by knowing the distance to the marke .

Question .

A Cannon or Culuering of 12 foote in length , hauing three inches more mettall at the breech on each side then at the mouth , shooting at a marke supposed to be within the leuell range , and 600 yeards from the mouth of the peece , being shot without her disparture , how much shall the shot flie ouer the marke ?

Resolution .

It is a generall rule , that looke how much the peece is thicker of mettall , in any one side at the breech , then at the thickest part at the mouth , as also looke how many times the length of the peece is to the marke , so many times that ouerplus of thicknesse shall the bullet flie ouer the marke , being no higher then the peece , and the said peece discharged without her disparture .

Example .

Deuide 600 yeards ( being the distance from the peece to the marke ) by 4 , ( the length of the peece ) your quotient is 150 , which multiplied by 3 inches the ouerplus of mettall , ariseth 450 inches : so much shall the bullet flie ouer that marke , the marke being placed on the side of a hill or bearing banke , and within the leuell range of the peece .

In like manner shooting at anie marke within ½ the vtmost range of the peece , and not disparting your peece , you shall ouer shoote somthing , giuing the peece her due length and due loading .

How you may lay your peece point blanke without instrument .

If you bring the height of the mettall at the mouth of the peece , and the height of the mettall at the breech , equall with the horison , the hollow cilinder of the peece will lie point blanke .

How you may Arithmetically dispart any peece of Ordinance truely diuers waies .

If you measure with a paire of Callapers the greatest height of mettall at the mouth of the peece , and likewise at the breech , abating the lesse out of the greater , ½ the remainder is the iust disparture .

Example .

A Culuering that is 19 inches high at the greatest part of mettall in the breech , will be 13 inches high at the greatest part of mettall at the mouth : which 13 inches abated from 19 , rests 6 , which deuided in 2 equall parts , the quotient being 3 inches sheweth the true disparture of that Culuering .

Another way to dispart any peece without Callapers .

Take a line and measure the greatest circumference of mettall in the breech , then multiply that measure by 7 , deuiding the product by 22 , the quotient is the diameter , or height of the circumference . Likewise measure the greatest circumference of mettall at the mouth , multiplying that measure by 7 , deuide by 22 as before , the quotient will shew the diameter of the mettall at the mouth : substract that diameter last found , from the diameter at the breech ½ , the remaine is the true disparture .

Example .

A Culuering whose greatest circumference of mettall at the breech containeth 66 inches , and at the mouth 44 inches , I demaund how high is the diameter of the mettall both at the breech and mouth , as also what is the true disparture of that peece ?

Resolution .

Multiply 66 by 7 , ariseth 462 , deuide by 22 , the quotient is 21 , the height of the mettall at the breech : likewise multiply 44 by 7 , you haue 308 , deuide by 22 , the quotient is 14 , the height of the mettall at the mouth , which 14 abated from 21 rests 7 , the which 7 deuided in 2 equall parts , yeelds 3 inches ½ for a part , the true disparture of that Culuering .

This is one of the principallest points belonging to a Gunner , to know truely how to bring the concaue of the mettall of his peece euen : diuers other waies there is to do the same . As for chambred peeces , there is no perfect or generall rule , but is to be considered according to the chamber or concaue of the peece . Euery reasonable Gunner may iudge in that case .

How by Arithmeticall skill you may mount any great peece of Ordinance by an inch rule vnto 10 degrees of the quadrant , if you want a quadrant or other instrument .

First you must measure the iust length of the Cannon or bore of the peece : reduce that measure into inches , and double the same : afterwards multiply the number of inches so doubled by 22 , and deuide by 7 , and note what the quotient number is , which quotient deuided by 360 the degrees contained in the whole circumference of euery circle , the last quotient number will shew you the number of inches , and parts of an inch , that will make a degree in the quadrant for that peece .

Example .

Admit there is a Saker or Fawcon , whose concaue or bore containeth iust 7 foote in length , and that you desire to know what parts of an inch rule will mount her to one degree of the quadrant , you must reduce 7 foote into inches , and you haue 84 inches , that 84 doubled is 168 , the which multiplied by 22 ariseth 3696 , the which deuided by 7 , the quotient will be 528 ; that quotient number being deuided againe by 360 , wil yeeld 1 7/15 ( that is ) one inch and ½ , wanting 1/15 part of an inch . So I affirme that any peece of Ordinance whose chase or bore is but 7 foote long , being mounted by an inch rule one inch and 7/15 parts , that peece shall lye iust the height she wold haue done if you would haue mounted her one degree of the quadrant . The like order is to be obserued in mounting any other peece of Ordinance by an inch rule , of what length soeuer . And note that in mounting any other peece of Ordinance , to any degree of the quadrant , by a Geometricall quadrant , you must put the rule of the quadrant into the peece mouth , lifting the peece vp or downe with a leauer or hand-spike towards the breech , till the plummet cut iust vpon that degree of the quadrant you desire .

But to mount her by an inch , you must place the rule vpon the highest part of the mettall at the breech of the peece , coyning the peece vp or downe , till through the sight or slit in your rule ( be lifted to that part or deuision in your rule that answereth the degrees you desire ) you espie the Carnoize or highest part of the mettall at the mouth of the peece , and the marke , all 3 in a streight line .

If you would mount the same peece to 2 degrees of the quadrant by an inch rule , you must multiply the measure in your rule last found , being 1 inch 7/15 parts by 2 , in the order of fractions , and you shall haue 44/15 , the which 44 being the numerator of the fraction deuided by 15 the denominator , the quotient being 2 inches 14/15 is your desire ; so may you affirme that 3 inches by the rule wanting 1/15 part of an inch , will make 2 degrees by the quadrant .

And note , that looke how much you would haue your peece mounted by an inch rule for to answer any number of degrees vnder 10 , either multiply that number by the number of inches and parts of an inch , that makes a degree of the quadrant , or else working as you did the first conclusion , multiplying the first product by the number of inches desired , and deuiding that product by the numbers afore mentioned , your last quotient will resolue you of your desire .

Example .

I demaund how much the peece afore mentioned should be eleuated by an inch rule , to answere to 8 degrees of the quadrant ?

Resolution .

Reduce the length of the bore of the peece into inches , as afore is shewed , doubling that measure , and it makes 168 , as you see in the 1 conclusion : which 168 inches multiplied by 22 , yeeldeth 3696 inches , the which product afterwards multiplied by 8 , ariseth 29568 , which summe deuided by 7 , the quotient is 4224 : the same deuided by 360 , yeelds in the quotient 11 inches 11/15 parts of an inch , so many inches and partes of an inch must the same peece be eleuated to with an inch rule , to answere to 8 degrees of the quadrant , as by triall you may find .

How by Arithmeticke skill you may know the true thicknes of mettall in any part of any peece of Ordinance .

Take a paire of callapers , and measure the height of the out side of the mettall in that place of the peece whereas you desire to know the thicknes of the mettall , then with an inch rule , or else a paire of streight compasses , measure the diameter of the bore , or concaue of the peece , abating the height of the said diameter from the height of the whole thicknes of that part of the peece so measured . And note the remainder , the which deuide in 2 equall parts , and the one of those parts is the iust measure of the thicknesse of the mettall in that part of the peece .

Example .

I prooued this conclusion with a Culuering , whose bore or concauity at the mouth was 5 inches ½ height , & I found that the thicknes or height of the whole circumference of the sayd peece at the touch-hole , was 16 inches ⅓ , from the which I abated 5 inches ½ ( fraction wise ) rests 10 inches ⅚ parts of an inch : that deuided in 2 equal parts , the quotient is 5 inches , and 5/12 or 5 inches ½ wanting the 1/12 part of halfe an inch , so thicke was the mettall of that Culuering at the touch-hole .

Likewise I searched for the thicknesse of mettall in the same peece at the end of the trunions , and I found that the thicknes or height of the superficies of all the mettall there contained 13 inches , from which I abated the diameter or concaue at the mouth , being 5 ½ inches , rested 7 ½ , which deuided in 2 equall parts , the quotient being 3 inches ¾ shewed the true thicknesse of the mettall at the trunions . In this order you may find the true thicknesse of mettall in any part of any peece of Ordinance .

Another way to know the thicknesse of mettall in any part of any peece of Artillerie .

Take a letherne girdle , and gird about that part of the peece you desire the thicknesse of mettall , lay the same measure to an inch rule , and note how many inches or other measure the same containeth : then multiply that measure by 7 , and deuiding the product by 22 , your quotient is the true measure of the whole thicknesse of the peece in that place . Then substracting the diameter of the bore or concauity of the peece from that quotient , note the remainder . Deuide that remaine in two equall partes , the one of those parts is the thicknesse of the mettall in that part of the peece so measured .

Example .

I prooued this conclusion with a demy Cannon of sixe inches diameter , in girding the same about with a line hard behind the trunions , and laying the same to an inch rule , it contained 44 inches , which summe multiplied by 7 , amounted to 308 inches : that summe deuided by 22 , my quotient was iust 14. And so many inches was the height of the whole mettall in that part of the peece , out of which quotient I did abate the diameter or bore of the peece being 6 inches , and the remaine was 8 inches , which deuided in 2 equall partes , my quotient being 4 inches , shewed the true thicknesse of mettall in that part of the peece , being hard behind the trunions towards the breech .

And it is to be noted , that euery peece of Ordinance if it be truly fortified with mettall , ought to containe as much mettall in thicknesse round about , so farre as the chamber where the powder and wad lyeth , as the bullet is in height .

How to make a good shot in a peece that is not truly bored : or to know how much any peece will shoote amisse , that is thicker of mettall on the one side then on the other , if you know the distance to the marke .

Question .

A certaine Gunner hauing shot diuers times in a Cannon at a marke supposed to be 500 paces from the peece , findeth she shooteth still towards the right hand , & searching whether the fault were in him selfe , or some impediment in the peece , he findeth that the peece is 2 inches thicker of mettall on the right side then on the left . And therefore requesteth how to lay the concaue of the peece ( being 9 foote in length ) equall with the marke , so as he may make a straight shot .

Resolution .

To do this or the like , there is a generall rule , that looke how oftentimes the length of the cilinder or concaue of the peece is to the marke , which is easily done by deuiding the distance to the marke , by the length of the concaue of the said peece . And knowing likewise how much the one side of the peece is thicker then the other , the one halfe of that ouerplus being multiplied by the quotient first found , the product will shew you how much the peece shooteth wide of the marke . And this is a generall rule : that looke which side of the peece is thickest of mettall , towards that side shall the bullet fall , for that the thinner side is more smart , and the thicke side more dull in heating .

Example .

The Cannon in this conclusion , is said to be 2 inches thicker of mettall more in thicknesse on the right side then on the left . And the distance to the marke is supposed to be 500 paces , ( that is , 2500 feete ) the which deuided by 9 feete , being the length of the hollow cilinder of the Cannon , yeeldeth in the quotient 277 feete 7/9 , the which multiplied by ½ the super fluitie of the mettall being one inch , makes 272 feete 7/9 still , and so much wide of the marke should the said peece haue shot at such a distance , although she had beene laid full against the mids thereof .

How to remedie your peece being thicker of mettall in one part then another to make her shoote streight .

You must first search your peece with an instrument , to know which is the thicker side , then deuide the ouerplus of mettall in 2 parts , setting the disparture of your peece one of those parts towards the thickest side of the peece mouth , and bringing the midle part of mettall at the taile of your peece , that disparture and the midle of the marke , all in one streight line , giue fire and you shall make a streight shot But beware of ouercharging of such peeces , for they are dangerous .

If the thickest part of the mettall be aboue , then you ought to make your disparture one inch more : if vnder ( I meane towards the carriage ) an inch lesse .

To know the different force of any two like peeces of Ordinance planted against an obiect , the one being further of from the said obiect then the other .

Question .

Admit there is a Castell or Fort to be battered , being situate vpon a hill , which hill is 50 paces in height , and that 140 paces from the said Castell there is another hill , of equall height to that hill whereon the Castell is built , and Ordinance planted thereon to beat or batter the Castell wall , and in the valley at the foote of the said hill 180 paces off from the Castell hill , there is Ordiance planted , and mounted at 20 degrees , to shoot and beat downe the said castell : I would know whether the Ordinance in the valley being 180 paces distance from the Castell , and mounted at 20 degrees , or the Ordinance on the height of the hill , lying leuell to shoote a litle aboue the base of the wall , being distant therefrom 140 paces , shall worke the greatest effect in battering downe the said Castell wall , the said peeces being of like length and height , and hauing like charge in powder and bullet ?

Resolution .

To resolue this or the like , a man would thinke that the peece planted on the height of the hill , lying leuell to shoote a litle aboue the ground-worke of the Castell , would batter sorest , because she is nearest : yet by experience we find the contrary , for the Castell being a great way within the reach of both the peeces , that peece shall not onely shoote much further , that is any thing eleuated , but also pierce much sorer , by so much as she is able to ouer shoot the other selfe like peece that lyeth leuell : albeit the said peece so eleuated , be planted furthest off from the said resisting obiect : for euery Gunner knoweth , and reason and experience doth teach euery reasonable man , that no peece of Artillerie will shoote so far at point blanke , as when the same is eleuated at any number of degrees ; because the bullet being ponderous , flieth more heauily and sooner declineth , being shot out of any peece lying leuell , then out of any such like peece mounted at any degree of the randon . So that of force it must needs follow , that the peece planted in the valley 180 paces off from the Castell , shall pierce and batter a great deale sorer then the like peece planted on the height of the hill being but 140 paces from it .

Example .

Suppose a Cannon or Culuering at point blanke shoot 240 paces , and being mounted at one degree outshoote the same 30 paces , what will the sayd peece do being mounted 20 degrees ?

By proportion I find , that if at the mount of one degree , any bullet range 30 paces beyond the leuell range , that at 20 degrees if shall outflie the same 600 paces : albeit the sayd bullet range not in euery degree a iust like number of paces , yet the proportion will be very neare thereto . And because the peece at the foote of the hill is sayd to be 40 paces further from the Castell , then the like peece planted on the height of the hill , I abate 40 out of 600 , rests 560 paces : so farre would the peece in the valley out shoote the other like peece on the hill ; so that it must needs follow , her bullet shall pierce sorest , for that it hath most strength to flie furthest .

Another exmaple or triall of the former conclusion .

The peece planted vpon the hill , is sayd to be 140 paces from the Castell , and the like peece at the soote of the hill 180 paces . Now suppose each of those peeces being layd at point blanke , would not range aboue 240 paces , abate 140 paces ( the length to the marke of the peece on the hill ) from 240 paces her leuell range , and the remaine is 100 paces ; and so many paces shall that peece strike the marke before the end of her leuell range .

Now to find the like in the peece planted in the valley 180 paces from the Castell , mounted at 20 degrees , I find by the conclusion afore set downe , that she shall out shoote the other 600 paces : so that abating the distance from the peece to the Castell , being 180 paces from 840 paces , her whole range mounted at those degrees , there remaines 660 paces . And forasmuch as the sayd peece eleuated at 20 grades , doth strike the marke 660 paces before the full end of the range of her bullet , it must of force pierce or batter sorer then the other peece whose bullet beates the marke but 140 paces before the full end of his range .

How you may hauing diuerse kinds of Ordinance to batter the wals of any Towne or Castell , &c. tell presently how much powder will loade all those Ordinance one or many times .

Question .

There is a Castell besieged , and to batter the same there is appointed 4 Cannons , 6 demy Cannons , 6 Culuerings , 8 demy Culuerings , and 5 Sakers : these peeces are charged euery time with corne powder , the whole Cannons shootes at euery shot 32 pound of powder a peece , the demy Cannons 18 pound , the whole Culuering 16 pound , the demy Culuering 12 pound , and the Sakers 6 pound a peece . All which peeces being 10 times discharged , did make a breach sufficient for 9 or 10 men to enter in by ranke ( a breach of such a widenesse is thought sufficient to be assaultable , ) I demaund how much powder was spent before the breach was made ?

Resolution .

To answere to this demaund , I multiply the number of euery sort of peeces , by the weight in powder that one of them shootes , and the product sheweth me how much powder euery sort of the said peeces did spend at one bout : then I adde euery number together , and the totall of that addition sheweth me how much powder will loade all those peeces one time , which addition multiplied by 10 , being the times they were supposed to be discharged , the product sheweth the iust quantitie of corne powder occupied at the said siege by the great Ordinance .

Example .

I multiply 32 pound the weight of powder due to loade euery Cannon by 4 the number of Cannons , ariseth 128. Likewise 18 pound of powder being the duety of euery demy Cannon multiplied by 6 the number of the same peeces , ariseth 108 , and 16 pound of corne powder being the duety of euery Culuering multiplied by 6 the number of those peeces , is 96. And 12 pound of powder being the due loading of euerie demy Culuering multiplied by 8. the number of the same is 96. And lastly 6 pound of powder the duety of euerie Saker , multiplied by 5 the number of that sort of peeces , is 30. These summes or additions put together makes 458 pound weight of powder : and so much will discharge all those peeces one time ; the which summe multiplied by 10 , is 4580 pound of powder , that is , two Last of powder wanting 220 pound . In this order if you haue 20 Last of powder , by knowing the number of euery sort of seuerall Ordinance , you may presently know how many shots , or how many times the said powder will loade all the said Ordinance , as this table sheweth .

Names of the Peeces . Number of each sort of Peeces . Powder due to loade each sort of Peeces one time . Cannons . 4. 128. Demy Can. 6. 108. Culuerings . 6. 96. Demy Culuer . 8. 96. Saker . 5. 30.

Summe 458 pound of powder , which multiplied by 10 , makes 4580 pound weight .

And it is worthy the noting , that in planting of Ordinance to batter or beate downe any curtaine , wall , or Cullion point , you must plant the same in 3 or 2 seuerall places at the least , from the thing to be beaten downe ; so as the said Ordinance be a pretty distance from other , vpon conuenient platformes , hauing Gabbions or Baskets , about 8 foote high , ramd full of earth conueniently placed betweene each peece , to saue the Gunners and Laborers from the danger of the enemies shot : which Ordinance would be planted within 200 or 240 paces of the obiect to be ouerthrowne , if it be possible to haue conuenient platformes and to bring them so nigh the said obiect . The which Ordinance ( if so you haue made 3 mounts or platformes , the Ordinance from the 2 side mountes doth coine or cut out that which the Ordinance from the midle mount doth batter or pierce , or shake , as this draught here drawne sheweth .

The best shooting to batter downe the broad side or curtaine of any wall , is to leuell something vnder the midle part of the wall , and after to shoote 2 or 3 foote higher : for the lower part being beaten downe , the height or vpper part of the said wall must fall of necessitie . And a speciall regard must be had to giue fire from each platforme or mount at one instant , for that the bullets beating all together , do more shake and batter the said wall , then lighting now one and then another .

In the figure or draught which I haue drawne shewing how Ordinance may be planted to ding downe or batter the broade side or curtaine of any wall , Castell or Fort , the middle Ordinance placed on the middle mount or platforme , directlie against the obiect to be beaten downe , are called the peircers , and are onely to shake and beate the wall , and the Ordinance on the two other side mounts , or platformes shooting something slanting , are to coyne or cut out that which the Ordinance from the middle platforme doth shake or loose . The Baskets ramd full of earth being placed betweene each peece of Ordinance are to defend the Gunners and Laborers from hurt of them that are besieged , as afore I haue said .

And further it is to be noted , that to batter the coyne or cullion point of any wall , two places is sufficient to plant your Ordinance in . Also you may batter and beate downe the wall of a Towne or Castell as well by night as day , so as the enemie shall haue no time to builde vp in the night that which was dung downe in the daie , as thus : Lay your peece or peeces , to the marke in the day light , and note well what degree of the quadrant she lieth at , which is soone done in putting the rule of your quadrant into the peece mouth , so laid against the marke , letting a line and plummet fall to the ground from the said point of your quadrant , and at the lighting of the plummet on the ground , there driue in a stake or wooden pin ; and letting a plumbe line fall likewise from the midle part of the taile or breech of your peece to the ground , driue therein another stake into the ground , then stretch a line from the said 2 pinnes , so as the ends of the said line may reach 2 or 3 yards further then the pinnes at each end . And there make them fast in driuing a pin of wood or yron into the ground at each end , then bringing your peece or peeces to lie streight aboue the said line or lines so drawne ( which is easily done hauing a lanterne with a close couer ) you may both charge and recharge , and shoote aswell by night as day , according to your desire .

How you may know the true weight of any number of shot , for seuerall peeces of Ordinance , how many soeuer they be , and how many Tun weight they do all weigh .

Question .

Suppose a Ship is loaden with Bullets to be caried to the siege of a Towne , &c. in which ship is 500 shot for whole Cannons , 800 demy Cannon shot , 900 Culuering shot , 1000 demy Culuering short , 1100 Saker shot 1200 Minion shot , and 1400 Fawcon shot , the question is to know the true weight of all the shot , and how many Tun they do all weigh .

Resolution .

In the beginning of this treatise , I shewed how to find out the weight of any vnknowne bullet , by the weight of a knowne bullet of the like mettall , so that multiplying the number of euery seuerall sort by the weight that one of them weigheth , and adding all the products into one summe ; and then deuiding that totall by 2240 pound , which is the pounds in a Tun , the quotient will shew you how many Tun all those bullets weigheth .

Example .

Admit the Cannon shot weigh 60 pound a peece , by which I multiply 500 ( the number of that kind of bullet ) so ariseth 30000 pound weight , and then there is 800 demie Cannon shot of 32 pound weight a peece , which multiplied as before , makes 25600 pound weight . And then there is 900 Culuering shot of 16 pound weight a peece , which makes 14400 pound weight . And then 1000 demie Culuering shot of 10 pound weight a peece , which makes 10000 pound weight . And then 1100 Saker shot of 5 pound weight a peece , which makes 5500 pound weight . And then 1200 Minnion shot of 3 pound weight a peece , which makes 3600 pound . And lastly , 1400 Fawcon shot of 2 pound weight a peece , which makes 2800 pound weight . All these summes added together makes 91900 pound weight , which deuided by 2240 , yeelds in the quotient 41 Tun , and 60 pound weight remaining .

In this order you may know how many Tun weight any number of shot weigheth , so that knowing how many Tun any ship is of burthen , you may easily know how many shot will loade the said ship .

How any Gunner or gunfounder may by Arethmiticke skill , know whether the trunions of the peece be placed rightly on the peece or not .

Measure the length of the bore of the peece , from the mouth to the breech , deuide that measure by 7 , and multiply the summe that commeth in the quotient by 3 , the product will shew you how many inches or other measure the trunions ought to stand from the end of the lowest part of the concauity of the sayd peece at the breech .

And note that the trunions ought so to be placed , as ⅔ parts of the circumference of the peece may be seene in that place whereas the trunions are set .

Example .

Admit the cilinder or concaue of a Cannon , or other peece of Ordinance be 10 foote ½ long , I demaund where the trunions of the sayd peece ought to stand ?

Answere .

Reduce the length of the concaue of the peece into inches , you haue 126 inches , the which deuided by 7 , the quotient is 18 , that multiplied by 3 , makes 54 inches , or 4 foote ½ , so farre ought the trunions to be placed from the breech or lowest part of the hollow concauity of the sayd peece .

Another way .

Or multiplying the length of the concaue of the peece by three , and deuiding the product by 7 , the quotient will shew the true place , how farre the trunions ought to stand from the lowest part of the bore or concauity of the peece .

Example .

126 inches the length of the concaue of the peece , multiplied by 3 , makes 378 inches , which number deuided by 7 , the quotient is 54 inches as before .

And note that the trunions of euery peece were inuented to hold the peece vp in her cariage , to moue her vp and downe to make a perfect shot , and to hold her fast in her cariage , after she is discharged : for if the trunions be placed too neare the mouth , the peece will be too heauy towards the breech , so as the Gunner appointed to serue with her , shall haue much adoe to raise her , to coyne her vp or downe , or being placed too neare the breech , the contrary will happen .

How you may know what empty caske is to be prouided to boy or carry ouer any peece of Ordinance ouer any riuer , if botes or other prouision cannot be gotten .

It is thought sufficient that 5 Tun of empty caske will swimme and carry ouer a Cannon of 8 or 9000 pound weight , 4 Tun will carry ouer a demy Cannon , 3 Tun a Culuering , and 2 Tun a Saker , accounting all prouisions to be made fast thereto , as plankes , ropes , &c. so that knowing what number of Ordinance is to be ferried or caried ouer any riuer , adding all their weights into one summe , by framing the Golden rule , you may presently know what empty caske is to be prouided to ferry ouer all the sayd Ordinance at one instant .

Example .

If a Cannon of 8000 weight require 5 Tun of empty caske , how much emptie caske is to be prouided to carry ouer so many Ordinance as is supposed to be 100000 weight ?

Resolution .

I multiply 100000 by 5 , so ariseth 500000 , the which being deuided by 8000 , the quotient is 62 ½ , so many Tun of emptie caske is to be prouided to carry ouer so many Ordinance as weigeth 100000 pound weight . The which empty caske made fast head to head a row on each side , by such as haue skill in such seruices , and planked aboue , would serue for a bridge to carry ouer a whole Army with all prouisions thereto belonging .

All which necessaries in time of seruice , and many more , belongeth to the Master of the Ordinance his office , to haue in readinesse , as also to be prouided of Trunkes , Arrowes , Balles , and all kind of fire-workes , wet or drie , and the receits for making thereof . As also engines for mounting or dismounting of Ordinance , wheeles , Axeltrees , Bullets , Powder , Ladles , Sponges , Ropes , Shouels , Anckors , &c. Also it is the duety of the Maister of the Ordinance , the Maister Gunner , and euery chiefe officer or quarter Maister vnder them , to be expert in the Arte of Gunnery , the better to teach and instruct their inferiors , the which without some practise in Arethmiticke and Geometry they cannot well accomplish . They ought to haue some sight in the Mathematicalles , the better to teach and instruct such as would shoote at all randons , to know what Ordinance is conuenient for an Army , or to batter or beate downe the walles of any Towne or Castell , to know what powder and shot is to be prouided for that or such like purpose , what cariage horses , labourers and other necessaries is to be allowed for the same . They ought to practise all Geometricall instruments , for the measuring of heights , lengthes , breadthes , depthes , &c. To practise how to conueigh mines vnder the ground , and how the same should be truely wrought , to blow vp any Towre , Castell , &c. To know what length the mine will containe with all his windings to and fro to the place appointed . To haue skill , in the handling of all engines and inuentions belonging to the Ordinance . To appoint to euery peece of Ordinance in time of seruice ' , Gunners that know perfectly how to mannage their peeces , to charge , shoot , clense , scoure , wad and ram the same , and what laborers are to attend thereon . To know in euery platforme appointed , how to place the baskets or gabbions , and what proportion of widenesse , height , or thicknesse they ought to containe : and that the loopes haue their due proportion of widenesse . To see that euery Gunner be able to discharge his duety , and not for fauour or affection to preferre such as can say most , and doe least : but that euery man be preferred to place of credite , and esteemed according to his honest behauiour and skill in this singular Arte. That none be permitted to the profession of a Gunner , but that he be first truely instructed in the principals of the Arte , by such as haue skil therein . And not to make or suffer euery tagge and rag to be a Gunner , as is too much vsed in these daies in Townes of garrison , who was neuer practised in the Arte , nor bath discretion nor desire to practise therein : a great number of such haue but onely the bare name of a Gunner , although their standing hath bene of long time : for as a great many of Marriners haue saild 7 or 8 yeeres and yet farre from a Nauigator , so a great many such haue continued in pay a large prentise-hood , and yet farre from a good Gunner . Such in time of seruice would worke as the builders did at the Towre of Babell , when one cald for one thing , he had deliuered a contrary thing . In seruice the Prince by such is not truely serued , the Arte lesse esteemed , and themselues discredited .

The Arte is like to a circle without end , or like to a Laberinth , wherein a man being well entred in , knoweth not how to get out againe , and therefore it must be exercise and industrie that must make a perfect Gunner . Many things here could I write pertaining to the duety of a Gunner , and euery officer pertaining to the Ordinance , but for as much as the same is not peculiar to this Arethmeticall treatise , and sufficiently handled by other Authors , I omit .

How to know the true time that any quantitie of Gunmatch being fiered , shall burne , to do an exploit at any time desired .

Take common match , and rub or beat the same a litle against some post or stoole to soften it , and then either dip the same in salt-peter water and drie it againe in the sun , or else rub it in a litle powder and brimstone beaten very small and made liquid with a litle Aqua vitae , and dried afterwards . Now when you would occupie the same , trie how long one yarde will burne , which suppose to be ¼ part of an houre , then 4 yards will be a iust houre in burning . Now suppose you haue laid some powder or balles of wilde fire to burne some house , ship , mine , corne-stacks , &c. or that you haue placed the said powder or balles in some secret place to burne some thing you are desirous to spoile , and that you would be going from the place 3 houres before it effect , then binding the one end fast to the balles , laying loose powder vnder & about the same , or wrapping the one end like a wreath amongst the powder loosely , draw out the other end , or lay it crookedly , or wrap it softly about something , so as one part doe not touch another , and fire it at the other end : which match so drawne or rolled , being iust 12 yards in length , shall kindle the thing you would burne at the end of 3 houres , according to your desire : for the rule of proportion sheweth , that if one yard require a quarter or ¼ of an houre , that 12 yards of match will burne out in 3 houres . The like order you may obserue , to answer to any time appointed .

How by Arithmeticall skill you may know what number of men , horses , or oxen , is sufficient to draw any peece of Artillerie , and how much euery one draweth a peece , so as they all draw together .

Question .

If 90 men be able sufficiently to draw a Cannon of 9000 pound weight , accounting carriage and all , I demaund how many men is able to draw a Culuering of 2500 pound weight , and how much euery man drew for his part ?

Resolution .

I answere : If a Cannon of 9000 pound weight , require 90 men , the quotient sheweth me that a Culuering of 2500 weight requireth 25 men to draw the same : and deuiding the weight of the peece to be drawne by the number of men appointed to draw the same , the quotient will shew you how much euery man drew to his part ( to wit ) 100 weight .

To know how many horses is to be prouided to draw any Peece of Ordinance , and how much euery one draweth .

Question .

If three horses draw a Fawcon of 900 weight , how many horses will draw a Culuering of 3000 weight ?

Resolution .

I say as before , if a peece of 900 weight require 3 horses , what will a peece of 3000 weight ? and in working according to the rule , the quotient is 10 , shewing that 10 horses must be prouided to draw a Culuering of 3000 weight . Also deuiding 3000 , the weight of the said peece , by 10 ( being the number of horses ) there will stand in the quotient 300 , shewing the draught of each horse .

To know how many Oxen is to be prouided to draw any peece of Artillerie .

It is to be noted that 3 yoake of oxen is thought to draw as much as three horses , and that 3 yoake of oxen is sufficient to draw a Saker of 1400 weight .

Question .

How many oxen must be prouided for a Cannon of 8000 weight ?

Resolution .

In working as before , I find that 34 oxen , or 17 yoake of oxen , will serue to draw a Cannon of 8000 pound weight . And note that whereas there doth remaine 2/7 parts of a whole number , neither men , horses , nor other cattell , can in any such millitare questions be brought into a fraction , but yet the rule it sheweth that 17 yoake of oxen is sufficient for the draught of a Cannon of 8000 pound weight , when 3 yoake of oxen serue for to draw a Saker of 1400 pound weight .

If you deuide the weight of the whole Cannon being 8000 pound weight by 34 , the oxen appointed to draw the same , the quotient is 235 pound 5/17 : so much did euery oxe draw .

How you may wanting both oxen and horses to draw any peece of Ordinance , know presently how many men is able sufficiently to draw the same , either on plaine or marrish ground .

Question .

I shewed in a conclusion before , that 3 yoake of oxen would draw a peece of 1400 pound weight , and that 90 men would draw a Cannon of 9000 pound weight ; now if there want both horses and oxen , or that you are occasioned to draw the said peece through some marrish ground , whereas horses and oxen cannot passe , I demaund how many men is sufficient to hale a Saker of 1400 pound weight through the said marish ground ?

Resolution .

If a Cannon of 9000 weight require 90 men to draw the same , I find that a Saker , weighing 1400 pound weight must haue 14 men to draw the same , and euery one shall draw 100 weight for his part .

In drawing Artillery through any soft marrish ground it is requisite to haue in readinesse , in the Maister of the Ordinance his carts , which carrieth the prouisions for the Ordinance certaine hurdels of boords , or rather flat bottomed boates , in which any peece of Ordinance may be placed carriage and all , and by force or strength of men may be drawne as easily , as to draw the said peece on the firme land , for that the said boate is apt to slide or swimme on the soft owish , the ropes being made fast to the forestearne or sides of the sayd boates , which boates do serue also for cariage of the Ordinance , and all things thereto belonging , ouer any riuer or soft owish ground , &c.

How you may by the rule afore , know how many oxen will draw any peece of Ordinance , if you want men and horses .

I shewed that 90 men is able to draw a Cannon of 9000 pound weight , and that three yoake of oxen will serue to draw a Cannon of 1400 pound weight : now wanting men and horses , I say if a Saker of 1400 pound weight require 6 oxen , what will a Cannon of 9000 ? and in multiplying the weight of the Cannon by 6 , the number of oxen appointed to draw the Saker , and deuiding that product by the weight of the lesser peece , the quotient is 38 oxen or 19 yoake , so many must be prouided to draw a Cannon of 9000 pound weight , which weight deuided by the 38 oxen appointed to draw the same , the quotient sheweth that euery oxe drew 236 pound weight .

How you may wanting men and oxen to draw any peece of Ordinance , know how many horses is requisite to draw the same .

Also I noted before , that 3 horses would serue to draw a Fawcon of 900 pound weight : I demand how many horses will serue to draw a Cannon of 9000 pound weight ? In working as before , the quotient is 30 , so mamy horses is requisite for that purpose : which peece , her weight deuided by the number of horses appointed to draw the same , the quotient sheweth that euery horse drew 300 pound weight . In this order you may know what number of men , horses , or oxen , is able to draw any peece of Ordinance , and what euery one seuerally doth draw .

How to know how many 100 of Haberdepoize weight any peece of Ordinance , or other grosse weight containeth .

In the conclusions afore set downe , thou must note gentle reader , that euery 100 weight of most things , is accounted after fiue score to the hundreth : but if thou be desirous to know how many hundreth of Haber depoize weight any peece of Ordinance or other grosse weight containeth , thou mayst by Arithmetike soone be resolued , for euery 100 of Haberdepoize weight containeth 112 pound , the halfe hundreth 56 pound , the quarter 28 pound , and the pound 16 ounces : so that deuiding the weight of any great peece by 112 , thou mayst easily know how many hundreth of Haberdepoize weight the same containeth .

I would know how many hundreth of Haberdepoize weight is in a Cannon of 9000 pound weight , I deuide the same by 112 as aforesayd , and the quotient being 80 40/112 , sheweth that a Cannon of 9000 pound weight containes 80 hundreth of Haberdepoize weight , one quarter and 12 pound .

A Tun containeth 2000 of Haberdepoize weight .

How you may proportionally prooue all sorts of peeces of Artillerie for seruice whether they will hold or no.

All peeces that shoote a bullet vnder 10 pound weight , and duely fortified with mettall , being shot 3 times , first with the whole weight of the yron bullet . Secondly with 5/4 partes thereof , and lastly with 3/2 partes of the same , will hold for any seruice , being charged with her ordinarie charge , albeit the said peece were discharged 100 times in one day .

How you may find out the proportionall charge afore named as thus .

Suppose a peece shoote a bullet of 6 pound weight , and that you desire to know what 5/4 partes in powder of the weight of the bullet is : multiply the weight of the said bullet by the numerator 5 , and deuide by the denominator 4 , the quotient is your desire .

Example .

6 multiplied by 5 , is 30 : the same deuided by 4 , the quotient is 7 ½ . The like order you must vse in giuing her 3/2 parts in powder to the weight of the shot , and your quotient is 9 pound .

How to prooue any peece that shooteth a bullet vnder 50 pound weight , and aboue 10 pound weight .

Any peece that shooteth a bullet aboue 10 pound in weight , and vnder 50 pound , would for the first shot be charged with ⅔ parts in powder of the pellets weight : for the second shot with ⅚ partes , and lastly with the whole weight of the bullet .

Example .

Admit a peece shoote a bullet of 40 pound weight , the ⅔ partes thereof is 26 pound ⅔ , and ⅚ partes thereof is 33 pound ⅔ parts .

And note that in proouing any peece of Ordinance , whether she be seruiceable or not , her mouth would be mounted to 20 or 30 degrees of the quadrant , or thereabout .

To know how much one coyler rope , for the draught of any peece of Ordinance is bigger then another , and how thicke any of them is .

Take the compasse of the lesser , and likewise the circumference of the greater , abating the lesser out of the greater , the remaine is your desire , which knowne by the rule of proportion you may find out the height or thicknesse of the lesser .

Example .

Suppose you haue a coyler rope of 6 inches compasse , and another of 4 inches compasse , abating 4 inches from 6 inches the compasse of the greater , rests 2 inches , the diameter or height of the greater : which knowne , frame the rule of proportion saying : If 6 yeeld 2 , what 4 ? the quotient is one inch ⅔ parts , shewing the true thicknesse or height of the lesser .

To know how much one coyler rope is more then another .

Take the compasse of your rope , and multiply it in it selfe , and looke how much you would haue the other greater , augment your product by the same proportion , extract the square roote , you haue your desire .

Example .

A coyler rope of 6 inches compasse squared , makes 36 inches . Now if you would haue one 3 times as much , then multiply 36 by 3 , the product is 108 , the square roote thereof is 10 inches and something better , and so thicke ought a rope to be that is 3 times the compasse of the other .

How by knowing the waight of a faddome of one rope , to know the weight of a faddome of any other .

A cable or coyler rope of 10 inches compasse weighing 16 pound euery faddom , how much will a faddom of that rope weigh , that is 15 inches compasse , and made of the same stuffe ? I multiply the greater in it selfe , ariseth 225 , and that multiplied by 16 pound the weight of a faddom of the lesser rope , ariseth 3600 , the which deuided by 100 , being the square of the lesser rope , the quotient is 36 pound , and so much will euery faddome of the greater rope weigh . In this order by knowing what a faddome of the greater rope weigheth , you may soone find what a faddome of the lesser rope weigheth .

How by knowing the quantity or compasse of any small rope , to find out the same in another that is many times that bignesse .

Admit I haue a small rope of 3 inches compasse , and that it is required to know the height of another that is 5 times that compasse . I square the number 3. ariseth 9 , which multiplied by 5 makes 45 , the square roote thereof is 6 inches ¼ so high is the greater . The like is to be done of all such like demaunds .

To know the weight of a whole coyler rope for the draught of any peece of Ordinance .

Question .

There is a coyler rope of 8 inches compasse weighing 12 pound euery faddome , I demaund the whole weight of that rope being 20 faddome long ?

Resolution .

Multiply the number of faddoms in the rope ( being 20 ) by the weight of one faddome , the product is 240 pound weight , your desire .

The length of a coyler rope for a whole Cannon ought to be 70 faddome or thereabouts .

For an ordinary Cannon 64 or 66 faddome , and for a demy Cannon 60 faddome or thereabouts .

For a Culuering 40 faddome , a demy Culuering 36 faddome , and a Saker 30 faddome , &c.

To find out the superficiall content of the hollow concauity of any peece

If you multiply the length of the cilinder or bore of the peece , by the circumference of the hollow concaue about the mouth , the product will shew you the superficiall content of the cilinder of the said peece .

Example .

A Cannon of 7 inches diameter hauing her concaue or hollow cilinder 12 foote in length , how much is the superficiall content thereof ?

Resolution .

Reduce the length of the hollow concaue of the pecce into inches , ariseth 144 inches , which multiplied by 22 inches , the circumference of the concaue at the mouth of the peece , ariseth 3168 inches , the superficiall content of the mettall compassing the concaue of the peece .

To find out the crassitude or solid content of the cilinder or concaue of any peece .

First you must by the rules taught in the beginning of the booke , find out the content of the base or plaine of the concauity at the mouth of the peece , in multiplying ½ , the diameter in halfe the circumference or else squaring the diameter and multiplying that product by 11 , and deuiding the result by 14 , the quotient will also shew you the content , the which multiplied in the length of the cilinder of the peece , the product is your desire .

Example .

The Cannon aboue named of 7 inches diameter , wrought as is shewed , yeeldeth 38 inches ½ at the base or circular content of her mouth , which multiplied by 144 inches , the length of the cilinder , yeeldeth 8280 inches , the solid content of the concaue of the said peece .

If you desire to know how many foote in square measure the solide content of the empty or hollow concauity of the peece aforenamed or any other doth containe , you must worke thus ; deuide the number of inches in the solide content thereof by the number of inches in a foote square being 1728 , the quotient is your desire .

Example .

The solide content of the peece of 7 inches diameter aboue named , containeth 8280 inches , which deuided by 1728 , the quotient is 4 57/72 , that is 5 feete in square measure wanting 15 inches . The like is to be done in any other peece , or in measuring the cilinder or Cone in any other solide body .

How you may Arithmetically know how much any peece of Ordinances is taper-bored , or whether the same be taper-bored or not .

Put vpon your rammer staffe a tampion of wood , that is iust the height of the hollow concaue of your peece , and thrust the same home into the peece ; if it go not home to the breech , then the peece is taper-bored , if it go home the peece is not taper bored : if she be taper-bored , then put on such a tampion of wood vpon your rammer staffe , as may fill the concaue of the peece in the narrowest part where she is taper-bored , and be sure that it go home to the breech of the peece , and afterwards with your compasses , measure the diameter of either tampion , abating the lesser measure out of the greater , the remaine is your desire .

And note that the tampion at the end of euery rammer staffe , is to thrust home the wad and bullet close to the chamber or place where the powder lyeth , and euery rammer staffe ought to haue a sponge at the one end , to cleanse the peece with , and a tampion of wood at the other end , to put home the bullet and wad with , in the center of which ought to be a hollow screw wherein the Gunner may screw in a wad hooke to vnloade any peece at his pleasure .

How to shoote in any morter peece .

Morter peeces were inuented onely to annoy the enemy , when other Ordinance cannot be vsed against them , as being charged with stone to beate down the houses of the enemy , or to fal amongst men being assembled together , or charged with balles of wild-fire to burne the enemies ships , houses , or corne . To make a perfect shot in one of these peeces , it is requisite you know 2 things belonging to the same ( that is to say ) how farre your morter peece will carry a bullet , or a ball of fire-worke , as she is to shoote at the best of the randon : and likewise how far it is from your peece to the marke you intend to shoot at , which knowne you may make a perfect shot , as thus .

Example .

If a morter peece shoot a bullet or fire-worke 700 paces , and that the marke which you intend to shoote at is but 500 paces ; I demand at what degree of the quadrant , shall the peece be layd at , to make a good shot ?

Resolution .

To answer this and all such like , reason and experience teacheth , that the lesser ground you intend to shoot , you must raise the mouth of your morter peece so many degrees aboue the best of the randon , as is sufficient to reach the marke desired : and therefore I say if 700 paces require 45 degrees of the quadrant , what will 500 ? and the quotient tels me , that at 63 degrees of the quadrant the mouth of the sayd peece must be eleuated at , to cause the bullet or fire-ball to light accordingly .

If you abate 45 degrees ( being the best of the randon ) from 63 degrees , that the peece was eleuated at , the remaine is 18 degrees , & so many degrees of the quadrant was the mouth of the morter peece eleuated at to reach the marke .

To know how farre or short any morter peece will shoote further or shorter , at the mount or dismount of one or many degrees .

Question .

A morter peece that shoots 450 paces at the best of the randon , I would know how much shorter shall she shoote , being eleuated one degree aboue the vtmost range ?

Resolution .

Deuide the distance of the vtmost range being 450 paces , by 45 the degrees in the best of the randon , the quotient is 10 , so many paces will the sayd peece shoote shorter , her mouth eleuated one degree .

How you may know verie neare how farre from your peece the bullet shall light , the sayd morter peece mouth being raised to what degree you thinke good .

Question

Suppose there is a Castell &c. besieged , and that the Gunners had brought their Ordinance as neare as they would wish , so that hauing discharged the morter peece in the former conclusion , at the mount of 60 degrees , they find that the bullet fals in or about the mids of the sayd Castell or Fort. The question is how farre it is betweene the peece and the fall of the sayd bullet ?

Resolution .

You must first seeke what difference of degrees is betweene 60 and 45 , and you shall find 15 , then by the rule of proportion say , if one degree abate 10 paces , what will 45 ? and you shall find 150 paces in your quotient . And in this order by the help of Arithmeticke you may find how farre it is from the peece to the marke .

Also it is possible to shoote so directly vpright in a quiet , faire , and calme day , that the bullet shot out of your morter peece , shal fall into the peece mouth againe or hard besides the same , if you raise the peece mouth iust to 90 degrees of the quadrant , which albeit it be not seruiceable , yet it is possible to be done : For this is a generall rule , that no peece of Ordinance whatsoeuer can shoote a bullet to continue still in a streight line , during the motion of the said bullet , except you eleuate or raise the concaue of the said peece directly towards the zeneth of the skie , or else plumbe downe towards the center of the earth .

The diameter of the chamber mouth in euery morter peece , ought to be equall to the semi-diameter in the mouth of the said morter .

The length of euery chamber in a morter peece , ought to be once and a halfe the diameter of the chamber .

The mettall at the breech of euery morter peece , ought to be fortified equall in thicknesse to the diameter of the mouth of the chamber within , and at the trunions to the semi-diameter , and at the fore-part or necke of the peece , to the ⅓ part of the diameter of the chamber mouth .

To mount a morter peece by the quadrant , some vse to put the rule of the quadrant into the peece mouth , close to the mettall , or inside of the peece , noting at what degree the plummet hangs ; but for as much as there be many morter peeces a little taper-bored at the mouth , ( I meane the diameter at the mouth is something wider then it is within ) therefore it is the best to haue a rule made for the purpose , which among the experienced Gunners is common , the said rule being about 18 inches length , at the middle point or pricke whereof is another shorter rule , framed artificially about a foote long , ioyned close , and falling perpendicularly on the longer rule , whose containing angle lighteth iustly on the middle point or mids of the longer rule , from which point is drawne by Arte the ⅛ part of a circle , and deuided into 45 equall deuisions or degrees , so as the 90 degree stands iust on the center or middle point of the longer rule : so that laying the longer rule crosse the mouth of the peece , you shall presently know at what degre the said morter peece is eleuated at by the plummet , the peece being mounted at any grade aboue 45. And thus may you mount your morter peece , to shoote at what degree you thinke good . The patterne of the rule this figure sheweth , plainely drawne .

The orderly flight or motion of the bullet or fire-ball shot out of any morter peece , by the figure or draught hereunder may be perceiued .

Hauing planted Ordinance vpon any mount or platforme , to besiege any Towne , &c. and that you desire to make some little trench or ditch about the same for the defence thereof , how you may know how much the earth and turfe that is cast out of the said ditch , shall raise a wall in height , being laid orderly at the brim of the said ditch , on the inside thereof , making the same wall to any proportion assigned .

Question .

Suppose the Generall commaund the captaine of the Pyoners , that a ditch be made about the mounts or platformes where the Ordinance plaies , making the same 18 foote in bredth at the brim , 12 foote in bredth at the bottome , and 8 foote in depth , and that the earth and turfe digged out of the said trench be laid orderly in the inside thereof at the brim of the said ditch , so as a wall may be made in bredth at the bottome 12 foote , and at the top 8 foote , I demaund how high shall that wall be when it is finished ?

Resolution .

To worke this , there is a generall rule , ( as thus . ) Adde the widenesse or breadth of the brim , and the breadth or widenesse at the bottome together , the ½ of that addition multiplied by depth of the ditch the product of that multiplication shall be your deuident , or number to be deuided . Now to find the height of the wall , adde the thicknesse of the bottome of the wall which you meane to make , to the thicknesse or bredth that you intend to make it at the head ; the ½ of that addition shall be your deuisore , which deuident deuided by the deuisor , the quotient will shew you the height of the wall .

Example .

The trench in this conclusion is said to be 18 foote broad at the mouth or brim thereof , and 12 foote at the bottome , which 2 numbers being added , makes 30 , the halfe whereof is 15 feete , which 15 feete multiplied by 8 feete being the depth , ariseth 120 feete for my deuident . Likewise , adde twelue foote ( the thicknesse of the wall at the bottome ) to 8 foote the bredth you meane to make it at the head , so ariseth 20 feete , the ½ thereof is 10 feete for my deuisor , ( and so thicke the said wall will be in the mids : the which deuident being 120 , being deuided by the deuisor 10 , the quotient is 12 , and so many foote in height shall the earth and turfe casten out of the trench aforesaid , make a wall being 12 foote broade at the bottome , 8 foote at the head , and 10 foote in breadth at the mids : the said trench being 18 foote broad at the brim , 12 foote broad at the bottome , and 8 foote deepe .

In this order you may find out the height , bredth , or depth of any such like wall or ditch , in making the same after any proportion assigned .

Briefe obseruations of certaine principals in the Arte of Gunnery , for euery Gunner to consider of , to practise and learne , viz.

To know what disparture euery peece of Ordinance ought to haue in shooting either at or within point blanke , or with an inch rule at any aduantage .

To vse a mediocrity in ramming and wadding , and in giuing euery peece her due loading in powder and bullet .

To know the goodnesse and badnesse of powder , and how to mixe and make perfit good powder , and how to fine the peter , &c.

To consider the wind , whether it blow with you or against you , or on any side of the peece , and how to weather your peece to make a good shot .

To consider the platforme , whether it be flat , or else declining for the recoile of your peece , and whether the marke be higher or lower then your platforme , as also to know the distance thereto .

To know whether your peece be truly bored or not , and how to make a perfect shot in a peece that is not truly bored .

To consider whether the one wheele be more glad or reuerse faster vpon the axle-tree then the other , or whether the one wheele stand higher then the other , lest you do shoote wide .

To know whether a short peece will outshoote a long peece or not , keeping the length of the marke by the like degrees of the quadrant .

To know that leuelling with the quadrant towards a hill ( the marke standing higher then your platforme ) you shall shoote short : and shooting into a valley , you do ouershoote the marke , but shooting on a leuell ground you keepe the length with the quadrant , and how you ought to lay your peece to make a perfect shot with ●he quadrant at euery marke .

To know that giuing leuell with an inch rule ( which some call the rule of flat ) it is erronious in shooting in ●eeces of contrary length , as also at seuerall markes : obseruing one method .

To learne to know the distance to the marke , and what distance your peece will shoote at point blanke , or mounted from degree to degree ( which is the best rule to snoote by .

To know whether the cariage or stocke of your peece haue her due length or not , and whether the peece be truly placed therein or not .

To consider that in shooting diuerse peeces from one platforme , to discharge that peece which stands to the ley wards first , and to set your match or fire euer on the ley side , and your powder on the wind hand .

To know the true order in mixing and making all kind of fire-workes , wet and dry .

To know the height and weight of all peeces of Ordinance , and whether the same lye streight in the cariage or not .

To know the height and weight of all bullets of like mettall , and the circumference thereof : and what proportion a bullet of one mettall beareth to the like or vnlike bullet of a contrary mettall .

To know how much Serpentine or corne powder is requisite to charge any peece of Artillery .

To know what necessaries belongeth to any peece of Ordinance , being in seruice by land or sea ; as ladles , sponges , hand-spikes , ropes , coines , &c. and what labourers should attend the same .

To know likewise what men , horses , or oxen , is able to draw any peece of Ordinance in seruice , or on the sudden .

To be circumspect of lighted matches and candles &c. for feare of powder , being in sea-seruice : and to keepe a perfect register of euery thing pertaining to your Ordinance , both what you haue present , and what you haue spent , to keepe your Ordinance drie within , and to haue in readinesse all kind of seruiceable fire-workes , which fire-workes ought to be made either in the boate or on land , but not in the ship for feare of had I wist .

To know the vse of all Geometricall instruments belonging to the profession of a Gunner , as also to haue some sight in Arithmeticke and Geometry , thereby to shoote at all randons , and how to mannage and handle all engines , for the mounting or dismounting of any peece of ordinance , in or out her cariage , &c.

To know that euery peece ought to be as thicke of mettall in euery part from the lowest part of the concaue at the breech , to that part of the chamber that holds the powder , as the bullet due to that peece is in height .

A breuiary of certaine secrets in the Art of Gunnery .

A bullet violently driuen out of any peece of Ordinance by the force of the powder , flieth swiftest and streightest from the mouth , till it be past ½ the distance of the leuell range .

The great noise or rore that the peece makes in deliuering the bullet ( or discharged without bullet ) ariseth betweene the ayre within the peece , violently driuen out into the open aire by the force of the fire ( the Petre or Maister being resolued into a windie exhalation . ) And according to the quantity of the fire and aire , bursting out of the peece , so is the cracke more or lesse .

Any bullet shot out of a peece lying leuell , doth flie more heauily , and worketh lesse effect in piercing an obiect , then when the peece is eleuated at any degree or degrees of the randon .

A heauy bullet violently mouing pierceth sorer then a lighter bullet , hauing the like motion .

A bullet of lead shall worke as great effect against an obiect , as the like bullet of yron , hauing the like motion , by reason of his ouerplus of weight .

A bullet shot out of any peece of Artillery , will pierce more against any thing standing firme , then against a moueable obiect , and shot at an obiect a reasonable distance from the peece , will pierce more effectually , then shot at the same nearer hand .

Euery bullet doth make a long or short range , according to the eleuation of the peece out of which it is shot .

A bullet flieth euer furthest in his streight motion ( or in an insensible streight line ) the higher that the peece is eleuated at the mouth .

Any peece discharged twise with one and the selfe like quantity of powder , wad , and bullet , hauing one and the selfe like proportion in ramming and wadding , and shot at one like degree of randon , the peece of like temper at either shot shall make like ranges , but the sayd peece discharged as afore , but not of like temper , shall make seuerall grazes .

Two peeces in all respects equall , saue onely that the one is something longer then the other , discharged with one like quantity in powder and bullet , shall make seuerall grazes , according to the length of the cilinder of the peece , the longer shall outshoote the shorter .

Two peeces in all respects equall , saue onely in length , discharged at a marke of equall distance from each peece , and being within the range of both peeces , the bullet shot out of the shorter peece , shall graze or beate the marke , before the bullet shot out of the longer peece .

Two peeces proportionall in all respects , being discharged with one like quantity and kind of powder , but differing in bullet , as the one yron : the other lead , and both bullets of like height , shall make seuerall ranges , the yron bullet shall outflie the leaden bullet , but discharged with a bullet of mettall , and afterwards with a like bullet made of wood , obseruing one and the like quantity in powder at euery shot , the bullet of wood shall not flie so farre as the like bullet of mettall .

A peece any whit eleuated at the mouth , will shoote further in an insencible steight line , then lying leuell : and by how much more any bullet is driuen more swifter through the ayre , by so much it is made the more lighter in the mouing or drift thereof .

Two peeces a like in euery respect , shot with one like bullet , but different quantity of powder , shall make seuerall ranges . Also the sayd peeces and bullets equall in all respects , and the powder also in quantity equall , sauing that the mixtures of the sayd powder is not alike , shall make seuerall ranges .

One peece discharged diuerse times with one like bullet , first with the quarter of the weight of the bullet in powder , after with halfe the weight , thirdly with ⅔ parts of the weight , and lastly with the whole weight of the bullet in corne powder , and the ranges differing at point blanke noted , the ranges at the vtmost randon differing , shall be proportionall , one method in charging , &c. being obserued .

To euery peece of Ordinance , according to the proportion of the diameter , length of the cilinder , and weight of the bullet belonging thereto , there is a due quantity of powder to be allowed , so that charging the peece with more or lesse then the sayd due proportion , shall rather hinder then further the bullet in his furthest range .

By how much the mettall of any peece is made hotter by often shooting , then it was before you made the first shot , by so much is the concaue or bore of the peece made more attractiue , the mettall more dulled and the peece worketh lesse effect then in the beginning .

All peeces in whose mettall is mingled most tin , lead , or copper , is more attractiue a great deale then those peeces in whom is put most bel-mettall .

A brasse peece made hote with often shooting , is more apt to breake then when it is cold ; and any peece of Artillery is more apt to breake at the first or second shot in a hard frost being cold , then made hote with often shooting .

Any peece of Ordinance discharged , hauing her full charge in powder , will range and pierce further , then wanting any part thereof ; and hauing a little quantity more then her due charge in powder , will ouershoot the other , but it will daunger the peece ; but doubling the weight of the bullet in powder , shall shoote lesse ground then hauing a meane proportionall charge in powder ( to wit betweene ⅔ parts and the whole weight of the bullet ) for that the cilinder of the peece is too much choked , and the bullet driuen out into the open aire before the powder be all fired .

Euery peece of Artillery ought to haue her conuenient length and weight of mettall , according to the proportion of the diameter or bore of the same , and being made longer or shorter then her sayd due length , will rather hinder then further her vtmost range .

Any peece of Ordinance made hote through much shooting , will neither range so farre , nor pierce so deepe , as being temperatly cold .

No peece of Artillery can shoot a bullet to range still in a perfect streight line , except you shoote the same either directly vpright towards the zeneth of the skie , or else directly plumme downe towards the center of the earth .

The right line of the vtmost randon in all peeces , is more then the right line of the leuell range ; and the right line of the vtmost range , is not so much as the right line of 90 degrees .

The vtmost range in all sorts of peeces , is not at iust 45 degrees of randon , as Tartallia and diuerse others do affirme , but shooting with the wind in a quiet or calme day , is at or about 45 degrees , but the wind against , or on any side , or rough , or the aire thicke , &c. will range as farre at or about 40 degrees .

Two peeces in all respects equall saue onely in length , discharged with a like quantity in powder , wad , & bullet , and shot at a marke within the reach of both peeces , mounted at like degrees of randon with the quadrant , the shorter peece shall outshoote the longer .

The right lines made by any 2 peeces at one degree of randon discharged , are proportionall to the ranges of their bullets at the same degrees of randon , and the right lines made by any 2 peeces at any randon , are proportionall to their vtmost ranges .

Any peece of Ordinance first discharged with the whole weight of the bullet in Serpentine powder , & after discharged with ½ the weight of her bullet , in such corne powder as shall cause the peece to range the same ground : and lastly discharged with halfe the quantity of either sort of powder , the second ranges shall not be equall , although the manner of charging and temper of the peece be all alike .

Three peeces in all respects equall , saue euery one exceeds other in like proportion in length , the vtmost ranges of their bullets shall not be alike proportionall , although the forme of charging be vniforme and alike .

A peece twise charged , first with an yron bullet fit for the same peece , and after with a leaden bullet of the like weight , but differing in height , and with one and the like quantity in powder and wad , at either time the yron bullet shall outflie the leaden bullet .

A peece discharged first with an yron bullet , and after with a leaden bullet of like height , and at either time discharged with the weight of the bullet in Serpentine powder , shall make vnequall ranges .

A peece twise discharged at like degree of randon , first with an yron , and then with a leaden bullet , and after discharged with any other quantity of powder , the ranges of the bullets shall not retaine the same proportion .

If 2 peeces of one length but differing in bore , the one discharged with an yron , the other with a leaden bullet at one like randon , hauing the weight of either bullet in course powder , do range both alike ground , and the sayd peeces after discharged with halfe the weight of their bullets , of the same or any other powder , shall not range one like distance of ground .

Two peeces of one mettall and length , but of different bullets equally mounted , discharged with any like quantity of one powder , shall not range iustly one distance of ground .

The proportion of the different ranges , that yron and leaden bullets make , being found by experience in any one peece of Ordinance , the same proportion will not hold in all other peeces of Ordinance of contrary length , that shootes the same like bullet .

Any peece of Ordinance being thicker of mettall on the one side then on the other , discharged at a marke , will cast the bullet towards that side , that is thickest of mettall .

Two peeces of contrary length , but of like diameter , hauing both one like charge , being shot off at a marke within the reach of both peeces giuing leuell with an inch rule , at one like height of the rule , shall make seuerall grazes , the shorter peece shall outshoote the longer .

Any peece of Ordinance will conuey the bullet more ground , her mouth eleuated at 18 or 20 degrees , then from the sayd grades to the best of the randon , although there be 7 degrees vantage in the latter .

Any peece of Ordinance hauing her due loading will conuey the bullet more then fiue times the distance of her leuell range .

A Table shewing the contents of this booke . A Table of the deminite parts vsed in mensurations . 1. A table shewing how to weigh any great quantity with few weights . 2. How to extract the cubicke radix or roote of any number , and how to find a true denominator to the cubicke remaine , and how to proue if you worke right or not . 4.5 Theoremes , shewing the proportion betweene a bullet of one mettall , to a bullet of contrary mettall , and betweene the diameter and circumference thereof , &c. 8. How by knowing the true weight of any bullet , and diameter of the peece due for the same , to find the weight of any other bullet of like mettall belonging to a contrary peece of Ordinance . 8. How by the knowne weight of any small bullet , you may find out the weight of a greater , and how to proue if you worke right or not . 9. By knowing the weight of any bullet , whose diameter containeth both whole numbers and broken , how to find the weight of any other of like mettall . 10 By knowing the diameter height and weight of an iron bullet , to find the height and weight of a bullet of marble stone : or contrariwise , by knowing the height and weight of a bullet of marble stone , to find the weight of the like bullet of iron . 11. By knowing the weight and diameter of an iron bullet , to find the height and weight of a leaden bullet of the same proportion : or contrariwise , by knowing the weight of a leaden bullet , to find the weight of an iron bullet of like height . 12. To find out the weight of any bullet made of marble stone , by knowing the weight of the like bullet of lead , or else by knowing the weight of any leaden bullet , to find out the weight of a bullet of marble of like diameter . 12. To find out the circumference of any bullet or round body , &c. 13. By knowing the circumference of any bullet , how to find out the diameter thereof . 14. To find the solid content of any bullet or globe . 15. To find the superficiall content of any bullet , &c. 15. To find out the circumference of any circular body diuerse waies . 16. How to find the superficiall content of any round body , as bullet or globe diuerse waies . 17. How to find the crassitude or solid content of any bullet , &c. diuerse wayes . 18. By knowing the diameter and weight of any bullet , &c. to find the diameter of another of like mettall , that is twice the weight of the first . 18. How you may diuerse waies Geometrically find out the weight of any vnknowne bullet , that is double the weight of a knowne bullet , and how to proue the same conclusions by numbers . 19. 20. By knowing the superficiall content of the flat or plaine of any circle , to find out the superficiall content of another , that is twise the diameter of the first . 23. By knowing the weight and height of any one bullet , to find out the true weight of another that is twise the height of the former . 24. How you may Arithmetically find the diameter or height of any bullet , and to know how much any one bullet is higher then another , without any callapers . 25. A table shewing the weight of all iron bullets , from the Fawconet to the Cannon , in Haberdepoize weight . 26. How you may Arithmetically know the true bredth of the plate of any ladle due to any peece of Ordinance , by knowing the diameter of the bullet fit for the peece . 27. How to make a ladle for a chamber-bored peece . 28. To find out the height of the diameter of the chamber in any chamber-bored Cannon , or other peece : and how to find out the thicknesse of mettall , round about the chamber thereof . 29. How you may Arithmetically know whether the cariage for your peece be truly made or not , and how the cariage for any peece of Ordinance ought to be made . 30. By knowing the weight of any one peece of Ordinance , to find the weight of any other . 31. How by Arithmeticke skill you may know how much of euery kind of mettall is in any brasse peece of Ordinance . 33. How to know how farre any peece of great Artillery will conuey her bullet at the best of the randon , by knowing the vtmost range and point blanke of another peece , and by the same rule how you may know how farre any great peece will range at point blanke and vtmost randon . 35. To know how much a bullet of yron will flie further then the like bullet of lead , being discharged the one after the other out of any great peece , with one like quantity in powder . 36. By knowing how much powder is sufficient to charge any one peece of Ordinance , to know how much of the same powder will charge any other peece of Ordinance . 37. By knowing how much Serpentine powder will charge any peece of Ordinance , to know how much corne powder will do the like : or contrariwise , by knowing how much corne powder will charge any peece , to know how much Serpentine powder will serue . 38. By knowing how farre any peece shootes with her due charge of powder , to giue a neare estimate how farre the sayd peece will shoote , with a charge more or lesse in powder then the other . 39. How by knowing how much powder a few peeces of Ordinance hath spent , being but a few times discharged , to know how much powder a great number of the like peeces will spend to be often discharged . 40. How to know how much powder euery little caske or firken ought to containe , and how many of those caskes doth make a Last of powder , and how many shootes any quantity of powder will be for any great peece of Artillery . 41. By knowing how many shootes a firken of powder will make for a Culuering , to know how many shootes a Last of powder will make for a Cannon . 41. To know how many shootes of powder will be in a graund barrell , for any peece of Ordinance . 42. How you may Arithmetically know what proportion of euery receipt is to be taken to make perfect good powder : what quantity soeuer you would make at a time . 43. How by Arithmeticke skill you may know how with one and the selfe like charge in powder and bullet , how much farre or short any peece of Ordinance will shoote , in mounting or dismounting her any degree , whereby you may know how farre your peece will shoote at any degree of the randon , by knowing how farre she will reach at the vtmost randon . 46. By knowing the distance to the marke by the conclusion aboue , you may know whether your peece will shoote short or ouer the marke , or you may know how farre any marke is from your platforme , being within the reach of your peece , onely by knowing the distance of the vtmost range of your peece , and the degrees she is eleuated at . 46. 47. How to make a table of randons , or go very neare to know the true range of the bullet , out of all sorts of great peeces of Artillery , being mounted from degree to degree . 48. How you may Arithmetically know how much wide , ouer , or short any peece will shoote from the marke , by knowing the distance to the marke , and how your peece is layd to shoote at the sayd marke . 51. How to lay your peece to make a streight shot at any marke . 51. In shooting at any marke within point blanke , not disparting your peece , to know how farre the bullet will flie ouer the sayd marke , onely by knowing the distance to the marke . 53. How to lay your peece point blanke without iustrument . 54. How you may Arithmetically dispart any great peece of Artillery diuerse waies . 54. How by Arithmeticall skill you may mount any great peece by an inch rule to 10 degrees of the quadrant , if you want a quadrant or other instrument . 55. How you may know the true thicknesse of mettall in any part of any great peece of Ordinance diuerse waies . 58. 59. How to make a good shot in a peece that is not truly bored , or to know how much any peece will shoote amisse , that is thicker of mettall on the one side then on the other , if you know the distance to the marke : & how to remedy your peece , being thicker of mettall in one part then another , to make her shoote streight . 60. To know the different force of any 2 like peeces of Ordinance planted against an obiect , the one being further off from the sayd obiect then the other . 62. How you may hauing diuerse kinds of Ordinance to batter the wals of any Towne or Castell , &c. tell presently how much powder will loade all those Ordinance , one or many times . 65. How you may know the true weight of any number of shot for seuerall peeces of Ordinance , how many soeuer they be , and how many Tun weight they do all weigh . 70. How any Gunner or gunfounder may by Arithmeticke skill know whether the trunions of any peece be rightly placed on the peece or not . 72. How you may know what empty caske is to be prouided to boy or carry ouer any peece of Ordinance ouer any riuer , if boates or other prouision cannot be gotten . 73. How to know the true time that any quantity of gun-match , being fired shall burne to do an exploite , at any time desired . 76. How by Arithmeticke skill you may know what number of men , horses , or oxen , is sufficient to draw any great peece of Artillery , and how much euery one draweth , so as they all do their indeuor . 77. To know how many hundreth of Haberdepoize weight any peece of Ordinance or other grosse weight containeth . 81. How you may proportionally proue all sorts of peeces of Artillery for seruice , whether they will hold or not . 82. To know how much one coyler rope is more then another , for to draw any great peece of Ordinance . 83. By knowing the weight of a faddome of one coyler rope , to know the weight of a faddome of any other . 84. By knowing the quantity or compasse of any small rope , to find out the same in another that is many times that bignesse , and how to find out the weight of a whole coyler rope , for the draught of any peece of Ordinance . 84. To find out the superficiall content of the hollow concauity of any peece . 85. To find out the crassitude or solid content of the cilinder or concauity of any peece , and how much the same containeth in square measure . 86. How you may know how much any peece of Ordinance is taper-bored by Arithmeticke skill , or whether any great peece of Ordinance be taper-bored or not . 87. A table wherein you may know the names of all peeces of Artillery , their height and weight , and thicknesse of mettall in any part of them , and what men , horses , or oxen , is sufficient to draw the same , and the height , weight , and compasse of the bullet belonging to euery peece : and how much powder will charge euery of the sayd peeces , and the length and breadth of the ladle fit for any peece , and how thicke , broade , long , or deepe , the cariage of euery peece should be , and how long euery coyler rope should be , for the draught of any great peece of Ordinance . 87. Conclusions for shooting in morter peeces . 87. To know how much further or shorter any morter peece will shoote at the mount or dismount of one or many degrees . 88. To know very neare how farre from your peece the bullet shall light , the morter peece raised at what degree you thinke good . 89. Notes to be learned concerning morter peeces . 89. To know how much the earth and turfe that is digged & throwne out of any ditch , shall make a defencible ramper or wall at the brim of the sayd ditch , making the same to any proportion assigned for the better defence of the Ordinance in time of seruice . 93. Certaine briefe obseruations of certaine principals of the Art of Gunnery , to be knowne of euery gunner : with a breuiary of certaine secrets of the same Art , very necessary for all professors of the Art of Gunnery . 94. FINIS .

Notes, typically marginal, from the original text

Notes for div A12531-e6290