The scales of commerce and trade: ballancing betwixt the buyer and seller, artificer and manufacture, debitor and creditor, the most general questions, artificiall rules, and usefull conclusions incident to traffique: comprehended in two books. The first states the ponderates to equity and custome, all usuall rules, legall bargains and contracts, in wholesale ot [sic] retaile, with factorage, returnes, and exchanges of forraign coyn, of interest-money, both simple and compounded, with solutions from naturall and artificiall arithmetick. The second book treats of geometricall problems and arithmeticall solutions, in dimensions of lines, superficies and bodies, both solid and concave, viz. land, wainscot, hangings, board, timber, stone, gaging of casks, military propositions, merchants accounts by debitor and creditor; architectonice, or the art of building. / By Thomas Willsford Gent.

THE THIRD PART, Consisting of Military Propositions.

PROPOSITION I.

By the Diameter and weight of any Bullet known, with the Diameter of another to find the second Bullets weight.

IT is a common received opinion, 〈 math 〉〈 math 〉 that an iron bullet of 4 inches diameter will weigh 9 lb, which if it be true, and that all iron will weigh alike in equal magnitudes then this rule is a positive truth, viz. as the cube of 4 is to 9 lb weight, so shall the cube of any iron bullets diameter, be proportionable to the weight thereof, according to the last Theorem: as for ex∣ample,, an iron bullet, whose diameter is 6 inches,

the cube of it 216; so the proportion is as 64 is to 9 lb. so will 216 be to 30 ⅜ lb, as in the table is e∣vident, which ⅜ is 6 ounces.

PROPOSITION II.

By knowing the weight of two bullets, and diameter of one to find the other diameter.

For illustration of this Propositi∣on, 〈 math 〉〈 math 〉 I will reverse the last question, viz. if a bullet of 9 lb weight shall contain 64 inches, in the diameters cube, then a bullet 30 ⅜ lb or 243/8 will require 216 inches, whose cubique root is 6 inches for the bul∣lets diameter; these 2 examples are sufficient for any question of this kind: but observe, if by the di∣ameter of the guns concave you would find what the bullet belonging unto it will weigh, the diame∣ter of it must be ¼ of an inch less then the diame∣ter within the muzzle, although it be not a taper bor'd gun.

To find what thickness they are in metal, their Cylindars, concaves with their bullets diameters; Galaper Compasses are held the best for expedite∣ness, especially those that open with a quadrant divided proportionally in inches, and to 1/10, com∣monly known to every Engineer: as for propo∣sals of this art, there be divers books extant, to which I refer you, they belonging more to the pra∣ctise then any Theory; besides doubtful queries are made by them, viz. as whether the quantitie of

powder can be proportioned by Arithmetick to the weight of bullets, or whether they move in a right line, or circular; or if a Canon be more for∣tified in metal upon one side then the other, wherefore the gun discharged shall convey the bullet wide from the mark, and the concaves cy∣lindar incline to that side on which the metal is thickest, because most resisted, or wherefore a piece of great Artillary mounted at 18 or 20 degrees of the quadrant shall convey a shot the farthest, and almost twice the level range, also how good pow∣der is known; all which must be referred to expe∣rience. This we know, that the sulphur makes it quick to fire, the Charcole maintains it, and the Salt-peter turning into a windy exhalation by re∣percussion of the aire, causeth such violent effects to amaze the world, as if ambitious to imitate the thunder and lightning, from which good Lord deli∣ver us. This tract I will leave, and return to such Propositions as may be exactly peformed by Arith∣metick, and founded upon Demonstration.

Compendious Rules for marshalling of Souldi∣ers in any rectangular form of battel, either in one body, or in several Squadrons or Re∣giments.

Definitions.

Battels are considered in two several respects, one depending upon the number of men to be put into a Military array, the other reflects upon the ground, on which the Battalios are to be orde∣red.

A square battel of men hath an equal number both in rank and file, yet the ground in such cases lon∣ger on the file then upon the rank.

A square battel in respect of the ground hath the rank and file equal in length, yet the number of men in rank exceeding those in file.

In respect of the men to be drawn forth in bat∣talio, it is either termed a square battel, or in pro∣portion, as the men in rank to the number of those in file.

PROPOSITION III.

If a square battel of men be required of any number whatsoever, the Quadrat root extracted from the list, or number of Souldiers delivered in, shews the number to be marshalled either in rank or file: As for example, a Serjeant Major delivers

in a list of 22500 souldiers to be ordered in a square battel of men, the quadrat root of that number is 150, and so many must there be placed in rank, and so many likewise in file, lib. 2. parag 1. examp. 2. Arith.

PROPOSITION IV.

If the difference of the men in rank to those in file should be in any proportion required, observe these Rules.

RULE 1.

As the term which is given for men in file

Shall be to the term propounded for the rank,

So will the number marshalled in this Array

Be in proportion to the root, or men in rank.

RULE 2.

As the term propounded for men in rank

Shall be unto the term which is for the file,

So will the whole number of souldiers marshalled

Be in proportion to the square root for those in file.

As for example, 〈 math 〉〈 math 〉 20184 souldiers are to be ordered in battel of array, & in such propor∣tion between the rank and file, as 8 to 3, that is as 8 men in rank for 3 in file therefore 20184 (the number of souldiers)

multiplied by 8 and divided by 3, the quotient will be 53824, the quadrat root of it will be 232, as in the first table, for the number of men to be placed in rank, the number for the file is found, if you di∣vide 20184; by 232 the quotient will be 87, or by the second rule to find the men in file; as 8 the term for the rank unto 20184, the number of soul∣diers, so will 3 the term for the file be in a direct proportion unto 7569, the quadrat root will be 87 for the number of men in file, according to the se∣cond table in the margent.

PROPOSITION V.

To marshal in battalio any number of Souldiers, when there is a double proportion stated, as in respect of the men and ground both for the rank and file.

The RULE.

As the product of the two terms for the Rank

Shall be in proportion to the number of Souldiers,

So will the product of the two for the File

Be to a fourth number, whose square root is the File.

For the illustration of 〈 math 〉〈 math 〉 this Proposition, sup∣pose the number of sol∣diers to be marshalled near 41160 in this or∣der and proportion, viz. as for 3 in rank, 7 in file, and in respect of the ground as 2 is in propor∣tion to 5; the products of these terms are 6 and 35. then say as 6 is to 35, so will 41160 be unto 240100, whose square root will be 490, as in the first table, with which divide the list of souldiers given, viz, 41160, the quotient will be 84, the true number of men both in rank and file, or by the second table, as 35 is to 6, so will 41160 be in proportion to 7056, the root 84; or with this di∣vide the list of souldiers, the quotient will be 490, as before, and so in all questions of this kind, by the rank found you may find the file and the con∣trary.

PROPOSITION VI.

If an Army were drawn out in their particular Re∣giments, and those again divided into several squa∣drons with their depth and proportion both in rank and file.

This Proposition (although of most use) depends upon the former; for having the number of Regi∣ments, or lift of the army, they may be reduced into little squadrons, as the Maj. Gen. shall think fit, and then marshal those according to order in what proportion shall be required, betwixt the rank

and file, by one of the 3 last Propositions, of which I have given you examples, not according to custom, or the Military Discipline practised in any place, but whereby you may solve any question of this kind, and not as precedents, but rules onely; for the Foot squadrons 10 deep is the most that I have heard of, the usuall custom in Europe is 6 deep for the Foot and 3 for the Horse, when they charge the Enemy.

PROPOSITION VII.

For the incamping of Souldiers in their severall quarters.

For a quartering of Souldiers in the field, it is performed by the common rule of Three; as for example, suppose a Regiment of 1000 men may be quartered in a square of ground containing 20 per∣ches, what shall the side of a square be to lodge a greater or lesser number; the proportion will be, as 1000 to 400 perches, so shall any number of soul∣diers be to a proportional square of ground, whose quadrat root is the side required: and for example, admit the number of souldiers were 24000, then say as 1000 men is to 400 the square of 20 pole, so 24000 men wil be in proportion to 9600, the quad. root of it in a decimal fraction is 97 ••/10 perches, the side of a square that will incamp those men accor∣ding to the proportion given: but here are sundry occurrences to be consulted on, which must be re∣ferred to the experienced Master de Campo to mar∣shal up together, as in respect of the enemy, the

Campanio, the advantage of ground, the securing of passages; and multitudes of other things to be considered, in preserving the Army so well as when ingaged in fight, by reserves, or how to draw off and make retreats, &c. depending more upon the practise then any Theory, or prescription of Rules.

PROPOSITION VIII.

The perpendicular height of any Tower or other place being given, to find at any distance (appointed from the basis thereof) how long any scaling ladder or rope extended must be to reach the top or summity of it.

According to the state of this question, the rope or ladder will include a right-angled triangle, and by the second Probleme of this book, the quadrat root extracted from the summe of the squares made of the two containing sides will be equal to the Hypothenusal, which is the ladder or length of the rope: as for example, admit there were a Turret in height 45 feet, there was a Moat before it in breadth 22 feet, the square of 45 is 2025, and the square of 22 is 484, the summe of these squares is 2509, the quadrat root of it is 50 feet for the length of the ladder or rope that will reach unto the summity or top of it, if the remainder had been considerable, you might have extracted the root with a fraction, as in lib. 2. parag. 1. examp. 4 or 5. Arith. but some (where ignorance hath got the up∣per

hand of their reason) will say (peradventure) what care they for this; give them rope enough; and so say I with all my heart.

PROPOSITION IX.

To find the height of an accessible Fort, Turret, or any other place, by a common square, or with two sticks of equal length artificial joyn'd together at right angles.

[illustration] diagram

Admit the height required were the Tower C D, I move my station from F towards D, holding the triangle or square parallel with the ground-line and perpendicular by help of a plummet, as at K, where by both ends of the little square I behold the Towers summity, as at C. Now by the 19 Proposi∣tion of my Geometry, A B must be equal to B C, and A B or L D is found by measure 48 feet, the t••ue height of B C, to which adde B D or A L (the

height of the square above ground) viz. 3 feet, the summe 51 feet, for the altitude of C above the Ho∣rizontal plain F D, the proposition answered.

PROPOSITION X.

To finde the distance unto any Fort or place, al∣though not accessible, yet discovered by this square or triangle.

Erect a staff perpendicular, whose height is exactly known, as in the last Scheme E G, which admit 6 feet, or 72 inches; upon the top of it cut a notch, so that the square may fall down in it something straight, yet so as to turn at E. suppose the distance required were G D, place your eye at E, then turn the square upwards or downwards, un∣till by the edge of it you see the basis of the Tow∣er, or place at D. the square being fixt, look down from E to F. at which place a mark upon the ground, and measure the distance F G, which is he••e 8 inches: here you have 2 equiangled trian∣gles, viz. G E F, and G E D, and by the 19 pr••∣position of my Geometry the sides are proportio∣nal: now admit this little triangle were delinea∣ted in the greater, as G H L, then is G L equal to G E, and G H to G F. thus are they in the rule of proportion, as G H 8 inches is to G L 72 inches, so will E G 72 be in proportion unto G D 648 inches or 54 feet, the true distance required.

PROPOSITION IX.

To finde the height of any place approachable by the shadow which it makes, with the help of a Pike erected perpendicular to the horizontall plane, or by any Tur∣ret, whose height is directly known, or by the height of any Tower, to finde the distance, though not approa∣chable.

[illustration] diagram

The height of the Tower A B is required, to be found by the shadow which it makes upon the Ho∣rizontal place, as in this figure; suppose B D by measure found to be 12 per 6 3/10 in. upon the same horizontall plane, I measure the shadow of some other body, or erect a Pike perpendicular, as C D, whose height above ground is 7 feet, and the length of the shadow which it makes extends it self from

D to E, by measure 12 feet 3 ⅓ inches: this known, the proportion is, as E D 12 feet 3 ⅗ inches is to C D 7 feet, so the shadow B D 198 feet 6 ••/10 inches unto 112. 98 for the height of the Tower A B, which caused the shadow, that is 113 feet 3 in∣ches and more, much exactness is required (in que∣stions of this nature) or else little truth to be expe∣cted, and shadows commonly falling in broken parts, which made me herein use the Decimals, yet with more exactness performed by Natural Arith∣metick, and vulgar fractions, and so found 112 F. 11 inches 28/100, if the height A B had been known 113 feet, and the distance or extent of the shadow B D required, the proportion would have been, viz. as C D 7 feet to E D 12.3, so will A B 113 feet be unto 198 feet 5 4/7, that is 198 feet 6 inches and ••/7 for B D, the distance required, which is very near the truth.

PROPOSITION XII.

To discover the altitude of an accessible place by a mirrour or looking-glass, or by a Towers height known to find the distance unto it.

Let the position of your Glass or Mirrour be ho∣rizontally plac'd at some convenient distance; from thence go backward into a direct line, untill you can descry in the glass the top of the Tower, or object whose height is required; then will the di∣stance from your body to that part of the glass (where the summity of the Turret was represented) be in proportion to the distance from the glass to the ••er••endicula•• basis of the Tower or Sconce, as

the height of your eye is to the perpendicular height required; for by the Optick Science it is an apparent Maxime, that the angles of Incedence and Reflection are equal, as A D B, and F D E, and your body being parallel with the Tower, the Ra∣dius of your sight incloseth a triangle, equiangled with that of the Turrets shadow, as by the 8, 9, or 10 Proposition of my Geometry, and consequently by the 19 proposition of the same book those tri∣angles are proportional in all their sides; this is so visible that it needs no explanation, if they can see themselves from their shadows, or shall ever be∣hold my Trigonometry, to which I refer them for 2 more ample satisfaction; and this to their im∣partiall and judicious censures, yet wishing a legal trial to answer unto my charge (if there shall be any fomented) in the mean time, hopes of a can∣did construction from a serene verdict free from all obstructions of malice to obtenebrate my intenti∣ons, bids me with comfort to proceed.

PROPOSITION XIII.

A Captain of a Castle expecting to be beleagured, makes good his out-works, and having fortified those best where he conceived most danger of be∣ing stormed; he over-looks the inventory of his Magazine, and takes a list of his Souldiers, with the supernumerary persons, in all 800. by which he findes his provision of victuals good but for 3 moneths & 3 weeks, that is 105 dayes: having more

men then were necessary, and expecting no relief under 6 moneths, or 168 dayes, the question is' how many men must be dismiss'd this fort (before the enemies approach) whereby the same victuals might last the just time required?

The Solution.

The rule thus stated, accor∣ding 〈 math 〉〈 math 〉 to lib. 2. parag. 8. Can. 9. Arith. in a reverss'd proporti∣on, viz. as the provision of victuals for 105 dayes allowed for 800 men, what number will 168 dayes require, allowed in the same proportion, which according to the rule, will prove 500 men, as in the operation of the margent is made evident, and consequently there must be 300 ment dismiss'd of the supernumeraries, ••nd Souldiers not able to perform their duties, or of those least serviceable to defend the works.

PROPOSITION XIV.

The Castellain commanded the Master of the great Artillary (or chief Gunner) to render a strict account of all the guns (mounted upon this Fort-royal) whether offensive or defensive, with the dia∣meters of each bullet belonging to every piece of Ordnance, with the weight of the said bullet, and quantity of powder; also the distances in Geome∣tricall paces, that each piece will convey the shot, so laden, both at point-blank, and at the utmost randome; which accordingly was thus delive∣red in.

The Table or Inventory.

The number of Guns	The names of this Ar∣tillary	Each bullets diame∣ter	Every bullets weight	The due charge of pow∣der.	The distan∣ces in paces at pointblank and random
10	Canon	7	48 15/64	26 lb	340 & 1600
10	Dem. can.	6	30 ⅜	18 lb	350 & 1700
8	Culvering	5	17 37/64	15 lb	420 & 2100
8	Dem. culv.	4 inch.	9 lb	8 lb	320 & 1600
6	Sakers	3 ½	6 15/512	5 lb	300 & 1500
6	Minions	3	3 51/64	3 ½ lb	280 & 1400
4	Faulcons	2 ½	2 101/512	2 ¼ lb	260 & 1200
4	Faulconets	2	1 ⅛	1 ½ lb	220 & 1000

PROPOSITION XV.

There are in this Fort 56 pieces of great Artilla∣ry, as are specified in the Table, viz, the whole Canon hath a bullet of 7 inches diameter, in weight 48 lb and 4 ounces almost; to which there is al∣lowed for every shot 26 lb of powder: this Gun discharg'd will carry on the level-range 340 paces, and shot at random 1600. and note that all pieces for battery ought to be planted within ½ or ⅔ parts at most of the paces they carry point-blank; but as for our present purpose, the magazine of powder was found here 10 T. 11 C. 4 st. and 12 lb weight, and the question propounded is, whether this quan∣tity of corn-powder will discharge the 10 whole Canons 20 times round, the 10 Demi-canons 30 times, the 8 Culverins 40 times, the 8 Demi-culv. 50 times, the 6 Sakers 60 times, the 6 Minions 70 times, the 4 Faulcons 80 times, and the 4 Faulco∣nets 100 times.

The Solution.

Reduce first the gross weight of powder deliver∣ed into pounds subtil, and you will find 21100 gross to be 23632, to which adde 4 st. and 12 lb, that is 68 lb, the total is 23700 lb.

(1)	(2)	(3)	(4)	(5)	(6)
20	10	Cannon	26	260	5200
30	10	Dem. Can.	18	180	5400
40	8	Culverin	15	120	4800
50	8	Dem. culv.	8	64	3200
60	6	Sakers.	5	30	1800
70	6	Minions	3 ½	21	1470
80	4	Faulcons	2 ¼	9	720
100	4	Faulconets	1 ½	6	600
450	56	Totals	79 ¼	690	23190

In the first column stands the number of charges impos'd upon every Gun, in the second the num∣ber of each piece, in the third are inscrib'd the Ord∣nance, in the fourth each particular charge, in the fifth is placed the whole quantity of powder that charges all the guns of each sort, the sixth and last column contains the whole quantity of powder ac∣cording to the number of their several charges, whose totall is 23190 lb.

PROPOSITION XVI.

The last Proposition was not judg'd convenient, being but 510 lb of powder remaining: upon which, by order there was deducted from the Ma∣gazine 3000 lb, viz. for small shot, for Grana∣does, for murdering shot (in case there should be any breaches made) for wast and priming powder, the query next stated was how many shot about

the 20700 lb will make, which according to lib. 2. parag. 11. Arith. may be thus stated: The fifth column (in the last table) contains the quantity of powder that charges all this great Artillary once round about, whose total at the bottom of the same column is 690 lb for the first number, the magazine or whole stock of powder is the second number, viz. 20700, which in this may be total∣ly divided by the first, that is by 690. so the first and second numbers are now 1 and 30. each par∣ticular for every species is comprehended in the fifth column, to a proportional allowance, that each piece shall spend, being once discharged, and then the fourth proportional number found shall be the quantity that every kind shall spend at an equal number of shot to be made, whose total (if the operation be true) shall be equal to the second number in the rule, or the Ammunition delivered in for this purpose, as by the following rule is made conspicuous.

	260		7800
	180		5400
As 690 lb is to 20700, so	120		3600
	64	unto	1920
or reduced	30		900
	21		630
As 1 in proportion to 30, so	9		270
	6		180
	690	totals	20700

By this it is made aparent the Canon must be allow'd 7800 weight of powder, whereof there are 10 in number, so each whole Canon must have 780 lb, which divided by its proper charge, viz. 26 lb, as in the fourth column and former table, the quotient will be 30 shot, again, for the 4 Faulcons there is allowed 270 lb, so for one of those guns 67 ½ lb, which divided by its allowance of powder the quo∣tient will shew 30 charges; and so many shot eve∣ry piece of Ordnance will make round with the al∣lowance according to the last table. A Castle that is fortified both by is Nature and Art, provided with Ammunition, Mann'd, and victualled well, and all things necessary for a defensive War closely be∣leaguered; if the men stand sound, and yet sur∣render it to the enemy before 6 moneths being ex∣pired, it will be conceived the souldiers are better fed then taught.

Ladles for guns are proportioned according to the bullets, the plate made plain at first, and in breadth ⅗ parts of the bullets circumference, the ⅖ abated, whereby to empty the Ladle in the Guns chamber.

Generall Rules and Observation, or a Contexture of various questions.

Iron bullets are proportioned to Lead, made in equal moulds, as 5 to 7, and iron to equall bullets of marble, 7 to 3, the proportioning of their weights is uncertain, when bullets of iron will dif∣fer, though cast in the same mould, one metall be∣ing more pory then another.

The Demi-culvering bullet 4 inches diameter, is generally received as a gage for the rest, whereby to find their weights or magnitudes: this bullet made of some iron, will be just 9 lb weight, and it is a medium almost betwixt the least and greatest sort of Guns upon carriages usually made, yet I have seen and measured one, the diameter of whose concave Cylinder was above 20 ¼ inches, the cube of the bullet 20 inches is 8000; then say as 64 is to 9 lb, so will 8000 be unto 1125 lb for the weight of such a bullet.

Gun-founders of brass pieces use an allay of cop∣per and tinne, proportioned to every 100 lb weight of brass, but the mixture various; which you may find in any piece of Ordnance, having the true weight of the gun and the allay, as thus; sup∣pose a Cannon to weigh 7000 lb subtile, and the allay for every 100 lb of brass 40 lb copper, and 10 lb tinne, then state your question in form, according to the rules of society, as in lib. 2. parag. 11. Arith. as thus:

	lb			lb
As 150 is to 7000, so	100	Brass		4666 ⅔
or reduced by 50	40	Copper	unto	1866 ⅔
As 3 shall be to 140, so	10	Tinne		466 ⅔
	150	totals		7000

All Guns more fortified with metall on the one side then on the other, if discharged at a mark, the bullet will fall wide from the object, inclining to that side which is most fortified or thickest in me∣tal: the reason (I conceive) is, that the thinnest part is soonest hot (by the agility of the fire) and so from thence dismisses the bullet with the greater force, or else in imitation of sulphurious Meteors fir'd in the wombs of clouds, break forth in their deliverance with amazement to mortals, and strikes most at that which is strongest, or most fortified to resist.

Two pieces equal in all things, but their length, and if charged and levelled alike, the longest will convey its bullet farthest; yet if discharged toge∣ther at a mark within distance of either gun, the bullet from the shortest piece will be at the place first, or the object aimed at.

Three Cannons discharged

G	lb	G
3	156	10
2		30
6	156	300
Facit		7800

twice spent 156 lb of powder, how much will 10 equal guns to them spend? if 30 times dis∣charged with the same allow∣ance, which stated, as in the margent, according to lib. 2. pa∣rag. 10. quest. 3. Arith. or may be reduced to 1. 26. 300, or thus, 1. 156. 50. the fourth proportionall found will be 7800 lb.

Most guns will shoot at random 4 times so far and more then their level-range, and some of the great Artillary 5 times; the best random of a piece is held when elevated 22 or 23 degrees of the qua∣drant above the horizontal plain.

Alwayes observe to keep your link, stock, match or fire, to lee-ward of the gun or powder.

An iron bullet will flie farther then one of lead, but the greatest batters most at 80 or 90 paces; and either of them with most force from a gunne a little elevated, then on the level range, although within distance, and the heavier bullet will raze a work the soonest.

No bullet from a gun, that is levelled and dis∣charged, does move in a direct straight line, but cir∣cular, ascending first with the violence of there, and over-shoots the mark within the level-range; and as the heat lessens it tends towards the seat of gravity, and at point-blank crosses the line of level, protracted from the center of its concave cylindar, which arch is greater accordingly as the gun is ele∣vated from 1 to 45 degrees of the quadrant, and lesser if discharged below the level-range.

All guns, if over heated with often shooting, are apt to break; those perpetrated with cold and fro∣sty weather are most subject to an eruption at the first shot; the reason is, that in all metal there is a radical humor, which connexes and keeps the parts together, and is made weak by being dilated with over-much heat, or contracted with too much cold, leaves the parts enervated, and each mem∣ber of that body dissoluble, or easily discorporated, and the sooner by opposition of its contrary, the agile and penetrating fire invading the condensed cold.

Any bullet discharged from a gun does strike most violently against that which is hard, firm and strong to resist, and soonest deaded where it wants opposition, as being shot against wooll, sand, any soft earth or moveable object, and hath more vio∣lence at reasonable distance then near the Cannons mouth which delivered it unto a convoy of the subtile Air; the greatest force of any bullet for battery, is generally conceived from ¼ to ½ of the level-range of that Piece which made the shot, and from ••/2, the force of the bullet lessens in its raptile or violent motion.

The weight and content of Caskes for Powder.

A Barrel or empty Firken ought to weigh 12 lb, and should contain 100 lb of powder neat; so the weight of a Firken thus filled is 112 lb, that is 100 grosse, and 24 of such Firkens makes one last containing 2400 lb neat.

The common ingredients and quantities in making of Powder.

The ordinary Powder is composed of these 3 Simples, viz. to every lb of Salt-peter adde ¼ of Charcoal & ⅙ part of Sulphur: To these take any 3 proportional numbers at pleasure, according to the 15 Axiom, lib. 2. parag. 7. or 14 Arith. As for ex∣ample, 60, 15, 10, or 12, 3, 2, the least without fractions is usually best, as the least trouble, and ad∣mit one Last of Powder were the quantity to be made, as 2400 lb, the proportion will be thus:

lb	12	Peter	1614 3/17
As 17 is to 2400—so	3	Coale	423 ••/17
	2	Sul••hur	282 6/17
	17	Totals	2400

As for the quality and goodnesse of Powder, ex∣perience is the best tutor; yet as a common and general observation, good powder will be bright in colour, tart upon the tongue, and very salt in tast, apt to burn, and quick in being fired: the Brim∣stone makes it apt to kindle, the cole continues the inflammation, by which the Salt-peter is resolv'd into a windy exhalation, and strives to dilate it self restrained in the concave of the gun, vents it self at the mouth of the Piece, being the easiest passage; but if the bullet be rusted in, or over charged, and cannot get out, it will force a passage through the weakest part, as subterranean Meteors do when

much rarified, and restrained in the concaves of the solid earth.

PROPOSITION XVII.

If a barrel of powder will charge a Demiculve∣rin 12 times, burning 8 lb at a discharge, how ma∣ny shot wil a Last of powder make for a Canon that spends 26 lb at every charge?

This question is sol∣ved, 〈 math 〉〈 math 〉 as in lib. 2. parag. 10. quest 4. Arith. in the first rule; stand the quantities of Powder propounded, and the Shot made in the Demi∣culverin in a proportion direct; the second is each charge of powder reverst, and by their products made direct, as against III. in the last it is reduced to 13. 6. and 192. these by the common rule of 3 will produce 88 8/13. so in a Last of powder there will be 88 shot good for discharging the Cannon, and the 8/13 is 16 lb of powder over, for pri••••ing and waste, &c. the question answered, the thing r••qui∣red.

PROPOSITION XVIII.

There is a Rope 3 inches in compasse, and one 4 ••imes so big is required: the greatnesse of these is ••ccording to the squares made of their diame∣ters, as in the Problemes of this book, their cir∣cumferences

being also in the same proportion; so the square of 3 is 9, which multiplied by 4 produ∣ceth 36, the quadrat root of it is 6 inches, the cir∣cumference of the Rope required.

PROPOSITION XIX. By knowing the weight of a fathom of any Rope, to finde the weight of another either greater or lesser.

A Rope in compasse 4 inches, and every fathom of it admit does weigh 3 lb, how much shall a Coy∣ler Rope weigh that is 6 inches in circumference; which two circumferences, if multiplied by 7, they would retain the same proportion: and so likewise if those products were divided by 22, as in lib. 2. parag. 7. axiom 13. Arith. then institute the rule of Three, with the circumferences given and squared, viz. as 16 shall be to 3 lb weight for a fathom of that Rope, so will the greater square 36 be in pro∣portion unto 6 ¾ lb, that is 6 lb and 12 ounces for the weight of each fathom of that Coyler rope, whose circumference was 6 inches, the thing requi∣red. Many such propositions the Gunners do us••, which for brevity I here omit, supposing these m••y suffice young practitioners, so with strong hopes, and a slight fortification, I will conclude this work.

PROPOSITION XX.

A Mount or Plat-form is to be raised for battery, on which the great guns are to be mounted; the General commands the Captain of the Pyoners to draw a trench about it, as he and the Engineers should conceive convenient; which according to order was thus design'd: the Plat-form set out 4 square, 70 ••aces on every side: at the line or verge of this Trench (where the labourers first break ground) 16 feet over, to be 10 feet broad at the bottom, and 8 feet deep; the turf to be orderly laid at the brim, and the earth digg'd out of the Trench disposed of within that, for a wall to raise the Ordnance, and defend the men within the Wo••ks; which wall is ordered to be made 21 feet at bottom, and 18 feet broad at the top. The query is, how high the wall will be made of the earth digg'd out? and how many cubical yards is in the said Trench? and what the labourers work may be worth if paid by the great, or task-work.

The breadth of this Trench at the brim is 16 feet, at the bottom 10, the summe 26 feet, the half 13, which multi••lied by 8 feet (the depth of the said Trench) the product will be 104 superficial square feet; the wall to be made is to be in thickness 18 and 21 feet, the Arithmetical medium 19 ½ or ••••/2, a•• in lib. 2. parag. 5. theo••em 1. Arith. with which divide 104, the quotient will be 5 feet 4 inches, the true height of the wall required.

The square of this Platform is 72 Geometrical ••aces, that is, 350 feet, and the 4 sides contains

in extent 1400 feet, which multiplied by the square made of the Trenches breadth and depth, as before found 104 feet, the product will be 145600 cubical feet; at each corner of these Trenches there will be a Pyramidal Segment reverst with the greater end upwards, whose mean square is 13 feet, as is the Trench, the quadrat of it 169, which multiplied by 8 the depth, produceth 1352 cubical feet, 4 times that (for the 4 cor∣ners) will be 5408. this added to the former summe 145600, the total will be 151008 cubi∣cal feet, which divided by 27, the quotient proves 5592 cubical yards contained in the whole trench; which at 6 D the yard to dig and carry on to the works comes unto in money 139 L 16 ss, the true manner of measuring a Segment, and likewise the fraction in the last division was neglected, as unnecessary in these gross works. Each of these Segments contains 50 cubical yards of earth, which may raise a Rampire, Sconce, or Bulwark at each angle of the Platform 6 feet higher, 16 feet square at the bottom, an•• 14 f••e•• a•• the top.

[illustration] diagram

As for the marshalling and quartering of Souldi∣ers, with sundry other Military Propositions, I have here instated proposals, and delivered Examples for speculation onely, and transferred the forme to the judgement of experienced Commanders; since most Propositions (depending on this Subject) are undeterminable, but according to the custome of the Country, the advantage of the Place, the num∣ber of Horse or Foot, the Enemies condition, with multitudes of occurences intervening every day, & those circumspectly to be cons••derd by the field-Officers, or Council of war sitting upon this Tragick Scene, as Germany hath learned by sad experience under the Swords tuition this later age, whose Di∣sci••les have been generally Separates, oppugnant in opinion, yet united and armed with factions have commenced War under specious colours to

procure Peace, oppress'd the Truth to support Reli∣gion, supprest Kings to establsh Monarchy, and by rude Anarchy, pretending to introduce Civility, with divers such zealous Paradoxes by an Hyperbo∣lical Faith. But leaving all to God, whose De∣crees are inscrutable, his Wisdome infallible, his Justice certain, his Mer••y without limit; Infinite and Omni••otent in all his works. To whom be all Honour, Praise and Glory world without end.