Mathematical Instruments
Book V. Ch. IV.

# Of the Construction and Uses of Instruments for Gunnery.

Instrument is made of two Branches of Brass, about six or 7 Inches long when shut, each Branch being four Lines broad, and three in Thickness. The Motion of the Head thereof is like that of the Head of a two-Foot Rule, and the Ends of the Branches are bent inwards, and furnished with Steel at the Extremes.

There is a Kind of Tongue fastened to one of the Branches, whose Motion is like that of the Head, for raising or lowering it, that so it’s End, which ought to be very thin, may be put into Notches made in the other Branch, on the Inside of which are marked the Diameters answerable to the Weights of Iron Bullets, in this Manner: Having gotten a Rule, on which are denoted the Divisions of the Weights, and the Bores of Pieces (the Method of dividing which will be shewn in speaking of the next Instrument), open the Callipers, so that the inward Ends may answer to the Distance of each Point of the Divisions shewing the Weights of Bullets: And then make a Notch at each Opening with a triangular File, that so the End of the Tongue entering into each of these Notches, may fix the Opening of the Branches exactly to each Number of the Weights of Bullets. We commonly make Notches for the Diameters of Bullets weighing from one fourth of a Pound to 48 Pounds, and sometimes to 64 Pounds. And then Lines must be drawn upon the Surface of this Branch against the Notches, upon which must be set the correspondent Numbers denoting the Pounds.

The Use of this Instrument is easy, for you need but apply the two Ends of the Branches to the Diameter of the Bullet to be measured; and then the Tongue being put in a convenient Notch, will shew the Weight of the Bullet.

There ought always to be a certain Proportion observed in the Breadth of the Points of this Instrument; so that making an Angle (as the Figure shews) at each Opening, the Inside may give the Weight of Bullets, and the Outside the Bores of Pieces; that is, that applying the outward Ends of those Points to the Diameter of the Mouths of Cannon, the Tongue, being placed in the proper Notch, may shew the Weights of Bullets proper for them.

## Construction of the Gunners Square.

This Square serves to elevate or lower Cannons or Mortars, according to the Places they are to be levelled at, and is made of Brass, one Branch of which is about a Foot long, 8 Lines broad, and one Line in Thickness; the other Branch is 4 Inches long, and of the same Length and Breadth as the former. Between these Branches there is a Quadrant divided into 90 Deg. beginning from the shortest Branch, furnished with a Thread and Plummet.

The Use of this Instrument is easy, for there is no more to do but to place the longed Branch in the Mouth of the Cannon or Mortar, and elevate or lower it, ’till the Thread cuts the Degrees necessary to hit a proposed Object.

There are likewise very often denoted, upon one of the Surfaces of the longest Branch, the Division of Diameters and Weights of Iron Bullets, as also the Bores of Pieces.

The making of this Division is founded upon one or two Experiments, in examining, with all possible Exactness, the Diameter of a Bullet, whose Weight is very exactly known. For Example, having found that a Bullet, weighing four Pounds, is three Inches in Diameter, it will be easy to make a Table of the Weights and Diameters of any other Bullets, because, per Prop. 18. lib. 12. Eucl. Bullets are to one another as the Cubes of their Diameters; from whence it follows, that the Diameters are as the Cube Roots of Numbers, expressing their Weights.

Now having found, by Experience, that a Bullet, weighing four Pounds, is three Inches in Diameter; if the Diameter of a Bullet weighing 32 Pounds be required, say, by the Rule of Three, As 4 is to 32, So is 27, the Cube of 3, to a fourth Number, which will be 216; whose Cube Root, 6 Inches, will be the Diameter of a Bullet weighing 32 Pounds.

Or otherwise, seek the Cube Root of these two Numbers 4 and 32, or 1 and 8, which are in the same Proportions, and you will find 1 is to 2, As 3 is to 6, which is the same as before.

But since all Numbers have not exact Roots, the Table of homologous Sides of similar Solids (in the Treatise of the Sector) may be used. If now, by help of that Table, the Diameter of an Iron Bullet, weighing 64 Pounds, be required, make a Rule of Three, whose first Term is 397, the Side of the fourth Solid; the second 3 Inches, or 36 Lines, the Diameter of the Bullet weighing four Pounds; and the third Term 1000, which is the Side of the 64th Solid: the Rule being finished, you will have 90$$\frac{3}{4}$$ Lines for the Diameter of a Bullet weighing 64 Pounds. Afterwards to facilitate the Operations of other Rules of Three, always take, for the first Term, the Number 1000, for the second 90$$\frac{3}{4}$$ Lines, and for the third the Number found in the Table, over against the Number expressing the Weight of the Bullet. As to find the Diameter of a Bullet weighing 24 Pounds, say, As 1000 is to 90$$\frac{3}{4}$$ Lines, So is 721, to 65 Lines, which is 5 Inches and 5 Lines for the Diameter sought. By this Method the following Table is calculated.

A Table, containing the Weights and Diameters of Iron Bullets used in the Artillery.
Weights of Bullets Diameters
Pounds Inches Lines
$$\frac{1}{4}$$12$$\frac{1}{4}$$
$$\frac{1}{2}$$16
1110$$\frac{5}{8}$$
224$$\frac{1}{2}$$
328$$\frac{2}{3}$$
430
532$$\frac{3}{4}$$
635
737$$\frac{1}{4}$$
839$$\frac{3}{8}$$
9311
1040$$\frac{3}{4}$$
1243$$\frac{3}{4}$$
1649
18411$$\frac{1}{3}$$
2051$$\frac{1}{2}$$
2455
2758$$\frac{7}{8}$$
30510$$\frac{1}{2}$$
3360$$\frac{3}{4}$$
3662$$\frac{3}{4}$$
4065$$\frac{1}{2}$$
48610
50611$$\frac{1}{2}$$
6476$$\frac{3}{4}$$
A Table, containing the Bores of the most common Pieces used in the Artillery.
Bores of Pieces Inches Lines
$$\frac{1}{4}$$13
$$\frac{1}{2}$$16$$\frac{3}{4}$$
1111$$\frac{6}{8}$$
225$$\frac{3}{4}$$
3210
431$$\frac{1}{4}$$
534$$\frac{1}{4}$$
636$$\frac{7}{8}$$
739$$\frac{1}{8}$$
8311$$\frac{1}{8}$$
941$$\frac{1}{4}$$
1042$$\frac{3}{4}$$
1245$$\frac{3}{4}$$
16411$$\frac{1}{2}$$
1851$$\frac{2}{3}$$
2054
2458
27510$$\frac{2}{3}$$
3061$$\frac{1}{3}$$
3363$$\frac{1}{2}$$
3665$$\frac{3}{4}$$
4068$$\frac{1}{2}$$
4871$$\frac{3}{4}$$
5072$$\frac{3}{4}$$
64710$$\frac{1}{4}$$

## Of the Curved-Pointed Compasses.

These Compasses do not at all differ in Construction from the others, of which we have already spoken, excepting only that the Points may be taken off, and curved ones put on, which serve to take the Diameters of Bullets, and then to find their Weights, by applying the Diameters on the Divisions of the before-mentioned Rule. But when you would know the Bores of Pieces, the curve Points must be taken off, and the strait ones put on, with which the Diameters of the Mouths of Cannon must be taken, and afterwards they must be applied to the Line of the Bores of Pieces, which is also set down upon the aforesaid Rule; by which means the Weights of the Bullets, proper for the proposed Cannon, may be found.

## Construction of an Instrument to level Cannon and Mortars.

This Instrument is made of a Triangular Brass Plate, about four Inches high, at the Bottom of which is a Portion of a Circle, divided into 45 Degrees; which Number is sufficient for the highest Elevation of Cannon or Mortars, and for giving Shot the greatest Range, as hereafter will be explained. There is a Piece of Brass screwed on the Center of this Portion of a Circle, by which means it may be fixed or movable, according to Necessity.

The End of this Piece of Brass must be made so, as to serve for a Plummet and Index, in order to shew the Degrees of different Elevations of Pieces of Artillery. This instrument hath also a Brass Foot to set upon Cannon or Mortars, so that when the Pieces of Cannon or Mortars are horizontal, the whole Instrument will be perpendicular.

The Use of this Instrument is very easy; for place the Foot thereof upon the Piece to be elevated, in such manner that the Point of the Plummet may fall upon a convenable Degree, and this is what we call levelling of a Piece.

## Of the Artillery Foot-Level.

The Instrument S is called a Foot-Level, and we have already spoken of it’s Construction; but when it is used in Gunnery, the Tongue, serving to keep it at right Angles, is divided into 90 Degrees, or rather into twice 45 Degrees from the Middle. The Thread, carrying the Plummet, is hung in the Center of the aforesaid Divisions, and the two Ends of the Branches are hollowed, so that the Plummet may fall perpendicular upon the Middle of the Tongue, when the Instrument is placed level.

To use it, place the two Ends upon the Piece of Artillery, which may be raised to a proposed Height, by means of the Plummet, whose Thread will give the Degrees.

Upon the Surface of the Branches of this Square, which opens quite strait like a Rule, are set down the Weights and Diameters of Bullets, and also the Bores of Pieces, as we have before explained in speaking of the Gunner’s Square.

The Instrument T is likewise for levelling Pieces of Artillery, being almost like R, except only the Piece, on which are the Divisions of Degrees, is movable, by means of a round Rivet: that is, the Portion of the Circle (or Limb) may be turned up and adjusted to the Branch, so that the Instrument takes up less room, and is easier put in a Case. The Figure thereof is enough to shew it’s Construction, and it’s Uses are the same as those of the precedent Instrument.

## Explanation of the Effects of the Cannon and Mortars.

The Figure V represents a Mortar upon it’s Carriage, elevated and disposed for throwing a Bomb into a Citadel, and the Curve-Line represents the Path of the Bomb thro’ the Air, from the Mouth of the Piece to it’s Fall. This Curve, according to Geometricians, is a Parabolic Line, because the Properties of the Parabola agree with it; for the Motion of the Bomb is composed of two Motions, one of which is equal and uniform, which the Fire of the Powder gives it, and the other is an uniform accelerate Motion, communicated to it by it’s proper Gravity. There arises, from the Composition of these two Forces, the same Proportion, as there is between the Portions of the Axis and the Ordinates of a Parabola; as is very well demonstrated by M. Blondel in his Book, entitled, The Art of throwing Bombs.

Maltus, an English Engineer, was the first that put Bombs in practice in France, in the Year 1634, all his Knowledge was purely experimental; he did not, in the least, know the Nature of the Curve they describe in their Passage thro’ the Air, nor their Ranges, according to different Elevations of Mortars, which he could not level but tentively, by the Estimation he made of the Distance of the Place he would throw the Bomb to; according to which he gave his Piece a greater or less Elevation, seeing whether the first Ranges were just or not, in order to lower his Mortar, if the Range was too little; or raise it, if it was too great; using, for that effect, a Square and Plummet, almost like that of which we have already spoken.

The greatest Part of Officers, which have served the Batteries of Mortars since Maltus’s Time, have used his Elevations; they know, by Experience, nearly the Elevation of a Mortar to throw a Bomb to a given Distance, and augment or diminish this Elevation in proportion, as the Bomb is found to fall beyond or short of the Distance of the Place it is required to be thrown in.

Yet there are certain Rules, founded upon Geometry, for finding the different Ranges, not only of Bombs, but likewise of Cannon, in all the Sorts of Elevations; for the Line, described in the Air by a Bullet shot from a Cannon, is also a Parabola in all Projections, not only oblique ones, but right ones, as the Figure W shews.

A Bullet going out of a Piece, will never proceed in a straight Line towards the Place it is levelled at, but will rise up from it’s Line of Direction the Moment after it is out of the Mouth of the Piece. For the Grains of Powder nighest the Breech, taking fire first, press forward, by their precipitated Motion, not only the Bullet, but likewise those Grains of Powder which follow the Bullet along the Bottom of the Piece; where successively taking fire, they strike as it were the Bullet underneath, which, because of a necessary Vent, hat not the same Diameter as the Diameter of the Bore: and so insensibly raise the Bullet towards the upper Edge of the Mouth of the Piece, against which it so rubs in going out, that Pieces very much used, and whose Metal is soft, are observed to have a considerable Canal there, gradually dug by the Friction of Bullets. Thus the Bullet going from the Cannon, as from the Point E, raises itself to the Vertex of the Parabola G, after which it descends by a mixed Motion towards B.

Ranges, made from an Elevation of 45 Deg. are the greatest, and those made from Elevations equally distant from 45 Deg. are equal; that is, a Piece of Cannon, or a Mortar, levelled to the 40th Deg. will throw a Bullet, or Bomb, the same Distance, as when they are elevated to the 50th Degree; and as many at 30 as 60, and so of others, as appears in Fig. X.

The first who reasoned well upon this Matter, was Galilæus, chief Engineer to the Great Duke of Tuscany, and after him Torricellius his Successor.

They have shewn, that to find the different Ranges of a Piece of Artillery in all Elevations, we must, before all things, make a very exact Experiment in firing off a Piece of Cannon or Mortar, at an Angle well known, and measuring the Range made, with all the Exactness possible: for by one Experiment well made, we may come to the Knowledge of all the others, in the following Manner.

To find the Range of a Piece at any other Elevation required, say, As the Sine of double the Angle under which the Experiment was made, is to the Sine of double the Angle of an Elevation proposed, So is the Range known by Experiment, to another.

As suppose, it is found by Experiment that the Range of a Piece elevated to 30 Deg. is 1000 Toises: to find the Range of the same Piece with the same Charge, when it is elevated to 45 Deg. you must take the Sine of 60 Degrees, the double of 30, and make it the first Term of the Rule of Three; the second Term must be the Sine of 90, double 45; and the third the given Range 1000: Then the fourth Term of the Rule will be found 1155, the Range of the Piece at 45 Degrees of Elevation.

If the Angle of Elevation proposed be greater than 45 Deg. there is no need of doubling it for having the Sine as the Rule directs; but instead of that, you must take the Sine of double it’s Complement to 90 Degrees: As, suppose the Elevation of a Piece be 50 Degrees, the Sine of 80 Degrees, the double of 40 Deg. must be taken.

But if a determinate Distance to which a Shot is to be cast, is given (provided that Distance be not greater than the greatest Range at 45 Deg. of Elevation), and the Angle of Elevation to produce the proposed Effect be required; as suppose the Elevation of a Cannon or Mortar is required to cast a Shot 800 Toises; the Range found by Experiment must be the first Term in the Rule of Three, as for Example 1000 Toises; the proposed Distance 800 Toises, must be the second Term; and the Sine of 60 Degrees, the third Term. The fourth Term being found, is the Sine of 43 Deg. 52 Min. whose half 21 Deg. 56 Min. is the Angle of Elevation the Piece must have, to produce the proposed Effect; and if 21 Deg. 56 Min. be taken from 90 Deg. you will have 68 Deg. 4 Min. for the other Elevation to the Piece, with which also the same Effect will be produced.

For greater Facility, and avoiding the Trouble of finding the Sines of double the Angles of proposed Elevations, Galilæus and Torricellius have made the following Table, in which the Sines of the Angles sought are immediately seen.

The Use of this Table is thus: Suppose it be known by Experience, that a Mortar elevated 15 Degrees, charged with three Pounds of Powder, throws a Bomb at the Distance of 350 Toises, and it is required with the same Charge to cast a Bomb 100 Toises further; seek in the Table the Number answering to 15 Degrees, and you will find 5000. Then form a Rule of Three, by saying, As 350 is to 450, So is 5000 to a fourth Number, which will be 6428. Find this Number, or the nighest approaching to it, in the Table, and you will find it next to 20 Deg. or 70 Deg. which will produce the required Effect, and so of others.