Chapter D: Caldwell, (44)

Archimedes took up _mensuration_ where Euclid left it, and carried it a great length. He was the first who squared a curvilineal space, unless Hypocrates must be excepted on account of his lunes. In his times the conic sections were admitted in geometry, and he applied himself closely to the measuring of them as well as other figures. Accordingly he determined the relations of spheres, spheroids, and conoids, to cylinders and cones; and the relations of parabolas to rectilineal planes whose quadratures had long before been determined by Euclid. He hath left us also his attempts upon the circle: he proved that a circle is equal to a right angled triangle, whose base is equal to the circumference, and its altitude equal to the radius; and consequently that its area is found by drawing the radius into half the circumference; and so reduced the quadrature of the circle to the determination of the ratio of the diameter to the circumference; but which however hath not yet been done. Being disappointed of the exact quadrature of the circle, for want of the rectification of its circumference, which all his methods would not effect, he proceeded to assign an useful approximation to it: this he effected by the numerical calculation of the perimeters of the inscribed and circumscribed polygons; from which calculations it appears, that the perimeter of the circumscribed regular polygon of 192 sides is to the diameter in a less ratio than that of 3¹⁄₇ (3¹⁰⁄₇₀) to 1, and that the inscribed polygon of 96 sides is to the diameter in a greater ratio than that of 3¹⁰⁄₇₁ to 1; and consequently much more than the circumference of the circle is to the diameter in a less ratio than that of 3¹⁄₇ to 1, but greater than that of 3¹⁰⁄₇₁ to 1: the first ratio of 3¹⁄₇ to 1, reduced to whole numbers, gives that of 22 to 7, for 3¹⁄₇ : 1 ∷ 22 : 7, which therefore will be nearly the ratio of the circumference to the diameter. From this ratio of the circumference to the diameter he computed the approximate area of the circle, and found it to be to the square of the diameter as 11 to 14. He likewise determined the relation between the circle and elipsis, with that of their similar parts. The hyperbola too in all probability he attempted; but it is not to be supposed, that he met with any success, since approximations to its area are all that can be given by all the methods that have since been invented.

Besides these figures, he hath left us a treatise on the spiral described by a point moving uniformly along a right line, which at the same time moves with an uniform angular motion; and determined the proportion of its area to that of its circumscribed circle, as also the proportion of their sectors.

Throughout the whole works of this great man, which are chiefly on _mensuration_, he every where discovers the deepest design and finest invention; and seems to have been (with Euclid) exceedingly careful of admitting into his demonstrations nothing but principles perfectly geometrical and unexceptionable: and although his most general method of demonstrating the relations of curved figures to straight ones, be by inscribing polygons in them, yet to determine those relations, he does not increase the number and diminish the magnitude of the sides of the polygon _ad infinitum_; but from this plain fundamental principle, allowed in Euclid’s elements, viz. that any quantity may be so often multiplied, or added to itself, as that the result shall exceed any proposed finite quantity of the same kind, he proves that to deny his figures to have the proposed relations, would involve an absurdity.

He demonstrated also many properties, particularly in the parabola, by means of certain numerical progressions, whose terms are similar to the inscribed figures: but without considering such series to be continued _ad infinitum_, and then summing up the terms of such infinite series.

He had another very curious and singular contrivance for determining the measures of figures, in which he proceeds, as it were, mechanically by weighing them.

Several other eminent men among the ancients wrote upon this subject, both before and after Euclid and Archimedes; but their attempts were usually upon particular parts of it, and according to methods not essentially different from theirs. Among these are to be reckoned Thales, Anaxagoras, Pythagoras, Bryson, Antiphon, Hypocrates of Chios, Plato, Apollonius, Philo, and Ptolomy; most of whom wrote of the quadrature of the circle, and those after Archimedes, by his method, usually extended the approximation to a greater degree of accuracy.

Many of the moderns have also prosecuted the same problem of the quadrature of the circle, after the same methods, to greater lengths: such are Viera, and Metius, whose proportion between the diameter and circumference is that of 113 to 355, which is within about ³⁄₁₀₀₀₀₀₀₀ of the true ratio; but above all, Ludolph van Ceulen, who with an amazing degree of industry and patience, by the same methods extended the ratio to 20 places of figures, making it that of 1 to 3.14159265358979323846+.

The first material deviation from the principles used by the ancients in geometrical demonstrations was made by Cavalerius: the sides of their inscribed and circumscribed figures they always supposed of a finite and assignable number and length; he introduced the doctrine of indivisibles, a method which was very general and extensive, and which with great ease and expedition served to measure and compare geometrical figures. Very little new matter however was added to geometry by this method, its facility being its chief advantage. But there was great danger in using it, and it soon led the way to infinitely small elements, and infinitesimals of endless orders; methods which were very useful in solving difficult problems, and in investigating or demonstrating theories that are general and extensive; but sometimes led their incautious followers into errors and mistakes, which occasioned disputes and animosities among them. There were now, however, many excellent things performed in this subject; not only many new things were effected concerning the old figures, but new curves were measured; and for many things which could not be exactly squared or cubed, general and infinite approximating series were assigned, of which the laws of their continuation were manifest, and of some of which the terms were independent on each other. Mr. Wallis, Mr. Huygens, and Mr. James Gregory, performed wonders. Huygens in particular must be admired for his solid, accurate, and very masterly works.

During the preceding state of things several men, whose vanity seemed to have overcome their regard for truth, asserted that they had discovered the quadrature of the circle, and published their attempts in the form of strict geometrical demonstrations, with such assurance and ambiguity as staggered and misled many who could not so well judge for themselves, and perceive the fallacy of their principles and arguments. Among those were Longomontanus, and the celebrated Hobbes, who obstinately refused all conviction of his errors.

The use of infinites was however disliked by several people, particularly by sir Isaac Newton, who among his numerous and great discoveries hath given us that of the method of fluxions; a discovery of the greatest importance both in philosophy and mathematics; it being a method so general and extensive, as to include all investigations concerning magnitude, distance, motion, velocity, time, &c. with wonderful ease and brevity; a method established by its great author upon true and incontestible principles; principles perfectly consistent with those of the ancients, and which were free from the imperfections and absurdities attending some that had lately been introduced by the moderns; he rejected no quantities as infinitely small, nor supposed any parts of curves to coincide with right lines; but proposed it in such a form as admits of a strict geometrical demonstration. Upon the introduction of this method most sciences assumed a different appearance, and the most abstruse problems became easy and familiar to every one; things which before seemed to be insuperable, became easy examples or particular cases of theories still more general and extensive; rectifications, quadratures, cubatures, tangencies, cases _de maximis & minimis_, and many other subjects, became general problems, and delivered in the form of general theories which included all particular cases: thus, in quadratures, an expression would be investigated which defined the areas of all possible curves whatever, both known and unknown, and which, by proper substitutions, brought out the area for any particular case, either in finite terms, or infinite series, of which any term, or any number of terms could be easily assigned; and the like in other things. And although no curve, whose quadrature was unsuccessfully attempted by the ancients, became by this method perfectly quadrable, there were assigned many general methods of approximating to their areas, of which in all probability the ancients had not the least idea or hope; and innumerable curves were squared which were utterly unknown to them.

The excellency of this method revived some hopes of squaring the circle, and its quadrature was attempted with eagerness. The quadrature of a space was now reduced to the finding of the fluent of a given fluxion; but this problem however was found to be incapable of a general solution in finite terms; the fluxion of every fluent was always assignable, but the reverse of this problem could be effected only in particular cases; among the exceptions, to the great grief of the geometers, was included the case of the circle, with regard to all the forms of fluxions attending it. Another method of obtaining the area was tried: of the quantity expressing the fluxion of any area, in general, could be assigned the fluent in the form of an infinite series, which series therefore defined all areas in general, and which, on substituting for particular cases, was often found to break off and terminate, and so afford an area in finite terms; but here again the case of the circle failed, its area still coming out an infinite series. All hopes of the quadrature of the circle being now at an end, the geometricians employed themselves, in discovering and selecting the best forms of infinite series for determining its area, among which it is evident, that those were to be preferred which were simple, and which would converge quickly; but it generally happened, that these two properties were divided, the same series very rarely including them both: the mathematicians in most parts of Europe were now busy, and many series were assigned on all hands, some admired for their simplicity, and others for their rate of convergency; those which converged the quickest, and were at the same time simplest, which therefore were most useful in computing the area of the circle in numbers, were those in which, besides the radius, the tangent of some certain arc of the circle, was the quantity by whose powers the series converged; and from some of these series the area hath been computed to a very great extent of figures: Mr. Edmund Hally gave a remarkable one from the tangent of 30 degrees, which was rendered famous by the very industrious Mr. Abraham Sharp, who by means of it extended the area of the circle to 72 places of figures, as may be seen in Sherwin’s book of logarithms; but even this was afterwards outdone by Mr. John Machin, who, by means described in professor Hutton’s _Mensuration_, composed a series so simple, and which converged so quickly, that by it, in a very little time, he extended the quadrature of the circle to 100 places of figures; from which it appears, that if the diameter be 1, the circumference will be 3.1415926535, 8979323846, 2643383279, 5028841971, 6939937510, 5820974944, 5923078164, 0628620899, 8628034825, 3421170679+, and consequently the area will be .7853981633, 9744830961, 5660849819, 857210492, 9234984377, 6455243736, 1480769541, 0157155224, 9657008706, 3355292669+.

From hence it appears, that all or most of the material improvements or inventions in the principles or method of treating of geometry, have been made especially for the improvement of this chief part of it, _mensuration_, which abundantly shows, what we at first undertook to declare, the dignity of this subject; a subject which, as Dr. Barrow says, after mentioning some other things, “deserves to be more curiously weighed, because from hence a name is imposed upon that mother and mistress of the rest of the mathematical sciences, which is employed about magnitudes, and which is wont to be called _geometry_ (a word taken from ancient use, because it was first applied only to measuring the earth, and fixing the limits of possessions) though the name seemed very ridiculous to Plato, who substitutes in its place that more extensive name of _Metrics_ or _Mensuration_; and others after him gave it the title of _Pantometry_, because it teaches the method of measuring all kinds of magnitudes.” See SURVEYING, LEVELLING, and GEOMETRY.

MERHAU, _Ind._ A deduction or abatement is so called in India.

MERIT. Desert, excellence, deserving honor or reward.

MERIT, _Order of_, a military distinction given to officers or soldiers, for some signal service: the badge of which is generally expressive of the service. Such was the medal, or order of merit, presented by the Austrian emperor to the officers of the 15th British light dragoons, for their bravery in the affair of _Villers en Couché_, in 1794.

MERKIN. A mop to clean cannon.

MERLIN. Handspike.

MERLON. See FORTIFICATION.

MESS. It is usual and advantageous to discipline that the officers of a camp or garrison form one or more messes.

MESSENGERS of state in England, are officers under the direction of the secretaries of state, of whom there were 20 always in waiting, who were relieved monthly, and distributed in the following manner: four at court, five at each secretary’s office, two at the third office for North Britain, three at the council office, and one at the lord chamberlain’s office, who attended that office always in readiness to be sent with dispatches, either domestic or foreign; either to apprehend persons _accused_ or _suspected_ of high treason, or other offences against the state, being empowered by warrant from the secretaries; for the safe keeping of which, their houses are made a sort of confinement or prison; and for the maintenance of the prisoners they have a certain allowance from government. The number has been increased with the system of espionage since 1794.

_Military_ MESSENGERS. Confidential persons that are sent to and from head quarters, &c.

MESTRE _de_ CAMP, _Fr._ The commanding officer of a regiment of cavalry was so called in the old French service. He was distinguished by this appellation on account of there being a colonel-general in the cavalry. The duty of a _mestre de camp_ was principally confined to the following heads:--To see that the troops or companies were kept complete, that the arms were in good state and condition, the horses of a proper size, sound, and well trained. He had likewise the direction of the different guards, &c.

MESTRE _de_ CAMP _général_, _Fr._ The next officer in rank, in the old French cavalry service, to the colonel-general. This appointment was created under Henry II. in 1552.

MESTRE _de_ CAMP _général des dragons_, _Fr._ An appointment which first took place under Louis the XIVth. in 1684.

MESURES _à poudre_, _Fr._ Tin cases or vessels used in the artillery, to measure out gunpowder, according to the size and calibre of each piece of ordnance. See Powder MEASURES.

_Over_-METAL, (in gunnery,) when the mouth of a piece of ordnance, in disparting it, lies higher than the breech, it is then said to be _laid over metal_.

_Under_-METAL, (in gunnery) is when the mouth of a piece of ordnance lies lower than her breech.

_Right with_ METAL, (in gunnery.) When a piece of ordnance lies truly level, point blank, or right with the mark, she is said to lie _right with her metal_.

_Superficies of_ METALS, (in gunnery.) The surface or outside of a gun.

METIER, _Fr._ Means, literally, any calling or business. In a military sense, it is peculiarly applicable to those nations which keep up large standing armies, and make war their principal object and pursuit. In speaking of military matters, it is common among the French to say--_Guerre sur terre est notre métier; Guerre sur mer est le métier des Anglois_--The land service is our peculiar business or calling; the sea service is the peculiar business or calling of the English; meaning thereby to express their reciprocal superiority.

Chevalier Folard gives the following definition relative to the question which is often discussed on the subject of war, namely, whether war be a trade or a science? The English call it a profession. Folard, however, distinguishes it in this manner:--_La guerre est un métier pour les ignorans, et une science pour les habiles gens._ War in the apprehension, and under the management of ignorant persons, is certainly a mere trade or business, but among able men, it becomes an important branch of science.

METTRE _à la main_, _Fr._ To grasp or take hold of any thing.

METTRE _l’épée à la main_, _Fr._ To draw swords. _Ils mirent l’épée à la main_, a figurative expression, signifying, they took their ground, and stood prepared to fight.

METTRE _les armes à la main de quelqu’un_, _Fr._ To teach a person the first rudiments of war, or lead him for the first time into action. _C’est lui qui m’a mis les armes à la main._ He first taught me how to fight, or I fought the first campaign under his orders.

METTRE _aux arrêts_, _Fr._ To put under arrest.

METTRE _sur pied_, _Fr._ To arm, to equip, to put troops upon an established footing.

MEURTRIERES, _Fr._ Small loop holes, sufficiently large to admit the barrel of a rifle gun or musquet, through which soldiers may fire, under cover, against an enemy. They likewise mean the cavities that are made in the walls of a fortified town or place. See MURDRESSES.

MICHE. See MALINGERER.

MICROMETER, (_Micromêtre_, _Fr._) an instrument contrived to measure small spaces, as in the divisions of the worm of a screw.

MIDI, _Fr._ the South.

MILE, in _geography_, a long measure, whereby the English, &c. express the distance between places: it is of different extent in different countries. The geometrical mile contains 1000 geometrical paces, or _mille passus_, from whence miles are denominated.

We shall here give a table of the miles in use among the principal nations of Europe, in geometrical paces, 60,000 of which, according to the _English Military Dictionary_, make a degree of the equator.

Geometrical
paces.
Mile of Russia 750
Italy 1000
England 1200
Scotland and Ireland 1500
The old league of France 1500
The small ditto 2000
The great ditto 3000
Mile of Poland 3000
Spain and Portugal 3428
Germany 4000
Sweden 5000
Denmark 5010
Hungary 6000
Holland 3500

MILE. Comparison of the different miles, in geometric paces, each of which is equal to 5 feet French royal, 5·6719 feet Rhinland, or 6·1012 English feet.

Geometric
paces.
The mile of Sweden 5761
Switzerland 4512
Denmark 4071
Common, of Germany 4000
Holland 3158
League of France 2400
Spain 2286
Scotland 1500
Mile of Italy 1000
England 868
Werste of Russia 575

MILICE, _Fr._ soldiery, but more particularly the militia or trained bands.

MILICES _gardes-côtes_, _Fr._ A militia, somewhat similar to our sea fencibles, which existed during the old French government, and whose services were confined to the coast. Every province, contiguous to the sea, was obliged to furnish a certain proportion of its male inhabitants, from 16 to 60 years old. This militia was exempted from the regulations which governed the land militia. It was under the admiralty.

MILITANT, the state of warfare, or business of war.

MILITAR, MILITARY, something belonging to the soldiery or militia, &c.

MILITARY _architecture_, the same with fortification. See FORTIFICATION.

MILITARY _ways_, the large Roman roads which Agrippa procured to be made through the empire in the reign of Augustus for the marching of troops and conveying of carriages. They were paved from the gates of Rome to the utmost limits of the empire. The British have constructed a military road throughout India; with wells and other accommodations at certain distances.

MILITARY _discipline_. Next to the forming of troops, military discipline is the first object that presents itself to our notice: it is the soul of all armies; and unless it be established amongst them with great prudence, and supported with unshaken resolution, soldiers become a contemptible rabble, and are more dangerous to the very state that maintains them, than even its declared enemies. See DISCIPLINE.

MILITARY _execution_, the ravaging or destroying of a country or town that refuses to pay the contribution inflicted upon them. Also the punishment inflicted by the sentence of a court-martial.

MILITARY _first principles_, is the bodily training for a soldier, to make him hardy, robust, and capable of preserving health amidst fatigue, bad weather, and change of climate; to march at such possible pace, and for such length of time, and with such burden, as, without training, he would not be able to do.

MILITARY REGULATIONS. The rules and regulations, by which the discipline, formations, field exercise, and movements of the whole army, are directed to be observed in one uniform system. The American military system is scarcely entitled to the name of a system; and as to _regulation_ that requires yet to be established, the worst of all is that there does not appear to be a suspicion in congress that any regulation is required. See REGULATIONS.

MILITIA. A force whose services, in general, do not exceed the boundaries of the nation, but which may volunteer beyond them. The American militia has no coherent system, every state has power to regulate its own, and the effect is, that there is either no regulation at all, or what is worst, an imbecile mockery, the only use of which is the preservation on the statute book that there is a power though there is not a will to regulate the militia. The militia among the Romans was frequently called Agrarian soldiers. The system of our revolution though it was not complete in general was the most effectual ever established; the French system of conscription was borrowed from America, who borrowed it from the Romans.

MILL, properly denotes a machine for grinding corn, &c. but more generally all such machines whose action depends upon a circular motion. There are various kinds, though foreign to this work.

_Gunpowder_ MILL, is that used for pounding and beating together the ingredients of which gunpowder is composed.

These ingredients being duly proportioned, and put into the mortars of the mills, which are hollow pieces of wood, each capable of holding 20 pounds of paste, are incorporated by means of the pestle and spindle. There are 24 mortars in each mill, where are made each day 480 pounds of gunpowder, care being taken to sprinkle the ingredients in the mortars with water, from time to time, lest they should take fire. The pestle is a piece of wood 10 feet high, and 4¹⁄₂ inches broad, armed at bottom with a round piece of metal. It weighs about 60 pounds.

MIM BASHY, _Ind._ A commander of one thousand horse.

MINE, in a _military sense_, implies a subterraneous passage dug under the wall or rampart of a fortification, for the purpose of blowing it up by gunpowder.

The excavation formed by the blowing up of a mine is found by experiment to be nearly a paraboloid. It was formerly supposed that the diameter of the entonnoir, or excavation, was always equal to only double the line of least resistance; but experiments have proved, that the diameter of the excavation may be increased to six times the line of least resistance; and that the diameter of the globe of compression may be increased to eight times that line; this is called the _maximum_ of a mine, or the greatest effect that can be produced by a globe of compression. In any mine intended to produce an effect within this extent, the effects will be nearly as the charges.

The globes are to each other as the cubes of their radii. Their radii are the hypothenuse of rightangled triangles, of which the line of least resistance, and the semi-diameter of the excavation, are the other two sides. Therefore, to find the charge to produce any required diameter of the excavation, the following will be the rule, the radius being found as above:

As the cube of the radius of the globe of compression in the
following table, (having the same line of least resistance as the
required globe,)

Is to the cube of the radius of the required globe;

So is the charge corresponding in the following table,

To the charge required.

_Table for the Charges of Mines, according to Valliere._

-------------+----------++-------------+----------
Line of least|Charge for||Line of least|Charge for
Resistance. | the Mine.|| Resistance. | the Mine.
-------------+----------++-------------+----------
Feet. | lbs. oz.|| Feet. | lbs. oz.
1 | 0 2 || 21 | 868 3
2 | 0 12 || 22 | 998 4
3 | 2 8 || 23 |1140 10
4 | 6 -- || 24 |1296 --
5 | 11 11 || 25 |1558 9
6 | 20 4 || 26 |1647 12
7 | 32 2 || 27 |1815 4
8 | 48 -- || 28 |2053 --
9 | 68 5 || 29 |2286 7
10 | 93 12 || 30 |2530 4
11 | 124 12 || 31 |2792 4
12 | 162 -- || 32 |3072 --
13 | 205 15 || 33 |3369 1
14 | 257 4 || 34 |3680 12
15 | 316 4 || 35 |4019 8
16 | 384 -- || 36 |4374 --
17 | 460 9 || 37 |4748 11
18 | 546 12 || 38 |5144 4
19 | 643 -- || 39 |5561 2
20 | 750 -- || 40 |6000 --
-------------+----------++-------------+----------

This table is calculated upon a supposition that the excavation of the mine is a paraboloid, having a base double the line of resistance; and that 10 lbs. 10 oz. of powder is sufficient for raising one cubic fathom of earth. By the rule above given may be found the charge for any mine, that shall only shake the ground, without making any excavation, by making the line of least resistance of the required globe only equal to the radius of the globe of compression.

The charges thus found by means of this table, being only for one nature of soil; _viz._, light earth and sand, (that for which the table is calculated) must be augmented according to the following table of Vauban’s, by one, four, five, seven, or nine elevenths of the charge found.

_Table of the quantity of powder required to raise a cubic fathom, according to the soil._

1 Light earth, mixed with sand 11 pounds.
2 Common earth 12
3 Strong sand 15
4 Clay, or fat earth 16
5 Old and good masonry 18
9 Rock 20

The following rule is however laid down by Belidor, and generally adopted, if it be intended that the mine shall produce its maximum or greatest effect: multiply the line of least resistance, expressed in feet, by 300, the product will be the charge in pounds.

In making mines of any kind, the following remarks may be of service.

The best form for the chamber would be spherical; but from the difficulty of its construction, it is always made a cube, of one inch larger dimensions than the box to contain the powder.

The chamber must not be made in the prolongation of the branch of the mine, but at one side, and lower than the level of the branch, if the soil be dry; but higher if it be wet.

One cubic foot will contain 75 lbs of powder; upon which principle the size of the case to contain the powder must be regulated. The auget is generally one inch square interior dimensions, and the end of it must reach the centre of the chamber; where the saucisson must be fastened, to prevent its being easily pulled out.

The branch of the mine to be sprung must be closed in the strongest manner by doors well secured by props, and must be stopped with earth or rubbish to a distance, taken in a straight line, equal to 1¹⁄₂ times the line of least resistance.

In proportioning the length of saucisson, in order that any number of mines may be fired at the same instant, a return of a right angle is generally reckoned equal to 4 inches in a right line.

The first step in making a mine, whether for attack or defence, is to sink a shaft to the depth of the bottom of the gallery, having two of its sides in the direction of the sides of the gallery. These shafts should be where the galleries are to cross each other, or in the centre of the length of gallery to be made. These shafts should never be further apart than 40 or 50 fathoms; for it is found, that the air is not fit for respiration in the larger galleries at a greater distance from the shaft than 25 fathoms; at 20 fathoms in those of medium dimensions; and at 15 in the smallest.

The rectangular frames used in sinking a shaft are commonly placed 4 feet asunder; and in the galleries they are only 3 feet. A gallery intended to be lined with masonry, must be 7 feet high and 6 feet wide, in order that it may be when finished, 6 feet high and 3 feet wide.

Temporary galleries are only made 4¹⁄₂ feet high, and 2¹⁄₂ or 3 feet wide.

The branches, at the ends of which the chambers are to be placed, are only made 2¹⁄₂ or 3 feet high, and 2 feet, or 2 feet 3 inches wide.

The first of these is dug on the knees; the second sitting or lying.

The miners are divided into squads of 4 each; and the rate of the work for each squad is 3 feet of the temporary gallery in 4 hours. The first squad is relieved by a second, after having worked 4 hours, or laid one frame; which second squad is again relieved by the first, at the expiration of the same time.

In the most easy ground to work, a miner may be heard to the distance of 14 or 15 fathoms under ground; and the noise made by fixing the frames of the galleries may often be heard as far as 20 or 25 fathoms. A drum braced, standing on the ground, with a few peas or other round substances on the head, will be very sensibly affected by an approaching miner.

It is of the most essential consequence to place the entrances to the countermines beyond the reach of any surprise from the enemy.

To prevent an enemy gaining possession of the galleries of the countermines they should be well secured by strong doors, at every 15 fathoms. These should be musquet proof.

A glacis, properly countermined, and every advantage taken of it to retard the besiegers, may, with proper management, prolong a siege at least 2 months; and if the rest of the works are also countermined, and properly defended, they may add another month to the siege. Every system of countermines must depend upon the system of fortification to which they are to be adapted; the general principle for their regulation is, that the galleries should occupy situations, from which branches can be most readily run out under the most probable points of the besieger’s batteries and approaches. The general system of countermines commonly used in a place prepared before hand, is as follows: the principal or _magistral_ gallery runs all round the work, under the banquette of the covert way, and across the places of arms, having the entrances at the re-entering places of arms. Nearly parallel to this at 20, 25 or 30 fathoms distance is another gallery, called the _envellope_. These two galleries are connected by galleries of _communication_, under the gutters of the re-entering parts of the glacis, and under the ridges of the salient parts. From the envellope are run out about 15 or 16 fathoms, galleries in directions parallel to the capitals of the works, and at 23 fathoms distance from each other. These are called _listeners_.

Sometimes, shafts are sunk from the end of these listeners, and by connecting these shafts, a second envellope formed. Behind the escarps of the different works, galleries are likewise made, about the level of the bottom of the ditch; from whence branches may be run out into or under the foundations of the walls; and if the ditch be dry, galleries of communication may be made from these to the magistral gallery; and from which communications branches may be run out for chambers to annoy the besiegers in their passage of the ditch. The entrances to the escarp galleries are by means of posterns, which descend from behind the interior slope of the rampart.

If a place be not countermined before hand, a great deal may be done even after the investment of the place, to prolong the siege by countermines. In this case, the first thing to be done immediately that the place is invested, is to sink a shaft in each of the places of arms of the covert way; one in each branch of the covert way opposite that part of the bastion where the breach will most probably be made; and one in the flanked angle of each bastion. Those on the covert way will be on the banquette, and sunk to about 18 inches below the bottom of the ditch. Those in the bastions to about 12 feet below the bottom of the ditch. Thus prepared, the moment the side on which the attack is to be made can be ascertained, galleries must be carried on from these shafts on the side attacked along the capitals, in the form of trefles, or double T; and advanced as far into the country as the time will admit. Communication galleries may likewise be driven between these different works on the covert way, and from them to the work in the bastion; which will prevent the enemy gaining possession of their entrances. All these works may be carried on after the investment of the place; and be in sufficient forwardness by the time the enemy gains the third parallel.

The following rules are given by Vauban for fougasses, or small mines, having the diameter of the excavation equal to double the line of least resistance. The side of the chamber must be exactly a sixth part of the depth of the shaft. The side of the box to hold the powder exactly a ninth part of the depth of the shaft.

These remarks respecting mines are principally extracted from the General Essay on Fortification before mentioned, written in French and published at Berlin, 1799.

_Counter_-MINES, are those made by the besieged, whereas mines are generally made by the besiegers. Both mines and counter-mines are made in the same manner, and for the like purposes, viz. to blow up their enemies and their works; only the principal galleries and mines of the besieged, are usually made before the town is besieged, and frequently at the same time the fortification is built, to save expence.

_Eventer la_ MINE, _Fr._ to spring a mine. When used figuratively, this expression signifies to discover a plot, or make it known. It is likewise used to express the failure of any expedition or undertaking.

_Definitions of_ MINES. A mine is a subterraneous cavity made according to the rules of art, in which a certain quantity of powder is lodged, which by its explosion blows up the earth above it.

It has been found by experiments, that the figure produced by the explosion is a _paraboloid_, and that the centre of the powder, or charge, occupies the _focus_.

The place where the powder is lodged is called the _chamber_ of the mine, or _fourneau_.

The passage leading to the powder is called the _gallery_.

The line drawn from the centre of the chamber, perpendicular to the nearest surface of the ground, is called the line of least resistance.

The pit or hole, made by springing the mine, is called the _excavation_.

The fire is communicated to the mines by a pipe or hose, made of coarse cloth, whose diameter is about one and a half inch, called a _saucisson_, (for the filling of which near half a pound of powder is allowed to every foot) extending from the chamber to the entrance of the gallery, to the end of which is fixed a match, that the miner who sets fire to it may have time to retire, before it reaches the chamber.

To prevent the powder from contracting any dampness, the saucisson is laid in a small trough, called an _auget_ made of boards, three and a half inch broad, joined together, lengthwise, with straw in it, and round the saucisson, with a wooden cover nailed upon it.

_Foyer_, _Fr._ _Focus_ or _centre of the chamber_, some authors call the end of the saucisson that comes within the work, and which is to be set fire to, the foyer, or focus: but by most people, this is generally understood to be the centre of the chamber.

_Galleries and chambers of_ MINES. Galleries made within the fortification, before the place is attacked, and from which several branches are carried to different places, are generally 4 or 4¹⁄₂ feet wide, and 5 or 5¹⁄₂ feet high. The earth is supported from falling in by arches and walls, as they are to remain for a considerable time; but when mines are made to be used in a short time, then the galleries are but 3 or 3¹⁄₂ feet wide, and 5 feet high, and the earth is supported by wooden frames or props.

The gallery being carried on to the place where the powder is to be lodged, the miners make the chamber. This is generally of a cubical form, large enough to hold the wooden box, which contains the powder necessary for the charge: the box is lined with straw and sand-bags, to prevent the powder from contracting dampness.

The chamber is sunk something lower than the gallery, if the soil permits; but where water is to be apprehended, it must be made higher than the gallery; otherwise the besieged will let in the water, and spoil the mine.

_Quantities of powder to charge_, MINES. Before any calculation can be made of the proper charge for a mine, the density and tenacity of the soil in which it is to be made, must be ascertained, either by experiment, or otherwise; for, in soils of the same density, that which has the greatest tenacity, will require the greatest force to separate its parts. The density is determined by weighing a cubic foot (or any certain quantity) of the soil; but the tenacity can only be determined by making a mine. The following table contains experiments in 6 different soils, which may be of some assistance to form a judgment of the nature of the soil, when an actual experiment cannot be had.

+-------------------------------+--------+-------------+
| |Density.| Tenacity. |
| +--------+-------------+
| | Weight |Quantity of |
| | of |power to |
| | 1 cubic|raise 1 cubic|
| Nature of the soil. | foot. |fathom. |
+-------------------------------+--------+-------------+
|1. Loose earth or sand | 95 pds.| 8 pds. |
+-------------------------------+--------+-------------+
|2. Common light soil |124 | 10 |
+-------------------------------+--------+-------------+
|3. Loam, or strong soil |127 | 12¹⁄₄ |
+-------------------------------+--------+-------------+
|4. Potter’s clay, or stiff soil|135 | 13¹⁄₂ |
+-------------------------------+--------+-------------+
|5. Clay, mixed with stones |160 | 16 |
+-------------------------------+--------+-------------+
|6. Masonry |205 | 21¹⁄₂ |
+-------------------------------+--------+-------------+

All the requisites in mining may be determined by the following problems, which admit of 4 cases; for any 3 of the articles below being given, the 4th may thence be found.

1. The nature of the soil,

2. The diameter of the excavation,

3. The line of least resistance,

4. The charge.

PROBLEM. I.

Given the nature of the soil, the diameter of the excavation, and the
line of least resistance, to find the charge.

RULES.

1. To the square of the diameter of the excavation, add the square of double the line of least resistance, and reserve the said sum.

2. Multiply the square root of the reserved sum by double the line of least resistance, and subtract the product from the same sum.

3. Multiply half the remainder by the line of least resistance, and 1.57 times the product, will give the solidity of the excavation.

4. The charge will then be determined from the nature of the soil, as in the following example.

_Example_ I.

It is required to make a mine in the second sort of soil, mentioned in the foregoing experiments, which shall have a line of least resistance of 10 feet, and the diameter of its excavation 20 feet; what will be the proper charge?

The nature of this soil, by the table, requires 10 pounds of powder to 216 cubic feet.

_Calculation._

1. The diameter of the excavation is 20, and its square 400
Double the line of least resistance is 20, and its square 400
---
Therefore the sum to be reserved is 800

2. The square root of 800 is 28.3
Double the line of least resistance is 20 566
---
Which leaves the remainder 234
---
3. Half the remainder is 117
Which multiplied by the line of least resistance, 10
---
Gives the product 1170
Which multiplied by 1.57
------
Gives the solidity of the excavation feet 1836.9

feet. lb. feet. lb.
4. If 216 : 10 ∷ 1836.9 : 85 which is the charge required.

_By Logarithms._

1. Diam. of excavation is = 20 1.301030
Diameter squared is 2.602060 400
Double the line of least resistance is = 20
and its square 400
---
The sum to be reserved is 2.903090 800
--------
2. Square root of sum is 28.3 1.451545
Double the line of least resistance is = 20 1.301030
--------
Product to be subtracted is 2.752575 566
-------- ---
Remainder is 2.369216 234
Line of least resist. = 10 1.000000
10 pounds of powder 1.000000
To 216 cubic feet, _compl. arith._ 7.665546
To which add the _const. log._ 9.894870
--------
And the sum is the logarithm charge required 1.929632 = 85 lb.

_Problem_ II.

Given the nature of the soil, the line of least resistance, and the
charge, to find the diameter of the excavation.

RULES.

1. Find the solidity of the earth to be raised, by a proportion from the nature of the soil, and multiply it by 1.27.--Divide the product by the line of least resistance, and to the quotient add the square of the line of least resistance: reserve the sum.

2. Multiply the square root of the sum reserved by twice the line of least resistance, and add the product to the said sum, and from the result subtract 3 times the square of the line of least resistance; so will the square root of the remainder be the diameter of the required excavation.

_Example_ I.

Let a mine be charged with 100 pounds of powder in a soil which requires 11 pounds of powder to raise 216 cubic feet, and let its line of least resistance be 10 feet: what will be the diameter of the excavation?

By the nature of the soil 11lb. : 216 feet ∷ 100lb. : 1964 feet, which is the solidity of the earth to be raised.

1. Therefore multiply 1964
By 1.27
-------
The product is 2494.28
-------
Which divided by the line of least resistance, 10, is 249.428

To which add the square of the line of least resistance 100.000
-------
And the sum to be reserved is 349.428

2. The square root of 349.428 is 18.7, which multiplied by
twice the line of least resistance, 20, gives 374.
-------
This added to the sum reserved gives 723.428

From which subtract 3 times the square of least resistance 300.
-------
And there will remain 423.428

The square root of which is, 20.5 feet, being the required diameter of the excavation.

_By Logarithms._

Numb. Logar. Numb.
Cubic feet = 216 2.334454
Powder 11lb. _co. ar._ 8.958607
Charge = 100 2.000000
Line of least resist. 10 _co. ar._ 9.000000
Constant logarithm 0.103804
--------
2.396865 249.4
-----
To which add the square of line of
least resistance 100.0
-----
Sum to be reserved is 2.543323 349.4
--------
Half of which logar. 1.271661
Twice line of least resistance, 20, 1.301030
--------
Product to be added is 2.572691 373.8
-----
The result is 723.2

From which subtract thrice the square
of the line of least resistance 300.0
-----
And there remains 2.626546 423.2

Half of which logar. is 1.313273 20.57 feet, the diameter of the
excavation required.

_Loading and stopping of_ MINES. The gallery and chamber being ready to be loaded, a strong box of wood is made of the size and figure of the chamber, being about ¹⁄₃d or ¹⁄₄th bigger than is required for containing the necessary quantity of powder: against the sides and bottom of the box is put some straw; and this straw is covered over with empty sand bags, to prevent the powder from contracting any dampness: a hole is made in the side next the gallery, near the bottom for the saucisson to pass through, which is fixed to the middle of the bottom, by means of a wooden peg, to prevent its loosening from the powder: or that, if the enemy should get to the entrance, he may not be able to tear it out. This done, the powder is brought in sand bags, and thrown loose in the box, and covered also with straw and sand bags; upon this is put the cover of the box, pressed down very tight with strong props; and, to render them more secure, planks are also put above them, against the earth, and wedged in as fast as possible.

This done the vacant space between the props are filled up with stones and dung, and rammed in the strongest manner: the least neglect in this work will considerably alter the effect of the mine.

Then the auget is laid from the chamber to the entrance of the gallery, with some straw at the bottom; and the saucisson laid in it, with straw over it: lastly, it must be shut with a wooden cover nailed upon it. Great care must be taken, in stopping up the gallery, not to press too hard upon the auget, for fear of spoiling the saucisson, which may hinder the powder from taking fire, and so prevent the mine from springing. The gallery is stopped up with stones, earth, and dung, well rammed, 6 or 7 feet further from the chamber than the length of the line of least resistance.

_Globe of compression in_ MINES, _from Belidor_. If you imagine a large globe of earth homogeneous in all its parts, and, a certain quantity of powder lodged in its centre, so as to produce a proper effect without bursting the globe; by setting fire to the powder, it is evident, that the explosion will act all round, to overcome the obstacles which oppose its motion; and as the particles of the earth are porous, they will compress each other in proportion as the flame increases, and the capacity of the chamber increases likewise; but the particles of earth next to the chamber will communicate a part of their motion to those next to them, and those to their neighbors; and this communication will thus continue in a decreasing proportion, till the whole force of explosion is entirely spent; and the particles of earth beyond this term, will remain in the same state as they were at first. The particles of earth that have been acted upon by the force of explosion will compose a globe, which Mr. Belidor calls the _globe of compression_.

MINERS, in a _military sense_, are generally soldiers: most of the European regiments of artillery have each a company of miners, commanded by a captain and two lieutenants. When the miners are at work in the mines, they wear a kind of hood, to keep the earth that falls out of their eyes. In the English service the artificers are ordered for that purpose.

MINERS _tools_, consist in several sorts of spades, wheel-barrows, axes, hand-levers, chissels, sounding-augres, sledge-hammers, masons’ hammers, mattocks, augets, plummets, miner’s rule, and miner’s dial, &c.

_Different sorts of_ MINES, _are as follows_:

_Fougasses_, are a sort of small mines, frequently made before the weakest parts of a fortification, as the salient angles and faces, not defended by a cross fire.

_Treffle_ MINES, are mines with two chambers only.

T-MINES, so called from their great resemblance to that letter. They are double mines, having four lodgments.

_Double_ T-MINES, have eight lodgments, and four doors.

_Triple_ T-MINES, have twelve lodgments, and six doors.

_Double Treffle_ MINES, have four lodgments, and eight doors.

_Triple Treffle_-MINES, have six lodgments, and twelve doors.

MINING, in the _art of war_, is become one of the most essential parts of the attack and defence of places; so much artillery is used, that nothing above ground can withstand its effects; the most substantial ramparts and parapets can resist but a short time; the outworks, though numerous serve only to retard for a time the surrender of the place.

History informs us, that mines were made long before the invention of gun powder; for the ancients made galleries or underground passages, much in the same way as the moderns, from without, under the walls of the places, which they cut off from the foundation, and supported them with strong props; then they filled the intervals with all manner of combustibles, which being set on fire burnt their props and the wall being no longer supported, fell, whereby a breach was made.

The besieged also made under-ground passages from the town under the besieger’s machines, by which they battered the walls, to destroy them; which proves necessity to have been the inventress of mines, as well as of other arts.

The first mines, since the invention of gunpowder, were made in 1487, by the Genoese, at the attack of Serezanella, a town in Florence; but these failing, they were for some time neglected, till Peter Navarro, being then engineer to the Genoese, and afterwards to the Spaniards in 1503, against the French, at the siege of the castle del Ovo, at Naples, made a mine under the wall, and blew it up. In consequence of which the castle was taken by storm.

M. Valliere relates the same story, but differs in the name of the engineer; he says it was Francis George, an Italian, who, serving at Naples in quality of architect, proposed to Peter Navarro, the Spanish governor to take this castle by mines.

_Names of every thing used in_ MINING.

_Auget_, a kind of small trough, made of strong inch boards, about 4 inches square, in which the saucisson is laid in straw, to prevent the powder from contracting any dampness.

_Chamber_, the place where the powder is lodged, being first put in cubical boxes made for that purpose.

_Excavation_, _Entonnoir_, the pit or hole made by a mine when sprung.

_Focus_, the centre of the chamber where the powder is lodged.

_Fougas_, a kind of small mine.

_Fourneau_, See CHAMBER.

_Miners’ Tools_, are augers of several sorts, levers of different sorts, needles for working in rocks, rakes, spades, shovels, sledge-hammers, masons’ hammers, pick-axes, picks, mattocks, chissels, plummets, rules, a miner’s dial, &c.

_Line of least resistance_, is a line drawn from the centre of the space containing the powder, perpendicular to the nearest surface.

_Gallery_, the passage leading to the powder.

_Saucisson_, is a pipe or hose made of coarse cloth, whose diameter is about an inch, and filled with gunpowder; then laid in a trough or auget, which extends from the chamber to the entrance of the gallery, that the miner who sets fire to it, may have time to retire before it reaches to the chamber.

MINING, in _military affairs_, is the art of blowing up any part of a fortification, building, &c. by gunpowder. The art of _mining_ requires a perfect knowlege both of fortification and geometry; and by these previous helps, the engineer may be qualified to ascertain correctly the nature of all manner of heights, depths, breadths, and thicknesses; to judge perfectly of slopes and perpendiculars, whether they be such as are parallel to the horizon, or such as are visual; together with the true levels of all kinds of earth. To which must be added, a consummate skill in the quality of rocks, earths, masonry, and sands; the whole accompanied with a thorough knowlege of the strength of all sorts of gunpowder

MINION, a piece of ordnance, of which there are two kinds, the large and ordinary: the large minion has its bore 3¹⁄₄ inches diameter, and is 1000 pounds weight; its load is 3¹⁄₄ pounds of powder; its shot three inches in diameter, and 3³⁄₄ pounds weight; its length is eight feet, and its level range 125 paces. The ordinary minion is three inches diameter in the bore, and weighs about 800 pounds weight: it is seven feet long, its load 2¹⁄₂ pounds of powder, its shot near three inches in diameter, and weighs three pounds four ounces, and shoots point blank 120 paces.