Part 301

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.