A military dictionary

by William Duane

BATTALION-MEN. All the soldiers belonging to the different companies of an infantry regiment are so called, except those of the two flank companies.

Camp-Color MEN. Soldiers under the immediate command and direction of the quarter-master of a regiment. Their business is to assist in marking out the lines of an encampment, &c. to carry the camp colors to the field on days of exercise, and fix them occasionally for the purpose of enabling the troops to take up correct points in marching, &c. So that in this respect they frequently, indeed almost always, act as guides, or what the French call Jalonneurs. They are likewise employed in the trenches, and in all fatigue duties.

Drag-rope MEN. In the old artillery exercise, the men attached to light or heavy pieces of ordnance, for the purpose of advancing or retreating in action, were so called; the drag rope being exploded for the bricole, the term is preserved merely for explanation. The French servans à la prolonge are of this description.

MENACE, an hostile threat. Any officer or soldier using menacing words or gestures in presence of a court-martial, or to a superior officer, is punishable for the same.—See the [Articles of War].

MENSURATION, in general, denotes the act or art of measuring lines, superficies, and solids.

Mensuration, in military mathematics, is the art or science which treats of the measure of extension, or the magnitude of figures; and it is, next to arithmetic, a subject of the greatest use and importance, both in affairs that are absolutely necessary in human life, and in every branch of mathematics: a subject by which sciences are established, and commerce is conducted; by whose aid we manage our business, and inform ourselves of the wonderful operations in nature; by which we measure the heavens and the earth, estimate the capacities of all vessels and bulks of all bodies, gauge our liquors, build edifices, measure our lands and the works of artificers, buy and sell an infinite variety of things necessary in life, and are supplied with the means of making the calculations which are necessary for the construction of almost all machines.

It is evident that the close connection of this subject with the affairs of men would very early evince its importance to them; and accordingly the greatest among them have paid the utmost attention to it; and the chief and most essential discoveries in geometry in all ages, have been made in consequence of their efforts in this subject. Socrates thought that the prime use of geometry was to measure the ground, and indeed this business gave name to the subject; and most of the ancients seem to have had no other end besides mensuration in view in all their labored geometrical disquisitions. Euclid’s elements are almost entirely devoted to it; and although there be contained in them many properties of geometrical figures, which may be applied to other purposes, and indeed of which the moderns have made the most material uses in various disquisitions of exceedingly different kinds; notwithstanding this, Euclid himself seems to have adapted them entirely to this purpose: for, if it be considered that his elements contain a continued chain of reasoning, and of truths, of which the former are successively applied to the discovery of the latter, one proposition depending on another, and the succeeding propositions still approximating towards some particular object near the end of each book; and when at the last we find that object to be the quality, proportion or relation between the magnitudes of figures both plane and solid; it is scarcely possible to avoid allowing this to have been Euclid’s grand object. And accordingly he determined the chief properties in the mensuration of rectilineal plane and solid figures; and squared all such planes, and cubed all such solids. The only curve figures which he attempted besides, are the circle and sphere; and when he could not accurately determine their measures, he gave an excellent method of approximating to them, by shewing how in a circle to inscribe a regular polygon which should not touch another circle, concentric with the former, although their circumferences should be ever so near together; and, in like manner, between any two concentric spheres to describe a polyhedron which should not any where touch the inner one: and approximations to their measures are all that have hitherto been given. But although he could not square the circle, nor cube the sphere, he determined the proportion of one circle to another, and of one sphere to another, as well as the proportions of all rectilineal similar figures to one another.

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.