Men uration

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 nece ary 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 busine , and inform ourselves of the wonderful operations in nature; by which we measure the heavens and the earth, estimate the capacities of all ve els 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 nece ary in life, and are supplied with the means of making the calculations which are nece ary 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 e ential discoveries in geometry in all ages, have been made in consequence of their efforts in this subject. So crates thought that the prime use of geometry was to measure the ground, and indeed this busine 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 succe ively 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 po ible 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. Arch i me des took up mensuration where Euclid left it, and carried it a great length. He was the first who squared a curvilineal space, unle Hypo crates 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 a ign 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 le 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 le 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 succe , 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 progre ions, 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 Arch i me des; but their attempts were usually upon particular parts of it, and according to methods not e entially different from theirs. Among these are to be reckoned Thales, Anaxagoras, Pythagoras, Bryson, Antiphon, Hypo crates of Chios, Plato, A poll on i us, Philo, and Ptolomy; most of whom wrote of the quadrature of the circle, and those after Arch i me des, 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 a ignable 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 endle 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 a igned, 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, a erted that they had discovered the quadrature of the circle, and published their attempts in the form of strict geometrical demonstrations, with such a urance 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, . 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 a umed 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 maxim is & minim is , and many other subjects, became general problems, and delivered in the form of general theories which included all particular cases: thus, in quadratures, an expre ion would be investigated which defined the areas of all po ible 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 a igned; and the like in other things. And although no curve, whose quadrature was unsucce fully attempted by the ancients, became by this method perfectly quadrable, there were a igned 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 eagerne . 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 a ignable, 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 expre ing the fluxion of any area, in general, could be a igned 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 a igned 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 profe or 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 mistre 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 po e ions) 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 Pan to me try , because it teaches the method of measuring all kinds of magnitudes.” See Surveying , Levelling , and Geometry .