After Euclid, Archimedes took up the theory of mensura tion, and carried it to a much greater extent. Ile first found the area of a curvilinear space, unless, indeed, we except the Imittles of Hippocrates, which required no other aid than that of the geometrical elements. Archimedes found the area of the parabola to be two-thirds of its circumscribing rectangle, which, with the exception above stated, was the first instance of the quadrature of a curvilinear space. The Conic sections were at this time but lately introduced into geometry, and they did not fail to attract the particular attention of this celebrated mathematician, who discovered many of their very curious properties and analogies, lie likewise determined the ratio of spheres, spheroids, and co Bids, to their circumscribing cylinders, and has left us his attempt at the quadrature of the circle. He demonstrated that the area of a circle is equal to the area of a right-angled triangle, of mhich one of its sides about the right angle is equal to the radius, and the other equal to the circumference, and thus reduced the quadrature of the circle to that of determining the ratio of the circumference to the diameter, a problem which has engaged the particular attention of the most celebrated mathematicians of all ages, but which remains at present, and in all probability ever will remain, the desi deratum ofgeometricians, and, at the same time, a convincing and humiliating proof of the limited powers of the human mind.
But, notwithstandino. Archimedes failed in establishing the real quadrature of the circle, it is to him we are indebted for the first approximation towards it. Ile foin.d the ratio between the diameter of a circle, and the periphery of a circumscribed polygon of 96 sides, to be less than 7 to .32, or less than 1 to 314 ; but the ratio between the diameter, and periphery of an inscribed polygon of the same number of sides, he found to be greater than 1 to 314 ; whence is fortiori, the diameter (if a circle is to its circumference in a less ratio than 1 to :11, or less than 7 to !2:2. Having thus established this ap proximate ratio between the circumference and diameter, that of the area of the circle to its oh:mused bed square, is found to be nearly as 11 to 14. Archimedes, however, makes the latter the leading proposition. These, it is true, are but rude approximations, compared with those that have been since discovered ; but, considering the state of science at this period, particularly or arithinetic, we cannot hut admire the genius and perseverance of the man, who, notwithstanding the difficulties that were opposed to him, succeeded in deducing this result, which may be considered as having led the way to the other more accurate approximations which followed, most of which, till the invention of fluxions, were obtained upon similar principles to those employed by this eminent geometrician. Archimedes also determined the relation between the circle and ellipsis, as well as that of their similar parts ; besides which figures, he has left us a treatise on the spiral, a description of which will be given under that article. See SPIRAL.
Some advances were successively made in geometry and mensuration, though hut little novelty was introduced into the mode of investigation till the time of Cavalerius. Till his time, the regular figures circumscribed about the circle, as well as those inscribed, were always considered as being limited, both as to the number of their sides, and the length of each. He first introduced the idea of a circle being a polygon of an infinite number ot' sides, each of which was, of course, indefinitely small ; solids were supposed to he made up of an indefinite number of sections indefinitely thin, &c.
This was called the doctrine of indivisibles, which was very general in its application to a variety of difficult problems, and by means of it many new and interesting properties were discovered : but it unfortunately wanted that distinguishing characteristic which places geometry so pre-eminent amongst the other exact sciences. In pure elementary geometry, we proceed from step to step, with such order and logical pre cision, that not the slightest doubt can rest upon the mind with regard to any result deduced from those principles ; but in the new method of considering the subject, the greatest possible care was necessary in order to avoid error, and fre quently this was not sufficient to guard against erroneous conclusions. But the facility and generality which it pos sessed, when compared with any other method then dis covered, led many eminent mathematicians to adopt its principles, and of these, Huygens, Dr. Wallis, and James Gregory, were the most conspicuous, being all very fortunate in their application of the theory of indivisibles. Huygens, in particular, must always be admired f.nr his solid, accurate, and masterly performances in this branch of geometry. The theory of indivisibles was. however, disapproved of by many mathematicians, and particularly by Newton, who, amongst his numerous and brilliant discoveries, has given us that of the method of fluxions, the excellency and generality of which immediately superseded that of indivisibles, and revived some hopes of squaring the circle, and accordingly its quadrature was again attempted with the greatest eagerness. The quadrature of a space, and the rectification of a curve, was now reduced to that of finding the fluent of a given fluxion ; but still the problem was found to be incapable of a general solution in finite terms. The fluxion of every fluent was found to be always assignable, but the converse proposition, viz., of finding the fluent of a given fluxion, could only be etR.cted in particular cases, and amongst these exceptions, to the great disappointment and regret of geometricians, was included the case of the circle, with regard to all the forms of finxions under a hich it could be obtained.
At length, all hopes of accurately squaring the circle and some other curves being abandoned, mathematicians began to apply themselves to finding the most convenient series for approximating towards their true length and quadrature ; and the theory of mensuration now began to make rapid progress towards perfection. Many of the rules, however, were given in the transactions of learned societies, or in sepa rate and detached works, till at length Dr. Hutton formed them into a complete treatise, intitled, A Treatise on Mensu ration, in which the several rules are all demonstrated, and some new ones introduced. Mr. Bonnycastle also published a very neat work on this subject, intitled An Introduction to Mensuration. Several books on the subject have been pub lished since the above, which, notwithstanding, still maintain their high reputation.
Particular rules for measuring the various kinds of geome trical figures and solids will be found under their respective heads; but as a collection of examples of the mensuration of distances capable of application to the purposes of enginery and architecture is a desideratum in science, the following, likely to occur in general practice, are here inserted.
MENsurtATioN OF LINES.