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tig 8 . ^withstanding we cannot attribute the invention ofJet w 6nCe mei 'suration to any particular person or nation,i ato a e m . a y discover it in an infant state, rising, as it were,heu Ce ® cier >tific form, amongst the ancient Egyptians; andt° lle honour of the discovery has frequently been giventh e Tfl..P eo Pi e ) and to the circumstance of the overflowing oferi(j s ■ > which takes place about the middle of June, andSeptember. It is, however, to the Greeks that wethe ourselves indebted for having first embodied
■^ u clid’ "" P r ' nc iples of this art into a regular system.dj re , ® Elements of Geometry were probably first whollye kgant t0 t *” S su ' 3 j ect » an< i many of those beautiful andjus,! c geometrical properties, which are so much and sogation v? lre< i> it is not unlikely, arose out of simple investi-of S .
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These collateral properties, when once dis-geo me T> Soon gave rise to others of a similar kind; and thushaiited^’ w ^*ch was first instituted for a particular andsci ence P u ’P ose , became itself an independent and importantatuj ’ "'hich has perhaps done more towards harmonizingUQjt e( j P an< iing the human faculties, than all the other sciences^ *
>0 g eoi ^ 0,; withstanding the perfection which Euclid attainedadv an j tr 7’ theory of mensuration was not in his timethis, Sq beyond what related to right-lined figures ; andm eag a f as regards surfaces, might all be reduced to thatte dti Ce( j Ul In S a triangle ; for as all right-lined figures may be}. t0 a number of trilaterals, it was only necessary toa Hy ot j w to measure these, in order to find the surface of^Usvn. T" ^ ure whatever bounded only by right lines. Thea M Co 10n °f solid bodies, however, was of a more variedS f eat er P ° x nat ure, and gave this celebrated geometrician aSC ?P e f°r the exercise of his superior talents, and, still^’jnself to bodies bounded by the right-lined planebiis j GS ’ lu was able to perform all than can be done even°»ly tl, e a ^‘ ^>th regard to curvilinear figures, he attempted1 ° Se > h e C f IC ^ e ari 'l the sphere, and if he did not succeed inP^t th e . ed only where there was no possibility of success;6 a eci,„,. \° t ' mt su °h surfaces and solids have to each other. Aft^'y ^termined.
f ° n ’ and Uc Archimedes took up the theory of mensura-3 0 aad t j ie c arried it to a much greater extent. He first? c «pt tj le , area °f a curvilinear space, unless, indeed, weti l han ti, Unu ^ es of Hippocrates, which required no other
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°t the geometrical elements. Arclqmedes foundv P P llra bola to be two-thirds of its circumscribingc ^ hi8(7 v n ch, with the exception above stated, was thet'c S o ct - e the quadrature of a curvilinear space. The
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e - Sect" l,lc quaarature of a curvilinear space.•H 0tt| etry Iotls were at this time but lately introduced intolotion’ 0 f .they did not fail to attract the particularIj/hty 0 f this celebrated mathematician, who discovered^ise ( j^"' 7 er y curious properties and analogies. He
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latl ° of the circumference to the diameter, ap r ^ebratgu ms en 8 a ged the particular attention of the< ta„. e8 ®nt. „ J 5 , ptuthematicians of all ages, but which remains
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. all probability ever will remain, the desi-
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in establishing
the real quadrature of the circle, it is to him we are |indebted for the first approximation towards it. He found .the ratio between the diameter of a circle, and the periphery Iof a circumscribed polygon of 96 sides, to be less than 7 to j22, or less than 1 to 3$; 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 3 %; whence a fortiori, \the diameter of a circle is to its circumference in a less ratio than I1 to 3\, or less than 7 to 22. Having thus established this ap- jproximate ratio between the circumference and diameter, thatof the area of the circle to its circumscribed square, is found to .be nearly as 11 to 14. Archimedes , however, makes the latter jthe leading proposition. These, it is true, are but rude \approximations, compared with those that have been sincediscovered; but, considering the state of science at this jperiod, particularly of arithmetic, we cannot but admire the jgenius and perseverance of the man, who, notwithstanding thedifficulties that were opposed to him, succeeded in deducingthis result, which may be considered as having led the wayto the other more accurate approximations which followed,most of which, till the invention of fluxions, were obtainedupon similar principles to those employed by this eminentgeometrician. Archimedes also determined the relationbetween the circle and ellipsis, as well as that of their similarparts; besides which figures, he has left us a treatise on thespiral, a description of which will be given under that article.See Spiral. j
Some advances were successively made in geometry and imensuration, though but little novelty was introduced intothe mode of investigation till the time of Cavalerius. Tillhis time, the regular figures circumscribed about the circle,as well as those inscribed, were always considered as beinglimited, both as to the number of their sides, and the lengthof each. He first introduced the idea of a circle being apolygon of an infinite number of sides, each of which was, ofcourse, indefinitely small; solids were supposed to be madeup 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 werediscovered : but it unfortunately wanted that distinguishing jcharacteristic which places geometry so pre-eminent amongstthe other exact sciences. In pure elementary geometry, we ;proceed from step to step, with such order and logical pre- jcision, that not the slightest doubt can rest upon the mind jwith regard to any result deduced from those principles ; but :in the new method of considering the subject, the greatest Ipossible 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 itsprinciples, and of these, Huygens, Dr. Wallis, and JamesGregory , were the most conspicuous, being all very fortunatein their application of the theory of indivisibles. Huygens ,in particular, must always be admired for his solid, accurate, •and masterly performances in this branch of geometry. Thetheory of indivisibles was, however, disapproved of by many \mathematicians, and particularly by Newton, who, amongsthis 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 somehopes 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 j