M E N 
and what feveral menftrva will diffolve any metal. JBticoii. 
■—White metalline bodies, by reafon of their exceflive 
denfity, feem to refleft almoft all the light incident on 
their firft luperficies, unlefs by folution in meitjlrunms 
they be reduced into very fmall particles, and then they 
become tranfparent. Newton's Optics. 
MENSU'RA, [Latin.] A meafure ; a ftandard. 
MENSU'RA REGA'LIS, the king's ftandard meafure, 
kept in the Exchequer, according to which all others are 
to be made. See ’flat. 16 Car. I. c. 19 ; and thev article 
Measure. 
MENSURABIL'ITY, f Capacity of being meafured. 
MEN'SURABLE, adj. [incvjura, Lat. meafure.] Mea- 
furable ; that may be mealured.—We meafure our time 
by law and not by nature. The folar month is no pe¬ 
riodical motion, and not eafily menfurablc; and the months 
unequal among themfelves, and not to be meafured by 
even weeks or days. Holder. 
MEN'SURABLENESS, f. The ftate or quality of 
being meafurable. 
MEN'SURABLY, adv. With a capacity of being mea¬ 
sured. 
MEN'SURAL, adj. Relating to meafure. 
To MEN'SURATE, t>. a. [Latin.] To meafure ; to take 
the dimenfion of anything. 
MENSURA'TION, J'. [from menfura, Latin.] The a6t 
or practice of meafuring; refult of meafuring.—After 
giving the menfuration and argumentation of Dr. Cum¬ 
berland, it would not have been fair to have fuppreffed 
thole of another prelate. Arbuthnot. 
Mensuration is that branch of mathematics which 
is employed in afeertaining the extenfion, folidities, and 
capacities, of bodies ; and, in confequence of its very ex- 
tenfive application to the purpofes of life, it may be con- 
fiderecl as one of the moft ufeful and important of all the 
mathematical fciences ; in fadt, menfuration, or geometry, 
which were anciently nearly fynonymous terms, feem to 
have been the root whence all the other exadt fciences, 
with the exception of arithmetic, have derived their origin. 
Menfuration is, next to arithmetic, a fubjedf of the 
greateft ufe and importance, both in affairs that are abfo- 
lutely neceftary in human life, and in every branch of 
mathematics 5 a fubjedf by which fciences are eftablilhed, 
and commerce is conducted; by whofe aid we manage 
our bufinefs, and inform ourfelves of the wonderful ope¬ 
rations in nature ; by which we meafure the heavens and 
the earth, eftimate the capacities of all veffels, and bulks 
of all bodies, gauge our liquors, build edifices, meafure 
our lands, and the works of artificers, buy and fell an in¬ 
finite variety of things neceftary in life, and are fupplied 
with the means of making the calculations which are ne- 
ceflary for the conftrudfion of almoft all machines. 
It is evident, that the clofe connedtion of this fubjedf 
with the affairs of men, would very early evince its im¬ 
portance to them ; and accordingly the greateft among 
them have paid the utmolf attention to it: and the chief 
and moft effential difeoveries in geometry in all ages, have 
been made in confequence of their efforts in this fub- 
jedt. Socrates thought that the prime ufe of geometry 
was to meafure the ground and indeed this gave name 
to the fcience (viz. yyu ptr^eu-,') and moft of the ancients 
feem to have had no other endbefides menjuration in view 
, in all their laboured geometrical difquifitions. 
Although we cannot attribute the invention of the 
fcience of menfuration to any particular perfon or na¬ 
tion, yet we may dilcover it in an infant ftate, riling as it 
were into a fcientific form amongft the ancient Egyp¬ 
tians ; and hence the honour of the difeovery has fre¬ 
quently been given to this people, and to the circum- 
dfance of the overflowing of the Nile. 
It is, however, to the Greeks that we mull confider 
ourfelves indebted for having firft embodied the leading 
principles of this art into a regular fyftem. Euclid’s Ele¬ 
ments of Geometry were probably firft wholly diredted 
to this fubjedf ; and many of thofe beautiful and elegant 
-- V-GL.XV. N0..IOZ8. 
MEN" 109 
geometrical properties, which are fo much and fo juftly 
admired, it is not unlikely arofe out of Ample inveftiga- 
tions, diredted folely to the theory and practical applica¬ 
tion of menfuration. Thefe collateral properties, when 
once difeovered, foon gave rife to others of a fimilar kind, 
and thus geometry, which was firft inlfituted for a parti¬ 
cular and limited purpofe, became itfelf an independent 
and important fcience, which has perhaps done more to¬ 
wards harmonizing and expanding the human faculties 
than all the other fciences united. But, notwithftanding 
the perfection which Euclid attained in geometry, the 
theory of menfuration was not in his time advanced be¬ 
yond what related to right-lined figures; and this, fo far 
as regards furfaces, might all be reduced to that of mea¬ 
furing a triangle; for, as all right-lined figures maybe 
reduced to a number of trilaterals, it was only neceftary 
to know how to meafure thefe, in order to find the fur- 
face of any other figure whatever bounded only by right 
lines. The menfuration of folid bodies, however, was of 
a more varied and complex nature, and gave this cele¬ 
brated geometrician a greater fcope for the exercile of 
his fuperior talents; and, ftill confining liimfelf to bodies 
bounded by right-lined plane fuperficies, he was able to 
perform all that can be done even at this day. With re¬ 
gard to curvilineal figures, he attempted only the circle 
and the fphere, and if he did not fucceed in thofe, he 
failed only where there was no poflibility of fuccefs ; but 
the ratio that fuch furfaces and folids have to each other 
he accurately determined. 
After Euclid, Archimedes took up the theory of men¬ 
furation, and carried it to a much greater extent. He 
firft found the area of a curvilinear fpace, unlefs indeed 
we except the lunules of Hippocrates, which required no 
other aid than that of the geometrical elements. Archi¬ 
medes found the area of the parabola to be two-thirds of 
its circumfcribing redtangle, which, with the exception 
above ftated, was the firft inftance of the quadrature of a 
curvilinear fpace. The conic feftions were at this time 
but lately introduced into geometry, and they did not 
fail to attraft the particular attention of this celebrated 
mathematician, who difeovered many of their very curious 
properties and analogies. He likewife determined the 
ratio of fpheres, fplieroids, and conoids, to their circum¬ 
fcribing cylinders, and has left us liis attempt at the 
quadrature of the circle. He demonftrated that the area 
of a circle is equal to the area of a right-angled triangle, 
of which one of its fides 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 diame¬ 
ter, a problem which has engaged the particular attention 
of the moft celebrated mathematicians of all ages, but 
which remains at prefent, and in all probability ever will 
remain, the defideratum of geometricians, and at the fame 
time a convincing and humiliating proof of the limited 
powers of the human mind. 
Some advances were fucceffively made in geometry and 
menfuration, though but little novelty was introduced 
into the mode of inveftigation till the time of Cavalerius. 
Till his time the regular figures circumfcribed about the 
circle, as well as thofe inlcribed, were always confulered 
as being limited, both as to the number of their tides, and 
the length of each. He firft introduced the idea of a 
circle being a polygon of an infinite number of fides, each 
of which was of courfe indefinitely fmall; folids were 
fuppofed to be made up of an infinite number of fe&ions 
indefinitely thin. See. This was called the doctrine of in- 
divifibles, which was very general in its application to a 
variety of difficult problems, and by means of it many 
new and interefting properties were difeovered; but it 
unfortunately wanted that diftinguifhing charadleriftic 
which places geometry fo pre-eminent amongft the other 
exaft fciences. In pure elementary geometry we proceed 
from ftep to ftep, with fuch order and logical precifion s 
that not the flighteft doubt can reft upon the mind with 
F f regard 
