M E 1ST 
Tt 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 geo- 
metry was to measure the ground, and in- 
deed this business gave name to the subject ; 
and most of the antients seem to have had 
no other end besides mensuration in view in 
all their geometrical disquisitions. Euclid’s 
Elements are almost entirely devoted to it ; 
and although there are contained in them 
many properties of geometrical figures, 
which may be applied toother 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 is 
considered that his Elements contain a con- 
tinued chain of reasoning, and of truths, of 
which the former are successively applied 
to the discovery of the latter, one proposi- 
tion depending on another, and the succeed- 
ing propositions still approximating towards 
.some particular object near the end of each 
book ; and when at the last we find that ob- 
ject 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: 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, bv 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, be- 
tween any two concentric spheres to de- 
scribe a polyhedron which should not any 
where touch the inner one : and approxima- 
tions to their measures are all that have hi- 
therto 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 simi- 
lar 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 Hippocrates 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 mea- 
suring of them as well as other figures. Ac- 
cordingly he determined the relations of 
spheres, spheroids, and conoids, to cylin- 
ders and cones ; and the relations of para- 
bolas to rectilineal planes whose quadratures 
had long before been determined by Euclid. 
He has 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 
MEN 
circumference; and so reduced the quadra- 
ture ot the circle to the determination cl the 
ratio ot the diameter to the circumference ; 
but which, however, has not yet been found. 
Being disappointed of the exact quadrature 
ot the circle, tor want of the rectification o! 
its circumference, which all his methods 
would not effect, he proceeded to assign 
an useful approximation to it : this he effect- 
ed by the numerical calculation of the pe- 
rimeters of the inscribed and circumscribed 
polygons ; from which calculations it ap- 
pears that the perimeter of the circumscrib- 
ed regular polygon of 192 sides is to the 
diameter, in a less ratio than that of 3 1 . (3|§) 
to L, and that the inscribed polygon of9t> 
sides is to the diameter in a greater ratio 
than that of 3y° to 1 ; and consequent- 
ly much more than the circumference 
of the circle is to the diameter in a 
less ratio than that of 31. to 1, but great- 
er than that of 3i5. to 1 : the first ratio of 3^ 
to 1, reduced to whole numbers, gives 
that of 22 to 7, for 3-i. : 1 : : 22 : 7, which 
therefore will be nearly the ratio of the cir- 
cumference 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 likew ise de- 
termined the relation between the circle and 
ellipse, with that of their similar parts. The 
hyperbola too, in all probability, he attempt- 
ed ; but it is not to be hoped, 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 lias left ns a 
treatise on the spiral described by a point 
moving uniformly along a right line, which 
at the same time moves with an uniform an- 
gular motion ; and determined the propor- 
tion of its area to that of its circumscribed 
circle, as also the proportion of their sec- 
tors. 
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 admit- 
ting into his demonstrations nothing but 
principles perfectly geometrical and unex- 
ceptionable : and although his most general 
method of demonstrating the relations of 
curved figures to straight ones, is by inscrib- 
ing polygons in them, yet to determine 
those relations, he does not increase the num- 
ber and diminish the magnitude of the sides 
ad infinitum ; but from this plain fundamen- 
tal principle, allowed in Euclid’s Elements, 
viz. that any quantity may be so often mul- 
tiplied, 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 progressions, whose terms 
are similar to the inscribed figures ; but 
without considering such series to be con- 
tinued 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 
M E E I5fj 
fi qures, in which he proceeds as it were me- 
• nsnically bv weighing them. 
Several other eminent men among the 
indents w rote upon (his subject, both before 
and after Euclid and Archimedes ; bat their 
attempts were usually upon particular parts 
of it, and according to methods not essen- 
tially different from theirs. Among these 
are to be reckoned Thales, Anaxagoras, Py- 
thagoras, Bryson, Antiphon, Hippocrates of 
Chios, Plato, Apollonius, Philo, and Ptole- 
my ; most of whom wrote of the quadrature 
of the circle : and those after Archimedes, by 
his method, usually extended the approxi- 
mation to a greater degree of accuracy. 
Many cf the moderns have also prosecut- 
ed the same problem of the quadrature of 
the circle, after the same methods, to great- 
er lengths: such are Vieta and Metius, 
whose proportion between the diameter and 
circumference is that of 113 to 355, which 
is within about of the true ratio ; 
but above all Lndolph van Ceulen, who, with 
an amazing degree of industry and patience, 
bv the same ratio to 20 places of. figures, 
making it that of 1 to 3.1415926535897932 
3846 -f-. See Circle. 
Hence it appears, that all or most of the 
material improvements or inventions in the 
principles or methods of treating of geometry, 
have been made especially for the improve- 
ment of this chief part of it, mensuration, 
which abundantly shews the dignity of the 
subject ; a subject which, as Dr. Barrow' 
says, after mentioning some other things, “ de- 
serves to be more curiously weighed, because- 
from hence a name is imposed upon that 
mother and mistress of the rest of the mathe- 
matical 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 measur- 
ing the earth, and fixing the limits of pos- 
sessions) : though the name seemed very ridi- 
culous to Plato, who substitutes in its place 
the more extensive name of metrics or men- 
suration ; and others after him give it the’ 
title of pantometry, because it teaches the 
method of measuring all kinds of magni- 
tudes.” See Heights, Surveying, Le- 
velling, Geometry, and Gauging. 
MEItCURIALIS, mercury, a genus of 
the enneandria order, in the dicecia class of 
plants, and in the natural method ranking 
under the 3 S t h order, tricocceax The calyx 
of the male is tripartite ; there is no corolla, 
but 9 or 12 stamina ; the anthers globular and 
twin. The female calyx is tripartite ; there is 
no corolla, but two styles ; the capsule is bi- 
coccous, bilocular, and monospermous. 
There are six species. 
Of these, the perennis, according to Mr. 
Lightfoot, is of a soporific deleterious nature, 
noxious both to man and beast. There are 
instances of those who have eaten it by mis- 
take, instead of the chenopodium "bonus 
Henricus, or English mercury, and have 
thereby slept their last. Toumefort informs 
vis, that the French make a syrup , of the juice 
of the annua, another species, two ounces of 
which are given as a purge; and that they 
use it in pessaries and clysters, mixing one 
part of honey to one and a half of the 
juice. Dr. Withering differs greatly from 
Lightfoot concerning the qualities of the per- 
enuis. “ This plant, (says he), dressed like 
spinach, is very good eating early in tli<^ 
