MEA 
M E A 
M E A 
diameter is 1 ) = c ; and AC, the altitude of the 
cone, = .v ; then CD = a — x. By El. iii. 3.5, 
AC X CD = CB 2 , that is, x X a — a- — ax — x 2 
~ CB 2 ; but the square cf the diameter is four 
times the square ef the radius ; therefore, by Ei. 
xii. 2 , 4ucx — 4-c-x 2 — • the area of the cone's base, 
which, by EI. xii. 10 , drawn into f x, is d. dcx 2 __ 
= the cone’s solidity, which is a maximum ; 
therefore, by taking- away what is common, we 
get ax 2 — jr a maximum, the fluxion of which 
is = 0 , that is, 2 axx — 3x 2 x — 0 , or 2 a — 3 a -, 
2a 
I and a- z=: . So that the cone will be a maxi- 
3 
: mum, when its altitude is equal to two-thirds 
I of the sphere’s diameter. 
MEAD, an agreeable liquor made of ho- 
, ney and wafer. See Ho new. 
1 here are many receipts for making mead, 
: of which the. following is one of the best. 
Take four gallons of water, and as much ho- 
| ney as will make it bear an egg; add to this 
| the rind of three lemons, boil it, and scum it 
i well as it rises. Then take it off (he lire, and 
| add the three lemons cut in pieces; pour it 
| into a clean tub or open vessel, and let it w ork 
| for three days, then scum it well, and pour 
! off the dear part into a cask, and let it stand 
I open till it ceases to make a hissing noise; 
; then stop it up dose, and in three months 
I time it will be fine and fit for bottling. If 
j .you would give it a finer flavour, take cloves, j 
mace, and nutmeg, ol each four drams; beat 
them small, tie the powder in a piece of 
cloth, and put it into the cask. 
MEADOW. See Husbandry. 
MEAN, a middle state between two extremes; 
as a mean motion, mean distance, arithmetical 
j mean, geometrical mean, &c. 
Arithmetical Mean, is half the sum of the ex- 
| tremes. So, 4 is an arithmetical mean between 
2 and 6 , or between 3 and 5, or between 1 and 
7 ; also an arithmetical mean between a and b is 
! _ 
| — , or -f- •§ b. , 
Geometrical Mean, commonly called a mean I 
| proportional, is the square root of the product j 
r of the two extremes ; so that, to find a mean | 
j proportional between two given extremes, mul- 
! tiply these together, and.extract the square root j 
; °f the product. Thus, a mean proportional be- 
tween 1 and 9, is yT " x~9~= y/9 = 3 ; a mean 
between 2 and 4\ is y 2 x H — y'D = 3 also; 
the mean between 4 and 6 is x 6 — ^/24; 
and the mean between a and b is ^/ab. 
The geometrical mean is always less than the 
arithmetical mean between the same two ex- 
, tremes. So the arithmetical mean between 2 
j and 4\ is 3-1, but the geometrical mean is only 3. 
| To prove this generally, let a and b be any two 
; terms, a the greater, and b the less ; then, uni- 
versally, the arithmetical mean ^ shall be 
2 
greater than the geometrical mean *Jab, or 
a l> greater than 2 *Jab. For, by 
squaring both, they are a 2 2ab -\- b 2 ~7 4 ab \ 
subtr. 4 ab from each; then a 2 — 2 ab -}- b 2 ~y 0 , 
i that is, - - - [a — b) 2 -7 0 . 
j To find a Mean Proportional geometrically, be- 
tween two given lines M and N. Join the two 
given lines together at C, in one continued line 
AB ; upon the diameter AB describe a semi- 
circle, and erect the perpendicular CD ; .which 
will be the mean proportional between AC and 
CB, or M and N. 
| . To find tnvo Mean Proportionals between two 
■given extremes. Multiply each extreme by the 
fcquare of the other, viz. the greater extreme by 
the square of the less, and the less extreme by 
! the square of the greater ; then extract the cube 
root out of each product, and the two roots 
will be the two mean proportionals sought. 
That is, 3 / a 2 b and if air are the two means be- 
tween a and b. So, between 2 and 16, the two 
mean proportionals are 4 and 8 ; for^/2 2 x 16 
= — 4, and yf> x W — {/ 51 2 = 8 . 
In a similar manner we proceed for three 
means, or four means, or five means, &c. ; from 
all which it appears, that the series ef the several 
numbers of mean proportionals between a and 
b will be as follows : viz. 
1 mean, ^/ab ; 
2 means, a 2 u, 1/ ab 1 ; 
3 means, a' b, if a b 2 , \/ ab x \ 
4 means, \/ d'b, s / a'b 1 ,^/ a 2 b\\/ ah ' ; 
5 means. £/ a'L t \/ a s b 2 ,l/ a'b\ \/a 2 b', \/ aT : 
&c. &c. 
_ Hatmontcal Mean, is double a fourth propor- 
tional to the sum of the extremes, and the two 
extremes themselves a and b : thus, as a 4 - b ‘ a 
.. Tab 
. . 2 b l — | — - — m, the harmonical mean be- 
a -p b 
tween a and b. Or it is the reciprocal of the 
arithmetical mean between the reciprocals of 
the given extremes ; that is, take the reciprocals 
of the extremes a and b, which will be — - and 
^ a 
y ? then take the arithmetical mean between 
these reciprocals, or half their sum, which will 
be— 1 
2 a ~ 
of this is 
1 
W 
2 ab 
a -J- b 
a -{- b 
2 ab 
; lastly, the reciprocal 
the harmonical mean : 
for arithmeticals and harmomcals are mutually 
reciprocal^ of each other; 
so that if a, m, b, &c. be arithmeticals, 
then shall — , , — , &c. be harmonieals ; 
a m b 
or if the former be harmonieals, the latter will 
| be arithmeticals. 
for example, to find a harmonical mean be- 
J tween 9 and 6 : here « = 2, and b = 6; there- 
: ~ ab _ 2 X 2 X 6 24 
a -f- b 2 — (— 6 8 ‘ 3 
| harmonical mean sought between 2 and 6. 
Pappus has shewn a curious similarity that 
subsists between the three different sorts of 
mean: a, m.- /k being three continued terms, 
either arithmeticals, geometricals, or harmoni- 
eals, then in the 
A.nthmeticals a * a ", a — m * m b 
Geometricals a * m * * a — m * m — b 
Harmonieals a ; b * * a — m ] m — b. 
MEASLES. See Medicine. , 
Measure of an angle, is an arch describe 
ed from (he vertex in any place between its 
legs. Hence angles are di stinguished by the 
ratio or the arches, described from the vertex 
between the legs to the peripheries. Angles 
then are distinguished by those arches ; and 
the arches are distinguished by their ratio to 
the periphery: thus an angle 'is said to be so 
many degrees as there are in the said arch. 
See Angle. 
Measure of a figure, or plane surface, is 
a square whose side is one inch, one foot, 
one yard, or some other determinate length. 
Among geometricians, it is usually a rod call- 
ed a square rod, (Jrivided into 1 0 square feet, 
and the square feet into 1 0 square digits.- 
Measure of a solid, is a cube whose side 
is one inch, foot, yard, or any other deter- 
minate length. In geometry it is a cubic 
perch, divided into- cubic feet, digits, &c. 
119 
Hence cubic measures, or measures of capa- 
city. See Sphere, Cube, & c. 
Measure of velocity, in mechanics, the 
space passed over By a moving hotly in a 
given time. To measure a velocity therefore; 
the space must be divided into as many equal 
parts as the time is conceived to be divided 
into; the quantity of space answering to such 
a part of time is the measure of the. velo- 
city. 
Measure, in geometry, denotes any 
quantity assumed ' as one, or unity, to which 
ttie ratio of the other homogeneous or similars- 
quantities is expressed. 
Measures in a legal and commercial sense 
are various, according to the various kinds 
and dimensions of the things measured. 
Hence arise lineal or longitudinal m -asures, 
for lines or lengths; square measures, for 
areas or superficies ; and solid or cubic mea.- 
I sures, for bodies and their capacities ; all 
! which again are very different in different 
; countries and in different ages, and even . 
many ot them for diiferent commodities. 
Whence arise other divisions of ant lent and 
modern measures, domestic and foreign, ones, 
j dry measures, liquid measures, Ac. . 
I. Long measures, or measures of applications 
1. The English and Scotch standards. 
The English lineal standard is the yard, 
containing 3 English feet, equal to 3 Paris 
feet 1 inch and of an inch, or 2. of a Paris. 
ell. The use ot this measure was established 
by Henry I. of England, and the standard 
taken from the length of his own arm. 
It is divided into 36 inches, and each inch is 
supposed equal to 3 barley-corns. When 
used lor measuring cloth, it is divided into 
4 quarters, and each quarter subdivided into 
4 nails. The English ell is equal to a yard 
and a quarter, or 45 inches, and is used in 
measuring linens imported from* Germany 
and the Low-countries. 
The Scots elwand was established by king 
David I. mad divided into 37 inches. The 
standard is kept in the council-chamber of 
Edinburgh, and being compared with the 
English yard, is found to measure 37 ± inches; . 
and .therefore the Scots inch and foot are 
larger than the English, in the proportion of 
180 to 185; but this difference being so in- 
considerable, is seldom attended to in. prac- - 
tice. The Scots ell, though forbidden By 
law, is still used for measuring some coarse 
commodities, and is the foundation of the 
land-measure of Scotland; 
Itinerary measure is the same both in Eng- 
land and Scotland. The length of the chafn 
is 4 poles, or 52 yards; 80 chains make a mile. 
The old Scots computed miles were generally 
about a mile and a half each. 
The reel for yarn is 2| yards, or 10 'quar- 
ters, in circuit; 120 threads make a cut, 12- 
cuts make a ha spor hank, and 4 hanks make 
a spindle. 
2. The French standard was formerly the 
aune or ell, containing 3 Paris feet, 7 inches, 8 
lines, or 1 yard y English ; the Pali's foot 
royal exceeding the English by . A 1 parts 
as in one of the following tables. r J his ell is 
divided two ways, viz, into halves,, thirds, 
sixths, and twelfths; and into quarters, Lalf- 
quarters, and sixteenths. 
The French, however, have also formed 
an entirely new sv stem, of weights and mea-** 
sures, according to the following tables 
