120 M E N 
poflible, finch numbers have been feleCied, for diameters, 
as form either multiples or divifors of moll: other num¬ 
bers, even to fractions, to appreciate the odd numbers by. 
And, as all numbers found m the fuperficies are reiterated 
in the products of double thofe diameters in the areas, 
fo any number fought for (not in the table) may be 
found thus : If for areas, look for the number given in 
the column of diameters, which {hall be either a multiple or 
divifor of the number fought: fay 20, which is not there ; 
therefore, take the product of 10=78-5400X4— 3 1 A" 1 600 : 
conlequently 314-1600 is the area of 20, the fum lought. 
Again, to divide 10 by 2, (admitting 5 were not in the 
table,) fay ic=78'54.oo~-4=i9'6 3 50, therefore 19-6350 
is the area of 5, as given in the table. The fame mode 
of operation holds good refpefting fuperficies, as may 
be leen by films found in the table, and may be aCled 
upon by iiims not found there. For Inftance, if we feek 
the refuit of 3-t, the number is not in the table ; we mull, 
therefore, fay for fuperficies 3*X2=7=153-9384^-4= 
38-4846, the fum fought. Now, although this number 
is not in the column of fuperficies, we know, by what we 
have leen, (i. e. by fums found in the table,) that it mull 
be right, becaule 7 is double 3^, and (which having been 
fquared) its refuit is 4 times as much as 3^ ; therefore, 
38.4846 is the fuperficies of 3^, and fo on for any other 
number, either of areas or luperficies. One example, 
each way, for the folids, will make this equally plain : 
indeed the operation is exaClly the fame ; only, inftead of 
taking 4 for a multiplier, when we wilh to double a fum, 
we mult take 8, and the fame for a divifor, if we wifii to 
halve it. Suppole 14 were the number we fought for; 
it is not in the table, but 7 is, whofe folid content is 
179-5948x8=1436-7584; therefore, 1436-758455 the fo¬ 
lid content of a fphere whole diameter is 14. Laltly, 
let it be required to fhow the folid content of a fphere, 
whofe axis or diameter is 21. This number is not in 
- the table any more than its binary divifor, but its mul- 
ple 42 is, whofe lolidity 13=38792-4768-7-8=4849-0596 ; 
therefore, 4849.0596 is the cubic or folid dimenlion of a 
fphere, whofe axis or diameter is 21. No matter whether 
thefe numbers mean miles, feet, inches, or any thing 
elfe ; only the anlwers mult bear the lame denomination. 
A lingle glance at the fquare dimenfions will Ihow that 
all that has been laid and demonlfrated on the amplifica¬ 
tion or reduction of fpheres, areas, and fuperficies, is 
equally applicable to them alfo ; only their rel'ults are 
given in whole numbers, which it was impoluble to do in 
curvilineal figures. 
It may, perhaps, be neceffary to account for the num¬ 
ber one appearing to deviate from that rule by which all 
other numbers (as diameters for fpheres or cubes) are 
governed ; that is, of having their circumferences invol¬ 
ved in their refpeClive fuperficies, and which is always 
in the exaCt and direCt ratio of the numbers themfelves, 
or identical with their diameters. This feeming ano¬ 
maly, and real folicifm, arifes from (though not a very 
obvious, yet) a very rational caufe, that “ the fquare of 
one is but one, and can be no more, from the very nature 
and e {fence of that number, it being the loweft denomi¬ 
nation that man can conceive, either when a whole num¬ 
ber or a decimal; therefore, the circumference, whofe 
diameter is one, and the fuperficies, mull appear the fame, 
(as in the table,) which feems very ftrange, if not confi- 
dered in the abliraCl, w-hile all other numbers have their 
circumferences involved in their fuperficies as many 
times as there are units, or ones, in their diameters. In¬ 
deed, it is upon the fquares of their diameters being 
multiplied by the circumferating faCtor, that their fu¬ 
perficies are found, and not upon the fimple diameters 
themfelves, conlequently, a number that does not increale 
by being lquared, or multiplied by itfelf, (which one 
does not,) mull be the fame in the circumference as in the 
fuperficies, becaufe the circumference is found by mul¬ 
tiplying the fimple number, or diameter, by the fame 
factor, (3-1416 ;) and, therefore,.mull appear equal to the 
MEN 
fum of the fuperficies, which is found by one, the fquare 
of one multiplied by the fame number, (3-1416.) The 
apparent error may, however, be eafily reconciled, by 
adding a nought to the one and making ten of it, when 
the fuperficies will be immediately ten times the circum¬ 
ference, as feen in the table. 
The Table and its explanations are the work of Mr. 
Snart, optician, in Tooley-ftreet, to whom the public are 
indebted for an original and ufeful aerometer, and alfo 
of a chondrometer. See the article Mechanics, vol.xiv. 
p. 709, 10. In the calculation of the Table, Mr. S. claims 
the invention or original application of the following 
new principles: that is to fay, the two modes of finding 
proportions both under and over 6 by indices inftead of 
elaborate fums ot operations ; the two kinds of involu¬ 
tion refulting therefrom ; the facile and economical me¬ 
thod of anfwering ten queftions at once fimply from the 
diameters; and the explanation of the folecilin of the 
unit. The making the numbers chofen divifors or mul¬ 
tiples, if not quite original, will be found extremely ufe¬ 
ful. Thefe principles and calculations have met with the 
approbation of that enlightened and fcientific nobleman, 
earl Stanhope; and the Table with its explanations has 
been accordingly publfthed on a large ftieet, convenient 
for framing or rolling, and dedicated, by permiftion, to 
his lordlhip. Mr. Snart has alfo in the prefs a much lar¬ 
ger and more general exemplification of the fame princi¬ 
ples ; in which cylinders, frufta of cones, pyramids, &c. 
will be demonftrated, not as by a Ready Reckoner, but 
in a fcientific way by principles much eafier and far more 
brief than any heretofore fhown. In that work he pro- 
pofes adding to, and re-conftruCling, the diameters, lb as 
to make them perfect multiples or divifors of all num¬ 
bers. It will contain a copious Table of Specific Gravi¬ 
ties, not only for folids and fluids, butforgafes alfo, from 
platinum to hydrogen; by which means the fubjeCt Of 
Menfuration is made a general vehicle of fcientific infor¬ 
mation, and adapted to all the purpoles of life. From 
this volume we hope to prefent the reader with fome ex¬ 
tracts under the article Number. 
MENT, one of the participles pajjive of To MeiNe; 
Mingled : 
Bufie bent 
To fight with many foes about him merit. Spcnjcr. 
MENTA'GRA, f. [from the Lat. mentum, the chin, 
and the Greek aypa, a feizure.] A tetter or ringworm 
beginning on the chin, and lpreading itlelf over the face 
and breaft. 
MEN'TAL, adj. Intellectual; exifting in the mind.—• 
If the ideas be not innate, there was a time when the 
mind was without thofe principles; for, where the ideas 
are not, there can be no knowledge, no aflent, no mental 
or verbal propofitions about them. Locke. 
She kindly talk'd, at leaft three hours, 
Of plaltic forms and mental pow’rs. Prior. 
MEN'TALLY, adv. Intellectually; in the mind ; not 
praCtically or externally, but in thought or meditation.— 
Ifweconflder the heart the firft principle of life, and 
mentalhj divide it into its conftituent parts, we find no¬ 
thing but what is in any mulcle of the body. Bentley. 
MENTASTRUM, f. in botany. See Bystropogon. 
MEN'TCHIKOF (Alexander), a ftatefman and gene¬ 
ral under Peter I. of Rufiia, and a remarkable example of 
a rife to a high fortune from the meaneft origin, was the 
fon of peafants who were vaffals of the monaltery of Cof- 
mopoli. At the age of thirteen he went to Mofcovr, 
where he w-as taken into the lervice of a paftry-cook, and 
employed to cry his wares about the ftreets. The czar, 
happening one day to hear him, was ftruck with the plea- 
fant long which he annexed to his cry, and entered into 
-converlation with him. The fmartnels of the boy’s re¬ 
plies pleafed him fo well, that he ordered him to come to 
court, where he was at firft placed in a very low- ftation ; 
but the quicknefs of his parts, and the facility with which 
