CUB 
of any cube is found by multiplying the su- 
perficial area of one of the sides by the 
height. Cubes are to one another in the 
triplicate ratio of their diagonals; and a 
cube is supposed to be generated by the 
motion of a square plane, along a line equal 
to one of its sides, and at right angles there- 
to ; whence it follows, that the planes of all 
sections, parallel to the base, are squares 
equal thereto, and, consequently, to one 
another. See Body. 
Cube, duplication of, is the finding the 
side of a cube that shall be double in so- 
lidity to a given cube, a problem of great 
celebrity, first proposed by the oracle of 
Apollo at Delphos, which, being consulted 
about the mode of stopping a plague then 
raging at Athens, returned for answer, that 
the plague should cease when Apollo’s al- 
tar, which was cubical, should be doubled. 
Hence it is called the Delian problem. 
This problem cannot be effected geome- 
trically, as it requires the solution of a 
cubic equation, or requires the finding 
of two mean proportionals, viz. between 
the side of the given cube, and the dou- 
ble of tbe same, the first of which two 
mean proportionals is the side of the double 
cube, as was tiist observed by Hippocra- 
tes. Let a be the side of the given cube, 
and X the side of the double cube sought, 
then x’ z=: 2 o’ or : a;* :: a? : 2 a, so that, if 
a and x be the first and second terms of a 
set of continued proportionals, then o’ : ar’ 
is the ratio of the square of the first, to the 
square of the second, which, it is known, is 
the same as the ratio of the first terra to the 
third, or of the second to the fourth, that is 
of x : 2 o ; therefore x being the second 
term, 2 a will be the fourth : so that x, the 
side of the cube sought, is the second of 
four terms in continued proportion, the 
first and fourth being a and 2 a ; that is, the 
side of the double cube is the first of two 
mean proportionals between a and 2 a. 
Cube, or Cubic number, in arithmetic, 
that which is produced by the multiplica- 
tion of a square number by its root ; thus, 
64 is a cube number, and arises by multi- 
plying 16, the square of 4, by the root 4. 
Cube, or Cubic quantity, in algebra, the 
thii’d power in a series of geometrical pro- 
portionals continued; as a is the root, a a 
the square, and a a a the cube. All cubic 
numbers may be ranged into the form of 
cubes ; as 8 or 27, whose sides are 2 and 3, 
and their bases 4 and 9 ; whence it appears, 
that every true cubic number, produced 
from a binomial root, consists of these 
cue 
parts, viz. the cubes of the greater and 
lesser parts of the root, and of three times 
the square of the greater part multiplied by 
the lesser, and of tliree times the square of 
the lesser multiplied by the greater, as, 
aa-j-2a6-|-6J 
a-j-6 
aaa-|-2au6-|-a66 
aab-\-^abb-\-bbb 
aaa-\-oaab-\-3abb-^bbb 
From hence it is easy to understand both 
the composition of any cubic number, and 
the reason of tlie method for extracting the 
cube root out of any member given. 
Cube root of any number or quantity, 
such a number, or quantity, which, if mul- 
tiplied into itself, and then, again, the pro- 
duct thence arising by that number or 
quantity, being the cube root, this last pro- 
duct shall be equal to the nmnber or quan- 
tity whereof it is the cube root, as 2 is the 
cube root of- 8, because two times 2 is 4, 
and two times 4 is 8 ; and a -(- 6 is the cube 
root of a’ -J- 3 a’ ft -|- 3 4" 
Every cube number has three roots, one 
real root, and two imaginary ones, as the 
cube number 8 has one real root 2, and 
two imaginary roots, viz. \/ — 3 — 1 and 
a/ — 3 -|- 1 ; and generally if a be the 
real root of any cube number, one of the 
imaginary roots of that number will be 
l +V-3 . «a andthe other 
2 
a — a/ — S aa 
2 
CUBEBS. See Materia Medica. 
CUBIC, or Cubical, Equation, in alge- 
bra, one whose highest power consists of 
three dimensions, as x’ = a’ — 6’, or x’ -}- 
rxx=p^, &c. See Equation. 
Cubic / oof of any substance, so much of 
it as is contained in a cube, whose side is 
one foot. See Cube. 
CUBIT, in the mensuration of the an- 
cients, a long measure, equal to the length 
of a man’s arm, from the elbow to the tip 
of the fingers. Dr. Arbuthnot makes the 
English cubit equal to 18 inches; the Ro- 
man cubit equal to 1 foot, 5,406 inches ; 
and the cubit of the Scripture equal to 1 
foot, 9,888 inches. 
CUCKOW. See Cuculus. 
CucKow spit. See Cicada. 
CUCUBALUS, in botany, a genus of the 
Decandria Trigynia class and order. Natu- 
ral order of Caryophyllei. Essential charac- 
