CUB 



<he measuring the space contained in it ; 

 or finding the solid content of it 



CUBE, in geometry, a solid body, con- 

 sisting of six equal square sides. The 

 solidity of any cube is found by multiply- 

 ing the superficial area of one of the sides 

 by the height. Cubes are to one another 

 in their 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 thereto ; whence it follows, that 

 the planes of all sections, parallel to 

 the base, are squares equal thereto, 

 and, consequently, to one another. See 

 BOUT. 



Cr UK, 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 consult- 

 ed about the mode of stopping a plagtie 

 then raging at Athens, returned for an- 

 swer, that the plague should cease, when 

 Apollo's altar, which was cubical, should 

 be doubled. Hence it is called the Delian 

 problem. This problem cannot be ef- 

 fected geometrically, as it requires the 

 solution of a cubic equation, or requires 

 the finding of two mean proportionals, 

 vis. between the side of the given cube 

 and the double of the same, the first of 

 which two mean proportionals is the side 

 of the double cube, as was first observed 

 by Hippocrates. Let a be the side of the 

 given cube, and x the side of the double 

 cube sought, then x3 = 2 a3 or a 1 : x 1 :: 

 x: 2 a, so that, if a and x be the first and 

 second terms of a set of continued pro- 

 portionals, then a> : x* 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 term to the third, 

 or of the second to the fourth, that is, of 

 x -. 2 a -, 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 multi- 

 plication of a square number by its root ; 

 thus, 64 is a cube number, and arises by 

 multiplying 16, the square of 4, by the 

 root 4. 



CUBE, or Cubic quantity, in algebra, 

 the third power in a series of geometri- 

 cal proportionals continued ; as a is the 

 jroot, a a the square, and a a a the cube. 



CUB 



AH 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 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 three times the square of the lesser 

 multiplied by the greater, as, 



-j- b.b 



aaa-\-2aab-\-abb 

 gab + 2ab 



-\-bbb 



_ 

 aaa-\-3aab-}-3abb-\-bbb 



From hence it is easy to undersand 

 both the composition of any cubic num- 

 ber, and the reason of the method for ex- 

 tractingthe cube root out of any member 

 given. 



CUBE root of any number or quantity, 

 such a number, or quantity, which, if 

 multiplied into itself, and then, again, 

 the product thence arising by* that num- 

 ber or quantity, being the cube root, this 

 last product shall be equal to the number 

 or quantity 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 -+- b is the cube root of a* -j- 3 a* b + 

 3 a 63 -f fc, 



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 ^/ 3 -f- 1 ; and generally, if a be 

 the real root of any cube number, one of 

 the imaginary roots of that number will be 



a -f- v/ 



and the other 



a >/ 3ao 



2 



CUBEBS. See MATEHIA MEDICA. 



CUBIC, or Cubical Equation, in al- 

 gebra, one whose highest power con- 

 sists of three dimensions, as x> = a' 

 A3 or x3 -f- r x x = p 6 , &c. See EU.UA- 

 TIOK. 



CUBIC foot 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 Roman cubit equal to I 



