xiv.] UNITS AND .STANDARDS OF MEASUREMENT. 325 



one-millionth part, and temperature to less than the 

 thousandth part of a degree Fahrenheit, and to less there- 

 fore than the five-hundred thousandth part of the absolute 

 temperature, whereas the mechanical equivalent of heat is 

 probably not known to the thousandth part. Hence the 

 need of a provisional unit of heat, which is often taken as 

 that requisite to raise one gram of water through one degree 

 Centigrade, that is from o to i. This quantity of heat is 

 capable of approximate expression in terms of time, space, 

 and mass ; for by the natural constant, determined by Dr. 

 Joule, and called the mechanical equivalent of heat, we 

 know that the assumed unit of heat is equal to the energy 

 of 42 3 '5 5 gram-metres, or that energy which will raise 

 the mass of 423-55 grams through one metre against 9-8... 

 absolute units of force. Heat may also be expressed in 

 terms of the quantity of ice at o Cent., which it is capable 

 of converting into water under inappreciable pressure. 



Theory of Dimensions. 



In order to understand the relations between the quan- 

 tities dealt with in physical science, it is necessary to pay 

 attention to the Theory of Dimensions, first clearly stated 

 by Joseph Fourier, 1 but in later years developed by several 

 physicists. This theory investigates the manner in which 

 each derived unit depends upon or involves one or more of 

 the fundamental units. The number of units in a rectan- 

 gular area is found by multiplying together the numbers 

 of units in the sides ; thus the unit of length enters twice 

 into the unit of area, which is^therefore said to have two 

 dimensions with respect to length. Denoting length by L, 

 we may say that the dimensions of area are L x L or 

 L' 2 . It is obvious in the same way that the dimensions of 

 volume or bulk will be L s . 



The number of units of mass in a body is found by mul- 

 tiplying the number of units of volume, by those of density. 

 Hence mass is of three dimensions as regards length, 

 and one as regards density. Calling density D, the dimen- 

 sions of mass are I?D. As already explained, however, 

 it is usual to substitute an arbitrary provisional unit of 



1 Theorie Analytique de la Chakur, Paris; 1822, 157162. 



