The unit of heat ought to be simply the unit of energy, already described. But a weight can be measured to the one-millionth part, and temperature to less than the thousandth part of a degree Fahrenheit, and to less therefore 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 0° to 1°. 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 423·55 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 0° Cent., which it is capable of converting into water under inappreciable pressure. *Theory of Dimensions.* In order to understand the relations between the quantities dealt with in physical science, it is necessary to pay attention to the Theory of Dimensions, first clearly stated by Joseph Fourier,[228] 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 rectangular 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* × *L* or *L*^{2}. It is obvious in the same way that the dimensions of volume or bulk will be *L*^{3}. [228] *Théorie Analytique de la Chaleur*, Paris; 1822, §§ 157–162. The number of units of mass in a body is found by multiplying 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 dimensions of mass are *L*^{3}*D*. As already explained, however, it is usual to substitute an arbitrary provisional unit of mass, symbolised by *M*; according to the view here taken we may say that the dimensions of *M* are *L*^{3}*D*.