The principles of science

by William Stanley Jevons

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 L². It is obvious in the same way that the dimensions of volume or bulk will be L³.

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 L³D. 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 L³D.

Introducing time, denoted by T, it is easy to see that the dimensions of velocity will be L/T or LT^-1, because the number of units in the velocity of a body is found by dividing the units of length passed over by the units of time occupied in passing. The acceleration of a body is measured by the increase of velocity in relation to the time, that is, we must divide the units of velocity gained by the units of time occupied in gaining it; hence its dimensions will be LT^-2. Momentum is the product of mass and velocity, so that its dimensions are MLT^-1. The effect of a force is measured by the acceleration produced in a unit of mass in a unit of time; hence the dimensions of force are MLT^-2. Work done is proportional to the force acting and to the space through which it acts; so that it has the dimensions of force with that of length added, giving ML²T^-2.

It should be particularly noticed that angular magnitude has no dimensions at all, being measured by the ratio of the arc to the radius (p. [305]). Thus we have the dimensions LL^-1 or L⁰. This agrees with the statement previously made, that no arbitrary unit of angular magnitude is needed. Similarly, all pure numbers expressing ratios only, such as sines and other trigonometrical functions, logarithms, exponents, &c., are devoid of dimensions. They are absolute numbers necessarily expressed in terms of unity itself, and are quite unaffected by the selection of the arbitrary physical units. Angular magnitude, however, enters into other quantities, such as angular velocity, which has the dimensions 1/T or T^-1, the units of angle being divided by the units of time occupied. The dimensions of angular acceleration are denoted by T^-2.

The quantities treated in the theories of heat and electricity are numerous and complicated as regards their dimensions. Thermal capacity has the dimensions ML^-3, thermal conductivity, ML^-1T^-1. In Magnetism the dimensions of the strength of pole are M^1/2L^3/2T^-1, the dimensions of field-intensity are M^1/2L^-1/2T^-1, and the intensity of magnetisation has the same dimensions. In the science of electricity physicists have to deal with numerous kinds of quantity, and their dimensions are different too in the electro-static and the electro-magnetic systems. Thus electro-motive force has the dimensions M^1/2L^1/2T^-1, in the former, and M^1/2L^3/2T^-2 in the latter system. Capacity simply depends upon length in electro-statics, but upon L^-1T² in electro-magnetics. It is worthy of particular notice that electrical quantities have simple dimensions when expressed in terms of density instead of mass. The instances now given are sufficient to show the difficulty of conceiving and following out the relations of the quantities treated in physical science without a systematic method of calculating and exhibiting their dimensions. It is only in quite recent years that clear ideas about these quantities have been attained. Half a century ago probably no one but Fourier could have explained what he meant by temperature or capacity for heat. The notion of measuring electricity had hardly been entertained.

Besides affording us a clear view of the complex relations of physical quantities, this theory is specially useful in two ways. Firstly, it affords a test of the correctness of mathematical reasoning. According to the Principle of Homogeneity, all the quantities added together, and equated in any equation, must have the same dimensions. Hence if, on estimating the dimensions of the terms in any equation, they be not homogeneous, some blunder must have been committed. It is impossible to add a force to a velocity, or a mass to a momentum. Even if the numerical values of the two members of a non-homogeneous equation were equal, this would be accidental, and any alteration in the physical units would produce inequality and disclose the falsity of the law expressed in the equation.

Secondly, the theory of units enables us readily and infallibly to deduce the change in the numerical expression of any physical quantity, produced by a change in the fundamental units. It is of course obvious that in order to represent the same absolute quantity, a number must vary inversely as the magnitude of the units which are numbered. The yard expressed in feet is 3; taking the inch as the unit instead of the foot it becomes 36. Every quantity into which the dimension length enters positively must be altered in like manner. Changing the unit from the foot to the inch, numerical expressions of volume must be multiplied by 12 × 12 × 12. When a dimension enters negatively the opposite rule will hold. If for the minute we substitute the second as unit of time, then we must divide all numbers expressing angular velocities by 60, and numbers expressing angular acceleration by 60 × 60. The rule is that a numerical expression varies inversely as the magnitude of the unit as regards each whole dimension entering positively, and it varies directly as the magnitude of the unit for each whole dimension entering negatively. In the case of fractional exponents, the proper root of the ratio of change has to be taken.

The study of this subject may be continued in Professor J. D. Everett’s “Illustrations of the Centimetre-gramme-second System of Units,” published by Taylor and Francis, 1875; in Professor Maxwell’s “Theory of Heat;” or Professor Fleeming Jenkin’s “Text Book of Electricity.”