To the events we assign four identification-numbers or coordinates according to a plan which is arbitrary within wide limits. The connection between our physical measurements of interval and the system of identification-numbers is expressed by the general quadratic form~\Eq{(2.1)}. In the particular case when these identification-numbers can be so assigned that the product terms in the quadratic form disappear leaving only the four squares, the coordinates have the metrical properties belonging to rectangular coordinates and time, and are accordingly so identified. If any such system exists an infinite number of others exist connected with it by the Lorentz transformation, so that there is no unique space-time frame. The relations of these different space-time reckonings have been considered in detail. It is \PageSep{42} shown that there must be a particular speed which has the remarkable property that its value is the same for all these systems; and by appeal to the Michelson-Morley experiment or to Fizeau's experiment it is found that this is a distinctive property of the speed of light. But it is not possible throughout the world to choose coordinates fulfilling the current definitions of rectangular coordinates and time. In such cases we usually relax the definitions, and attribute the failure of fulfilment to a field of force pervading the region. We have now no definite guide in selecting what coordinates to take as rectangular coordinates and time; for whatever the discrepancy, it can always be ascribed to a suitable field of force. The field of force will vary according to the system of coordinates selected; but in the general case it is not possible to get rid of it altogether (in a large region) by any choice of coordinates. This irreducible field of force is ascribed to gravitation. It should be noticed that the gravitational influence of a massive body is not properly expressed by a definite field of force, but by the property of irreducibility of the field of force. We shall find later that the irreducibility of the field of force is equivalent to what in geometrical nomenclature is called a curvature of the continuum of space-time. For the fuller study of these problems we require a special mathematical calculus which will now be developed \Foreign{ab initio}. \PageSep{43} \Chapter{II}{The Tensor Calculus} \Section{19.}{Contravariant and covariant vectors} \index{Contravariant vectors}% \index{Transformation of coordinates!general}% We consider the transformation from one system of coordinates $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$ \index{Coordinates!general transformation of}% to another system $x_{1}'$, $x_{2}'$, $x_{3}'$,~$x_{4}'$.