A four-dimensional continuum obeying Riemannian geometry can be represented graphically as a surface of four dimensions drawn in a Euclidean hyperspace of a sufficient number of dimensions. Actually $10$~dimensions are required, corresponding to the number of the~$g_{\mu\nu}$. For let $(y_{1}, y_{2}, y_{3}, \dots\Add{,} y_{10})$ be rectangular Euclidean coordinates, and $(x_{1}, x_{2}, x_{3}, x_{4})$ parameters on the surface; the equations of the surface will be of the form \[ y_{1} = f_{1}(x_{1}, x_{2}, x_{3}, x_{4}),\ \dots\dots,\ y_{10} = f_{10}(x_{1}, x_{2}, x_{3}, x_{4}). \] For an interval on the surface, the Euclidean geometry of the~$y$'s gives \begin{align*} -ds^{2} &= dy_{1}^{2} + dy_{2}^{2} + dy_{3}^{2} + \cdots + dy_{10}^{2} \\ &= \biggl\{\left(\frac{\dd f_{1}}{dx_{1}}\right)^{2} + \left(\frac{\dd f_{2}}{dx_{1}}\right)^{2} + \cdots + \left(\frac{\dd f_{10}}{dx_{1}}\right)^{2}\biggr\} dx_{1}^{2} + \cdots \\ &\qquad + \biggl\{\frac{\dd f_{1}}{dx_{1}}\, \frac{\dd f_{1}}{dx_{2}} + \cdots + \frac{\dd f_{10}}{dx_{1}}\, \frac{\dd f_{10}}{dx_{2}}\biggr\} 2\, dx_{1}\, dx_{2} + \cdots. \end{align*} Equating the coefficients to the given functions~$g_{\mu\nu}$, we have $10$~partial differential equations of the form \[ \frac{\dd f_{1}}{dx_{\mu}}\, \frac{\dd f_{1}}{dx_{\nu}} + \cdots + \frac{\dd f_{10}}{dx_{\mu}}\, \frac{\dd f_{10}}{dx_{\nu}} = g_{\mu\nu}, \] to be satisfied by the $10$ $f$'s. Clearly it would not be possible to satisfy these equations with less than~$10$ $f$'s except in special cases. When we use the phrase ``curvature'' in connection with space-time, we always think of it as embedded in this way in a Euclidean space of higher dimensions. It is not suggested that the higher space has any existence; the purpose of the representation is to picture more vividly the metrical properties of the world. It must be remembered too that a great variety of four-dimensional surfaces in $10$~dimensions will possess the same metric, i.e.\ be applicable to one another by bending without stretching, and any one of these can be chosen to represent the metric of space-time. Thus a geometrical property of the chosen representative surface need not necessarily be a property belonging intrinsically to the space-time continuum. A four-dimensional surface free to twist about in six additional dimensions has bewildering possibilities. We consider first the simple case in which the surface, or at least a small portion of it, can be drawn in Euclidean space of five dimensions. \PageSep{150}