To explain all the changes that are legitimate in your co-ordinates, suppose you take a large piece of soft india-rubber. While it is in an unstretched condition, measure little squares on it, each one-tenth of an inch each way. Put in little tiny pins at the corners of the squares. We can take as two of the co-ordinates of one of these pins the number of pins passed in going to the right from a given pin until we come just below the pin in question, and then the number of pins we pass on the way up to this pin. In the figure, let =O= be the pin we start from and =P= the pin to which we are going to assign co-ordinates. =P= is in the fifth column and the third row, so its co-ordinates in the plane of the india-rubber are to be 5 and 3. [Illustration: Fig. 1.] [Illustration: Fig. 2.] Now take the india-rubber and stretch it and twist it as much as you like. Let the pins now be in the shape they have in Fig. 2. The divisions now no longer represent distances according to our usual notions, but they will still do just as well as co-ordinates. We may still take =P= as having the co-ordinates 5 and 3 in the plane of the india-rubber; and we may still regard the india-rubber as being in a plane, even if we have twisted it out of what we should ordinarily call a plane. Such continuous distortions do not matter. To take another illustration: instead of using a steel measuring rod to fix our co-ordinates, let us use a live eel, which is wriggling all the time. The distance from the tail to the head of the eel is to count as one from the point of view of co-ordinates, whatever shape the creature may be assuming at the moment. The eel is continuous, and its wriggles are continuous, so it may be taken as our unit of distance in assigning co-ordinates. Beyond the requirement of continuity, the method of assigning co-ordinates is purely conventional, and therefore a live eel is just as good as a steel rod.