Suppose, then, we wish to express the distance between two points and , defined by the intersections of a certain line with two successive lines (Fig. II). The co-ordinates of the two points are () and (), respectively, where represents the increase in the value of as we pass from to on the line. On no account may we say that the distance (which we shall call ) is given by the value , for is nothing but a difference between numbers serving to localise points; it has nothing in common with a distance. In order to dispel any doubts on this score, we may notice that if we compare the lines to parallels and the lines to meridians, the points and will be given by the intersections of two consecutive meridians [Pg 86] with the same parallel. In this case would correspond to the difference in longitude between the two points; and obviously a difference in longitude is no criterion of a distance, since for the same difference in longitude the distance decreases from equator to pole. In short, the distance between and cannot be fully determined by . Yet, on the other hand, just as the distance between two points lying on the same parallel is affected by their difference in longitude, so now the distance between and must be a function of the co-ordinate difference . A sufficiently general expression of a quantity when defined in terms of another quantity on which it depends is given by where , , are appropriate magnitudes. If, however, we consider points exceedingly close together, becomes exceedingly small in value, and as a result and , etc., are many times smaller still. At the limit, therefore, when the two points are at an infinitesimal distance apart, the higher powers of become so insignificant that they can be neglected in comparison with . We thus obtain , and in order to specify that we are considering infinitesimal distances we replace the symbol by and obtain It is customary to designate by so that, using squares, our formula becomes [Pg 87] And now we have to consider an important question. What is ? What does it represent? We do not propose to enter into its full mathematical significance, but we may mention certain of its important characteristics. In our illustration of meridians and parallels, let us consider the various points of intersection defined by the intersections of the successive parallels with two fixed meridians. The difference in longitude between these various point pairs will, of course, be constant and will be given by the same invariable quantity . Inasmuch as the distance between these point pairs varies from pole to equator, we see from