to the bar scale does not give a distance related to the true distance in any simple way. (But see addendum). Even if a distance along this line of constant planar angle could be worked out, its relevance is questionable since such a line on the projection corresponds to a complicated curve on the sphere. For these reasons bar scales on small-scale maps must be used with extreme caution. Mercator projection The Mercator projection maps the sphere to a rectangle (of infinite extent in the -direction) by the equations where a, and are as in the previous example. Since the scale factors are: parallel scale meridian scale In the mathematical addendum it is shown that the point scale in an arbitrary direction is also equal to so the scale is isotropic (same in all directions), its magnitude increasing with latitude as . In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases. Lambert's equal area projection Lambert's equal area projection maps the sphere to a finite rectangle by the equations where a, and are as in the previous example. Since the scale factors are parallel scale meridian scale The calculation of the point scale in an arbitrary direction is given below. The vertical and horizontal scales now compensate each other (hk=1) and in the Tissot diagram each infinitesimal circular element is distorted into an ellipse of the same area as the undistorted circles on the equator. Graphs of scale factors The graph shows the variation of the scale factors for the above three examples. The top plot shows the isotropic Mercator scale function: the scale on the parallel is the same as the scale on the meridian. The other plots show the meridian scale factor for the Equirectangular projection (h=1) and for the Lambert equal area projection. These last two projections have a parallel scale identical to that of the Mercator plot. For the Lambert note that the parallel scale (as Mercator A) increases with latitude and the meridian scale (C) decreases with latitude in such a way that hk=1, guaranteeing area conservation. Scale variation on the Mercator projection The Mercator point scale is unity on the equator because it is such that the auxiliary cylinder used in its construction is tangential to the Earth at the equator. For this reason the usual projection should be called a tangent