This system is that which was adopted in 1803 by the "Dépôt de la Guerre" for the map of France, and is there known by the title of _Projection de Bonne_. It is that on which the ordnance survey map of Scotland on the scale of 1 in. to a mile is constructed, and it is frequently met with in ordinary atlases. It is ill-adapted for countries having great extent in longitude, as the intersections of the meridians and parallels become very oblique--as will be seen on examining the map of Asia in most atlases. If [phi]0 be taken as the latitude of the centre parallel, and co-ordinates be measured from the intersection of this parallel with the central meridian, then, if [rho] be the radius of the parallel of latitude [phi], we have [rho] = cot [phi]0 + [phi]0 - [phi]. Also, if S be a point on this parallel whose co-ordinates are x, y, so that VS = [rho], and [theta] be the angle VS makes with the central meridian, then [rho][theta] = [omega] cos [phi]; and x = [rho] sin [theta], y = cot [phi]0 - [rho] cos [theta]. The projection has the property of equal areas, since each small element bounded by two infinitely close parallels is equal in length and width to the corresponding element on the sphere or spheroid. Also all the meridians cross the chosen parallel (but no other) at right angles, since in the immediate neighbourhood of that parallel the projection is identical with the simple conical projection. Where an equal-area projection is required for a country having no great extent in longitude, such as France, Scotland or Madagascar, this projection is a good one to select. [Illustration: FIG. 19.--Sinusoidal Equal-area Projection.] _Sinusoidal Equal-area Projection._--This projection, which is sometimes known as Sanson's, and is also sometimes incorrectly called Flamsteed's, is a particular case of Bonne's in which the selected parallel is the equator. The equator is a straight line at right angles to the central meridian which is also a straight line. Along the central meridian the latitudes are marked off at the true rectified distances, and from points so found the parallels are drawn as straight lines parallel to the equator, and therefore at right angles to the central meridian. True rectified lengths are marked along the parallels and through corresponding points the meridians are drawn. If the earth is treated as a sphere the meridians are clearly sine curves, and for this reason d'Avezac has given the projection the name sinusoidal. But it is equally easy to plot the spheroidal lengths. It is a very suitable projection for an equal-area map of Africa.