meridian; 0° ♊ therefore falls on the point K, at the distance of 13 equatorial degrees before the meridian or mid-heaven, and will be three temporal hours past the meridian (the position of 0° ♉) on arriving at L, distant 51 equatorial degrees from the mid-heaven: the whole distance from K (the first position of 0° ♊) to L, its second position, equal to 64 degrees of the equator, is therefore the prorogation between 0° ♉ and 0° ♊, when 0° ♉ is past the meridian at the distance of three temporal hours. Ptolemy has also instanced two other positions for 0° ♉; viz. at two temporal hours past the meridian, and at two temporal hours before the occidental angle; or, in other words, on the cusp of the 9th house, and on that of the 8th. Now, if 0° ♉ be on the cusp of the 9th house, it must be at M, and 0° ♊ will be at N, distant 62 equatorial degrees from Q, which is also on the cusp of the 9th. If 0° ♉ be on the cusp of the 8th, it must be at O, and 0° ♊ will be at P, distant 66 equatorial degrees from R, which is also on the cusp of the 8th: these two several numbers of degrees will be the respective prorogations between 0° ♉ and 0° ♊, when 0° ♉ is placed on the 9th and 8th houses. Ptolemy’s “Exemplification” has been followed thus minutely in order to show how perfectly Mr. Ranger’s invention is adapted to assist (if not to supersede) arithmetical calculation; for, after the Planisphere has once been accurately laid down, a line drawn parallel to the equator, from the significator to the promittor, or to the promittor’s pole of position, and measured by degrees of the equator, will accomplish the whole operation of ascertaining the amount of prorogation. Fig. 2 is the Equator extended, _in plano_, on a scale proportionate to the planispheres in Figs. 1 and 3: it is divided into 360 degrees, and into equal time, as measured by the 24 hours of the earth’s daily rotation on its axis, and by smaller portions of four minutes each, corresponding with degrees of the equator.