Log. 0.8726650 (= 50° to radius 1) -1.9408476 Log. MF (sin. 35°) -1.7585913 Log. MT (tang. 55°) 0.1547732 ---------- Take the sum -1.8542121 from log. N_n_ (= .6923772) -1.8403427 ---------- the remainder -1.9861306 is the logarithm of C_x_. And because 1: C_x_ ∷ MT : _xt_, to this adding the log. MT 0.1547732 ---------- The sum 0.1409038 is the log. of _xt_ = 1.383260; and _x_R (= _xr_ = ½ L_l_) being .4363325, R_t_ will be 0.9469275, _rt_ = 1.8195925. Whence having fixed upon any convenient size for our map, the center _t_ is easily found. As, allowing an inch to a degree of a great circle, or 50 inches to the line R_r_, R_t_ the semidiameter of the least parallel will be 54.255 inches, and that of the greatest parallel 104.255 inches. Again, making as radius to MF so the longitude 110° to the angle S_t_V, that angle will be 63° 5´ ⅗. Divide the meridians and parallels, and finish the map as usual. _Note_, The log. MT being repeated in this computation with a contrary sign, we may find _xt_ immediately by subtracting the sum of the logarithms of L_l_ and MF from the log. of N_n_. VI. A map drawn by this rule will have the following properties: 1. The intersections of the meridians and parallels will be rectangular. 2. The distances north and south will be exact; and any meridian will serve as a scale. 3. The parallels thro’ _z_ and _y_, where the line R_r_ cuts the arc L_l_, or any small distances of places that lie in those parallels, will be of their just quantity. At the extreme latitudes they will exceed, and in mean latitudes, from _x_ towards _z_ or _y_, they will fall short of it. But unless the zone is very broad, neither the excess nor the defect will be any-where considerable. 4. The latitudes and the superficies of the map being exact, by the construction, it follows, that the excesses and defects of distance, now mentioned, compensate each other; and are, in general, of the least quantity they can have in the map designed.