5. If a thread is extended on a plane, and fixed to it at its two extremities, and afterwards the plane is formed into a pyramidal or conical surface, it may be easily shewn, that the thread will pass thro’ the same points of the surface as before; and that, _conversely_, the shortest distance between two points in a conical surface is the right line which joins them, when that surface is expanded into a plane. Now, in the present case, the shortest distances on the conical surface will be, if not equal, always nearly equal, to the correspondent distances on the sphere: and therefore, all rectilinear distances on the map, applied to the meridian as a scale, will, nearly at least, shew the true distances of the places represented. 6. In maps, whose breadth exceeds not 10° or 15°, the rectilinear distances may be taken for sufficiently exact. But we have chosen our example of a greater breadth than can often be required, on purpose to shew how high the errors can ever arise; and how they may, if it is thought needful, be nearly estimated and corrected. Write down, in a vacant space at the bottom of the map, a table of the errors of equidistant parallels, as from five degrees to five degrees of the whole latitude; and having taken the mean errors, and diminished them in the ratio of radius to the sine of the mean inclination of the line of distance to the meridian, you shall find the correction required; remembering only to distinguish the distance into its parts that lie _within_ and _without_ the sphere, and taking the difference of the correspondent errors, in _defect_ and in _excess_. But it was thought needless to add any examples; as, from what has been said, the intelligent reader will readily see the use of such a table; and chiefly as, whenever exactness is required, it will be more proper, and indeed more expeditious, to compute the distances of places by the following canon. _Multiply the product of the cosines of the two given latitudes by the square of the sine of half the difference of longitude; and to this product add the square of the sine of half the difference of the latitudes; the square root of the sum shall be the sine of half the arc of a great circle between the two places given._ Thus, if we are to find the true distance from one angle of our map to the opposite, that is, from S to Q, the operation will be as follows: L. sin. 30° = -1.6989700 L. sin. 80° = -1.9933515 2 L. sin. 55° = -1.8267290 ---------- -1.5190505 = log. of 0.330408 and 2 L. sin. 25° = -1.2518966 = log. of 0.178606 ---------- -------- Log. of the sum 0.509014 is -1.7067297 Whose half is -1.8533648 the L. sin. of 45° 31´, the double of which is 91° 2´, or 5462 geographical miles.