It will easily be conceived by the Reader, that all roads, except such as are made through a country, in which the public works are in the highest state of improvement, and where also the face of it is perfectly level (the curvature of the earth excepted) must have some degree of inflexion, both horizontally and vertically; and either of these will occasion a Traveller to trace a line of greater length by the road, than can be measured on a straight line, drawn from the point of commencement, to that of the termination of his journey. The quantity of the difference must vary with the nature of the country; but in ordinary cases, still more with the extent of the line of distance: for a different ratio between the road distance and horizontal distance must obtain, as the line of distance is increased. Let it be admitted, that in stages of ten or twelve miles, the winding of the road occasions a loss of only one-tenth part, which may be termed the _simple_ winding: yet as the different stages in an extent of 100 or 150 miles, do not lie in a straight line, drawn through the whole extent, but often very far to the right and left of it, a _compound_ winding arises: and I have found by long experience, that one mile in eight must be deducted, to reduce the road measure, on such a length, to horizontal measure. When a line of distance is extended to 500 miles and upwards, the rule becomes much more vague than when applied to moderate distances; because it often happens (and more particularly in unimproved countries) that obstacles present themselves, and give an entire new direction to the course of the road; although the two parts of it, considered separately, may have only an ordinary degree of crookedness. However, as some of the lines of distance applied to the present subject, are from thirty to forty days journey, it becomes necessary that some general rules should be adopted. It happens that examples are furnished, in two cases, on very long journies, where the real distances between the terminating points of the routes are nearly known: such is that of fifty-three days journey, between the Capital of _Fezzan_ and _Cairo_; and the mean horizontal distance for each day, is fourteen and a half geographic miles, or those of sixty to a degree. I confess I should have expected much less. The other example is between _Arguin_ and _Gallam_: there forty days produce thirteen miles for each day; and this is conformable to my expectations. In the examples of small distances, such as six days journey, sixteen miles per day is the result, and is consistent. For a caravan journey, taken at twenty-two British miles of _road_ distance, will produce, when the allowance for winding is deducted, and the remainder reduced to geographic miles, about sixteen and a half such miles for a single day. The following are the proportions which I have established, for the application of a scale, to the different degrees of distance.