VI. this right line makes with the meridian, and, consequently, BOOK we may fix its place with respect to the meridian, or construct, by means of the given angle, the meridian of the plan. One may recover also, by similar means, the scale of a topographical map which is wanting; for, knowing the distance of two points of this map, we have only to divide the line which joins those two points into parts proportional to the itinerary measures contained in this distance; then it becomes the scale of the map, and shews the respective distances of all the other points. Chorographic maps are reduced into general maps by a General process analogous to that we have just explained; we bring maps. on the quadrilaterals formed by the meridians and the parallels of the general map, what is contained in the corresponding quadrilaterals of the chorographic maps we wish to assemble. But here the necessity of geographical criticism appears; here it is that the designer, quitting the humble. part of a copyist, should, by his knowledge and researches, and especially by his intelligence, supply the imperfections of topographical data. Sometimes it is errors he has to correct, sometimes deficiencies to fill up; most commonly these two inconveniences are combined together. of errors of It may happen that, in the topographic plans employed Correction? in the construction of chorographic maps, there may be cr- topograrors common to all the points, as distances too small or too phy. great in the same direction, and these errors may have been accumulated on the chorographic maps, and afterwards on the general map; the great spaces which it represents are then either considerably narrowed, or considerably lengthened, without the geographer himself being able to perceive it. In this case, the geographer will connect the details of the general map with the different points, the latitudes and longitudes of which are known by astronomical observations; these points determine on the map spaces into which the intermediary details will necessarily fit, sometimes the excess or defect that is found may be attributed to the imperfection of the mechanical processes employed for the as VI. BOOK Semblage of the maps; but then there is no other way but to divide the differences among all the points of each partial plan, which will render the errors less sensible. Employment of itinerary The geographer unfortunately is but too often deprived of astronomical observations and trigonometrical surveys; distances. there are only a few countries, as France, England, Denmark, Holland, and Hungary, which have been surveyed trigonometrically in their whole extent; there are still some European provinces where no astronomers have been, Geography is obliged therefore to have recourse to itinerary distances, always very difficult to estimate in a rigorous way, even when we know exactly the value of the measures in which they have been calculated. This science is as yet very little advanced, either on account of the immense number of measures to be compared, or on account of the variations to which they are subject, or, finally, with respect to many ancient measures, because the authentic modules are wanting.* Valuation sures. We have already seen that there are various opinions on of mea- the manner of estimating the stadia of the ancients, and that it is still doubtful if they should be considered as astronomical modules or local measures. We find in the ancients one passage in three which does not allow us to admit the first supposition, unless by adopting the most violent and innumerable corrections, or by the admission of an improbable mixture of different stadia; in the second hypothesis, which to us appears preferable, we do not yet perceive the basis from which to set out; we are on the right road but surrounded with profound darkness. However, this obscurity is better than the false glimmer of an hypothesis void of proof; and, besides, why should we be surprised at the *Traité des Mesures Itinéraires des Anciens, par d'Anville. Observations, &c. par Gosselin, en avant de la traduction Française de Strabon. Traité des Mesures, par Romé de l'Isle. Métrologie constitutionnelle, par Paucton, Traité des Monnaies, des Mesures, &c. par Gérard Kruse, en Allemand. See the Tables at the end of this Volume. See the Notes of the French Translation of Strabo. doubts in which ancient metrology is enveloped, when we know that even modern measures present cases in which it is difficult to reduce them? Undoubtedly we know exactly the relations of the measures most commonly used in capital towns, and cited in the works of the learned: we know, for example, the value of the English mile, and of the nautical mile, the degree of the meridian containing 69 of the former and 60 of the latter; we know also that the English foot, being equal to 0,9384 of the French foot, is 11 inches 3 lines, 1, and that the yard used in England for measuring short distances is three feet English; we consequently conclude that the yard represents 33 inches, 9 lines, 3 of France. Similar reductions give the means of converting into each other the measures generally in use in great states; but there are besides, in the provinces, local measures little known, and with respect to which multiplied researches must be made to obtain their relation with the others, either by comparing their components with the best fixed unities, or by setting out from some distance valued in local measure, and known in geographical measures. In France, for example, nothing formerly varied more, than the length of a league between one province and another. The perch even, which serves for surveying, was sometimes 22 feet, sometimes only 18. The new metrical system will prevent any such confusion in future.* BOOK VI. route. When we know the value of the measures in which an Tracing of itinerary is conceived, the direction of the route is marked a nautical after the points of the compass. When we have the length and the direction of a route, setting off from a point, the position of which is given, we very easily find that of the point where this route terminates. In the first place, when the route is not considerable, one may, in the space it traverses, neglect the curvature of the earth, that is to say, con Since this was written, M. Uckert, a professor at Gotha, has attempted to prove, in a learned manner, that the ancients only used one kind of studium, that of Olympia, and that all the differences and contradictions in point of measures proceeded from the imperfect means they were obliged to employ. Ep. VI. BOOK sider the meridians as parallel with each other, and, conscquently, the points of the compass as right lines. To construct this route on a flat map, it is then sufficient to draw, through the point of departure, a line which may make with the meridian of this point an angle equal to that which the point of the compass followed gives, and to carry on that line a number of parts of the scale equal to that of the itinerary measures gone over: the point where these parts terminate will be the point of arrival. Calculation may also be substituted for construction; if from the extremity of the route gone over we bring down on the meridian, which passes by the other extremity, a perpendicular, a rectangled triangle will result, in which the part of the meridian intercepted between the point of departure and the perpendicular brought from the point of arrival will indicate the distance of these points taken on the line north and south, or the difference of latitude expressed in itinerary measures; which are afterwards reduced, according to their value, into degrees of the meridian. And the perpendicular will express the distance of these same points taken on the line east and west, which is confounded in this case with the difference of longitude expressed in itinerary measures. If we wish to convert it into degrees, we must divide it by the number of those measures which a degree of the parallel of the point of departure ought to contain, or, if we wish for still more exactness, by the number of measures comprised in a degree of the parallel which holds the middle between that of the point of departure and that of the point of arrival. All this operation is nothing but dividing the number of the itinerary measures by the cosines of the latitude of the mean parallel. A second question may present itself, in the case where direction of the direction of the route is not known; the latitude of the point of arrival is then substituted for it. The construction on the flat map consists, in this case, in drawing through its latitude the parallel of the point of arrival; in taking on the scale of the map the number of measures assigned to the distance gone over, and to describe with this To find the a route. distance as radius, and from the point of departure as centre, a circle which will cut in the point of arrival the parallel previously drawn. If we wish to resolve this question by calculation, we must convert into itinerary measures the difference of latitude between the point of arrival and the point of departure; we have then in the rectangled triangle formed by the meridian of the point of departure, the perpendicular let down from the point of arrival and the route, two known sides, namely, the length of the route, or the hypothenuse, and the part of the meridian comprised between the point of departure, and the perpendicular of the point of arrival: by calculating the length of this perpendicular, we find the distance of the points of departure and of arrival, taken on the line east and west, whence we conclude, as above, the difference of longitude. BOOK VI. route. When the route gone over is of a considerable length, it Spherical becomes necessary to take the curvature of the earth into curve of a account. The construction of the two preceding problems requires, with respect to the reduction of leagues gone over in the direction east and west, in degrees of longitude, the employment for the tables for measuring latitudes, which contain the results of the trigonometrical calculation, by which the case may be resolved.* For the first ques * We have seen above, (p. 120.) that it is only by the help of the integral calculus that we can arrive at the exact construction of the tables of increas ing latitudes; but geographers commonly make use of a very simple approximative means to reduce the curve to a right line, by considering the route gone over as divided into parts small enough to be regarded as right lines. In fact, since the points of the compass cut all the meridians under the same angle, we may conceive that, through the extremities of all these subdivisions, may be brought meridians and parallels; thus will be formed, on each of these parts, a rectangled triangle, in which the sides of the right angle will be the differences of latitude and longitude, and the part of the route gone over will form the hypothenuse. But rectilinear trigonometry supplies the means of calculating this 1 triangle, since we know the hypothenuse and an angle. Let ABC, Fig. 47. be one of these triangles; we shall have by the principles of rectilinear trigonometry, AB AC: 1 cos. BAC; whence we shall conclude ACAB cos. BAC. Now, as the angle BAC is the same with respect to all the meridians which traverse its route, each difference of latitude, from one small triangle to VOL. I. 18 |