Having completed the construction of the itinerary, it remained to subject all these lines of route to the invariable data of which geography is already possessed. I first sought among these data for points common to M. Caillié’s march: they are unfortunately very few in number. How then could I flatter myself, whatever trouble I might take, with whatever care I might combine all the data, hazarding nothing without some authority to support it, that I should produce any thing beyond a mere essay? If it should be hereafter confirmed by the observations of travellers furnished with astronomical instruments, the only merits of this work will consist in fortunate combinations; if it should be falsified by future discoveries, still it will have called for the criticism of geographers, and will consequently not have been useless to science. In submitting to the reader results differing from those hitherto admitted, I wish to warn him against an error, too common, especially in map-making, that of giving the preference to the more recent publications, and to place confidence in them in proportion as they are so. I am far from desirous of usurping this species of interest, to the prejudice of geographical works in general estimation.
The points common to the route of M. Caillié and to the list of positions considered by geographers as perfectly or sufficiently established, are confined to the following: the points of the Western coast of Africa, Kakondy, Timbo, Sami and Yamina (for the latitude), Bakel, Elimané, Fez; and I shall add to them the position of Ain-Salah, though published here for the first time. With respect to the positions of Djenné, Timbuctoo, and the places situated in the great desert, the uncertainty is so great, that there is no resting any solid calculation upon them, and they are of no use in verifying the exactness of new itineraries.
Thus we are reduced, for a space which comprehends twenty five degrees of latitude and from ten to twelve degrees of longitude, to eight points in the interior.[77] Still, the point whence the traveller set out on the first part of his travels, the position of Timbo in the middle of this part, and the very probable knowledge of the parallel of Sego, a town which is connected with the itinerary and attaches itself to the fixed points of the Senegambia, with the almost certain situation of Fez, form a first basis, which may serve to verify as well the inflections of the route as the length of the lines travelled over. I began by establishing the lines from Kakondy to Timé, from Timé to Djenné and Timbuctoo, and from Timbuctoo to el-Araouan; first, by supporting them separately upon Timbo, the parallel of Sego, and the position of Fez; and secondly, by attending to the declination of the compass. These lines were at first formed without any other modification than the necessary substitution of the true north for the magnetic north, in the night journeys. The direction of the first line from Fez gave me a very fair position for Timbuctoo; and that of the second line from Timé furnished me with another but little different, and which the situation of the parallel of Sego brought considerably nearer to the first: whatever uncertainty still remained has been cleared up by new data, of which it would have been difficult not to have made some use.
Whilst at Timé, the idea struck M. Caillié of observing the length of the shadow of a style at midday; his long stay there gave him an opportunity of making the observation twice: the first time, which was on the 30th of October 1827, the height of the style, with every reduction, was 0,706 metre; that of the meridian shadow was equal to 0,2945 metre.[78] The second observation was made on the 1st of November 1827; but this measurement cannot have been taken with so much precision. It was the shadow, properly speaking, which was measured, that is to say, the shadow terminating distinctly and without the penumbra. The calculation gives for the latitude as nearly as possible nine degrees.[79]
This being admitted, I perceived that the construction (made in the manner before explained) of the line which represents the first part of the journey, gave to the situation of Timé the same latitude within a few minutes. This agreement convinced me that no change was required in the construction; so small a difference, considering the insufficiency of the means employed, might indeed be regarded as an entire concordance, and I could not but suppose that it was probably the effect of a fortunate compensation for many errors on the contrary side. I might therefore look upon Timé as a point nearly fixed, and leave it to establish the other two lines. The longitude of Timé, resulting from the preceding operations, is nine degrees two minutes west of Paris. Thus Timé would be at a nearly equal distance from the equator and from the meridian of Paris.
The bearing of the line from Kakondy to Timé, according to the travels of M. Caillié, having been confirmed, has given me confidence in the bearings of the rest of the route. I have therefore first laid down the line from Timé to Timbuctoo, and that from Timbuctoo to Fez, such as they result from the construction of the map of the route. The point of Fez being fixed, it became necessary to modify a little the absolute length and the direction of these lines, to confine myself between the two points of Timé and Fez, and I have proceeded upon a proportionate reduction. The difference was nothing extraordinary for so long a route, amongst so many obstacles and difficulties which the indefatigable traveller had to overcome. It amounted on the whole, upon near three thousand English miles,[80] to about one hundred and fifty, or a twentieth part of the space travelled over, and the total angular difference is less than six degrees upon the angle between the meridian of Kakondy and the direction upon Arbate. The latitude of Timbuctoo, obtained by this means, is near seventeen degrees fifty minutes north.
Possessing upon this latitude no geographical data properly so called, having only the routes of caravans, and not even the hours of march, but merely the reckoning of the day’s journey, so that to the uncertainty of the length of the journeys is added the still greater uncertainty of the pace of the caravans, according to whether they were more or less numerous, whether composed of camels more or less laden, or only of pedestrians; together with our ignorance as to the number and situation of the forced halts which they make in the desert, either on account of wells, or of those unforeseen accidents which will happen in these terrible peregrinations;[81] in the midst, I say, of so many causes of hesitation, which ought to warn geographers against the employment of the vague itineraries of the Arabs and the Moors, could I grant less confidence to the route of M. Caillié than to the marches of the Africans?
These routes are constantly divided by hours: the rests are noted with exactness, and they are never undecided with respect to the length of the marches: it only remains then to estimate the pace, and we are enlightened on this latter subject by the composition of the caravan. For these reasons, and others still which it would occupy too much time to explain; I have not thought it right to take preceding itineraries into account in combining the elementary facts necessary for ascertaining the position of Timbuctoo.
I should, however, have still remained in doubt, and have abstained from offering an opinion had there not been other new data susceptible of comparison with the itinerary of M. Caillié; I mean the measure of the meridian shadow which he took at Timbuctoo itself. This observation was made by the same method as that at Timé: this proceeding is undoubtedly very imperfect, but, for want of others I think it should not be entirely neglected. On the 1st of May 1828 our traveller planted a style of the height of 0,635 metre; he measured, at noon, the shadow of this style and found it equal to 0,030 metre.[82] The calculation gives seventeen degrees fifty one minutes north latitude. I must repeat here the reflection that this agreement may very probably result from contrary errors which have balanced each other: but as it is impossible to discover the points in which the errors lie, or the limit of their extent, the final result is all that can be obtained.
I will add one consideration which will not have escaped those geographers who have studied the calculation of probabilities. In a series of observations made under the same circumstances, and especially by the same observers, the greater the number, the more probable is it that their amount will approach to the total quantity required. When there is no reason that the errors committed should be on one side rather than another, they mutually destroy each other, and the more so, the more the observations are multiplied. There is even a rule which teaches us how much the sum found differs from the truth; its discovery belongs to the learned geometrician, who is at present the organ of the French Institute for mathematical science. Knowing the error which may have occurred in an observation, it must be multiplied by the square root of the total number of observations. Thus, instead of growing with this number the total possible error decreases proportionably. For example, for four observations it would be represented by two, and for a hundred observations, by ten only. The proportion of total errors is therefore as ¹⁰⁄₂ whereas the proportion of the number of observations is as ¹⁰⁰⁄₄: thus the error is but the fifth part of what it would be proportionally in four observations.[83] Hence it follows that the more observations are multiplied the more any imperfection in the processes by which they have been made will be corrected.