It is evident, that in all these calculations of Malines, Sir William Petty, and Gregory King, political arithme tic had derived no assistance from the more sure and ac curate sciences, but had been suffered to range unen lightened and uncontrouled, either by geography or geometry. At length, Dr Edmund Halley, in conse quence of the earnest desire of his industrious and in quisitive friend Mr Houghton, author of the Collections for the improvement of Husbandry and Trade, made a most elaborate calculation with respect to the contents of England and Wales, which lie found to contain in the gross—that is taking the whole from a single map,— 38,660,000 acres ; and on a strict computation of the se veral counties, each separately examined, he computed the total to be 39,938,500 acres ; and as these sums, attain ed to by these two different methods, so nearly approach ed each other, he concluded that neither of them could be very wide from the truth. He adds, that, in his judg ment, England and Wales might be esteemed the 3000th part of the whole globe of the earth, and the 1500th part of the inhabited world. Dr Halley's own account of his calculations is given in vol. i. p. 69, of Houghton's Col lections. Besides the source of error to which the cal culations of Dr Halley were exposed, from the inaccu racy of the maps of England, which existed in his time, (a circumstance which will be afterwards more particu larly noticed,) his estimate was rendered erroneous, by his having used (as most of our geographers still do) Norwood's measure of a degree of instead of the true measure; or sixty-nine miles and a half, instead of sixty-nine and one fifth : had he used the latter mea sure of a degree of latitude, the total would have been reduced about 370,000 acres, which would have nearly coincided with the area of South Britain, as given by Morden's map, when measured according to the true length of the successive degrees of latitude and longi tude :—the single map, from which Dr Halley took the gross amount, was Adam's, and the map from which he took the several counties, was the six-sheet map of Saxton's.
In the Philosofihical Transactions for the year 1711, (No. 330, page 266.) are given the calculations of Dr Nehemiah Grew on this subject, and the result of these calculations. According to him, South Britain contains 72,000 statute miles, and consequently 46,080,000 statute acres. As Dr Grew was at the trouble to obtain a wheel measure of those roads, which were necessary to supply him with what he conceived to be sufficient data on which to found his calculations, it might be supposed that the area he assigns to South Britain would approach very nearly to the truth; but he unaccountably forgot, that roads are not always straight or smooth; and assum ing, on the contrary, that the wheel measure of them gave him a straight line, he makes the length of South Britain, from Newhaven, in Sussex, to Berwick, 395 miles ; and its breadth, from the South Foreland of Kent to the Land's End, he makes 367 miles; both, as we shall afterwards see, considerably above the truth. Nor was this mistake respecting the roads the only source of his error ; for he very carelessly and inaccurately con jectures, that a right-lined triangle, formed by uniting the South Foreland and the Land's End with Berwick, is commensurate with the space contained by the irre gular outline of the country. This error would have given less than the real area to South Britain, had the sides of the triangle corresponded with the actual dis tances, though the amount was not sufficient to destroy the error arising from making these sides too long.
The next calculation respecting the area of South Bri tain, was given to the world by Mr Thomas Templeman of Bury, in his work entitled a New Survey of the Globe. According to him, it contains 49,450 square miles, or 31,648,000 statute acres. This extent, much more be low the truth than the calculation of Dr Grew raised it above it, Mr Templeman obtained, by adopting a mode of computation that is progressively more erroneous, as it is applied to countries more distant from the equator. Mr Arthur Young seems to have been the first who ob jected to the accuracy of Templeman's calculation; but not perceiving the source from which the error of this author took its rise, he endeavours to correct it on im proper grounds; and supposing that the error would be rectified by taking 691 miles instead of 60 miles to a degree, and moreover, supposing that Templeman erred as much in his calculation respecting the area of Eng land, as the calculation of Necker respecting the area of France proved he had done with regard to that kingdom, Mr Young assigns to England and Wales 46,915,933 acres.
Amidst this uncertainty of calculation and opinion on this subject, the trigonometrical survey of England was begun ; and, in its progress, it was abundantly and clear ly proved, that the geography of this country, in many respects, was very erroneous. Indeed, before it was commenced, it had been discovered, that in the older maps the counties in the vicinity of the metropolis were erroneously laid down, both with respect to the general amount of their area, and with respect to the particular areas of each; so that the distance from the South Fore land in Kent, to the Land's End, was ascertained to be less by about half a degree than had been formerly laid down.
For all the purposes of political arithmetic, however, it is not necessary to have recourse to that minute ac curacy which trigonometrical surveys afford. A suffi cient degree of accuracy will be afforded by a mode of computation, ingenious, easy, and satisfactory, which was published by Dr Becke, in his Observations on the Produce of the Income Tax. He was naturally led to the consideration of the subject of the extent of South Britain, by the computation of the income of Great Bri tain, as stated by Mr Pitt in the House of Commons. This statesman had assumed the truth of Mr Young's calculation respecting the area of England, (to which Mr Middleton, in his Survey of Middlesex, had also lent the authority of his name,) and, consequently, had rated the income of this county, in Dr Becke's opinion, much too high. Dr Becke formed a scale, which is given below, the construction of which depends on the length of the degrees of latitude and longitude, accord ing to their situation on the surface of the earth. In this scale, the number of acres and decimal parts in eve ry successive four-sided figure, or trapezoid, are given; these trapezoids being formed by the intersection of lines, or circles of longitude and latitude, whose distance from one another is a minute, or the sixtieth part of a degree. The length of the degree of latitude and longi tude, according to their situation, by which the following scale is calculated, is taken from the recent observations and measurements for that purpose, in this country, as they are detailed at large in the Philosofihical Trans actions.