that between Putchapoliam and Bomasundrum too great; ⚫ making, of consequence, the length of a degree too great in the first case, and too little in the second. Being,' he adds, ⚫ confident as to the accuracy of the observations at both places, in consequence of the circumstances just mentioned, I thought it reasonable to take the mean of the two degrees, which gave 60490 fathoms for the degree in latitude 11° 59′ 54′′.

In the conclusion of the paper, the Major reduces the degrees into a consistent form, and apparently cleared of all irregularity (p. 94), but on a principle of which we cannot entirely approve, as it involves in it too much theory. The mathematical reasoning is correct; but the introduction of a degree measured in another latitude, though it is quite legitimate in a general inquiry into the figure of the earth, prevents the results of the Indian measurement from appearing as independent facts, resting on the foundation of experiment alone.

The simplest and most unexceptionable way of deducing from a large arch, (the parts of which, as actually measured, are not perfectly consistent), the results that may be accounted the nearest approximation to the truth, is to consider, that if the elliptic hypothesis be true, whatever be the compression, the successive degrees of the meridian must increase, on receding from the equator, by a quantity proportional to the sine of the double latitude. Thus, if x be the degree in the latitude L, the next degree is an sin. 21; the next to that is x + n sin. 21 + n sin. (2 L +2°), &c. where n is a constant quantity, to be determined without the assistance of theory, by assuming different values for it, and adopting that which agrees nearest with the observations. This is easy, because n sin. 2 L is always a small quantity. In the southernmost point of Major Lambton's arch, it is between 2.4 and 3. fathoms: the value that seems to us to answer best, is 3.1 fathoms; and in this way we deduce the first degree of the arch, that which begins at Punnae, in lat. 8° 39' 38", and has its middle in 9° 9′ 38", equal to 60473 fathoms. This is derived from a comparison of the arch between Punnae and Putchapoliam, which consists of 2° 50' 10", and is certainly, as far as observation can go, very accurately determined. In this way, the successive degrees are as follows.

Mid. Lat.

9° 10' 10° 10' 11° 10'

Length. 60473 fath. 60476.1

60479.5

Mid. Lat.

12° 10'

Length. 60483.2 fath.

These are a little different from Major Lambton's results, to which they would have been brought nearer if we had employed the arch between Punnae and Dodagoontah, in the determina

tion of the first degree. But as the latitude of Dodagoontah is in all probability affected by the attraction of the plummet toward the north, so that its zenith is carried too far to the south, the arch between it and Punnae must be too small; and therefore we thought it best to avoid this arch in the fundamental determination. *

The anomalies which have occurred in the measures of degrees, and of which the appearances seem to increase in proportion as greater pains are taken to avoid inaccuracy, have naturally drawn the attention of mathematicians; and the question, what part of them is to be ascribed to error, and what to irregularities in the structure of the globe, has come, of course, to be considered. That a small part of them only can be ascribed to the former cause, is rendered probable by the very circumstance just stated; that they are not diminished, nay, that they even seem to be increased, by the care taken to avoid error. It seems clear from that consideration, that the irregularities are in the object sought for, and are only brought more in sight by more microscopical observation, by the excellence of the instruments, the accuracy of the computations, and the extent of the lines measured. No measurement was ever executed with greater care than that in France; and the great extent of the arch measured, as well as the ability and skill of the observers, make the mean result, the length of the degree in the parallel of 45°, the datum most perfectly ascertained of any that regards the figure of the earth. Yet even here, we find in the detail that there are great anomalies, and that the successive degrees increase with much irregularity.

The arch between Greenwich and Dunkirk gives the degree greater than that which is derived from the arch between DanKirk and the Pantheon at Paris by 7.23 toises; the next difference is 8.4; then 32.4, 12.9; and lastly-2 from the arch between Montjouy and Formentera. In this last case, there is an absolute retrogradation; and the degree increases on going to the south, just as it is observed to do in the arch measured in England, and in that measured in Hindostan.

The irregularities in the French measurement induced De

* To deduce the mean degree from a large arch, such as one of nearly three degrees, by dividing the length of the arch by its amplitude or number of degrees, is not exact, as the degrees increase each above the preceding by the quantity n sin. (2 + 2). The length of the arch ought to be diminished by the sum of all these quantities before it is divided by the amplitude; and this division gives not the degree in the middle of the arch, but that at the beginning of it, or the farthest to the south.

LAMBRE to scrutinize the latitudes of all the above places with the utmost care; but he could find nothing sufficient to account for the irregularities. (see Base Metrique, tom. III. p.84.) The observation of the latitude at Montjouy appeared exact; yet, when compared with one at Barcelona, very near to Montjouy, an error of 3".24 was discovered; and DE Lambre, apparently with much reason, considers this difference as a certain proof of the irregularities of the earth. To the same cause he ascribes the rest; and indeed, from the very progress which they hold, some local affection seems necessarily suggested.

The consequence of all this is, that for the whole of the arch in France, the degrees are best represented by supposing a compression of, or ; while, by taking in a greater range, and comparing the degrees in France with those in distant countries, the compression comes out less than the half of this, viz., or To reconcile the measures actually made with a compression of, it will be necessary to make the following corrections on the latitudes-For Paris, 0; Montjouy, +3.6; Carcassonne, + 0.88; Dunkirk, + 3".06; and for Evaux, +5.83. These are wholly improbable as errors of observation, and must be attributed to local attractions, which act irregularly on the plumb line.-Base Metrique, ib. p. 92.

The same thing may be said of the arc measured in England by Colonel MUDGE: the whole arc, taken together, agrees very well with the measures in France, and with that in Lapland, as lately ascertained by the Swedish academy. But if the parts of this arc be compared, an irregularity is found, and

We have compared together the five arches of the meridian, which from their extent, and all other circumstances, seem the best entitled to confidence, viz. that in Peru, by Bouguer and Condamine; in Hindostan, by Major Lambton; in France and England, comprehending the whole extent, from the parallel of Greenwich, to that of Formentera, by Delambre and Mechain, and in part by General Roy; that in England afterwards, by Colonel 'Mudge; and, lastly, that in Lapland, by M. Swanberg; and the results which we have found, are extremely consistent, and give, for the compression at the poles When this compression is adopted, there does not appear an error of more than 9 fathoms in the measure of any of the above degrees. The French, from their own measures in France and Peru, bring out a compression of nearly. Thus the results are consistent with the supposition that the earth is an elliptic spheroid, when the arches compared are large and distant from one another: when they are small, and near to one another, they do not agree with that hypothesis, nor indeed with any other single hypothesis that can be laid down. This is what might be expected, and does not invalidate the general conclusion.

the degrees appear to increase on going from the north to the south. In giving an account of Colonel Mudge's measurement in a former Number of this Journal, we ascribed the fact just mentioned, to local irregularities in the direction of gravity, and we still consider this as by far the most probable supposition. A paper, however, written with great knowledge of the subject, and full of sound mathematical reasoning, has been published by DoN RODRIGUEZ in the Philosophical Transactions for 1812, which is quite on the opposite side, and ascribes the irregularities in the arc to errors of observation. DON RODRIGUEZ, if we mistake not, is one of two Spanish gentlemen who accompanied MM. BIOT and ARRAGO, and assisted in the operations by which the meridian that had been traced through France was extended to the southermost of the Balearic isles. He seems perfectly acquainted with the methods of calculation, and all the most recent improvements which respect the problem of the figure of the earth. We do not think that he has proved that the irregularities in this measurement arise from errors of observation; and we are of opinion, though the amount of these irregularities may now be more exactly estimated than before, that with regard to their cause, the question rests precisely where it did. But though we are not convinced by DON RODRIGUEZ, we must do him the justice to say, that his argument is fairly conducted, and that he has displayed great knowledge of the subject, and perfect familiarity with the best methods hitherto employed in the solution of this difficult problem. We have therefore observed with regret, that this ingenious foreigner has been attacked in some of the English Journals, with a violence and asperity which the subject did not call for, and which his paper certainly did not authorise.

When there are unlooked for results in any system of experiments or observations, the errors into which the observer may have fallen, naturally come to be considered as affording one solution of the difficulty. We are not to suppose, that any man engaged in experimental investigations, can be exempted from such an inquiry; nor, when such inquiry is instituted, are we to suppose that he is subjected to a personal attack. The principle on which Don Rodriguez proceeds, though it may be erroneous, seems to be general; it is applied equally to the French and the English mathematicians; and the anomaly of more than 3" in the latitude of Montjouy, is ascribed by him, not to local irregularity, but to the mistake of MECHAIN, a man eminently skilled in the art of astronomical observation. The calm and dispassionate memoir of the Spanish mathematician,

VOL. XXI. NO. 42.

Y

does not therefore give any ground for supposing it to be meant as a personal attack, and still less as a national one.

We observe, with pleasure, however, that the true resolution of the difficulty is most probably at hand. The continuation of a meridional arch must afford the best means of discovering from what cause the irregularities observed in it arise. If they arise from physical irregularities in the structure of the globe, or in the direction of gravity, a compensation in the course of a great arch may be expected to take place. If a body of heavy matter, at any point, make the plummets on each side. of it converge more than they ought to do, the zeniths will be carried too far off from one another; the amplitude of the celestial arch will be increased; and the length of the terrestrial degree, will, of course, be diminished. But as the zenith on one side of this point was carried too far to the south, and on the opposite too far to the north, the degrees on either side will be rendered too great, the amplitudes of the celestial arches being made too small. Thus an opposite error will take place, and what is added to one degree will probably be taken from the next. This is not likely to happen if the errors arise from inaccuracy of observation: these errors will not be as any function of the distance, but, depending on accident, must be quite irregular in their distribution. It is with pleasure, therefore, that we see a meridian which has been extended from the shores of the British channel along the west side of England, viz. the meridian of Delamere now produced into Scotland, where it falls on the east side of the island, and is about to be continued till it intersect the shores of the Murray Firth, or the Northern Ocean. The combined arches in France and England will then extend nearly to 20 degrees; and in a few years we shall perhaps see the distance between the parallels of the Balearic and the Orkney Islands, ascertained by actual mensuration. We believe that this important operation could not easily be in better hands than those in which it is actually placed; and, when it shall be completed, the British army-in General Roy and the officers who have succeeded him in the conduct of the English survey-and in Major LAMBTON whose works we have been now treating of, will have the glory of doing more for the advancement of general science, than has ever been performed by any other body of military men.