rived from Legendre's theorem, that the arches which form the sides of small spherical triangles, are proportional to the sines of the opposite angles, when each of these angles is diminished by the third part of the spherical excess. By means of this proposition, the sides themselves may be directly computed, and the investigation of the chords avoided as unnecessary. From this it also appears, that the correction made on the observed angles, by taking from each of them one-third of the spherical excess, can be no longer viewed as an arbitrary correction, but as a legitimate and necessary inference from a geometrical theorem extremely curious in itself, as it marks the continuity of plane and spherical triangles; and extremely valuable, as it leads to the most accurate and simple rules of calculation. Though it is a theorem that, in strictness, is only an approximation to the truth, yet its accuracy, in all such cases as can come under consideration in a survey of any portion of the earth's surface, may be safely relied on, the quantities which it rejects being then really evanescent.

In reducing the observed angles in the manner of Major Mudge, (which is also that of Delambre,) there is always the inconvenience of an operose and unnecessary calculus; and, in certain cases, such as the computation of the distances of the stations from the meridian or the perpendicular to it, it is not quite evident but that inaccuracies of some

consequence may be introduced. If, on the other hand, in the solution of this last problem, since the hypothenuse and one of the oblique angles of a right-angled triangle are given to find the sides; if we first calculate the spherical excess, and proceed to find the other oblique angle by making the sum of it and the given angle=90°+that excess; if we then subtract one-third of the spherical excess from each of these oblique angles, and, with the angles so corrected, compute the sides by the rules of plane trigonometry; we shall obtain them with great ease, and with all the precision that the problem admits of. It would seem, then, that this last method of calculation is greatly preferable to the former.

We are perfectly aware of the caution with which theoretical men, sitting quietly in their closets, should offer advice to those who add the practice of art to the speculation of science; who sacrifice ease and comfort to literary pursuits, and earn their reputation not merely by deep study, but by the sweat of the brow and the labour of the hands. We feel the full force of this maxim, and are writing at the present moment under the strongest impression of its truth. Yet, when we venture to recommend the method of calculation just described, as fit to be employed in the trigonometrical survey, we are not much afraid that the person best able, and most interested to judge correctly on that subject, after

VOL. IV.

making the trial, will be inclined to censure our rashness.

The length to which these remarks have already extended, forces us to pass over the operations of several of the subsequent years when the survey was carried westward as far as the Land's-End, and again eastward to the remaining part of Kent, to Essex and the interior of the country, Oxfordshire, &c. Though these accounts are interesting from the importance of the places surveyed, and particularly from the drawing of four different meridians, and the determination of their difference of longitude, we shall pass to the consideration of the last thing performed in the survey, which is the measure of an arch of the meridian between Dunnose in the Isle of Wight, and Clifton near Doncaster. For the purpose of this measurement, Major Mudge was furnished with a new instrument, of the workmanship of Ramsden, viz. a zenith sector for the celestial observations, which were now required to be made with the greatest possible exactness. Though several instruments of this kind have been constructed by former artists, and many of them excellent, yet that which we have just mentioned seems greatly superior to them all. In it, the defects of former constructions are obviated, and many new improvements introduced. Among these must be reckoned the method of suspending

the instrument, of bringing it into the vertical plane, of turning it to face opposite ways, and, most of all, the contrivance for adjusting the plumbline perpendicularly over the centre of the sector, in which the skill of the optician and the mechanist are eminently combined. The instrument is supported by a strong pyramidal frame; the telescope is an achromatic, eight feet long; the radius of the sector is nearly the same; and the angles measured on its limb may be read off to decimals of a second. The whole is a masterpiece of original design and skilful execution; and, to its intrinsic value, adds that of being almost the last work of an artist, who was never equalled by those who went before him, and will not soon be surpassed by those who shall come after.

The meridian of Dunnose, in the Isle of Wight, was pitched on by Major Mudge to be extended northward, as affording a better opportunity than any other of measuring on it a large arch, free, as far as could be foreseen, from the action of any disturbing force. The meridians further to the west, though they might be produced to a greater length before they reached the sea, entered sooner into a mountainous, or at least a hilly country, where the direction of gravity must be affected by the inequalities of the surface. The meridian of Dunnose, traversing the plains of Hampshire, Berkshire, &c. and so on to Yorkshire, intersects the

sea-coast near the mouth of the Tees, without having passed over any high land, except on the confines of Northamptonshire and Warwickshire, near the sources of the Nen and the Avon, where the ground rises to the height of 800 or 900 feet above the level of the sea, with a gradual slope both to the south and to the north. The part of this line which has been actually measured, extends from Dunnose to Clifton, not far from Doncaster, a distance of more than 196 miles, the length of which was determined by a series of triangles carried from one end of it to the other, like those that have been already mentioned. The sides of these triangles were deduced from the bases on Hounslow-heath and Salisbury-plain, as in other parts of the survey; but, for the greater security, another base of about five miles was measured on Misterton Car, near the northern extremity of the chain of triangles; and this was done with the same precautions which have been before enumerated. The latitude of Dunnose and Clifton were then accurately determined by the sector, from stars near the zenith, observed in their passage over the meridian. The instrument was also carried to Ardbury-hill, (near Daventry in Northamptonshire,) not far from the middle of the line, the latitude of which was also exactly determined. Besides this, as Blenheim was not far distant from this meridian, and as its parallel of latitude had been accurately determined by