of readers, by taking them into the study of the mathematician, that he is or can be any thing but a being poring over diagrams and tables, and puzzling his brains over imaginary difficulties. We therefore propose not to enter into the study of the recluse, but rather to bring him out into the wide world, and shew him engaged in one of those works which he only is able to accomplish, in order that our readers may see that he is capable of entering with effect into the business which men regard as useful and important to them. Of all the practical works that are accomplished by the direct application of mathematical studies, the two greatest are astronomical and geographical. The latter, or rather one great department of the latter work, is that which is now to occupy our attention; and it is our purpose to give a very rapid view of the object to be accomplished in a trigonometrical survey, and then to make some remarks on the great trigonometrical survey that has for a long time been going on in British India.

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The idea of a great Trigonometrical Survey of a country, to be undertaken by the Government of that country, was first conceived by General Watson, at the suppression of the rising" in Scotland in 1745. The execution of it was committed to General Roy, and was originally intended to extend no farther than the disaffected districts of the Highlands. The design however was subsequently enlarged, and the grand Trigonometrical Survey of Great Britain and Ireland was projected-a Survey that has cost the country an enormous sum of money, which, albeit it has not been paid ungrudgingly, it is yet very creditable to the country to have paid. This Survey, begun just a hundred years ago, has been frequently suspended, but never wholly abandoned; and it is now, we believe, brought within a little of its termination.

ways.

This survey has effected much for science in various It was in the course of conducting it, that the real difficulties of the accomplishment were evolved, and the most scientific men in Europe were set to work to overcome them. The extreme accuracy that was required, gave an impulse to the efforts of our instrument-makers, and may be regarded as having given a beginning to the process of improvement in this department, which has advanced so stedfastly ever since, that now we have probably attained almost as near perfection as it is permitted to man to reach. It was in this Survey too, that the great accuracy of the instruments first brought into notice an element that had never been taken account of before, and whose treatment gave occasion to the discovery of some of the most elegant propositions in spherical trigonometry. As this element is a

very important one, we shall endeavour to explain its nature in a popular way, so that all may be able to apprehend its nature, and so to form some estimate of the extreme accuracy required, and happily attained, in such works as those in question. Every one knows that the three angles of every plain triangle are equal to two right angles. But this is not so in regard to spherical triangles. If, for example, we look at a common globe, and observe a spherical triangle formed by any arc of the equator, and by two quadrants of meridians, we at once see that the angles at the base of this isosceles spherical triangle are each of them right angles. Their sum therefore is two right angles, and the sum of the three angles of the triangle exceeds two right angles by the whole amount of the vertical angle, or the angle formed by the meridians with each other at the pole. A moment's thought will shew any one that this angle is the same part of four right angles that the portion of the globe's surface, included within the triangle, is of the hemisphere, or bears the same proportion to eight right angles, that the area of the triangle bears to the area of the sphere. Now this, which is evidently true of the particular triangle that we have selected as the simplest for illustration, is true of every spherical triangle. The angles of every such triangle are always together greater than two right angles, and the excess of their sum over two right angles will always be to four right angles, as the area of the triangle to the area of the hemisphere; and this is what geometers call the spherical excess.' Now when it is considered how very small a proportion the area of any triangle actually measured on the earth's surface bears to the whole of that surface, it must be evident that the spherical excess must, in all cases, be very small. In the English survey it seldom, or we believe never, exceeds four seconds; and it would, of course, be ridiculous to take such a quantity into account, were the observations not made with a degree of accuracy unknown before.

Perhaps a more important survey in some respects than the British one was that undertaken by the French nation at the period of the Revolution. The reasons which led to the undertaking are highly interesting, and are so germane to our subject that it can hardly be deemed a digression if we briefly state them.

In all ages and in all countries during which and in which a moderate degree of civilization has obtained, it has been felt to be a matter of great importance, as well as of considerable difficulty, to ascertain a uniform standard of measure, which may be easily verified and tested when necessary. The breadth of the human thumb, and that of the human hand, the length of the human foot, of the fore-arm, of the arm, and

of the distance to which a man can stretch with his extended arms, have been adopted, probably in every country, as rudimental standards; and have given rise to measures corresponding to the inch, the hand, the cubit, the yard and the fathom. Then in larger measures we have the distance that a camel can travel in a day, and, as a measure of surface, the quantity of land that a yoke of oxen can plough. Now it is with these measures as it is with human language. Plerumque ex captu vulgi induntur, atque per lineas, vulgari intellectui maxime conspicuas, res secant. Quum autem intellectus acutior, aut observatio diligentior, eas lineas transferre velit, ut illæ sint magis secundum naturam, verba obstrepunt.* As the stature of man and the developement of the several members of his body vary very considerably, it is evident that the measures thus ascertained cannot be regarded otherwise than as rude approximations, which may serve well enough for the ordinary purposes of half-civilized life, but are quite unfit for accurate, and especially for philosophical purposes. Accordingly we find that, just as the meanings of words are fixed down from time to time by definitions of encreasing accuracy, so the standards of measure have been attempted to be reduced to greater accuracy from time to time. For example the inch having been found to be about equal in length to three grains of barley, this came to be fixed upon as its standard length. This of course is but a clumsy correction, and some might suppose that it was rather a retrogression than an advance in the march of accuracy. As there is a principle involved in the matter which will perhaps be of some use to us in the sequel, we shall take the liberty of devoting a sentence or two to the exposition of the rationale of it. The length of a barley-corn is doubtless as variable as the breadth of a man's thumb or the length of his foot. Indeed it may be expected to be much more so, inasmuch as the developement of the human body takes place during a long period, and consequently the influences that affect it, being extended over many years, are more likely to give a uniform average result than those that affect the growth of a crop of corn, which is perfected in a single season, and the size of whose grains must be modified by countless varieties of soil, climate and exposure. How then can it be said that the length of three barley-corns is more likely to be

"Words are generally imposed according to vulgar conceptions, and divide things by lines that are most apparent to the understanding of the multitude; and when a more acute understanding, or a more careful observation, would remove these lines, to place them according to nature, words cry out and forbid it."-Bacon's Novum Organum, Aph, 59.

uniform than the breadth of a man's thumb? Simply because the average of three is more likely to be uniform than any one, however selected. Accordingly we believe it will be found that if thirty-six grains of barley be taken at random from a well-winnowed heap, and extended end to end in a straight line, and then thirty-six others be taken and extended in the same way along-side of the former line, the difference of length of the two lines will be very small indeed. We have heard that some of the Arab tribes have improved upon this method of correcting their measures, by greatly diminishing the length of the standard, and consequently encreasing the number of times that the standard must be taken in order to measure any considerable length. It is said that their standard is tested by laying a certain number of hairs taken from a horse's tail alongside of each other. Now although we do not doubt that the thickness of a horse's hair varies quite as much in proportion as the length of a grain of barley or the breadth of a man's thumb, yet it is probable that the united breadth of any thousand of such hairs, especially if taken from the tails of several horses, will differ very little indeed from the united breadth of any other thousand hairs similarly selected. This method we suppose must have been handed down from the days when the mathematical sciences flourished in Arabia.

But it is very evident that none of these methods would at all suffice for the construction of measures sufficiently accurate for the purposes of advanced science. In every country where science has made any considerable progress, there must be a standard measure, made with consummate exactness, and preserved with the greatest possible care, by which those measures made for ordinary use may be tested. Now if we possessed any material which was perfectly indestructible and unalterable by the progress of time, the friction of use and the variations of temperature, the actual length of this standard would be a matter of perfect indifference. Any one length would be quite as good as any other. All that would be necessary, would be arbitrarily to assume a particular length, and to give to it a particular name, to divide it in the most convenient manner, and to adhere fixedly to its length as a standard. But as all the materials within our reach are variable and perishable, it becomes necessary to have some mode of verifying the standard itself, in order that it may be with perfect confidence relied on as the instrument of verification for all the other measures. Such verification of the grand standard must evidently be derived from some great natural element, which may be assumed as invaria

able.* Two such elements have been suggested by mathematicians as the basis of a standard of length. It was proposed, we believe by Huyghens, that the length of the pendulum which vibrates seconds in any particular latitude, as for example in the latitude of 45°, or half way between the equator and the pole, should be adopted as the standard unit of measure. The other standard proposed is that of the length of some of the great lines that might be drawn through the earth's centre, or of one of the great curves that might be drawn over its surface; as the equatoreal or polar radius, or any other radius of the earth, or the equatoreal circumference of the earth, which may be regarded as a circle, or any of the meridians on the globe, which approach very nearly to ellipses.

It is evident that the adoption of a new standard of measurement cannot be effected without considerable inconvenience; the conservative principle is so very strong in the minds of the multitude in regard to those things that have to do with their daily and hourly habits. However certain classes of politicians may tell us that change in itself is neither good nor bad, but that change from a better to a worse is evil, while change from evil to better is good, the multitude will generally declare, in regard to those things, that change is always in itself an evil, or at least is always attended with various evils. Accordingly it is only in periods when men are in the spirit of change, when the stability of things is broken up for a time, that even improvements can be easily introduced. Such a period was that of the French Revolution, when the foundations of all things seemed to totter as with a fearful earthquake, when all social and civil institutions were overturned, and men's minds were so unhinged that they preferred adopting any thing that was recommended by novelty, rather than remaining satisfied with the things that had the sanction of usage. We presume our readers are aware that at an early stage of that fearful period, the philosophers of France undertook to introduce a great reformation in regard to

We are not entitled to conclude that any of the grand elements of our system are absolutely invariable. It is very possible that the radius and circumference of the earth for example may be subject either to fluctuations or to permanent changes. But if such changes do take place, their period must be so long, as to set us free from any danger of their discomposing our standard during any moderately long period.

This may be the proper place to remark, in case any of our readers should be ignorant of the fact, that it is quite sufficient to ascertain a fixed standard of lineal measure, all the other standards being made dependent on it. The unit of superficial measure is of course a square, whose side is the ascertained unit of lineal measures, that of solid content or measure of capacity, a cube whose edge is the said lineal unit; while the unit of weight may be fixed at the weight of such a quantity of pure or distilled water at a given temperature, as shall just fill the unit of solid content.