gebra, from treating only of particular questions, to be the most general and most perfect language that has yet been employed in science. The use of letters, instead of numbers, to express the powers of quantities, prepared the way for the discovery of the binomial theorem, and, of course, for the fluxionary calculus. Euler's arithmetic of the sines depended much on the use of a contraction that kept the relation between the sine, or the tangent and the arch to which it belonged, continually in view. The improvement we are going to mention in the notation of the trigonometric formula, is certainly not so great as any of the preceding, but is nevertheless much greater than could be expected from so small a matter. It consists in denoting the angles of a spherical triangle, by the capitals A, B, C, as usual, and the sides opposite to them, by the same letters in the small form, a, b, c. This notation keeps in view the relation between the sides and angles that enter into a trigonometric formula, and conveys the meaning more distinctly and immediately to the mind, than is done in the ordinary way. The advantage derived to the rules by this very simple contrivance, is therefore much greater than, without the experience of it, could easily be imagined. Mr Woodhouse has used it in his rules, which are laid down with great distinctness, and are rather the better for not being reduced into a table, but stated at full length as in other problems, and always expressed in the language of algebra. The rules for the solutions are indeed stated with remarkable clearness, and with such a degree of detail as, without being prolix, is sufficient to prevent error. There is hardly any one of all the different methods that have been invented for the resolution of spherical triangles, that is not brought under some one of the six cases into which the whole is divided. We would very much recommend the study of this part (the six cases of oblique spherical triangles) to all who would acquire a thorough knowledge, both of the principles and the practice of spherical trigonometry. The illustrations from the calculations of the Trigonometrical Survey of England, are well selected. Here we are glad to observe à just tribute to the memory of the late GENERAL ROY for his application of the theorem concerning the area of a spherical triangle to an important purpose in practical geometry. From that area being proportional to the excess of the three angles of the triangle above two right angles, and from the circumstance that the said area can, in triangles constructed on the earth's surface, be easily estimated, General Roy perceived that he had the means of verifying his observations of the angles of spherical triangles as much as of plane triangles, where all the angles ought to amount precisely to 180 degrees. The theodolite theodolite which he employed in his measurement of the distance between Greenwich and Dunkirk, and afterwards in the Survey of England, was of such exquisite construction, that it could measure an angle to a fraction of a second, and so could exhibit the spherical excess, that is, the excess of the three angles of the triangles in his Survey, (which were of course on the surface of a sphere) above two right angles. In the Philosophical Transactions for 1790, he gave the following rule for computing the spherical excess-From the logarithm of the area of the triangle, taken as a plane one, in feet, subtract the constant logarithm 9.3267737, and the remainder is the logarithm of the excess above 180°, in seconds nearly.' Mr Woodhouse has given the demonstration of this rule; and he adds, after explaining the numerical calculation- In this manner the spherical excess computed, will enable the observer to examine the accuracy of his observations, and, in some degree, to correct them. He may then proceed to calculate the sides of the triangles, by the rules of spherical trigonometry; but these rules, though they must give exact results, will not give results very expeditiously, when several hundred triangles are to be solved; and this M. Legendre has enabled us to do, who has very happily combined sufficient exactness with conciseness, by means of this theorem '-' A spherical triangle being proposed, of which the sides are very small relatively to the radius of the sphere, if from each of its angles one third of the excess of the sum of its three angles above two right angles be subtracted, the angles so diminishmay be taken for the sides of a rectilinear triangle, the sides of which are equal m length to those of the proposed spherical triangle. The demonstration of this very valuable rule is afterwards given in the Appendix. The theorem, as has been said above, was discovered by Legendre, and is demonstrated in the Notes on the Spherical Trigonometry annexed to his Elements. A demonstration of the same theorem, remarkable for its simplicity, has also been given by Professor Leslie in the Notes to his Elements of Geometry. ed The other method of calculation accommodated to trigonometrical surveys, and actually employed by Colonel Mudge in that of England, is that of reducing the angles of the spherical triangles to the angles contained by the chords., A theorem, by which the latter angles are inferred from the former, is investigated Lore. The formula which Mr Woodhouse has given for this purpose, expresses the value of the whole angle contained by the chord bat as, in such triangles as can occur in the practice of measurement, this angle can never differ much from that which is contained by the arches themselves, it is the simplest way to seek for their differ ence, or the small correction which is necessary for reducing the one of them to the other. He accordingly remarks, that Delambre, who has been so much engaged in surveys of this nature, has investigated a formula for this reduction, and constructed tables, in which its numerical value, in any given case, may readily be found. Professor Leslie, we must also observe, in the work already quoted, has deduced a very simple formula for the same purpose. As logarithms are the great instruments by which all trigonometrical calculation is now performed, our author, in an Appendix, has investigated the rules for their construction and use. Besides a very clear and distinct exposition of one of the most important branches of arithmetical science, we meet, here, with several views of the subject which have not before been sufficiently remarked. One of these consists in the explanation of the advantages which arose from the change which BRIGGS introduced in the construction of logarithms, when he took the number 10 for the base of the system. If any number were to be assumed for the base, different from that which is the root of the arithmetical scale, much inconvenience would result, or rather a great loss would necessarily take place of those advantages which we now possess. Thus, if we were to adopt the duodecimal scale of arithmetic, and to return in our notation from 12, as we now do from 10, the base of the logarithmic scale must be changed to 12 also, and 12 N, would be the equation from which the tables of logarithms must be constructed. After a full elucidation of the subject of logarithms, the Appendix and the volume conclude with the demonstration of some trigonometric theorems; an investigation of the reduction of an gles from an oblique to a horizontal plane, and of the property of the spherical excess already mentioned. On these we are not now to enter; but we cannot conclude, without again remarking the perspicuity, conciseness and extent, which diftinguish this treatife. The analytical method is well preferved, as much perhaps as is poffible in a work where many references to the fynthetical demonftrations of elementary geometry must neceffarily be made. One confequence of this is, that the perufal of this fhort analytical treatife will do more to make a perfon completely mafter of the principles and methods of trigonometrical calculation, than the study of many voluminous works drawn up. in the ordinary form." We must alfo commend our author, not only for what he has included, but for what he has left out. The omiffion, which may feem faulty to fome, of the folutions of spherical triangles by letting fall perpendiculars, feems to us quite judicious. The computations made by fuch means are perplexed by the continual infpection of figures which it is im poffible poffible to conftruct with accuracy, and favour much of that ftate of science, when men reforted to particular and numerical calculations, in order to avoid the investigation of general and algebraic theorems. We have already fuggefted that fome improvement might be made in the arrangement of the parts of this treatife; and we have particularly stated in what we conceive these improvements fhould confift. Thefe confiderations we are still more inclined to recommend to the author's attention, after having finished what we think is a very careful, and, what we are fure, is a very im partial, examination of his book. We shall be happy to fee these alterations introduced into a new edition; and we would be highly gratified if we could fuppofe that we have had any share in bringing this about, or that our remarks have in the leaft contributed to the perfection of a work which is already fo deferving of praise. ART. VII. A View of the antient and present State of the Zetland Islands, including their Civil, Political, and Natural History, Antiquities, and an Account of their Agriculture, Fisheries, Commerce, and the State of Society and Manners. By Arthur Edmondston, M. D. 2 vol. 8vo. pp. 709. Edinburgh, 1809. WE E shall know something of old Thulé at last. Since Dr Barry favoured the world with a quarto volume on Orkney, we have been illuminated by tours and travels to both clusters of our northern islands: we have now before us two goodly octavos on Zetland; and are credibly informed, that we may soon expect the publication of agricultural and mineralogical reports on both sets of islands. All this notwithstanding, our worthy author begins by complaining, that while the most trivial observation' concerning the South Sea islands is read with interest, and remembered with satisfaction, many valuable and useful communications which relate to our native country are soon overlooked and forgotten.' We sincerely hope that this is not said in anticipation of his own fate; and shall contribute all that lies in our power to avert it, by entering pretty fully into the subject of his erudite volumes. We cannot, however, agree with the Doctor in the propriety of proportioning the freedom and impartiality of our discussion to the importance of the subject;'t though it would have been a reai benefit, if he had thought I 4 Neill's Tour; Hall's Travels, &c. + Preface, p. ix, thought of measuring the number of his pages by that rule. In defiance, however, of all such maxims, the situation, general appearance, and climate of Zetland,' are all despatched in a dozen pages; while one hundred are rather unprofitably occupied with what is called a general history' of the country. The history of Zetland, however, is just the history of Orkney; and, having formerly * given our readers a short view of these edifying annals, we need not detain them long, either with the descriptive or the historical part of this book. The Zetland islands exceed one hundred in number; but thirty-four oniv are inhabited. The coast is rocky, and much indented. The hills are bleak and mossy: the highest is Rona's hill. Dr Traill informs us, that the barometer' indicated its height to be about 1400 feet. The Statistical Account says, that it was found, by geometrical mensuration,' to rise 3944 feet above the level of the sea. Dr Edmondston animadverts on these discordant statements, and proposes to settle the point, by a criterion which no philosopher, we presume, ever thought of,-his own bodily feelings. Were I,' he observes, to judge from my experience of the effects produced in similar situations, I should be disposed to believe that it does not exceed 2000 feet, if indeed it be so much. I. 5. The Zetland spring does not commence till April; there is little warmth till the middle of June; and the summer generally terminates with August. Nothing can equal the uncertainty of the weather' during the winter months. The medium temperature of winter may be taken at 38°; of summer at 65°. The islands have been represented as dismally dark in the winter sea son. But the Doctor repels this imputation with becoming spirit; and assures us, that even at the worst period of the year, they have a suit of daylight from 17 minutes past nine o'clock in the merring to 40 minutes past two in the afternoon ! As to the earliest inhabitants of Zetland, the author concurs with Mr Pinkerton in thinking, that they must have been Picts from Scandinavia. D. Barry conjectures, that these early navi gators went first to Scotland, and returned afterwards to these northern ishads; but Dr Edmondston is positive that the said Picts touched at Zetland and Orkney, and peopled these islands. in their way to Scotland. These Piks, or Peti, were, according to ir Pinkerton's exposition of a morkish legend, supplied from Ir. land with priests, or, as our author styles them, a species of clergy, called 'ape. * Vol. VIII. p. 93. et seqq. Harold |