plorers, Sir W. Hamilton, M. Ferber, Mr. Brydone, Spalanzanni, &c. may juftly give cause of apprehenfion to the inhabitants of the environs of Mount Mifery, as well as to thofe of the other volcanic iflands for as a period of feveral hundred years elapfed from the earliest tradition of an eruption of the former, till the famed one which deprived the elder Pliny of his life; and from this to the second on record; and as long intervals occurred between the eruptions of the latter, although from the veftiges of thofe which happened beyond the record of hiftory, had induced the natural hiftorian of Etna to calculate the existence of the world from a wonderful antiquity, (Brydone, let. vii.) may not the feemingly extinguished volcanoes of the western archipelago, when leaft feared, fuddenly burft their prefent bounds, and cover their vicinity with horror and déftruction. Montferrat, I am told, exhibited an alarming fpecimen about the beginning of this century, of what may be expected; Guadaloupe has very recently been diftinguished by the activity of its apparently extin guilhed volcano (Rapport fait aux Citoyens Victor Hugues et Lebas fur la fituation du volcan du Guadaloupe, 1798); and the inceffant earthquakes which terrified and endangered the lives of the inhabitants of St. Chriftopher's, during the whole of the year 1797, give folidity to apprchenfion. Heaven avert the evil! The form and ftructure of Brimstone Hill fanction the opinion however. In a curious map of this island prefixed to Du Tertre's Hiftoire Generale, and published in 1642, this hill is diftinguished by the name of " Mine de Souffre," It is nearly a truncated cone, terminating in two peaks, compofed of the most fingular congeries of different bodies, we can well imagine. Volcanic athes, confolidated by time into an immenfe calcareous mafs, form the bafis of this hill; but there are innumerable ftrata of fhells, of gravel, of pyrites, of lava, of pumice ftone, interpofed, and together with immenfe blocks of blue granite, and of argillaceous rock, evidently prove its volcanic nature and origin: and were more proofs wanting, the vicinity of the perfect crater of Mount Mifery, but more efpecially the exudation of fulphur from the hill itfelf, as well as the fulphureous fpiracula in thofe places where deep excavations have been made for the foundations of buildings, together with the exceffive heat of thofe fpots where fpiracles have been difcovered, would be decifive. Its pofition, with refpect to the adjacent heights, has fecuted to it a preference as a poft of defence; and it conftitutes the principal fortrefs of the island. It is dry and hot, notwithstanding the perflation of the trade wind; but it is also remarkably healthy, for the 9th regiment, ftationed on it from the year 1786 till the year 1794, loft no more than three men before the malignant peftilential fever appeared among them in July, 1793.

"In an island fo remarkable for the purity of its atmofphere, dif eafes of importance proceeding from endemic caufes, are not to be looked for. The yellow remittent fever feldom appears any where but in Baffeterre, during July. Auguft, and September; and then its violence is comparatively trifling. Simple remittents fometimes appear during the fame months; but intermittents, and difeafes depending on topical inflammation, fuch as hepatitis, are never met with," Vol. ii, P. 285.

Similar

Similar accounts are given of the rest of the islands, and of the interefting fettlement at Demerary, on the continent of South America. At the end of the fecond volume, we have an account of experiments with the nitric acid, and with the oxygenated muriate of pot-afh, in the yaws, leprofy, venereal, and other complaints; but we have already extended our view of the work beyond our ufual limits, and muft therefore refer our readers for these accounts to the volume.

ART. VI. A Treatise on Plane and Spherical Trigonometry with an Introduction, explaining the Nature and Ufe of Logarithms; adapted to the Uje of Students in Philofophy. By the Rev. S. Vince, A. M. F. R. S. Plumian Profeffer of Aftronomy and Experimental Philofophy in the University of Cambridge. 8vo. 4s. Deighton and Nicholfon, Cambridge; Lunn, London.

ASTRONOMY requiring the calculation of triangles from

various data, the progrefs of that science muft very soon have given rife to Trigonometry. The celebrated astronomer, Hipparchus, wrote twelve books on the Chords of circular Arcs, having relation to Trigonometry: and, foon after, Menelaus wrote fix books upon the fame subject, with three books upon Spherical Trigonometry; the latter only of which we now poilefs. The various calculations in Trigonometry were, at firit, performed by the Chords of Arcs; and this method was ufed till about the eighth or ninth century, when the Chords were changed into Sines by the Arabians, who introduced three or four new Theorems, which are now in ufe. In 1553. Erafmus Reinhold, Profeffor of Mathematics at Wirtemburg, published a Table of Tangents. Not long after this, Vieta published a Table of Sines, Tangents, and Secants, to every minute of the quadrant. But to avoid the trouble of long multiplications and divifions in the operations of this fcience, John Napier, Baron of Marchifton in Scotland, invented a fet of artificial numbers, called Logarithms, by means of which, multiplication and divifion are performed by addition and fubtraction, and thus the computations become very fimple and eafy. But the Scale of Logarithms, as computed by Napier, was changed to another more convenient, by Briggs, who made o the logarithm of 1; I the logarithm of 10; 2 the logarithm of 100; 3 the logarithm of 1000; and so on: and thefe are the Logarithms now in use.

In

In the Treatife of Trigonometry before us, the author has, very properly, begun with explaining the nature and use of Logarithms. He obferves, that the Logarithms of num bers below unity, may be written either with a negative index, or by adding 10 or 100 to it, fo as to keep the index always pofitive and here the Profeffor makes the following remark:

The negative index, however, is that which ftands in the regular fcale of Logarithms, and always reprefents the true Logarithm of a decimal, and of that one number only; whereas the Logarithms of a decimal, exp effed by adding 10 or 100 to the index, is 10 or 100 too great, and expreffes alfo the Logarithm of a number greater than unity: thus, 4,0827972 is the Logarithm of 76348; and confidering it as the Logarithm of a decimal, having 10 added to the index, it is also the Logarithm of 0,0000076348. By uting the negative index, there is no danger of a mistake, and every fource of error fhould be cut off; we fhall therefore derive all our conclufions in terms of the true Loga rithm."

In the utility of this, we agree entirely with the author.

He next explains the method of finding, from the Tables of Logarithms, the Logarithm of any number as far as eight figures, which is as far as the Tables will go; giving an example to each cafe: and he then goes on to how how the number may be found from any given Logarithm; exemplifying the different cafes, that the reader may be under no difficulty in applying the rules. After this, he explains the method of finding the Logarithm of a proper fraction, either by the negative index, or by adding 10 to the index.

When it is required to incorporate feveral Logarithms by addition and fubtraction, it will be more convenient to convert the fubtraction into an addition, for which purpose the arithmetic complement of the quantity to be fubtracted is taken; and this is found by writing down what the Logarithm wants of 10,0000000; for, as the author obferves,

"To add what a number wants of 10, muft evidently make a quantity greater by 10, than if you had fubtracted that number; for inftance, 14+6 is greater by 10 than 14-4. Subtracting therefore 10, after the addition of the arithmetic complement of a Logarithm, is the fame as fubtracting the Logarithm."

The decimal part of every Logarithm is naturally pofitive, even when the index is negative; but this author obferves,

"That it will be very often found convenient to change fuch a Logarithm into one which fhall have its decimal part alfo negative, and this he calls a negative Logarithm; this is done by fubtracting 1 from the Index, taking the arithmetic complement of the decimal part, and prefixing the fign-before the index (which otherwife ftands above it)

fo

fo as to effect the whole; for by this operation you increase the value of the index by unity, and diminish that of the decimal part by unity, and therefore the value of the fraction is not altered: thus, 3,5962748 =2,4038252."

The utility of this reduction appears afterwards in many inftances.

To a learner, difficulties frequently arife in the use of Logarithms, when the natural numbers become decimals; but this difficulty Mr. Vince avoids, by confidering the fignificant figures as whole numbers, and then making an allowance in the index for the number of decimals; and thus the computer has to take out the Logarithms of whole numbers only. His rule for multiplication is this. Let the product be a xbx c X, &c. containing n decimals in all the factors; and let A, B, C, &c. be the refpe&ive values of the factors, confidering the fignificant figures as whole numbers; then the Logarithm of a xbx c x, &c. = log. A + log. B+ log. C. +, &c. And tor divifion, he gives this rule. Let the value of

-n.

ax bxcx. &c.

be required, in which the dividend consxt xux, &c. tains n decimals, and the divifor contains m decimals and r factors; and let A, B, C, &c. S, T, U, &c. be the refpective values of a, b, c, &c. s, t, u, &c. confidering the fignificant figures as whole numbers; then the Logarithm of the quotiept = log. A log. B+ log. C +, &c. ar. co. log. Sar. co. log. Tar. co. log. U*+, &c. +m―n— 10 r. These rules are a confiderable improvement upon the common methods, tending greatly to facilitate, and render more certain, the operations.

The author next gives the rules for raifing powers and extraordinary roots; thefe he has inveftigated and expreffed fo clearly, that, with the examples annexed, we think there can now be no difficulty in making any fuch computations. He' then proceeds to a further exemplification of Logarithms in making various calculations.

The nature of Hyperbolic Logarithms, of Logistic Logarithms, and of Proportional Logarithms, are next explained, and their feveral particular ufes pointed out, with examples. Thus ends the Introduction, which we must confider as a very valuable addition to the work.

In this rule, p. 24, 1. 4, there is an erratum, not noticed by the author, where, for ar. co. U, read ar, co. log. U.

I

Plane

Plane Trigonometry becomes next in course the subject of confideration and in the definitions the author very properly obferves, that an Arc of 90° has not (according to the definition of those terms) either a tangent or a fecant. He inflances the abfurdity of the fuppofition, that such an Arc has a tangent or fecant, from a right-angled spherical triangle, where radius: cofine of the angle at the bafe :: tangent of the hypothenufe: tangent of the bafe; now when the base 90°, the hypothentic 90"; and therefore thefe Arcs being equal, if they have any tangen s, of whatever value they may be, they must be equal and erefore radius = cofine of the angle at the bafe, whatever that angle may be. This falfe conclufion arifes from the falfe fuppofition, that an Arc of 90° has a tangent. The amino: afterwards gives another inftance of a falfe conclufion arifing from the tame fuppofition.

=

In respect to the propofitions, the author appears to have selected a that are likely to be useful in philofophical or any other quiries. He has clearly pointed out the ambiguous cafes, and where there can be any poffible difficulty, he has fhown how the rules for computation are to be adapted to a logarithmic pe adion: and here he obferves, that if an Arc be found in terms of its cofine, and the Arc be very finall, or near 180°, the varia ion of the cofine will be fo fmall, that it will not vary for many feconds. Thus, if the log. cofine came out 9,9999998*, then in the tables this is the cofine of an Arc from 2' 52" to 3. 41"; here is therefore an Arc for 49", which has the fame coline in the tables, owing to their being continued to feven decimals only; it is impoffible therefore to say what Arc from 2′. 52" to 3" 41", we are here to take. In fuch cafes, it is here obferved, that the expreflion must be transformed into one where the fine enters inftead of the cofine. In like manner, if an Arc be near 90°, and be exprelled by the fine, the expreffion must be changed into one where the cofine enters.

The principles here delivered are next applied to find the heights of objects; to carry on a menfuration of a country by a furies of triangles; to find the length of an Arc of the meridian, &c. after which, examples are given of computing all the different cafes of triangles, by Logarithms. To this is added an Appendix, fhowing how to find the powers of the fine and coline of an Are; to conftruct a table of fines, cofines, &c. to exprefs the fine and cofine of an Arc in terms of the

*

This by miitake is printed 9,999998. See p. 72, line 4th from the bottom.

3

impoffible