Straits, because the enterprize is then commenced after a voyage of short duration, subject to comparatively few vicissitudes of climate, and with the equipments thoroughly effective, But important as these advantages are, they may, perhaps, be more than balanced by some circumstances which have been brought to light by our expedition. The prevalence of north-west winds during the season that the ice is in the most favourable state for navigation, would greatly facilitate the voyage of a ship to the eastward, whilst it would be equally adverse to her progress in the opposite direction. It is also well known, that the coast westward of the Mackenzie is almost unapproachable by ships, and it would, therefore, be very desirable to get over that part of the voyage in the first season. Though we did not observe any such easterly current as was found by Captain Parry in the Fury and Hecla Strait, as well as by Captain Kotzebue, on his voyage through Behring's Straits; yet this may have arisen from our having been confined to the navigation of the flats close to the shore; but if such a current does exist throughout the Polar Sea, it is evident that it would materially assist a ship commencing the undertaking from the Pacific. and keeping in the deep water, which would, no doubt, be found at a moderate distance from the shore.

"The closeness and quantity of the ice in the Polar Seas vary much in different years; but, should it be in the same state that we found it, I would not recommend a ship's leaving Icy Cape earlier than the middle of August, for after that period the ice was not only broken up within the sphere of our vision, but a heavy swell rolling from the northward, indicated a sea unsheltered by islands, and not much encumbered by ice. By quitting Icy Cape at the time specified, I should confidently hope to reach a secure wintering place to the eastward of Cape Bathurst, in the direct route to the Dolphin and Union Straits, through which I should proceed. If either, or both of the plans which I have suggested be adopted, it would add to the confidence and safety of those who undertake them, if one or two depots of provisions were established in places of ready access, through the medium of the Hudson's Bay Company." pp. 260-261.

ART. II.-1. An Elementary Treatise on Plane and Spherical Trigonometry, and on the application of Algebra to Geometry; from the Mathematics of Lacroix and Bezout. Translated from the French for the use of the Students of the University at Cambridge, New-England. Cambridge, N. E. At the University Press, 1820. pp. 162.

2. Essai de Géométrie Analytique appliquée aux courbes et aux surfaces du second ordre. Par J. B. BIOT. Sixième edition. Paris, 1823. pp. 447.

3. Application de l'Algèbre a la Géométrie. Par M. BOURDON, Chevalier, &c. Paris, 1825. pp. 624.

THE first of the volumes whose titles have been given above, is the fifth of a course of pure and applied mathematics, prepared by Professor Farrar, for the use of the University of Cambridge. The entire course consists of no less than eleven volumes, and is made up of translations from Lacroix, Euler, Legendre, Bezout, Francœur, and Biot. Occasional use is also made of the labours of Cagnoli, Bonnycastle, Puissant, Leslie, Poisson and Delambre. From this array of illustrious names, it is manifest, that the materials of the course are of the first order.

We take a strong interest in whatever pertains to mathematical learning, and we are convinced that the labours of Professor Farrar, considering his connexion with the oldest and best endowed of our colleges, will have an important influence on the fate of the exact sciences in this country. We therefore, propose in this, and if circumstances permit, in some of our future numbers, to examine the claims presented by this work to the public attention and confidence. The first four volumes have been examined by a cotemporary journal which enjoys an extensive circulation. And as it has become matter of serious complaint, that our various journals are too much in the habit of taking their readers in succession over the same ground, we shall abstain from all observations upon them, except so far as it may be necessary to refer to them for the sake of illustration or comparison.

The fifth volume consists of two parts, by different authors; one, on the two trigonometries by Lacroix, the other, on the application of algebra to geometry, by Bezout. The part exe

* Silliman's Journal of Science and Arts, vols. v. & vi.

cuted by Lacroix, leaves nothing to be desired; but along with Bezout, we have associated the very late treatises of Biot and Bourdon, that the deficiencies of the former writer may be made more manifest, by being brought into contrast with the merits of the authors last mentioned.

The "Traité de Trigonométrie Rectiligne et Sphérique," of Lacroix, the translation of which will first receive our notice, is a part of an extensive course of mathematics in nine volumes, which has, during some years, been very much used in the highest of the French literary institutions. It is a rich treasure of mathematical truth, drawn up with great care and in a uniform style. He appears to have formed himself on the model of Euler, and is a disciple every way worthy of his celebrated master. He sometimes goes beyond Euler in profoundness and reach of thought; but on the other hand, he is sometimes inferior to him in clearness. Still there is much difference in respect to clearness, among the various parts of the very extensive works which he has produced. His "Trigonométrie" is certainly one of the most luminous treatises which have been written on any department of the mathematics.

Elementary geometry makes known three parts of a triangle by means of three others; but it does this by constructions, whose accuracy finds narrow limits in the imperfection of our senses and our instruments. Instead of these geometrical constructions, rectilineal trigonometry substitutes calculations that are susceptible of any degree of approximation, and it accomplishes this end by determining in a circle of a given radius, a series of right-angled triangles, comprising all possible acute angles, so that the series may always furnish one similar to that which it is proposed to resolve. After this, by simple proportions between the sides of these two triangles, we may find the unknown parts of the triangle to be resolved, by their corresponding parts in the similar triangle furnished by the calculated series. The resolution of oblique-angled triangles is easily derived from that of right-angled triangles, since the former may always be resolved into the latter, and, therefore, every thing depends on the construction of the tables which contain the values of the parts of right-angled triangles. Accordingly, MLacroix has made it his first object to show how these tables may be constructed.

With this view, after giving definitions of the principal linearangular quantities, sine, cosine, tangent, cotangent, secant, &c. he proceeds to demonstrate the principal relations of these lines

* Essais sur l'Enseignement, p. 331.

to each other, and to show that it is only necessary to determine the sines, as the corresponding values of all the other lines which enter into the construction of the table, may be immediately deduced from them. He then, by an elegant and very luminous construction, obtains the well-known fundamental expressions for the sine and cosine of the sum and difference of two arcs; sin. a cos. b + sin. b cos. a; and

Sin. (a+b)=sin

R.

Cos. (a+b)=cos. a cos. b sin. a sin. b

R.

These equations involve all the properties of sines and cosines, and he immediately goes on to apply them to the finding of expressions for the sines and cosines of double and triple arcs, and the process for finding these, shows the method of doing the same for any multiple arcs whatever. The equation 2 sin. a cos. a or the expression of the sine of a

R.

sin. 2 a = double arc, when that of the original arc is known, is made to lead to the formula, sin. a=+√2R.2 + 2 R. cos. a which gives the sine of half an arc, when that of the whole arc is known. The same formula is obtained by a construction, in which it is shewn, that the two values of the positive solution belong to two arcs which are mutually the supplements of each other. Here he introduces the important observation, that it is not the absolute value of the sines which we have occasion to calculate, but only their ratio to radius. This is equally true with respect to the other trigonometrical lines, and on this acaccount we have called them, after Carnot, by a term which best expresses their nature, to wit, linear-angular quantities. Having proved that the length of an arc is always less than that of its coresponding tangent, and greater than that of its sine, and that the ratio between the tangent and sine of an arc, tends continually towards unity as the arc diminishes, it is inferred, that if the value of the tangent and that of the sine do not differ for a certain number of figures in a decimal series, these same figures may be taken as the value of the arc, sufficiently approximate. Taking, therefore, as an example of his reasoning, a sine which is only 0, 0001 part of radius, he calculates by means of the cosine, the corresponding tangent, and finds it decimally expressed only 0,0001000000005, which does not differ from the assumed sine except in the thirteenth figure, and this may evidently be taken as a sufficiently approximate value of the arc expressed in parts of radius or unity. In this connexion, he takes occasion to explain the sexagesimal and centesimal di

vision of the circle. The latter is a part of the celebrated French system of weights and measures, which is the only system founded on strictly scientific principles, all the parts of which have fixed relations to each other and to a common standard.

I 1

1 16'

Beginning with an entire quadrant, the formula, sin. a = √2 R2 − 2 R cos. a gives the sine of half of it, then that of a quarter, and thus in succession, of all the fractions of this arc comprised in the series, 4, 4, 1, 7', ', ', &c. At each term of this series, the cosine as well as the sine of the corresponding arc, is required to be calculated, and the approximation is carried to twelve decimal places, a degree of accuracy which we consider unnecessary in an elementary treatise, in which the object. rather is to show how trigonometrical tables may be constructed, than actually to construct them. At the fourteenth division in the series, we come to an arc which is only of a quad

rant, the size of which is 0,000095873799, less than 0,0001; consequently, as the sine 0,0001 was shown before not to differ from its corresponding tangent in the first twelve decimal figures, à fortiori, the sine 0,000095873799, does not differ from its corresponding arc in the same number of figures. Now it is evident, that all arcs which are confounded with their sines and tangents in the first twelve decimal places, may without sensible error, be considered proportional to these lines; whence, by a proportion founded on this principle, the sine of 1000 of a quadrant, or 0,000015707963 is obtained. Setting out with this, as the aproximate value of the sine of the smallest arc to which it is proposed to extend the calculation, the sines of the multiples of this arc are obtained by multiplying this sine by 1,2,3, &c. This simple method may be pursued as long as the arc has its sine and tangent confounded in the first twelve decimal figures, that is, as far as of a quadrant. If we are contented with approximate values to the eighth decimal only, we may extend this process to the of a quadrant. This method, which is only applied in the original to the centesimal division of the circle, is also applied by the translator to the sexagesimal division, in which he first finds the approximate sine of I", and from this commences the series for multiple

arcs.

100

1

000

1000

To calculate the sines and cosines of arcs greater than 1 of a quadrant, we use the equations, sin. 2 a. = 2 sin. a. cos. a, cos. 2 a cos. a2 sin. a2, together with the fundamental equations, sin. (a+b) = sin. a cos. b + sin. b cos. a cos. (a+b), =cos. a cos. b + sin. a sin. b The first two give, sin.