the 26th of September, after an absence of two years, seven months and a half. Lieutenant Back, who had been left in charge of the remainder of the party at Fort Franklin, travelled in the spring and summer by the usual route to York Factory, on Hudson's Bay, and reached Portsmouth on the 10th of October.

We shall now briefly notice the result of these expeditions: On the 18th of August, the day that Captain Franklin commenced his retreat from Beechy Point to the mouth of the Mackenzie, the party in the ship Blossom, under Captain Beechy, had passed Behring's Straits, and were then off Icy Cape, and on that very day, Mr. Elson, the master, with the barge, had quitted the ship, and were proceeding along the coast without interruption. It was not until the 22d, that he met any impediments from ice

"When he arrived off a very low sandy spit, beyond which, to the eastward, the coast formed a bay, with a more easterly trending than that on the west side; but it was so low that it could not be traced far, and became blended with the ice before it reached the horizon. It was found impossible to proceed round the spit, in consequence of the ice being grounded upon it, and extending to the horizon in every direction, except that by which the boat had advanced, and was so compact, that no openings were seen in any part of it. This point, which is the most northern part of the continent yet known, lies in latitude by meridian altitude of the sun, 71° 23' 39' N.; and is situated one hundred and twenty miles beyond Icy Cape. Between these two stations, and, indeed, to the southward of the latter, the coast is very flat, abounding in lakes and rivers, which are too shallow to be entered by any thing but a baidar. The greater part of the coast is thickly inhabited by Esquimaux, who have thin, winter habitations close to the beach." p.143.

The barge did not turn back before the 25th. From Beechy Point to the point where the Blossom's boat was detained, is but one hundred and sixty miles. If any provision had been made for an expedition overland, after the boats had proceeded as far as practicable, there might, during this summer, have been no difficulty in succeeding. For the party in the Blossom's boat did not commence their retreat until seven days after Captain Franklin had abandoned the expedition on his side, and might have continued somewhat longer on the coast. But the best arrangement would probably be the one we have formerly suggested. For the summers of this northern region, as every where else, vary greatly in temperature. Capt. Franklin found a considerable difference between the seasons of 1821 and 1826. The prevailing winds were different. The continuance

of fogs was much more obstinate during the latter summer than in the former.

"We were only detained three times in navigating along the coast that year (1821) to the east of the Coppermine River; but on this voyage, (says Captain Franklin) hardly a day passed that the atmosphere was not, at some time, so foggy as to hide every object more distant than four or five miles." p. 138.

"Could I have known, (says Captain Franklin) or by possibility have imagined that a party from the Blossom had been at the distance of only one hundred and sixty miles from me, no difficulties, dangers or discouraging circumstances, should have prevailed on me to return. It is useless now to speculate on the probable result of a proceeding which did not take place; but I may observe that, had we gone forward as son as the weather permitted, namely, on the 18th, it is scarcely possible that any change of circumstances could have enabled us to overtake the Blossom's barge." p. 145.

After Captain Franklin's return to England, he was informed by Captain Beechy, that the summer of 1827, was still less favourable to the navigation of those seas than that of 1826, the boats not being able to proceed within one hundred miles of the point they had reached the preceding season.

We will close this article with the observations which Capt. Franklin makes on the subject of a North-West Passage.

"The Northern Coast of America has now been actually surveyed from the meridian of 109° to 149° west; and from Icy Cape eastward to about 156° west, leaving not more than fifty leagues of unsurveyed coast, between Point Turnagain and Icy Cape. Further, the delineation of the west side of Melville Peninsula, in the chart of Capt. Parry's second voyage, conjoined with information which we obtained from the Northern Indians, fairly warrants the conclusion, that the coast preserves an easterly direction from Point Turnagain towards Repulse Bay; and that, in all probability, there are no insurmountable obstacles between this part of the Polar Sea and the extensive openings into the Atlantic, through Prince Regent Inlet and the Strait of the Fury and the Hecla.

"Whenever it may be considered desirable to complete the delineation of the coast of the American Continent, I conceive that another attempt should be made to connect Point Turnagain with the important discoveries of Captain Parry, by renewing the expedition which was undertaken by Captain Lyon, and which, but for the boisterous weather that disabled the Griper, must have long since repaid his well-known zeal and enterprize with discoveries of very great interest.

"In considering the best means of effecting the North-West Passage in a ship, it has hitherto been impossible not to assent to the opinion so judiciously formed, and so convincingly stated by Captain Parry, that the attempt should be made from the Atlantic rather than by Behring's

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, M. Lacroix 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.

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