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Again, if CF (c.) Periphery C: Perimeter F:: Per. F: Per. F'; whence, Per. C: Per. F':: Per.2 C: Per.2 F;

But

Per. C.

Per. F';

Therefore, Per. C Per. F;

And conseq. Per. C <Per. F.

Thus it appears that a circle has a periphery smaller than any regular figure of the same surface; and conversely, that a circle is greater than any regular figure of the same perimeter whence it follows from prop. (a) that a circle has a larger surface than any rectilinear figure of an equal perimeter.

Considering the proposition under the general enunciation of Professor Leslie, as applicable to all plane figures, his demonstration is totally irrelevant; and we know of none equal to T. Simpson's. But the one we have demonstrated is that which our author meant to establish. From the preceding induction of particulars, it is obvious that he flattered himself greatly, when he said respecting the subject of isoperimeters, I have treated it with the conciseness of the moderns, without departing, I hope, from the spirit of the ancient geometry.' (p. 471.)

We have so far exceeded the limits to which we wished to confine this article, that we must be very brief in our animadversions upon Mr. Leslie's Trigonometry. It exhibits his usual affectation and inaccuracy, and his usual superficial manner of discussing a topic. More instances of affectation we have not room for. As to inaccuracy, we may state that the demonstrations to the first six propositions are ALL unsatisfactory; they only apply to the case where the arcs in question, their sums, or differences, are less than a quadrant! With respect to deficiencies, we shall only specify here the treatment of one proposition, viz. the mutual distances of three remote objects being given, with the angles which they subtend at a station in the same plane, to find the relative place of that station. This problem, as originally proposed by Mr. Townley, in the Philosophical Transactions (Vol. I. p. 563, New Abridgment), is divided into six cases, of all which the consideration is absolutely necessary to its complete solution. Mr. Leslie presents his readers with one only: even of this, he has not given the analytical solution, much as its result simplifies the labour of computation. Let A, B, C, be the three points whose mutual distances are given. D the station at which the angles ADB, BDC, are taken, and 360°-(BDA+CDA+BAC.) AB sin CDA

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Then, cot ABD=cot AC sin BCA cos

+ 1):

whence ABD is known, and thence ACD, and the distance AD.

We have now reached the end of a task which has been more 6

VOL. IV. NO. VII.

than usually tedious and ungrateful. The Professor's book has greatly disappointed us: for, although our opinion of his talents was by no means so exalted as that of many of our contemporaries, yet we confidently expected, that in a work of which the topics (though connected) are so multifarious, he would have found one, at least, adapted to his abilities, and have treated it with elegance and perspicuity. But this is not the case. Mr. Leslie appears to have read much, thought little, and invented still less. In an elementary work where many subjects are introduced into a single volume, we do not imagine that every one of them is to be exhausted; but we certainly think that the most valuable particulars should be selected and exhibited with simplicity and perspicuity. Instead of which, the performance before us wants much that is important, contains not a little that is useless, and teems more with pedantry, error, and absurdity, than any book of geometry we have ever seen. If such a geometer, as this performance compels us to think Mr. Leslie, be really an honour to his country,' the descendants of Simson and Stewart must have miserably degenerated, and Scotland, whatever else may be its claims to distinction, can boast no extraordinary pretensions to mathematical knowledge.

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ART. III. Voyage de Découvertes aux Terres Australes, exéordre de sa Majesté L'Empereur et Roi, sur les Corvettes Le Géographe, Le Naturaliste, et la Goëlette Le Casuarina, pendant les Années 1800, 1801, 1802, 1803, et 1804, publié par Décret Impérial, et Rédigé, par M. F. Péron, Naturaliste, &c. &c. 4to. Tome premier, avec Atlas. A Paris. 1807.

FEW months after the retirement of Mr. Pitt, and the suc

resident commissary for French prisoners of war, addressed an application to the Lords of the Admiralty, to obtain the necessary passports for two armed vessels, Le Géographe and Le Naturaliste, which the French government had appointed for a voyage of discovery round the world, 'pour mettre le Capitaine Baudin à l'abri de toute attaque hostile, et lui procurer une reception favorable dans les établissemens Britanniques où il pourra être obligé de relâcher momentanément.' In consequence of this application, the good natured Minister, without farther inquiry into the tenor of

Captain Baudin's instructions, or the particular object of his mission, obtained his Majesty's commands that the French vessels 'should be permitted to put into any of his Majesty's ports, in case of stress of weather, or to procure assistance, if necessary, to enable them to prosecute their voyage.'

The perusal of M. Péron's book has convinced us that M. Otto's application was grounded on false pretences, and that the passport was fraudulently obtained; that there never was any intention to send these vessels on a voyage of discovery round the world, as stated by M. Otto, but that the sole object of it was to ascertain the real state of New Holland; to discover what our colonists were doing, and what was left for the French to do, on this great continent, in the event of a peace; to find some port in the neighbourhood of our settlements, which should be to them what Pondicherry was to Hindostan; to rear the standard of Buonaparte, then First Consul, on the first convenient spot; and, finally, that the only circumnavigation intended in this voyage d'espionage, was that of Australia.

If any doubt could be entertained that such was the sole intention of the French government, the heads of Captain Baudin's instructions, as stated by M. Péron, and indeed the whole proceedings of the voyage, are amply sufficient to set this point at rest. By these instructions, they were directed to touch, in the first instance, at the Isle of France; thence to proceed to the southern extremity of Van Dieman's land, visit Dentrecasteaux's channel, examine the eastern coast, enter the strait of Bass, through that of Banks, complete the discovery of Hunter's islands, examine the south-west coast of New Holland, penetrate behind the islands of St. Peter and St Francis, and visit that part of the continent concealed by those islands, where a strait was supposed to exist by which a communication was opened with the great gulph of Carpentaria. This being accomplished, they were to direct their course to cape Leuwen, examine the unknown parts of the coast to the northward, visit the coasts of the land of Edels and Endracht, make a particular survey of the island of Rottenest and Shark's bay, terminating their first campaign at the N. W. cape of New Holland.

From Timor, or Amboyna, (at one of which places they were to winter,) they were directed to proceed through Endeavour strait, to the eastern point of the great gulf of Carpentaria, to examine the whole circuit of its coast to the land of Arnheim, terminating the second campaign at the same north-west cape at which their first was completed; from hence they were to cross the Indian ocean to the Isle of France, and make the best of their way to Europe.

So much for this voyage of discovery round the world, of which

M. Péron has been employed to write the history. The perusal of his book has certainly afforded us considerable pleasure, although in the course of our examination of it we shall feel ourselves called upon to reprobate, in the strongest manner, the mean and illiberal conduct into which he must have been betrayed by superior influence. Of M. Péron, as a man of general science, we are disposed to think highly; but we repeat, that in the publication of the work before us, we do not and cannot consider him as a free agent. It is brought forward, in the first place, under the immediate sanction of Buonaparte, in consequence of a report of the Imperial Institute, which states,

That more than one hundred thousand specimens of animals, great and small, compose the zoological collection, and that the number of new species, according to the report of the professors of the Museum, amounts to above two thousand five hundred.-When it is recollected,' continues the reporter, that the second voyage of Cook, the most brilliant in this respect which has ever been made to this day, did not furnish more than two hundred and fifty new species, and that the combined voyages of Carteret, Wallis, Furneaux, Meares, and Vancouver, have not altogether produced so great a number; when it is observed that the case is the same with regard to all the French expeditions, it will follow, that M M. Péron and Lesueur alone have discovered more new animals than all the natural historians who have travelled in these latter times.

As a reward for this great exertion, the Institute accordingly resolves,

1. That the class should testify, in an authentic manner, how much it is satisfied with the zeal shewn by M. Péron to fulfil the functions with which he was charged.

2.That it should declare to government that he is deserving of those rewards usually granted to naturalists who travel; and that the works preparing by him must contribute to the progress of natural science.'

The reward, we are told in a note, was an order of Buonaparte for his works to be published at the expense of government.

Before we proceed to the examination of M. Péron's book, we shall extract one part of the report of the Institute, which, from the importance it attaches to the nascent colony in New South Wales, tends to corroborate what we have stated in regard to the real object of the voyage.

"In the midst of the regions which he has traversed, M. Péron has every where encountered the rivals of his country; in every place they have formed establishments which excite the greatest interest, of which we have hitherto in Europe received but imperfect and invariably false information. M. Péron has applied himself particularly to compre

hend, in detail, this vast system of colonization in Australia, which is exhibited at the same time on a great continent, and over an immense ocean. You will be enabled to observe, by that part only of his memoir on the seal-fishery, how far his researches on this subject are of importance, and with what sagacity the author of it has been able to develope them. His labours, in this respect, appear worthy, in every point of view, of the attention of the philosopher and the statesman. Never perhaps did a subject of greater interest or curiosity offer itself to their contemplation. Never perhaps was a more striking example afforded of the omnipotence of laws and institutions on the character of individuals and nations. To transform the most formidable robbers, and the most abandoned thieves of England, into honest and peaceable citizens, and into industrious planters; to operate the same revolution among the vilest prostitutes; to compel them, by infallible means, to become virtuous wives, and exemplary mothers; to bring under subordination and control a nascent population; to preserve it by assiduous care from the contagious example of its parents; and thus to lay the groundwork of a race more virtuous than that which at present exists-such is the affecting picture that the new English colonies present. But the statesman, in the very constitution of this new empire, and in the detail of its organization, too surely discovers the real views of the founder, and the formidable germ of those revolutions which must of necessity be produced.' Page 12.

6

This Voyage Historique' commences with observing that the efforts which England has made in scientific discoveries have been peculiarly distinguished in these latter times; and that in this glorious struggle among nations for promoting science, France alone has been able to dispute with advantage her superiority and her triumphs that, notwithstanding this, the numbers of enlightened Englishmen, placed on the immense theatre of a fifth part of the globe, might, perhaps, decide the opinion of Europe in favour of their country; that the national honour of France therefore called for an expedition of discovery to the South Seas, and that the Institute felt it a duty to propose the measure to gov

ernment.

The war at this period appeared to have redoubled its fury; the political existence of France was menaced; its territory was invadedbut Buonaparte was now First Consul. He received with eagerness the proposition of the Institute, which for many years before had to boast of his name in the list of its members. And at the very time when the army of reserve was put in motion to cross the Alps, he issued his orders to hasten the execution of this grand enterprise. an instant, three and twenty persons, nominated by him, on the presentation of the Institute, were destined for making scientific researches. Never was a display so considerable given to this department of a voyage of discovery; never were means so amply prepared for securing success! Astronomers, geographers, mineralogists, bo

In

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