or seven, did one feeble and sick man manage the affairs of a whole kingdom, and dispatch all business which related to foreign countries in the same time. The cabinet-secretaries then returned to their own houses at Potsdam, transcribed fairly the dictations of his Majesty, and brought the despatches back for his signature after dinner; Frederick making a point of reading himself every letter, and every order, before he would attach his signature to it. "I remember," says Count Hertzberg, "that a few days previous to his death, he prescribed to his aides-de-camp all the manoeuvres that were to be performed at the reviews in Silesia, adverting to the minutest circumstances of locality; and he sent also for General d'Anhalt, to direct some military arrangements for raising free battalions to expedite the movement of the army in case of a war. He also sent for the ministers of state, De Hoym and De Warder, to settle with them some new plans for the cultivation of land, and the improvement of manufactures, which he proposed to have executed in 1787 in different provinces ; and particularly to have new villages built, at his own expence, in all the districts where the husbandmen resided on plains of too great extent, and where the population appeared to be too scanty. On the 15th of August, 1786, he dictated and signed his despatches, as usual; and it was not till the 16th that he ceased to discharge the functions of a king and of a minister of state: but on that day he was entirely deprived of sense, and on the morning of the 17th he died, in the 75th year of his age, and the forty-seventh of his reign.

22

Frederick was a wit as well as a king; a poet as well as a philosopher; and a musician as well as a general. His talents, indeed, were of a high order; he had a taste for literature; and he courted the society of literary men, who flattered his vanity, and whom he bribed to trumpet forth his praises. Private individuals, however, and even princes, have possessed some, in various degrees, of these qualities and tastes, and perhaps all of them together, without obtaining that character for greatness which fell to the lot of Frederick. How happened this? He had a profound knowlege of the character and genius of the age in which he lived. This is the true solution of the enigma.

'Frederick never attempted to impose crude and ill-digested institutions, or any which were unsuited to the spirit of the times. He knew what that spirit required, and satisfied it. Thus, while innumerable armies assailed him, the ascendant genius of public opinion covered him with its buckler. If his legions were defeated and dispersed, or his treasures drained, his tutelary genius created new legions for him and supplied new treasures. He

Hh 4

might

might have had still more splendid qualities than those which he possessed: but, if he had attempted to impose on the age institutions which it was not prepared to receive, all his labours would have been fruitless. He effected great things with little means, while with greater means Charles the Fifth could not succeed in little measures: it was only on the verge of the grave, and when he could not synchronize the motions of two pendulums, that he perceived the folly of attempting to force on the age ideas which it was indisposed to receive. Joseph II., endowed with great qualities, and inspired with an ardent love of public good, after having spent his life in the most painful and useless efforts, left Belgium in open insurrection, Hungary in discontent, and the whole monarchy menaced with approaching troubles.

Erroneous judgments are frequently entertained concerning Frederick the Great; some persons imagining that his character would not have suited the times in which we live, while others maintain that he would have accomplished the greatest things in preserving the forms which he adopted. Both forget that in all ages the same elements constitute a great man, but that the first quality of superior genius is, not to endeavor to force our own views on the time in which we live, but to adapt our own views to those of the period. In the days of Gregory the Seventh, no Luther was to be found; in those of Luther, it would have been impossible even for a Gregory the Seventh. to have repressed the Reformation. In establishing order through the public finances, in the encouragement of agriculture and manufactures, in the suppression of arbitrary imposts and of the exercise of petty tyranny over his people by subordinate officers, and in raising the peasant and workman above the condition of mere brutes, Fre derick effected all that could be expected from him, and all that the spirit of the times would allow. Had he gone a step farther, he would have injured the existing edifice of social order. The wisdom of the statesman restrained the philosopher from carrying into execution all those projects which he had meditated in the retirement of his closet. A constitution, in which the people should participate in the management of public affairs, would, in his time, have been precocious.'

How much things have changed for the better in these enlightened times! Witness Italy, Portugal, and Spain !

Among the opuscula of Frederick which the present volume contains, we see nothing worthy of extended notice. The 'Theological Commentary of Don Calmet on Blue Beard' is a lively but profane jeu d'esprit; the liveliness of it consisting in the mock gravity with which the royal writer has quizzed the ponderous lucubrations of that learned Benedictine. The Account of the Siege of Asoph by the Russians, and of the Journey of Peter the Great to Holland,' was found among the King's papers at Potsdam, and will be consulted with advantage by the historian of the times.

ᎪᎡᎢ.

ART. III. Applications de Géométrie, &c.; i. e. Applications of Geometry and Mechanics to Marine Constructions, Bridges, Roads, &c. BY CHARLES DUPIN, Member of the Institute of France, &c. &c. 4to. pp. 360. Paris. 1822.

T

`HE name and reputation of M. DUPIN are both well known to the public. As a scientific traveller, and as an impartial historian and practical observer of the progressive improvements in the sciences and useful arts, he stands eminently conspicuous; and as a mathematician, although at present his labors in that department are not much known in this country, he occupies a rank equally respectable. The work which we now introduce to our readers is of considerable interest to naval, military, and civil engineers: but it is founded on a species of geometry (the descriptive) which is very little prosecuted among us. Descriptive geometry owes its birth to the inventive genius of Monge, and is perhaps capable of applications of which even its inventor had no adequate conception. As an élève of the polytechnic school, M. DUPIN had the advantage of studying under that able and distinguished philosopher; and he appears to have imbibed the pure geometrical spirit of his master, of whom, in his "Essais sur les Services et les Travaux Scientifiques de Gaspard Monge," he speaks in the highest terms of regard and respect.

M. DUPIN's first efforts in this department were made in five memoirs, intitled Dévellopements de Géométrie," presented to the class of the Institute previously to his being elected a member of that learned body; and which were approved by the commissioners, MM. Carnot, Monge, and Poisson, who were appointed to examine and report on them. They were afterward published in a quarto volume under the above title; and the present work, which treats of the application of the principles established in the former, may be considered as a continuation of it, and both as constituting part of a series of which the "Géométrie Descriptive et Analytique" of Monge is the first term.

The subjects in the volume now before us may be divided under five distinct heads; viz. 1. On the Stability of floating Bodies; 2. On the tracing of Roads for military and other Purposes, over Mountains and across mountainous Districts; 3. On the Deblai and Remblai, a Problem connected with military Constructions; the Earth removed, or excavated, being the Deblai, and the Mount, Rampart, &c. formed with it, the Remblai; 4. On the Track followed by a luminous Ray, and by elastic Bodies in general, in the Phænomena of Reflection and Refraction; and, lastly, a theoretical Examination of the Structure of English Ships of War.

It is difficult to give in a general form, without diagrams, any very explicit idea of a work which, from the beginning to the end, is dependent on figures and formulæ for illustration. We can therefore only attempt a very superficial sketch of its contents, and must refer our readers to the volume itself, if they are desirous of comprehending more distinctly the refined geometrical process on which the author founds his investigations.

The theory of floating bodies, their stability, instability, &c., are points of great importance in naval architecture, and have accordingly engaged the attention of several able mathematicians, among whom Bouguer and Euler stand most eminent. M. DuPIN, however, has taken an entirely new view of this subject, and his method of investigation is totally dif ferent from those of the able geometricians above named. Indeed, the theory which he pursues, and the geometry on which his investigations are founded, were altogether unknown in their time; and with this new instrument he has been led to new results, and has given additional extent to this interesting branch of hydrodynamics. He considers under one general point of view all the positions that a body can take while floating on the same fluid; its weight and form being supposed constant. The part of the floating body which is immersed in the fluid being termed the carène, we know that in order that the body itself may be in equilibrio, it is necessary that the centre of volume of this carène, and the centre of gravity of the body, must be in the same vertical; at the same time, the weight of the body being supposed constant, the mass or volume of the carène will be constant also but its figure will change, as well as the plane of floatation, as we vary the position of the several masses in the anterior of the body, so as to alter the position of the common centre of gravity; and consequently we shall find, for different states of the same body, an infinite number of planes of floatation, and a corresponding number of centres of carène : which centres connected form a surface which the author denominates the surface of the centres of carène. Again; all the planes of floatation are tangents to another surface, which, with reference to these planes, is of the same kind as those which M. Monge, in his Descriptive Geometry, calls enveloppes, and which therefore M. DUPIN denominates la surface enveloppe des flottaisons.

Our readers will conceive, from what is above stated, an idea of the great generality which the author thus introduces into this problem; and they will see also the impracticability of our giving any intelligible abstract of a solution so gener

alized,

6

alized, without the aid of diagrams, and will therefore require no apology from us for leaving the process thus imperfectly explained. It is indeed impossible to comprehend the process of investigation without consulting the book itself; the utmost that we can do is to state one or two of the conclusions; such, for example, as the following. The author having shewn that, if we place the floating body in a position of equilibrium, the centre of its carène will be in a certain point on the surface, which is the locus of the centres of carène; and that the plane tangent to the surface is necessarily parallel to the plane of floatation; he thence easily deduces, That, in any position of equilibrium, the right line drawn through the centre of gravity of the floating body, and through the centre of carène, is perpendicular in this point to the surface of the centres of carène. Hence he transfers the problem of finding the positions of equilibrium to the determination of right normals to the surface of the centre of carène, selecting among these normals those only which pass through the centre of gravity of the body. The positions of equilibrium, however, may be of three kinds stable, unstable, and mixed; and on this point M. DUPIN deduces a remarkably neat and curious result; viz. according as the position of a floating body is stable or unstable, the distance of the centre of gravity of this body from the centre of its carène is a minimum or a maximum, with respect to all the positions (indefinitely near to them) that the body can take in the fluid.'

We might select a variety of other theorems and corollaries, which flow with the greatest simplicity from the general principles first established: but we doubt much whether we could render them very intelligible, without extending our remarks beyond our proper limits.

The two following memoirs, on the tracing of roads and on the problem of Deblai and Remblai, terms employed equally by French and English military engineers, are treated in the same general way as the first, and equally bid defiance to perspicuous abridgment. The nature of the inquiry, however, is sufficiently obvious. In carrying a road over a mountainous district, we cannot follow the line which we should naturally adopt on a level plane, or on one slightly inclined: for the slope of the hill may in many cases be such as to be wholly impassable; and in others, although it be not absolutely so, yet to persevere in a direct ascent would be to throw away much time, and to exhaust the strength of the men or animals that are required to pass it. Our object, therefore, in such cases ought to be to proceed by such a degree of inclination that, without lengthening our road too much, we

may