severe reasoning, nor inured to the rigour of strict mathematical demonstration : but men possessing what is called natural sagacity, or a knowlege acquired irregularly, and with little formal study; who had not erected in their minds a gradual, regulat, and connected system of truth, but had taken hints from common objects, and bad derived their inventions from the forms of trees and the contrivances of animals. To prove à priori that the most able geometricians are not likely either to invent or improve what is uscrul in the arts, we must advert to the criterion of beauty in mathematical science. This criterion requires that the principles, on which the reasoning is to be founded, should be few in number, and the most simple and obvious truths; that the demonstration should be connected throughout ; that nothing should be assumed as axiomatical or evident, which admits of deduction from something more simple ; avd that the longest and most tedious operation is not to be shunned, in order to link together parts apparently but little disunited. Hence, he who loves to seek truth in abstract science combines slowly and scrupulously; examines an object on all its sides; makes only one step at a time; and makes that sure before he attempts a new one. To the inventions and improvement of machines, on the contrary, a different habitude of mind is requisite. Nature offers such a multiplicity and variety of circumstances, of which the separate and exact influence is unknown, that we cannot reason strictly from them, nor assign their precise .effect. We should combine more rapidly, and comprehend more largely, if we wonld add to the inventions of art. The mind must quit its slow, scrupulous, and sure operations, in order to make a large and sudden grasp. Although, however, men of speculative research have not always been what Bacon says they ought to be," the guardians of those stores from which men in active courses are furnished," yet, of late years, it must be observed, circumstances have determined them to apply their calculations to matters of practical - utility. The forms of lenses, on which the perfection of (telescopes depends, have been determined after an analytical investigation; the resistance and motion of Auids have been calculated; the figure of columns which support the greatest weight under a given volume has been assigned ; and navigation has been most essentially benefited by the methods which have been invented for determining the longitude. What degree of union has been already effected between science and experiment; what aid analytical research has given to mechanical contrivance, or may hereafter give; it is the 4 business business of M. PRONY's book now before us to shew. In his advertisement, he observes that the lines of demarcation between theory and practice begin to disappear; that it is necessary to hasten the epoch of their re- union, by making speculative and practical philosophers speak a common language; that the means of obtaining so desirable an end is to give to the former a taste for observation, and to make the latter sena sible of the dangers and inconveniences of mere triat. In thus making the procedures of the arts approach the sciences, and in diffusing over the fornier the light of the latter, the ob. jects of application which enrich the one will furnish matter to make the others more pure and perfect.' The first volume of this work is divided into five sections, which are preceded by a first part, containing the principles of mechanics : in the introduction to which is given the developement of those general notions on which the science of mechanics is founded. Here the author shews what are the phænomena which present themselves in considering motion, viz. the space and Time employed in describing it.. He remarks that, in a mathematical inquiry, the business is pot concerning the cause but the laws by which effects operate ; and that the determination of the measure of motion is an. affair of pure convention and commodiousness. · He then gives the true designation of the term velocity, and states that the theory of motion requires the dicussion of these two fundamental questions : ist, What is the velocity of a body at any instant of its motion; 2dly, What is the law of the variation of its motion. The first is thus determined: if, in an equation between the space and time, a curve be constructed to represent that equation, the time being computed on the axis of the abscissas, then the velocity for any instant will be equal to the corresponding ordinate divided by the subtangent. This method of representing the velocity, if thoroughly examined, is the same as the more common one, viz. v=- (« space, ť X time). The second question is reduced to this equation ® In the establishment of the laws of motion, M. Prony adopts the method given by D'Alembert in his Dynamique : which method is dependent on the principle of the sufficient reason of Leibnitz, which affirms a thing to be according to such a manner, because there is no reason why it should be according to any other. In the demonstration of the laws of impact, the author has also followed D'Alembert; whose method is grounded on this principle, that two equal bodies, meeting with equal velocities L13 in the direction of a line passing through their centre of gravity, will remain at rest after impact. This principle affords an easy solution of all cases in which the bodies are commensurable ; for the cases of incommensurable bodies, D'Alembert gave a particular demonstration *. The question concerning the forces vives, formerly discussed with so much zeal, is stated clearly and decided satisfactorily. The demonstration of the parallelogram of forces is taken from the Traité de Mecanique of the Abbé Marie. It has sufficient evidence when the two sides of a parallelogram represent uniform motions : but the case of the equilibrium of three forces, represented by the three sides of a triangle, requires a distinct proof, and cannot with rigor be deduced from the composition of motion. The demonstration of the lever is from Varignon, founded on the composition of forces: but it is faulty, inasmuch as it fails in the most simple case, viz. when the lever is straight, and the directions of the powers are parallel to each other : the objection is not removed by saying that parallel lines may be conceived to meet at an infinite distance, since on such an hypothesis no rigorous demonstration can be insti. tuted. The first part of volume I. is concluded by a recapitulation of the general principles of mechanics, which merits great encomium ; we know not where to point out a summary at once so luminous and so precise, so judiciously arranged and so logically consequent. The remarks of the author are not only apposite to the subject, but are largely impregnated with a philosophic spirit. In section I. on Statics, M. PRONY gives the ordinary theory of moments, and thence deduces the general conditions of equilibrium. The principles of this theory, the author observes, frequently lead in their application to long and tedious calculations. The principle of virtual velocity, devised by Le Grange, is more concise and easy in its application, and more extensive in its comprehension. Of this principle there is no direct and general demonstration, but its truth may be inferred from the exact conformity of its results with those which have been obtained by other methods. M. PRONY next proceeds to apply the general formulas of equilibrium to the determination of the centres of gravity; and to demonstrate the property of the centre of gravity discovered Instead of D'Alembert's demonstration, M. Prony proposes at the end of his second volume to give another; which, however, is delicient in mathematical rigour. by by Guildin, a Jesuit, and announced in a work intitled Centro Barica, published in 1640. The formulas deduced from the principle of the virtual ve• locities are applied to the six machines, the lever, pully, &c. The latter part of the section considers the equilibrium of arches. The author is of opinion that the art of constructing arches is one of the most difficult and important parts of hydraulic architecture. It proposes, as its object; the union of beauty and solidity, of decoration and durability. The architect must attempt to answer the ends of public safety, and to leave behind him a speaking and durable monument of the genius and taste of his age and nation. Although the antients have combined the forms of decoration in so great a variety, and with so much taste, that we must almost despair of producing a new beauty by a new combination; yet must they yield the palm of excellence in what regards scientific construction. The superiority of the moderns is abundantly evident, in the bridges which have been constructed since the beginning of this century. The second section relates to Dynamics; and here the au. thor adopts the principle followed by D'Alembert in his Dynamique, contained in this proposition: “ Whatever be the manner according to which any bodies change their actual mo. tion, if the motion that each body would have in the following instant were conceived to be compounded of two others, the first of which is what would really be after the change, then the second must be such that, if the bodies had not any other, they would all remain in equilibrium." This principle is evident, and requires little effort of the mind to comprehend it. It is applied to deduce the formulas of the velocities of bodies' mutually impelling; to determine the motion of the centre of gravity ; to assign the general formulas of a body solicited by any powers whatever, whether moving in a curve or revolving round an axis; to projectiles, and the simple pendulum ; to the curves of quickest descent and equal pressure; to the compound pendulum ; to the centres of oscillation and percussion; and to those of spontaneous rotation. In discussing the physico-mathematical theory of percussion, M. Prony explains what cause renders the ordinary theory of percussion insufficient and inapplicable to practice, and gives the sketch of a physical theory of percussion, which comprehends all the theory of motion as well during the shock as afterward. In this theory, the laws of the impulse of hard and soft bodies, perfectly or imperfectly elastic, will appear to flow from the same source. Don Gorges Juan, á Spanish author, first gave this theory, in a work intitled Examen maritime. A general A general theory of motion, built on the principle of virtual velocities, and applied to machines, terminates this section ; and the author notes the principal causes of error in the con. struction of machines, explaining under what point of view they ought to be considered, to the end that the labour destined to their perfection may not be fruitless. The third section is on the subject of Hydrostatics. ALthough the theory of fluids be derived from the same sources as that of solid bodies, yet it is necessary to assign to fluids a particular quality, which shall draw between them and solids a distinct line of demarcation. This property is the equality of the pressure of fluids in every sense, and is the principle adopt, ed by Euler in his memoir printed in the Petersburg Acts, and by D'Alembert in his treatise on the Equilibrium of Fluids. M. PRONY applies his theory, founded on this principle, to the propositions on the equilibrium and pressure of fluids, to levelling, &c. He afterward applies the theory of the equilibrium of heavy and incompressible fuids, to the stability of banks, dikes, &c. and adds the theory and description of Par. cieux's Areometer. Respecting elastic fluids, the general conditions of their equilibrium are given. The method of measuring the pressure of the air; the equation between the height in the barometer, and the difference of heights of levels; and the defects and degrees of utility of this instrument in measuring heights; are likewise considered. In the description of pumps, and of machines for elevating water, the author enters into a more particular detail, and uses a more diffuse explication. He begins from the most simple method of raising water, and proceeds through the se. veral states of modification and improvement to the most complicated contrivance. This order of explanation is not the one given by history, but is one which is scientific, and adopted for the purpose of perspicuity. Section IV. treats on Hydrodynamics. The mechanics of fluids is entirely due to modern research, and of all branches of science it is the most difficult, and the least advanced. The investigation of the nature and motion of fluids has given birth to various and important improvements in analysis. M. PROXY, in this section, relates the solutions of the problems concerning the flowing of a fluid through an orifice in a vessel, and the experiments made by the Abbé Bossut on the times of emptying vessels. In discussing the shock and resistance of fluids, the author explains why the common physico-mathematical theory of percussion is not applicable; and, in following a new route in order |