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 sensible of the dangers and inconveniences of mere trial. In thus making the procedures of the arts approach the sciences, and in diffusing over the former the light of the latter, the objects 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 not 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 equacion 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, t time). The second question is reduced to this equation par 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 allirms a thing to be according to such a manner, because there is no reason why it should be according other. In the deinonstration 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 to any in the direction of a 'line passing through their contre 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 Trnité 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 instituted. 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 1. 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 La Grange, is more concise and easy in its application, ari 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. Proxy proposes at the end of his second volume to give another; which, however, is deficient 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 velocities 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 con struction. 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 author adopts the principle followed by D'Alembert in his Dye namique, 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 conipounded 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, 11. PRONY explains what cause reuders 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, a Spanish author, first gave this theory, in a work intitled Examen maritime. A general L14 A general theory of motion, built on the principle of virtual velocities, and applied to machines, terminates this section ; and the author votes 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, hich 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 adopted by Eulen 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 fuids, te le• velling, &c. He afterward applies the theory of the equilibrium of heavy and incompressible fluids, to the stability of banks, dikes, &c. and adds the theory and description of Parcieux's Arrometer. Oi!!!!. Respecting elastic fluids, the general conditions of their .equilibrium are given. The method of measuring the pressare 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 heighệs; are likewise considered. In the description of pumps, and of machines for elevating water, the author enters into a more particuiart detail, and uses a more diffuse explication. He begins from the most simple method of raising water, and proceeds through the several states of morification 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 advancest, the investigation of the nature and motion of fluids bas given birth to various and important improvements in analysis. M. PRONY, in this section, relates the solutions of the problems concerning the flowing of a fluid through an orihce in a vessel, and the experiments made by the Abbé Bossut on the times of empty: ing vessels.'"*9.3 ks 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 order to arrive at the solution of questions relative to the shock of Auids, he uses as a guide M. Gorges Juan, author of the Examen maritime already mentioned. - In the latter part of the section, he gives the general and rigorous theory of the motion of fluids, and shews the insufficiency of the ordinary integral calculus for resolving problems appertaining to this cheory. The calculus essentially necessary to these problems is that of partial differences invented by Euler and D'Alembert. This calculus is so facilitated, as to be within the compreherssion of young students ; and its equations are applied to the motion of fuids i: tubes. Section V. treats of machines, and of moving powers to which they can be applied ; considering the different physical circumstances that affect their equilibrium and motion. The inanner in which adhesion, friction, and the stiffness of ropes and chains, operate, is explained; and the previous theory is applied to the force of men carrying, pushing, dragging, throu:ing, &c. and to the force of horses. In the latter part of the volume, the author treats, of the elastic fluid reduced to vapour, and of its application to the motion of machines. An account is also given of the discovery, use, and successive improvements of the steam engine, The Second Volume of the Hydraulic Architecture is destined to the description of steam engines. Since the great perfection, lately given to such engines, has determined their constructors to employ only those of double effect, the different met chanisms and contrivances which preceded the recent discoveries are to be considered only as objects of curious research and erudition. As it was the intention of the author to step beyond the strict limits which mere practical utility might'prescribc, in order to point out the progress of the human mind from one invention to another, by the combination of ideas, be has given the details relative to the antieat machines, and has disposed thein in such a manner that the connection between successive improvements may be traced, and the value of each improvement appreciated. Since, however, the machines of double effect were the primary objects of this work, the antient machines are treated only as aceessory. To have presented a full description of each machine would have extended the worla beyond all reasonable bounds : but sufficient particulars are: given of these machines, to make manifest the relatiou between the old and the new, and to point out when the former are only particular cases of the latter. • By this method (says M. PRONY) the progress of the human mind will be traced, which commences with isolated ideas, previously to passing to gene. **l nocions. It will appear that a machine of double effect, |