all such questions, we may remark, if there were no 'second Adam,' the Lord from Heaven, how can it be shewn to have been worthy of either the goodness or the wisdom of God to appoint a first Adam, who he foresaw would fall as the representative of his posterity? Nor can it be shewn to be consistent with a full display of his rectoral equity and sovereign mercy, that he should so have interposed as to secure Adam's continuance in the state in which he was first placed. As far, therefore, as the providence and government of God are concerned in the present state of things, we may safely assert, whatever is, is RIGHT." pp. 191-193.

"

After reviewing the different communications made to Adam, to Enoch, and to Noah, the Abrahamic covenant is particularly considered, where many remarks occur well worthy of attention. These are followed by an examination of the Mosaic dispensation, and a comparison of it with the Christian. It is observed, that the Mosaic dispensation was founded in favour,-that it had much of the nature of a strict covenant, that by implication, it contained an exhibition of sovereign grace, that it was intended but for a limited time,-and that it was in its design preparatory. Contrasted with this the Christian dispensation has greater amplitude and clearness in its revelations: particularly in its display of everlasting sanctions; in exhibiting not only an incomparably superior mediator, but also a surety; and in peculiarly promising the ministration of the Spirit. Each of these is illustrated in an interesting manner, and the chapter thus concludes,

From this detail, I hope it appears to the reader, that in each divine dispensation, sovereign mercy lays the foundation, equity presides to deter from unhallowed abuses, and efficacious grace raises the holy superstructure;-and when the top-stone, the Last of the building, is placed upon it, there will be abundant cause for a triumphant shout of "grace, grace unto it"-the beginning, the progress, and the end of this "habitation of God" was of grace in a manner wonderfully consistent with equitable government.'

In our next number we hope to complete our account of this valuable work.

:

ART. III. An Introduction to the Mathematical Principles of Natural Philosophy containing a series of Lectures upon the Rectilinear and Projectile Motion, the Mechanical Action, and the Rotatory and Vibratory Motion of Bodies. By the Rev. B. Bridge, B. D. F. R. S. Fellow of St Peter's College, Cambridge, and Professor of Mathematics and Natural Philosophy in the East India College. 2 vol. 8vo. pp xviii. 610. Cadell and Davies. 1813.

THE

HE general title of this work is not very correctly expressed; for the performance itself relates only to one branch of natural philosophy, viz. mechanics, comprising the theory of statics and of dynamics, though not according to such a classification as those scientific terms would naturally suggest. We readily overlook, however, any incorrectness in the title of this work, or irregularity in its arrangement, in consideration of the subjects to which it relates,-subjects of the utmost importance to men of science, but which, notwithstanding, have by no means so frequently exercised the talents of English mathematicians as might be expected or wished. In France, treatises on mechanics are very numerous: in England, much the contrary. The only work we have which deserves the name of a complete treatise on mechanics, even in theory, is Dr. Parkinson's; and the only treatises that comprehend both theory and practice, are those by Mr. Emerson and Dr. Gregory, both well-known and useful performances. Besides these we have the scientific treatise of Mr. Atwood on "rectilinear and rotatory motion;" a very excellent work, though unfortunately so defaced with press errors as to destroy nearly all reliance upon its formulæ, till they have been verified or corrected by a repetition of the investigations.

Mr. Bridge, then, has the advantage of entering a path which has not been too frequently trodden. It is our business to show how he proceeds after he has entered it. The work is thrown into the form of lectures, and is divided into four parts.

PART I. comprises six lectures, which relate to motion and the laws of motion,-the rectilinear ascent or descent of bodies acted upon by the force of gravity,-the composition and resolution of motion,-the centre of gravity, the collision of bodies, and the motion of projectiles. We confess we are less satisfied with the first of these lectures, than with many that follow it: but this, perhaps, is only saying, in other words, that on points where great diversity of opinion is known to exist, we should have recourse to different definitions and different illustrations. In our view, Mr. Bridge shines less as a logician and metaphy

sician than as a mathematician; and a like consciousness in his own mind, may sometimes have led him to be satisfied with a weak confirmation of a proposition, and passing on too hastily to illustration and exemplification. This is the case with regard to the laws of motion. Our author's examples in reference to them are excellent; but surely he might, in small compass, have established them much more decisively.

In the second lecture, which is devoted to the rectilinear ascent and descent of bodies acted upon by gravity, the principles first introduced by Galileo, in the theory of dynamics, are applied to the investigation of the chief theorems: and these again are applied to the solution of an interesting collection of problems. Indeed the chief novelty, not only in this lecture but in the whole work, arises from the problems which the author has sometimes selected, at others invented, for the purpose of showing the application and use of the several propositions and formulæ, as they arise in the order of the performance.

The third lecture relates to the composition and resolution of motion, and the investigation of the most useful formula that are derivable from what is usually denominated the parallelogram of forces. Here, again, the problems for illustration are very well selected: but the author, by omitting to establish the composition and resolution of forces, except by a bare inference, evades, in a way we cannot approve, one of the main difficulties which lie at the foundation of the theory of mechanics.

From the consideration of the operation of simultaneous forces, Mr. Bridge passes to that of the centre of gravity, the principal theorem relating to which he deduces from Galileo's demonstration of the fundamental property of the lever. The centro-beryc method he has thrown into a note at the end of the volume.

The collision of bodies is treated in the fifth lecture. This subject, discussed in all its generality, and with a due attention to the several particulars which necessarily enter into the disquisition, is an extremely difficult one. Even the elaborate theory of Don Juan, as given by M. Prony in his "Architecture Hydraulique", and by Dr. Gregory in his " Mechanics," is in some respects incomplete. Mr. Bridge satisfies himself with exhibiting the common theory, due to Wallis, Huygens, and Wren; applying it to the impact of hard bodies, and of bodies either perfectly or imperfectly elastic. This theory, however, under the assumed restrictions, is treated with considerable perspicuity and elegance, and so as to develope several curious results. It is shown, for example, that if there be a row of contiguous imperfectly elastic bodies, diminishing in magnitude by a constant ratio; if the first body impinge

upon the second with a given velocity, and the motion be propagated through the whole series; then when the common ratio by which the bodies decrease is the same fraction as that which expresses the degree of elasticity, the velocity communicated in each case will be that with which the first body struck the second, and with this velocity will the last body move off: So that in this case the same effect is produced upon the last body as when a row of equal perfectly elastic bodies are placed contiguous to each other; but the other bodies do not remain at rest after impact.'

Mr. Bridge gives, also, the proposition so much insisted upon by Bernoulli, namely, that in the collision of perfectly elastic bodies, the sum of the products formed by multiplying each body into the square of the velocity is not altered by the impact.' Then, in reference to this, he shews, with regard to imperfectly elastic bodies, that the sum of the products arising from multiplying each body into the square of its velocity before impact, is greater than the sum of the products arising from multiplying each body into the square of the velocity after impact. To complete this part of the theory, our author should have shewn, as Atwood has done at p. 45 of his "Analysis of a Course of Lectures," "what must be the force of elasticity, that the sums of the products formed by multiplying each body into any assumed power of the velocity, may not be altered by the impact."

In Lecture the sixth Mr. Bridge treats of the motion of projectiles in a nonresisting medium. Here the geometrical principles, and the construction of the general problem are neatly exhibited; the former, after the manner of Professor Robison, the latter agreeing with the construction originally given by Mr. Reuben Burrow. In the investigation of the Algebraic formulæ our author has not, we think, been quite so successful; his methods being rather tedious, and not always leading to the most commodious results.

Thus, in the problem where it is proposed, having given the proportion between the range and greatest altitude of a body projected with a given impetus, to find the angle of projection, Mr. Bridge's process is as follows.-R being the range, A the greatest altitude, p for the impetus, or height due to the velocity, a the angle of projection, or the elevation of the piecc; then

By art. 5. (page 206.)

"Ř: Д:: 4 px sin. a x cos. a: p x sin. 2a,

:: 4 cos, a sin a;

“ .. R2 : A2 : : 16 cos. 2a: sin. 2a :: 16 (1-sin. 2a): sin. a, " and R2: 16 A2

:: 16 (1.

--

2a, Hence, R2 + 16 A2 :: 16: 16 sin. a

... sin. a =

"Cor. If R = A, then sin. a

=

4 A

sin. a): 16 sin. : 1: sin. 2a,

i. e. in order that the greatest altitude of a projectile may be equal to its range, its direction must make an angle of about 76° with the

horizon.

Now, if in solving this problem our author had previously obtained a theorem like that given in Gregory's Mechanics, (Vol. I. p. 200,) or in Hutton's Course, (Vol. II. p. 161) namely 4 A a single step have obtained the 4 A

R =

tan a

he would, by

simple expression tan a = ; from which, when A and R

R

are equal, there would result tan a = 4 tan of 75° 58', the required angle of projection in the case of the corollary.

The SECOND PART of Mr. Bridge's work contains five lectures, of which the first three relate to the mechanical powers, the next to the pressure and tension of cords, and the last to the strength, stress, and pressure of beams. Here we have nothing particular to remark, except in reference to the last, or eleventh lecture. The subject of the strength and stress of materials was first handled scientifically by Galileo, in his Dialogues. He treated it with great elegance, but obviously simplified more than the nature of the enquiry would fairly allow. Mr. Bridge, however, has adopted the same principles, and shows in nearly a similar way to that celebrated philosopher, that the strength of a beam, placed horizontally, is inversely as its length, and directly as the product of its transverse section into the depth of the centre of gravity of that section, below the upper surface of the beam. This, for the purpose of obtaining ready practical estimates, may do very well; but it will not satisfy a man of rigid science: many of the later investigators have explored this business much more minutely and successfully we would, therefore, recommend Mr. Bridge, in the event of a new edition of these lectures, to examine Professor Robison's enquiries into the subject of the strength and stress, in the Encyclopædia Britannica; Dr. Thomas Young's, in the second volume of his Natural Philosophy; and those of M. Girard, in his "Treatise on the Resistance of Solids ;" and adopt their most curious and useful results. There is one par