the business of a scientific investigator is to establish some general principle of operation, which shall be universally applicable to all possible cases, the most complex as well as the most simple, and shall always ensure a result on which the practical architect may rely.

It was not till near the end of the seventeenth century, when the Newtonian mathematics opened the road to true me chanical science, that mathematicians directed any part of their attention to the theory of arches. Dr. Hooke gave the first hint of a principle, when he affirmed that the figure into which a chain or rope, perfectly flexible, will arrange itself when suspended from two hooks, becomes, when inverted, the proper form for an arch constituted of stones of uniform weight and size. The reasoning on which he grounded his assertion is, simply, that the forces with which the parts of the standing arch press mutually on each other in the latter case, are precisely equal and opposite to those with which they pull each other in the case of suspension. This principle, incontrovertibly true as far as it goes, was farther extended by Dr. David Gregory, in No. 231 of the Philosophical Transactions, for 1697, who also pointed out the way by which it may be applied to all possible cases. But the principle of the extension was misunderstood by some subsequent investigators, especially by Mr. Benjamin Martin, who, in the discussions that took place relative to the proper form to be given the arches of Blackfriar's bridge, contended that they should be simple catenaries. Martin did not consider, that though an arch of equal voussoirs might be thus balanced, it would be totally unfit for the purposes of a bridge, which requires much other masonry to be placed over the arch to fill up the space to the road-way; and that this superincumbent mass must necessarily destroy the equilibrium previously subsisting in the unloaded arch. It is hence obvious, that the theory of the simple catenary could never prevail much among real mathematicians.

The second method, which kept its reputation among theorists for a long time, though it was not acted upon by practical men, was deduced from the consideration of the archstones being frustums or parts of wedges. Accordingly, the mathematical properties of the wedge were introduced into the science, and employed to establish the theory of balanced arches. Yet, it is easy to perceive that, unless the various stones in the arch were perfectly smooth, and free from friction, this theory, however specious, would not admit of an application to real practice. For, so far from the archstones being kept in their places only by forces perpendi cular to their butting sides, and having full liberty to slide along those sides, as in the wedge theory, the sides are left

rough, so rough, indeed, that the friction between two contiguous blocks is at least equal to half their mutual pressure; and, further, are cemented and locked together by bars of iron, &c. so that they are prevented from the possibility of sliding, and sustained. in their places in the arch by forces that act in directions very oblique, nay often perpendicular, to those which the wedge theory requires. Besides, in the wedge method, as well as in the catenarian, since there is much ponderating matter above the arch, the balance must necessarily be destroyed, unless it be regulated by other principles. In order to obviate this difficulty, some theorists have conceived that the voussoirs should increase gradually from the crown of the arch to the abutments, so as to fill up all the necessary space between the intrados, and extrados: but an arch thus constructed is still liable to the objection stated above, arising from friction and adhesion of surfaces in addition to which there are many practical objections, upon which we cannot here enlarge. Notwithstanding all this, however, the wedge-theory was taught successively by La Hire, Parent, Varignon, Bellidor, Riou, Muller, and Samuel Clark; and prevailed, till its absurdity was shewn by Emerson in England, and by Bossut on the continent, nearly 40 years ago. A recent attempt was made to revive this exploded theory, by the late Mr. George Atwood, who, a short time before his death, after his faculties had been impaired by a paralytic stroke, published two inaccurate, obscure, immethodical, and excessively inelegant pamphlets on arches, according to this false theory. We mention the decay of Mr. Atwood's mental powers at this period, that no person, who reads these pamphlets and is able to appreciate their worthlessness, may suffer himself on that account to think meanly of Mr. Atwood's talents.

The third method, that has been devised for the theory of arches, establishes an equilibrium among all the vertical pressures of the whole fabric, contained between the soffit of the arch and the road-way, or other natural summit of the structure. The best judges and most skilful engineers and architects now acknowledge this theory to be the only true one, because it ensures an equilibrium in the whole of the ponderating matter, by making an equality at every point of the curve, between all the adjacent pressures, when reduced to the tangential directions, or those perpendicular to the joints, which are every where supposed at right angles to the curve, as the practice requires them to be. The most general deduction from this theory is, that if the height of the wall incumbent on any point of the intrados is directly as the cube of the secant of the curve's inclination to the hori zon at that point, and inversely as the radius of curvature

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there, all the voussoirs will endeavour to split the arch with equal forces, and will be in perfect equilibrium with each other. This flows directly from the principle laid down by David Gregory, when he affirmed that, if arches of other forms than the common catenary are supported, it is because in their thickness some catenary is included.' The same theorem has been deduced in various ways, and applied to almost all the modifications of cases, by Emerson in his Miscellanies: Dr. Hutton, in his little tract on Bridges; Bossut, in his Mecanique; Mascheroni, in his Nuove Ricerche sull' equili brio delle Volte; Prony, in his Architecture Hydraulique, tome 1; and O. Gregory, in the first volume of his Mechanics. The theory has likewise been correctly stated in the article Arch in the Supplement to the Encyclopædia Britannica, and in the article Bridge in Rees's New Cyclopædia.

Mr. Ware, to a critique on whose book we must now proceed, seems to have heard or read some very indistinct account of the three theories we have described; and, with a strange propensity toward error, to have indulged a disgust for the true theory, but a singular love for the two which are now exploded. These, if we rightly understand him, he wishes to incorporate into one. He does not seem aware that their materials are too heterogeneous to admit of union. This is far from being marvellous; for he has brought to the study of arches, no other of the essential requisites, than industry; of mathematical knowledge he is deplorably deficient. We regret that some friend did not whisper in his ear, that no smatterer in mathematics can safely approach the theory of arches.

Mr. Ware's book is divided into introductory definitions and remarks, and four sections. In the first are described the general laws of motion; in the second, we have propositions relative to arches of equilibration; the third relates to the catenary; and the fourth to abutment piers of equilibration, bridges of many arches, and the flying buttresses of cathedrals. In the introductory remarks, our author gives an extract from David Gregory's memoir on the catenary, to no other end, that we can perceive, than to prove that he does not comprehend the meaning of that illustrious author. He says that Gregory, in affirming that if an arch of any other figure than a common catenary is supported, it is because in its thickness some catenary is included', draws a true inference from false premises. Now, we affirm that the conclusion is drawn from true premises; and we are convinced that no man, who comes to the inquiry with the requisite preliminary knowledge, will agree with Mr. Ware on this point. He goes on, however, to say, that 'if the actions

of an inverted catenaria be equal by gravitation, as they must be to retain their situation, then every joint in a chain is equally liable to be broken by the gravitation of the parts; but the contrary is evident from experience: therefore, the inverted curve of a catenaria, composed of equal rigid polished spheres in a plane perpendicular to the horizon, cannot keep its figure. " Without stop ping to animadvert upon the remarkable peculiarity of a man, who adopts the catenary in his subsequent investiga tions, reasoning against it in his introductory observations, we may state that this reasoning is erroneous; the equal forces of which Mr. Ware speaks are vertical; while the forces tending to break the chain are in the tangent of the curve at the point of junction of the links, and therefore vary with the obliquity of the tangent. Any portion of a catenary is kept in its position by three forces, viz. the tensions at its extremities and its weight, the two former acting in the directions of the respective tangents, the lat ter vertically. Had Mr. Ware known this, he would have saved himself much trouble, and spared us the task of pointing out his blunders. The introduction contains several others; but we have not room. to enlarge upon them.

The first section contains five propositions, three of which exhibit errors. In prop. 3, our author says, "If one body acts against another body by any kind of force whatever, it exerts that force in the direction of a line perpendicular to the surface whereon it acts. Now we say, that when the direction is oblique to the surface, it is not that force' which is exerted, but a very different one. Mr. Ware's demonstration here contradicts his proposition. In the 4th proposition it is affirmed, that 'The force wherewith a rolling body descends upon an inclined plane is, to the force of its absolute gravity, by which it would descend perpendicularly in a free space, as the height of the plane is to its length. This is not true; the proposition does not apply to a rolling body, but to one sliding without friction: and we would advise Mr. Ware to hesitate about his proposition, till he can find such bodies with which to erect an arch.

We are next informed, that the difficulty of moving bodies on a horizontal line arises from the resistance of cohesion, and that of continuing the motion, from the resistance of friction. This is not correct. The difficulty of moving bodies arises from inertia: it is the difficulty of penetrating bodies which arises from cohesion. Before we can admit the remaining proposition in this section, our author must shew how he makes weights act in the oblique directions of which be speaks.

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But mistakes thicken as we proceed; and in the second

section on arches of equilibration, we are so surrounded with them we know not how to cut our way out. The demonstration of the 6th proposition (the first in this section) is the pure quintessence of absurdity. We would ask Mr. Ware, who can of course turn to his own diagrams,-Does de, fig. 14, express or represent the weight of the voussoir Vode? Does the line joining the centres of gravities of two contiguous voussoirs intersect the faces in contact perpendicularly to de, de, &c.? How can the pressures of the voussoirs be referred to their proper positions without knowing the places of their centres of gravity? Where would there be room for the piers, according to Mr. Ware's theory, which makes the distance (at the spring of the arch) between the extrados and the intrados, infinite in a horizontal direction? How happens it that what he here speaks of as theoretically. true' is not 'experimentally true'?

In the 7th proposition we are told, that certain voussoirs "the moment liberty of motion should be given them, would turn on de, as a centre, and fall in the direction ae.' Such is our obtundity of perception, that we cannot couceive how they could manage to turn on a centre and fall vertically at the same time. But again Mr. Ware informs us, that, according to the wedge theory, no arch of equilibration can have either a horizontal extrados or intrados.' This is contrary to his own hypothesis in this very propo sition; and contrary to fact. Mr. Atwood computed the size of the voussoirs for an arch with a horizontal extrados; and Professor Vince has a model made conformably to Atwood's computation with lubricous wedges, which he exhibits in his lectures, and the experiment succeeds perfectly. Prop. 8 we hardly know how to examine, on account of its obscurity but it must be erroneous, because it depends upon Prop. 6, which is egregiously incorrect. The author also reasons in a very loose illegitimate way from the circle to the other conic sections; and then informs us that his 'results are in opposition to those of Dr. Hutton', a circumstance, we conjecture, which will not very deeply mortify that distinguished mathematician.

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The third section is devoted to the catenary. And here we find that the author is still the same identical Mr. Ware, who has amused us with so many blunders in the preceding sections: he triumphantly establishes his claim to consis tency of character. First we learn, that when a chain is free to move at every connection, and forms a curve, a force acts at each extremity A,B, of the chain forming the catenaria, and there only. Of course, when such a chain breaks, as Mr. Ware has told us at page & it will do, it breaks without a cause. But this is not all the interesting information our