1. Arch, in building, is an artful disposition and adjustment of several stones or bricks, generally in a bow-like form, by which their weight produces a mutual pressure and abutment; so that they not only support each other, and perform the office of an entire lintel, but may be extended to any width, and made to carry the most enormous weights.

2. In those mild climates which seem to have been the first inhabited parts of this globe, mankind stood more in need of shade from the sun than of shelter from the inclemency of the weather. A very small addition to the shade of the woods served them for a dwelling. Sticks laid across from tree to tree, and covered with brushwood and leaves, formed the first houses in those delightful regions. As population and the arts improved, these huts were gradually refined into commodious dwellings. The materials were the same, but more artfully put together. At last agriculture led the inhabitants out of the woods into the open country. The connection between the inhabitant and the soil became now more constant and more interesting. The wish to preserve this connection was natural, and fixed establishments followed of course. Durable buildings were more desirable than those temporary and perishable cottages—stone was substituted for timber.

But as these improved habitations were gradual refinements on the primitive hut, traces of its construction remained, even when the choice of more durable materials made it in some measure inconvenient. Thus it happened, that while a plain building, intended for accommodation only, consisted of walls, pierced with the necessary doors and windows, an ornamented building had, superadded to these essentials, columns, with the whole apparatus of entablature borrowed from the wooden building, of which they had been essential parts, gradually rendered more suitable to the purposes of accommodation and elegance.

3. This view of ornamental architecture will go far to account for some of the more general differences of national style which may be observed in different parts of the world. The Greeks borrowed many of their arts from their Asiatic neighbours, who had cultivated them long before. It is highly probable that architecture travelled from Persia into Greece. In the ruins of Shushan, Persepolis, or Tchelminar, are to be seen the first models of every thing that distinguishes the Grecian architectures. There is no doubt, we suppose, among the learned, as to the great priority of these monuments to any thing that remains in Greece; especially if we take into account the tombs on the mountains, which have every appearance of greater antiquity than the remains of Persepolis. In these tombs we see the whole ordonnance of column and entablature, just as they began to deviate from their first and necessary forms in the wooden buildings. We have the architrave, frieze, and cornice; the far-projecting mutules of the Tuscan and Doric orders; the modillions no less distinct; the rudiments of the Ionic capital; the Corinthian capital in perfection, pointing out the very origin of this ornament, viz. a number of long, graceful leaves tied round the head of the column with a fillet (a custom which we know to have been common in their temples and banqueting rooms). Where the distance between the columns is great, so that each had to support a weight too great for one tree, we see the columns clustered, or fluted, &c. In short, we see every thing of the Grecian architecture but the sloped roof or pediment—a thing not wanted in a country where it hardly ever rains.

4. The ancient Egyptian architecture seems to be a refinement on the hut built of clay, or unburnt bricks, mixed with straw—every thing is massive, clumsy, and timid—small intercolumniations, and hardly any projections.

5. The Arabian architecture seems a refinement on the tent. A mosque is like a little camp, consisting of a number of little bell tents, stuck close together round a great one. A caravansery is a court surrounded by a row of such tents, each having its own dome. The Greek church of St Sophia at Constantinople has imitated this in some degree; and the copies from it, which have been multiplied in Russia as the sacred form for a Christian church, have added to the original model of clustered tents in the strictest manner. We are sometimes disposed to think that the painted glass (a fashion brought from the East) was an imitation of the painted hangings of the Arabs.

6. The Chinese architecture is an evident imitation of a wooden building. Sir George Staunton says, that the singular form of their roofs is a professed imitation of the cover of a square tent.

In the stone buildings of the Greeks, the roofs were imitations of the wooden ones; hence the lintels, flying cornices, ceilings in compartments, &c.

7. The pediment of the Greeks seems to have suggested the greatest improvement in the art of building. In erecting their small houses, they could hardly fail to observe occasionally, that when two rafters were laid together from the opposite walls, they would, by leaning on each other, give mutual support, as in Plate LII; fig. 1. Nor is it unlikely that such a situation of stones as is represented in fig. 2 would not unfrequently occur by accident to masons. This could hardly fail of exciting a little attention and reflection. It was a pretty obvious reflection, that the stones A and C, by overhanging, leaned against the intermediate stone B, and gave it some support, and that B cannot get down without thrusting aside A and C, or the piers which support them. This was an approach to the theory of an arch; and if this be combined with the observation of fig. 1, we get the disposition represented in fig. 3, having a perpendicular joint in the middle, and the principle of the arch is completed. Observe that this is quite different from the principle of the arrangement in fig. 2. In that figure the stones act as wedges, and one cannot get down without thrusting the rest aside. The same principle obtains in fig. 4, consisting of five archstones; but in fig. 3 the stones B and C support each other by their mutual pressure (independent of their own weight), arising from the tendency of each lateral pair to fall outwards from the pier. This is the principle of the arch, and would support the key-stone of fig. 4, although each of its joints were perpendicular, by reason of the great friction arising from the horizontal thrust exerted by the adjoining stones.

This was a most important discovery in the art of building; for now a building of any width may be roofed with stone.

8. We are disposed to give the Greeks the merit of this discovery; for we observe arches in the most ancient buildings of Greece, such as the temple of the sun at Athens, and of Apollo at Didymos—not indeed as roofs to any apartment, nor as parts of the ornamental design, but concealed in the walls, covering drains or other necessary openings; and we have not found any real arches in any monuments of ancient Persia or Egypt. Sir John Chardin speaks of numerous and extensive subterranean passages at Tchelminar, built of the most exquisite masonry, the joints so exact, and the stones so beautifully dressed, that they look like one continued piece of polished marble; but he nowhere says that they are arched—a circumstance which we think he would not have omitted: no arched door or window is to be seen. Indeed one of the tombs is said to be arch-roofed, but it is all of one solid rock. No trace of an arch is to be seen in the ruins of ancient Egypt; even a wide room is covered with a single block of stone. In the pyramids, indeed, there are two galleries whose roofs consist of many pieces; but their construction puts it beyond doubt that the builder did not know what an arch was; for it is covered in the manner represented in fig. 5, where every projecting piece is more than balanced behind, so that the whole awkward mass could have stood on two pillars. The Greeks, therefore, seem entitled to the honour of the invention. The arched dome, however, seems to have arisen in Etruria, and originated in all probability from the employment of the augurs, whose business it was to observe the flight of birds. Their stations for this purpose were temple, so called a templando, on the summits of hills. To shelter such a person from the weather, and at the same time allow him a full prospect of the country around him, no building was so proper as a dome set on columns; which accordingly is the figure of a temple in the most ancient monuments of that country. We do not recollect a building of this kind in Greece except that called the Lanthorn of Demosthenes, which is of very late date; whereas they abounded in Italy. In the later monuments and coins of Italy or of Rome we commonly find the Etruscan dome and the Grecian temple combined; and the famous Pantheon was of this form, even in its most ancient state.

9. It does not appear that the arch was considered as a part of the ornamental architecture of the Greeks during the time of their independency. It is even doubtful whether it was employed in roofing their temples. In none of the ancient buildings where the roof is gone can there be seen any rubbish of the vault or mark of the spring of the arch. It is not unfrequent, however, after the Roman conquests, and may be seen in Athens, Delos, Palmyra, Balbeck, and other places. It is very frequent in the magnificent buildings of Rome; such as the Coliseum, the baths of Diocletian, and the triumphal arches, where its form is evidently made the object of attention. But its chief employment was in bridges and aqueducts; and it is in these works that its immense utility is the most conspicuous: for by this happy contrivance a canal or a road may be carried across any stream, where it would be almost impossible to erect piers sufficiently near to each other for carrying lintels. Arches have been executed 130 feet wide, and their execution demonstrates that they may be made four times as wide.

10. As such stupendous arches are the greatest performances of the masonic art, so they are the most difficult and delicate. When we reflect on the immense quantity of materials thus suspended in the air, and compare this with the small cohesion which the firmest cement can give to a building, we shall be convinced that it is not by the force of the cement that they are kept together; they stand fast only in consequence of the proper balance of all their parts. Therefore, in order to erect them with a well-founded confidence of their durability, this balance should be well understood and judiciously employed. We doubt not but that this was understood in some degree by the engineers of antiquity; but they have left us none of their knowledge. They must have had a great deal of mechanical knowledge before they could erect the magnificent and beautiful buildings whose ruins still enchant the world; but they kept it among themselves. We know that the Dionysiacs of Ionia were a great corporation of architects and engineers, who undertook and even monopolized the building of temples, stadiums, and theatres, precisely as the fraternity of masons in the middle ages monopolized the building of cathedrals and conventual churches. Indeed the Dionysiacs resembled the mystical fraternity now called free-masons in many important particulars. They allowed no strangers to interfere in their employment; they recognised each other by signs and tokens; they professed certain mysterious doctrines, under the tuition and tutelage of Bacchus, to whom they built a magnificent temple at Teos, where they celebrated his mysteries as solemn festivals; and they called all other men profane because not admitted to these mysteries. But their chief mysteries and most important secrets seem to be their mechanical and mathematical sciences, or all that academical knowledge which forms the regular education of a civil engineer. We know that the temples of the gods and the theatres required an immense apparatus of machinery for the celebration of some of their mysteries; and that the Dionysiacs contracted for those jobs, even at far distant places, where they had not the privilege of building the edifice which was to contain them. This is the most likely way of explaining the very small quantity of mechanical knowledge that is to be met with in the writings of the ancients. Even Vitruvius does not appear to have been of the fraternity, and speaks of the Greek architects in terms of respect next to veneration. The Collegium Murarium, or incorporation of masons at Rome, does not seem to have shared the secrets of the Dionysiacs.

11. The art of building arches has been most assiduously cultivated by the associated builders of the middle ages of the Christian church, both Saracens and Christians, and they seem to have indulged in it with fondness; they multiplied and combined arches without end, placing them in every possible situation.

12. Having studied this branch of the art of building with so much attention, they were able to erect the most magnificent buildings with materials which a Greek or Roman architect could have made little or no use of. There is infinitely more scientific skill displayed in a Gothic cathedral than in all the buildings of Greece and Rome. Indeed these last exhibit very little knowledge of the mutual balance of arches, and are full of gross blunders in this respect; nor could they have resisted the shock of time so long, had they not been almost solid masses of stone, with no more cavity than was indispensably necessary.

13. Anthemius and Isidorus, whom the Emperor Justinian had selected as the most eminent architects of Greece, for building the celebrated church of St Sophia in Constantinople, seem to have known very little of this matter. Anthemius had boasted to Justinian that he would outdo the magnificence of the Roman Pantheon, for he would hang a greater dome than it aloft in the air. Accordingly he attempted to raise it on the heads of four piers, distant from each other about 115 feet, and about the same height. He had probably seen the magnificent vaultings of the temple of Mars the Avenger, and the temple of Peace at Rome, the thrusts of which are withstood by two masses of solid wall, which join the side walls of the temple at right angles, and extend sidewise to a great distance. It was evident that the walls of the temple could not yield to the pressure of the vaulting without pushing these immense buttresses along their foundations. He therefore placed four buttresses to aid his piers. They are almost solid masses of stone, extending at least ninety feet from the piers to the north and to the south, forming as it were the side walls of the cross. They effectually secured them from the thrusts of the two great arches of the nave which support the dome; but there was no such provision against the push of the great north and south arches. Anthemius trusted for this to the half dome which covered the semicircular east end of the church, and occupied the whole eastern arch of the great dome. But when the dome was finished, and had stood a few months, it pushed the two eastern piers with their buttresses from the perpendicular, making them lean to the eastward, and the dome and half dome fell in. Isidorus, who succeeded to the charge on the death of Anthemius, strengthened the piers on the east side by filling up some hollows, and again raised the dome. But things gave way before it was closed; and while they were building in one part, it was falling in another. The pillars and walls of the eastern semicircular end were much shattered by this time. Isidorus seeing that they could give no resistance to the push which was so evidently directed that way, erected some clumsy buttresses on the east wall of the square which surrounded the whole Greek cross, and was roofed in with it, forming a sort of cloister round the whole. These buttresses, spanning over this cloister, leaned against the piers of the dome, and thus opposed the thrusts of the great north and south arches. The dome was now turned for the third time, and many contrivances were adopted for making it extremely light. It was made offensively flat, and, except the ribs, was roofed with pumice stone; but, notwithstanding these precautions, the arches settled so as to alarm the architects, and they made all sure by filling up the whole from top to bottom with arcades in three stories. The lowest arcade was very lofty, supported by four noble marble columns, and thus preserved in some measure the church in the form of a Greek cross. The story above formed a gallery for the women, and had six columns in front, so that they did not bear fair on those below. The third story was a dead wall filling up the arch, and pierced with three rows of small, ill-shaped windows. In this unworkmanlike shape it has stood till now, and is the oldest church in the world; but it is an ugly mis-shapen mass, more resembling an overgrown potter's kiln, surrounded with furnaces pierced and patched, than a magnificent temple. We have been thus particular in our account of it, because this history of the building shows that the ancient architects had acquired no distinct notions of the action of arches. Almost any mason of our time would know, that as the south arch would push the pier to the eastward, while the east arch pushed it to the southward, the buttress which was to withstand these thrusts must not be placed on the south side of the pier, but on the southeast side, or that there must be an eastern as well as a southern buttress.

14. No such blunders are to be seen in a Gothic cathedral. Some of them appear, to a careless spectator, to be very massive and clumsy; but when judiciously examined, they will be found very bold and light, being pierced in every direction by arcades; and the walls are divided into cells like a honeycomb, so that they are very stiff, while they are very light.

15. About the middle, or rather towards the end, of the last century, when the Newtonian mathematics opened the road to true mechanical sciences, the construction of arches engrossed the attention of the first mathematicians. The first hint of a principle that we have met with is Dr Hooke's assertion, that the figure into which a chain or rope, perfectly flexible, will arrange itself when suspended from two hooks, is, when inverted, the proper form for an arch composed of stones of uniform weight. This he affirmed on the principle, that the figure which a flexible festoon of heavy bodies assumes, when suspended from two points, is, when inverted, the proper form of an arch of the same bodies, touching each other in the same points; because the force with which they mutually press on each other in this last case are equal and opposite to the forces with which they pull at each other in the case of suspension.

This principle is strictly just, and may be extended to every case which can be proposed. We recollect seeing it proposed in very general terms in 1759, when plans were forming for Blackfriars Bridge in London; and since it is perhaps equal in practical utility to the most elaborate investigations of the mathematicians, our readers will not be displeased with a more particular account of it in this place.

16. Let ABC (fig. 6) be a parcel of magnets of any size and shape; and let us suppose that they adhere with great force by any points of contact. They will compose such a flexible festoon as we have been speaking of if suspended from the points A and C. If this figure be inverted, preserving the same points of contact, they will remain in equilibrium. It will indeed be that kind of equilibrium which will admit of no disturbance, and which may be called a tottering equilibrium. If the form be altered in the smallest degree, by varying the points of contact (which indeed are points in the figure of equilibration), the magnets will no more recover their former position than a needle, which we had made to stand on its point, will regain its perpendicular position after it has been disturbed.

But if we suppose planes d e f g h i, &c., drawn through the points of mutual contact a b c, each bisecting the angle formed by the lines that unite the adjoining contacts (f g, for example, bisecting the angle formed by a b b c), and if we suppose that the pieces are changed for others of the same weights, but having flat sides, which meet in the planes d e f g h i, &c., it is evident that we shall have an arch of equilibration, and that the arch will have some stability, or will bear a little change of form without tumbling down; for it is plain that the equilibrium of the original festoon obtained only in the points a b c, of contact, where the pressures were perpendicular to the touching surfaces; therefore, if the curve a b c still passes through the touching surfaces perpendicularly, the conditions that are required for equilibrium still obtain. The case is quite similar to that of the stability of a body resting on a horizontal plane. If the perpendicular through the centre of gravity falls within the base of the body, it will not only stand, but it will require some force to push it over. In the original festoon, if a small weight be added in any part, it will change the form of the curve of equilibration a little, by changing the points of mutual contact. This new curve will gradually separate from the former curve as it recedes from A or C. In like manner, when the festoon is set up as an arch, if a small weight be laid on any part of it, it will bring the whole to the ground, because the shifting of the points of contact will be just the contrary to what it should be to suit the new curve of equilibration; but if the same weight be laid on the same part of the arch now constructed with flat joints, it will be sustained if the new curve of equilibration still passes through the touching surfaces.

17. These conclusions, which are very obviously deducible from the principle of the festoon, show us, without any further discussion, that the longer the joints are, the greater will be the stability of the arch, or that it will require a greater force to break it down. Therefore it is of the greatest importance to have the arch-stones as long as economy will permit; and this was the great use of the ribs and other apparent ornaments in the Gothic architecture. The great projections of those ribs augmented their stiffness, and enabled them to support the unadorned compartments of the roof, composed of very small stones, seldom above six inches thick. Many old bridges are still remaining, which are strengthened in the same way by ribs.

Having thus explained, in a very familiar manner, the stability of an arch, we proceed to give the same popular account of the general application of the principle.

18. Suppose it be required to ascertain the form of an arch which shall have the span AB (fig. 7), and the height F S, and which shall have a road-way of the dimensions C D E above it. Let the figure ACDEB be inverted, so as to form a figure A ede B. Let a chain of uniform thickness be suspended from the points A and B, and let it be of such a length that its lower point will hang at, or rather a little below, f, corresponding to F. Divide AB into a number of equal parts, in the points 1, 2, 3, &c., and draw vertical lines, cutting the chain in the corre-

sponding points 1, 2, 3, &c. Now take pieces of another chain, and hang them on at the points 1, 2, 3, &c., of the chain A f B. This will alter the form of the curve. Cut or trim these pieces of chain, till their lower ends all coincide with the inverted road-way c d e. The greater lengths that are hung on in the vicinity of A and B will pull down these points of the chain, and cause the middle point f (which is less loaded) to rise a little, and will bring it near to its proper height.

It is plain that this process will produce an arch of perfect equilibration; but some further considerations are necessary for making it exactly suit our purpose. It is an arch of equilibration for a bridge that is so loaded that the weight of the arch-stones is to the weight of the matter with which the haunches and crown are loaded, as the weight of the chain A f B is to the sum of the weights of all the little bits of chain very nearly. But this proportion is not known beforehand; we must therefore proceed in the following manner:—Adapt to the curve produced in this way a thickness of the arch-stones as great as are thought sufficient to insure stability; then compute the weight of the arch-stones, and the weight of the gravel or rubbish with which the haunches are to be filled up to the road-way. If the proportion of these two weights be the same with the proportion of the weights of chain, we may rest satisfied with the curve now found; but if different, we can easily calculate how much must be added equally to or taken from each appended bit of chain, in order to make the two proportions equal. Having altered the appended pieces accordingly, we shall get a new curve, which may perhaps require a very small trimming of the bits of chain to make them fit the road-way. This curve will be very near to the curve wanted.

We have practised this method for an arch of 60 feet span and 21 feet high, the arch-stones of which were only two feet nine inches long. It was to be loaded with gravel and shivers. We made a previous computation, on the supposition that the arch was to be nearly elliptical. The distance between the points 1, 2, 3, &c., were adjusted, so as to determine the proportion of the weights of chain agreeable to the supposition. The curve differed considerably from an ellipse, making a considerable angle with the verticals at the spring of the arch. The real proportion of the weights of chain, when all was trimmed so as to suit the road-way, was considerably different from what was expected. It was adjusted. The adjustment made very little change in the curve. It would not have changed it two inches in any part of the real arch. When the process was completed, we constructed the curve mathematically. It did not differ sensibly from this mechanical construction. This was very agreeable information; for it showed us that the first curve, formed by about two hours' labour, on a supposition considerably different from the truth, would have been sufficiently exact for the purpose, being in no place three inches from the accurate curve, and therefore far within the joints of the intended arch-stones. Therefore this process, which any intelligent mason, though ignorant of mathematical calculation, may go through with little trouble, will give a very proper form for an arch subject to any conditions.

19. The chief defect of the curve found in this way is a want of elegance, because it does not spring at right angles to the horizontal line; but this is the case with all curves of equilibration, as we shall see by and by. It is not material; for, in the very neighbourhood of the piers, we may give it any form we please, because the masonry is solid in that place; nay, we apprehend that a deviation from the curve of equilibration is proper. The construction of that curve supposes that the pressure on every part of the arch is vertical; but gravel, earth, and rub- blish, exert somewhat of a hydrostatic pressure laterally in the act of settling, and retain it afterwards. This will require some more curvature at the haunches of an arch to balance it; but what this lateral pressure may be, cannot be deduced with confidence from any experiments that we have seen. We are inclined to think, that if, instead of dividing the horizontal line AB in the points 1, 2, 3, &c., we divide the chain itself into equal parts, the curve will approach nearer to the proper form.

20. After this familiar statement of the general principle, it is now time to consider the theory founded on it more in detail. This theory aims at such an adjustment of the position of the arch-stones to the load on every part of the arch, that all shall remain in equilibrium, although the joints be perfectly polished and without any cement. The whole may be reduced to two problems. The first is to determine the vertical pressure or load on every point of a line of a given form, which will put that line in equilibrium. The second is to determine the form of a curve which shall be in equilibrium when loaded in its different points, according to any given law.

The whole theory is deducible from one principle, which will be found fully developed in the article Roof. It is this: when an assemblage of beams or other pieces of solid matter AB, BC, CD, DE, fig. 8, freely movable about its angles as so many joints, is retained in equilibrium by the joint effect of the pressures produced by the weight of its parts, the thrust at any angle, if estimated in a horizontal direction, is the same throughout, and may be represented by any horizontal line BT; and that if a vertical line QS be drawn through T, the thrust exerted at any angle D by the piece CD, in its own direction, will then be represented by BR, drawn parallel to CD; and in like manner, that the thrust in the direction ED is represented by BS, &c.; and, lastly, that the vertical thrust or loads at each angle B, C, D, by which all these other pressures are excited, are represented by the portions QC, CR, RS, of the vertical intercepted by those lines; that is, all these pressures are to the uniform horizontal thrust as the lines which represent them are to BT. The horizontal thrust, therefore, is a very proper unit, with which we may compare all the others. Its magnitude is easily deduced from the same proposition; for QS is the sum of all the vertical pressures of the angles, and therefore represents the weight of the whole assemblage. Therefore as QS is to BT, so is the weight of the whole to the horizontal thrust.

21. To accommodate this theory to the construction of a curvilinear arch vault, let us first suppose the vault to be polygonal, composed of the chords of the elementary arches. Let AVE (fig. 12) be a curvilinear arch, of which V is the vertex, and VX the vertical axis, which we shall consider as the axis or abscissa of the curve, while any horizontal line, such as HK, is an ordinate to the curve. About any point C of the curve, as a centre, describe a circle BLD, cutting the curve in B and D. Draw the equal chords CB, CD. Draw also the horizontal line CF, cutting the circle in E. Describe a circle BCDQ passing through B, C, D. Its centre O will be in a line COQ, which bisects the angle BCD; and Cc, which touches this circle in C, will bisect the angle bCd, formed by the equal chords BC, CD. Draw CLP perpendicular to cB, and DP perpendicular to CD, meeting CL in P. Through L draw the tangent GLM, meeting CD in G, and the vertical line CM in M. Draw the tangent Fa, cutting the chords BC, CD, in b and d, and the tangent to the circle BCDQ in e. Lastly, draw dN parallel to bc.

From what will be demonstrated in the article Roof, it appears that if BC, CD be two pieces of an equilibrated heavy polygon, and if CF represent the horizontal thrust in every angle of the polygon, Cd and Cb will severally represent the thrusts exerted by the pieces DC, BC, and that bd, or CN, will represent the weight lying on the angle BCD, by which those thrusts are balanced. In the mean time the reader may, without that article, understand the nature of the equilibrium in the following manner. Produce dC to o, so that Co may be equal to Cd. Draw bo to the vertical parallel to dC, and join ao. It is evident that bnoC is a parallelogram, and that nc (= bcl) = CN. Now the thrust or support of the piece BC is exerted in the direction Cb, while that of DC is exerted in the direction Co. These two thrusts are equivalent to the thrust in the diagonal Cn; and it is with this compound thrust that the load or vertical pressure CN is in immediate equilibrium.

22. Because bCL, NCF are right angles, and FCL is common to both, the angles bCF and MCL are equal; therefore the right-angled triangles bCF and MCL are similar. And since CF is equal to CL, Cb is equal to CM. It is evident that the triangles GCM and dCN are similar. Therefore CG : Cd = CM : CN = Cb : CN. Therefore we have CN = \(\frac{Cb \times Cd}{CG}\). But because CDP and CLG are right angles, and therefore equal, and the angle GCP is common to the two triangles GCL, PCD, and CD is equal to CL, we have CG equal to CP; therefore CN = \(\frac{Cb \times Cd}{CP}\). Also, since CDP is a right angle, DP meets the diameter in Q, the opposite point of the circumference, and the angle DQC is equal to DCe or cCb (because bCd is bisected by the tangent), that is, to PCQ (because the right angles bCP, cCO are equal, and cDP is common). Therefore PQ is equal to PC; and if PO be drawn perpendicular to CQ, it will bisect it; and O is the centre of the circle BCDQB.

Now let the points B and D continually approach to C (by diminishing the radius of the small circle), and ultimately coincide with it. It is evident that the circle BCDQ is ultimately the equicurve circle, and that PC ultimately coincides with OC, the radius of curvature. Also Cb × Cd becomes ultimately Ce². Therefore CN, the vertical load on any point of a curve of equilibration, is \(= \frac{Ce^2}{Rad. Curv.}\).

It is further evident that CF is to Ce as radius to the secant of the elevation of the tangent above the horizon. Therefore we have the load on any point of the curve always proportional to \(= \frac{Rad. Curv.}{Sec. Elev.}\).

This load on every elementary arch of the wall is commonly a quantity of solid matter incumbent on that element of the curve, and pressing it vertically; and it may be conceived as made up of a number of heavy lines standing vertically on it. Thus, if the element Ee of the curve were lying horizontally, a little parallelogram REer standing perpendicularly on it would represent its load. But as this element Ee has a sloping position, it is plain that, in order to have the same quantity of heavy matter pressing it vertically, the height of the parallelogram must be increased till it meets in s, the line Rs drawn parallel to the tangent EG. It is evident that the angle REg is equal to the angle AEG. Therefore we have ER : Eg = Rad. : Sec. Elev.

If therefore the arch is kept in equilibrium by the vertical pressure of a wall, we must have the height of the wall above any point proportional to \(= \frac{Rad. of Curv.}{Sec. Elev.}\).

23. Corol. I. If OS be drawn perpendicular to the Car... vertical CS, CS will be half the vertical chord of the equi-curve circle. The angle OCS is equal to eCF, that is, to the angle of elevation. Therefore \( \frac{1}{2} \text{Sec. Elev.} = \frac{CS}{CO} \), and the secant of elevation may be expressed by \( \frac{CO}{CS} \) and its cube by \( \frac{CO^3}{CS^3} \). Therefore the height of wall is proportional to \( \frac{CO^2}{CS^2} \) or to \( \frac{CO^2}{CS^2} \) or \( \frac{CO^2}{CS^2 \times CS} \), or to \( \frac{\text{Sec}^2 \text{Elev.}}{\text{Vert. Chord of Curve}} \).

Corol. 2. If we make the arch VC = z, the abscissa VH = x, the ordinate HC = y, the radius osculi CO = r, and the \( \frac{1}{2} \) vertical chord CS = s, the height of wall pressing on any point is proportional to \( \frac{dz^2}{dy^2} \); or to \( \frac{dz^2}{dy^2} \); or \( \frac{dz^2 + dy^2}{dy^2} \). Therefore, when the equation of the curve is given, and the height of wall on any one point of it is also given, we can determine it for any other point; for the equation of the curve will always give us the relation of \( dz, dx, dy \), and the value of \( r \) or \( s \). This may be illustrated by an example or two. For this purpose it will generally be most convenient to assume the height above the vertex V for the unit of computation. The thickness of the arch at the crown is commonly determined by other circumstances. At the vertex the tangent to the arch is horizontal, and therefore the cube of the secant is unity or 1. Call the height of wall at the crown H, and let the radius of curvature in that point be R, and its half-chord R (it being then coincident with the radius), and the height on any other point h; we have \( \frac{1}{R} : \frac{dz^2}{dy^2} = H : h \), and \( h = H \times \frac{dz^2}{dy^2} \times \frac{R}{r} \). The other formula gives \( h = H \times \frac{dz^2}{dy^2} \times \frac{R}{s} \).

Ex. 1. Suppose the arch to be a segment of a circle, as in fig. 10, where AE is the diameter, and O the centre. In this arch the curvature is the same throughout, or \( \frac{R}{x} = 1 \). Therefore \( h = H \times \frac{dz^2}{dy^2} \) or \( H \times \text{Cube Sec. Elev.} \).

This gives a very simple calculus. To the logarithm of H add thrice the logarithm of the secant of elevation. The sum is the logarithm of h.

It gives also a very simple construction. Draw the vertical CS, cutting the horizontal diameter in S. Draw ST, cutting the radius OC perpendicularly in T. Draw the horizontal line TZ, cutting the vertical in z. Join zO. Make Cc = Ve, and draw ux parallel to zO; Cc must be made = Cz. The demonstration is evident.

It is very easy to see that if CV is an arch of 60°, and Ve is \( \frac{1}{4} \)th of VO, the points v and c will be on a level; for the secant of CV is twice CO, and therefore Cc is eight times Ve, which is \( \frac{1}{4} \)th of VH.

The dotted line rcf is drawn according to this calculus or construction. It falls considerably below the horizontal line in the neighbourhood of c; and then, passing very obliquely through c, it rises rapidly to an immeasurable height, because the vertical line through A is its asymptote. This must evidently be the case with every curve which springs at right angles with a horizontal line.

It is plain that if ev be greater, all the other ordinates of the curve rcf, resting on the circumference AVE, will be greater in the same proportion, and the curve will cut the horizontal line drawn through v in some point nearer to v than c. Hence it appears that a circular arch cannot be put in equilibrium by building it up to a horizontal line, whatever be its span, or whatever be the thickness at the crown. We have seen that when this thickness is only \( \frac{1}{4} \)th of the radius, an arch of 120 degrees will be too much loaded at the flanks. This thickness is much too small for a bridge, being only \( \frac{1}{4} \)th of the span CM, whereas it should have been almost double of this, to bear the inequalities of weight that may occasionally be on it. When the crown is made still thinner, the outline is still more depressed before it rises again. There is therefore a certain span, with a corresponding thickness at the crown, which will deviate least of all from a horizontal line. This is an arch of about 45 degrees, the thickness at the crown being about one fourth of the span, which is extravagantly great. It appears in general, therefore, that the circle is not a curve suited to the purposes of a bridge or an arcade, which requires an outline nearly horizontal.

Ex. 2. Let the curve be a parabola AVE (fig. 14), of which V is the vertex, and DG the directrix. Draw the diameters DCF, GVN, the tangents CK, VP, and the ordinates VF and CN. It is well known that GV is to DC as VP² to CK², or as CN² to CK². Also 2 GV is the radius of the osculating circle at V, and 2 DC is one half of the vertical chord of the osculating circle at C.

Therefore CN² : CK² (or \( \frac{dy^2}{dz^2} : \frac{dz^2}{dy^2} \)) = R : s; and \( s = \frac{dz^2}{dy^2} \times R \). But Cc; or \( h = H \times \frac{dz^2}{dy^2} \times \frac{dz^2}{dy^2} \). Therefore \( h = H \times \frac{dz^2}{dy^2} \times \frac{dz^2}{dy^2} = H \times \frac{dz^2}{dy^2} \times \frac{dz^2}{dy^2} \). Therefore Cc = vV.

It follows from this investigation, that the back or extrados of a parabolic arch of equilibrium must be parallel to the arch or soffit itself; or that the thickness of the arch, estimated in a vertical direction, must be equal throughout; or that the extrados is the same parabola with the soffit or intrados.

We have selected these two examples merely for the simplicity and perspicuity of the solutions, which have been effected by means of elementary geometry only, instead of employing the analytical value of the radius of the osculatory circle, viz. \( \frac{dy^2}{dz^2} - \frac{dz^2}{dy^2} \), which would have involved us at last in the doctrine of second fluxions. We have also preferred simplicity to elegance in the investigation, because we wish to instruct the practical engineer who may not be a proficient in the higher mathematics.

25. The converse of the problem, namely, to find the form of the arch when the figure of the back of it is given, is the most usual question of the two, at least in cases where which are most important and most difficult. Of these, perhaps, bridges are the chief. Here the necessity of a road-way, of easy and regular ascent, confines us to an outline nearly horizontal, to which the curve of the arch must be adapted. This is the most difficult problem of the two; and we doubt whether it can be solved without employing infinite approximating series instead of accurate values.

Let arc (fig. 9) be the intended outline or extrados of the arch AVE, and let vQ be the common axis of both curves. From c and C, the corresponding points, draw the ordinates ch, CH. Let the thickness ev at the top be a, the abscissa vh be u, and VH = x, and let the equal ordinates ch, CH, be y, and the arch VC be z. Then, by the general theorem, \( cC = \frac{d^3}{dy^3} r \) being the radius of curvature. This, by the common rules, is

\[ \frac{d^3}{dy^3} x - \frac{d^2}{dy^2} y = \frac{dy^2}{dx} \times C \]

where \( C \) is a constant quantity, found by taking the real value of \( cC \) in \( V \), the vertex of the curve. But it is evident that it is also \( a + x - u \).

Therefore \( a + x - u = \frac{dy^2}{dx} - \frac{d^2}{dy^2} y \times C = \frac{C}{dy} \times \text{fluxion of } \frac{dx}{dy} \)

If we now substitute the true value of \( u \) (which is given because the extrados is supposed to be of a known form), expressed in terms of \( y \), the resulting equation will contain nothing but \( x \) and \( y \), with their first and second fluxions, and known quantities. From this equation the relation of \( x \) and \( y \) must be found by such methods as seem best adapted to the equation of the extrados.

Fortunately the process is more simple and easy in the most common and useful case than we should expect from this general rule; we mean the case where the extrados is a straight line, especially when this is horizontal. In this case \( u \) is equal to \( a \).

Ex. To find the form of the balanced arch AVE, having the horizontal line \( cv \) for its extrados.

Keeping the same notation, we have \( u = a \), and therefore

\[ a + x = \frac{C}{dy} \times \text{fluxion of } \frac{dx}{dy} \]

Assume \( dy = dx \); then \( \frac{dx}{dy} = v \), and \( \frac{C}{dy} \times \text{fluxion of } \frac{dx}{dy} = \frac{Cv}{dx} \), that is, \( a + x = \frac{Cv}{dx} \). Therefore \( adx + xdx = Cvdx \); and by taking the fluents, we have

\[ 2ax + x^2 = C^2v^2; \quad v = \sqrt{\frac{2ax + x^2}{C}} \]

Consequently,

\[ dy = \frac{\sqrt{Cvx}}{\sqrt{2ax + x^2}} \left( \text{being } \frac{dx}{v} \right) \]

Taking the fluent of this, we have \( y = \sqrt{C} \times L \left( 2a + 2x + \sqrt{2ax + x^2} \right) \). But at the vertex, where \( x = 0 \), we have \( y = \sqrt{C} \times L \left( 2a \right) \). The corrected fluent is therefore \( y = \sqrt{C} \times L \left( a + x + \sqrt{2ax + x^2} \right) \).

It only remains to find the constant quantity \( C \). This we readily obtain by selecting some point of the extrados where the values of \( x \) and \( y \) are given by particular circumstances of the case. Thus, when the span \( s \) and height \( h \) of the arch are given, we have

\[ s = \sqrt{C} \times L \left( a + h + \sqrt{2ah + h^2} \right) \]

and consequently \( \sqrt{C} = \frac{s}{L \left( a + h + \sqrt{2ah + h^2} \right)} \).

The general value of \( y = s \times \frac{L \left( a + x + \sqrt{2ax + x^2} \right)}{a} \)

\[ = \frac{s}{L \left( a + h + \sqrt{2ah + h^2} \right)} \times L \left( a + x + \sqrt{2ax + x^2} \right) \]

26. As an example of the use of this formula, we subjoin a table calculated by Dr Hutton of Woolwich, for an arch, the span of which is 100 feet, and the height 40, which are nearly the dimensions of the middle arch of Blackfriars Bridge in London.

| \( y \) | \( x \) | \( y \) | \( x \) | \( y \) | |-------|-------|-------|-------|-------| | 0 | 6,000 | 21 | 10,381| 36 | | 2 | 6,035 | 22 | 10,858| 37 | | 4 | 6,144 | 23 | 11,368| 38 | | 6 | 6,324 | 24 | 11,911| 39 | | 8 | 6,580 | 25 | 12,459| 40 | | 10 | 6,914 | 26 | 13,106| 41 | | 12 | 7,330 | 27 | 13,761| 42 | | 13 | 7,571 | 28 | 14,457| 43 | | 14 | 7,834 | 29 | 15,196| 44 | | 15 | 8,120 | 30 | 15,980| 45 | | 16 | 8,430 | 31 | 16,811| 46 | | 17 | 8,766 | 32 | 17,693| 47 | | 18 | 9,168 | 33 | 18,627| 48 | | 19 | 9,517 | 34 | 19,617| 49 | | 20 | 9,934 | 35 | 20,665| 50 |

The figure for this proposition is exactly drawn according to these dimensions, that the reader may judge of it as an object of sight. It is by no means deficient in gracefulness, and is abundantly roomy for the passage of craft; so that no objection can be offered against its being adapted on account of its mechanical excellency.

The reader will perhaps be surprised that we have made no mention of the celebrated Catenarian curve, which is commonly said to be the best form for an arch; but a little reflection will convince him, that although it is the only form for an arch consisting of stones of equal weight, and touching each other only in single points, it cannot suit an arch which must be filled up in the haunches, in order to form a road-way. He will be more surprised to hear, after this, that there is a certain thickness at the crown, which will put the Catenarian in equilibrio, even with a horizontal road-way; but this thickness is so great as to make it unfit for a bridge, being such that the pressure at the vertex is equal to the horizontal thrust. This would have been about 37 feet in the middle arch of Blackfriars Bridge. The only situation, therefore, in which the Catenarian form would be proper, is an arcade carrying a height of dead wall; but in this situation it would be very ungraceful. Without troubling the reader with the investigation, it is sufficient to inform him, that in a Catenarian arch of equilibration the abscissa VH is to the abscissa \( vh \) in the constant ratio of the horizontal thrust to its excess above the pressure on the vertex.

27. Thus much will serve, we hope, to give the reader a clear notion of this celebrated theory of the equilibrium of arches, one of the most delicate and important applications of mathematical science. Volumes have been written on the subject, and it still occupies the attention of mechanicians. But we beg leave to say, with great deference to the eminent persons who have prosecuted this theory, that their speculations have been of little service, and are little attended to by the practitioner. Nay, we may add, that Sir Christopher Wren, perhaps the most accomplished architect that Europe has seen, seems to have thought it of little value; for, among the fragments which have been preserved of his studies, there are to be seen some imperfect dissertations on this very subject, in which he takes no notice of this theory, and considers the balance of arches in quite another way. These are collected by the author of the account of Sir Christopher Wren's family. This man's great sagacity, and his great experience in building, and still more his experience in the repairs of old and crazy fabrics, had shown him many things very inconsistent with this theory, which appears so specious and safe. The general facts which occur in the failure of old arches are highly instructive; and deserve the most careful attention of the engineer; for it is in this state that their defects, and the process of nature in their destruction, are most distinctly seen. We venture to affirm, that a very great majority of these facts are irreconcilable to the theory. The way in which circular arches commonly fail, is by the sinking of the crown and the rising of the flanks. It will be found by calculation, that in most of the cases it ought to have been just the contrary. But the clearest proof is, that arches very rarely fail where their load differs most remarkably from that which this theory allows. Semi-circular arches have stood the power of ages, as may be seen in the bridges of ancient Rome, and in the numerous arcades which the ancient inhabitants have erected. Now, all arches which spring perpendicularly from the horizontal line require, by this theory, a load of infinite height; and even to a considerable distance from the springing of the arch, the load necessary for the theoretical equilibrium is many times greater than what is ever laid on those parts; yet a failure in the immediate neighbourhood of the spring of an arch is a most rare phenomenon, if it ever was observed. Here is a most remarkable deviation from the theory; for, as is already observed, the load is frequently not the fourth part of what the theory requires.

23. Many other facts might be adduced which show great deviation from the legitimate results of the theory. We hope to be excused, therefore, by the mathematicians for doubting of the justness of this theory. We do not think it erroneous, but defective, leaving out circumstances which we apprehend to be of great importance; and we imagine that the defects of the theory have arisen from the very anxiety of the mechanicians to make it perfect. The arch-stones are supposed to be perfectly smooth or polished, and not to be connected by any cement, and therefore to sustain each other merely by the equilibrium of their vertical pressure. The theory insures this equilibrium, and this only, leaving unnoticed any other causes of mutual action.

The authors who have written on the subject say expressly that an arch which thus sustains itself must be stronger than another which would not; because when, in imagination, we suppose both to acquire connection by cement, the first preserves the influence of this connection unimpaired; whereas in the other, part of the cohesion is wasted in counteracting the tendency of some parts to break off from the rest by their want of equilibrium. This is a very specious argument, and would be just, if the forces which are mutually exerted between the parts of the arch in its settled state were merely vertical pressures, or, where different, were inconsiderable in comparison with those which are really attended to in the construction.

But this is by no means the case. The forms which the uses for which arches are erected oblige us to adopt, and the loads laid on the different points of the arch, frequently deviate considerably from what are necessary for the equilibrium of vertical pressures. The varying load on a bridge, when a great waggon passes along it, sometimes bears a very sensible proportion to the weight of that point of the arch on which it rests. It is even very doubtful whether the pressures which are occasioned by the weight of the stuff employed for filling up the flanks really act in a vertical direction, and in the proportion which is supposed. We are pretty certain that this is not the case with sand, gravel, fat mould, and many substances in very general use for this purpose. When this is the case, the pressures sustained by the different parts of the arch are often very inconsistent with the theory; a part of the arch is overloaded and tends to fall in, but is prevented by the cement. This part of the arch, therefore, acts on the remoter parts by the intervention of the parts between, employing those intermediate parts as a kind of levers to break the arch in a remote part, just as a lintel would be broken. We apprehend that a mathematician would be puzzled how to explain the stability of an arch cut out of a solid and uniform mass of rock. His theory considers the mutual thrusts of the arch-stones as in the direction of the tangents to the arch. Why so? Because he supposes that all his polished joints are perpendicular to those tangents. But in the present case he has no existing joints; and there seems to be nothing to direct his imagination in the assumption of joints, which, however, are absolutely necessary for employing his theory, because, without a supposition of this kind, there seems no conceiving any mutual abutment of the arch-stones.

Ask a common but intelligent mason, what notion he forms of such an arch? We apprehend that he will consider it as no arch, but as a lintel, which may be broken like a wooden lintel, and which resists entirely by its cohesion. He will not readily conceive that, by cutting the under side of a stone lintel into an arched form, and thus taking away more than half of its substance, he has changed its nature of a lintel, or given it any additional strength. Nor would there be any change made in the way in which such a mass of stone would resist being broken down, if nothing were done but forming the under side into an arch. If the lintel be so laid on the piers that it can be broken without its parts pushing the piers aside (which will be the case if it lies on the piers with horizontal joints), it will break like any other lintel; but if the joints are directed downwards, and converging to a point within the arch, the broken stone (suppose it broken at the crown by an overload in that part) cannot be pressed down without forcing the piers outwards. Now, in this mode of acting, the mind cannot trace any thing of the statical equilibrium that we have proceeded on in the foregoing theory. The two parts of the broken lintel seem to push the piers aside in the same manner that two rafters push outwards the walls of a house when their feet are not held together by a tie-beam. If the piers cannot be pushed aside (as when the arch abuts on two solid rocks), nothing can press down the crown which does not crush the stone.

This conclusion will be strictly true, if the arch is of such a form that a straight line drawn from the crown to the pier lies wholly within the solid masonry. Thus, if the vault consist of two straight stones, as in Plate LII. fig. 1, or if it consist of several stones, as in Plate LIII. fig. 7, disposed in two straight lines, no weight laid on the crown can destroy it in any other way than by crushing it to powder.

29. But when straight lines cannot be drawn from the overloaded part to the firm abutments through the solid masonry, and when the cohesion of the parts is not able to withstand the transverse strains, we must call the principles of equilibrium to our aid; and, in order to employ them with safety, we must consider how they are modified by the excitement of the cohering forces.

The cohesion of the stones with each other by cement, or otherwise has in almost every situation a bad effect. It enables an overload at the crown to break the arch near the haunches, causing those parts to rise, and then to spread outwards, just as a Mansarde or Kirb roof would do if the truss-beam which connects the heads of the lower rafters were sawn through. This can be prevented only by loading that part more than is requisite for equilibrium. It would be prudent to do this to a certain degree, because it is by this cohesion that the crown always becomes the weakest part of the arch, and suffers more by any occasional load.

We expect that it will be said in answer to all this, that the cohesion given by the strongest cement that we can employ, nay the cohesion of the stone itself, is a mere nothing in comparison with the enormous thrusts that are in a state of continual exertion in the different parts of an arch. This is very true; but there is another force which produces the same effect, and which increases nearly in the proportion that those thrusts increase, because it arises from them. This is the friction of the stones on each other. In dry freestone this friction considerably exceeds one half of the mutual pressure. The reflecting reader will see that this produces the same effect, in the case under consideration, that cohesion would do; for while the arch is in the act of failing, the mutual pressure of the arch-stones is acting with full force, and thus produces a friction more than adequate to all the effects we have been speaking of.

30. When these circumstances are considered, we imagine that it will appear that an arch, when exposed to a great overload on the crown (or indeed on any part), divides itself into a number of parts, each of which contains as many arch-stones as can be pierced (so to speak) by one straight line, and that it may then be considered as nearly in the same situation with a polygonal arch of long stones abutting on each other like so many beams in a Norman roof; but without their braces and ties. It tends to break at all those angles; and it is not sufficiently resisted there, because the materials with which the flanks are filled up have so little cohesion, that the angle feels no load except what is immediately above it; whereas it should be immediately loaded with all the weight which is diffused over the adjoining side of the polygon. This will be the case, even though the curvilinear arch be perfectly equilibrated. We recollect some circumstances in the failure of a considerable arch, which may be worth mentioning. It had been built of an exceedingly soft and friable stone, and the arch-stones were too short. About a fortnight before it fell, chips were observed to be dropping off from the joints of the arch-stones, about ten feet on each side of the middle, and also from another place on one side of the arch, about twenty feet from its middle. The masons in the neighbourhood prognosticated its speedy downfall, and said that it would separate in those places where the chips were breaking off. At length it fell; but it first split in the middle, and about fifteen or sixteen feet on each side, and also at the very springing of the arch. Immediately before the fall a shivering or cracking noise was heard, and a great many chips dropped down from the middle, between the two places from whence they had dropped a fortnight before. The joints opened above at those new places above two inches, and in the middle of the arch the joints opened below, and in about five minutes after this the whole came down. Even this movement was plainly distinguishable into two parts. The crown sunk a little, and the haunches rose very sensibly, and in this state it hung for about half a minute. The arch-stones of the crown were hanging by their upper corners: when these splintered off, the whole fell down.

We apprehend that the procedure of nature was somewhat in this manner. Straight lines can be drawn within the arch-stones from A (Plate LIII. fig. 8) to B and D, and from these points to C and E. Each of the portions ED, DA, AB, BC, resist as if they were of one stone, composing a polygonal vault EDABC. When this is overloaded at A, A can descend in no other way than by pushing the angles B and D outwards, causing the portions BC, DE, to turn round C and E. This motion must raise the points B and D, and cause the arch-stones to press on each other at their inner joints b and d. This produced the copious splintering at those joints immediately preceding the total downfall. The splintering which happened a fortnight before arose from this circumstance, that the lines AB and AD, along which the pressure of the overload was propagated, were tangents to the soffit of the arch in the points F, H, and G, and therefore the strain lay all on those corners of the arch-stones, and splintered a little from off them till the whole took a firmer bed. The subsequent phenomena are evident consequences of this distribution and modification of pressure, and can hardly be explained in any other way, at least not on the theoretical principles already set forth; for in this bridge the loads at B and D were very considerably greater than what the equilibrium required; and we think that the first observed splintering at H, F, and G, was most instructive, showing that there was an extraordinary pressure at the inner joints in those places, which cannot be explained by the usual theory.

Not satisfied with this single observation after this way of explaining it occurred to us, and not being able to find any similar fact on record, the writer of this article got some small models of arches executed in chalk, and subjected them to many trials, in hopes of collecting some general laws of the internal workings of arches which finally produce their downfall. He had the pleasure of observing the above-mentioned circumstances take place very regularly and uniformly when he overloaded the models at A. The arch always broke at some place B considerably beyond another point F, where the first chipping had been observed. This is a method of trial that deserves the attention both of the speculative and the practitioner.

If these reflections are anything like a just account of the procedure of nature in the failure of an arch, it is evident that the ingenious mathematical theory of equilibrated arches is of little value to the engineer. We ventured to say as much already, and we rested a good deal on the authority of Sir Christopher Wren. He was a good mathematician, and delighted in the application of this science to the arts. He was a celebrated architect, and his reports on the various works committed to his charge show that he was in the continued habit of making this application. Several specimens remain of his own methods of applying them. The roof of the theatre of Oxford, the roof of the cupola of St Paul's, and in particular the mould on which he turned the inner dome of that cathedral, are proofs of his having studied this theory most attentively. He flourished at the very time that it occupied the attention of the greatest mechanicians of Europe; but there is nothing to be found among his papers which shows that he had paid much regard to it. On the contrary, when he has occasion to deliver his opinion for the instruction of others, and to explain to the dean and chapter of Westminster his operations in raising that collegiate church, this great architect considers an arch just as a sensible and sagacious mason would do, and very much in the way that we have just now been treating it. (See Account of the Family of Wren, p. 856.) Supported therefore by such authority, we would recommend this way of considering an arch to the study of the mathematician; and we would desire the experienced mason to think of the most efficacious methods for resisting this tendency of arches to rise in the flanks. Unfortunately there seems to be no precise principle to point out the place where this tendency is most remarkable.

31. We are therefore highly pleased with the ingenious contrivance of Mr Mylne, the architect of Blackfriars Bridge in London, by which he determines this point with precision, by making it impossible for the overloaded arch to spring in any other place. Having thus confined the failure to a particular spot, he with equal art opposes a resistance which he believes to be sufficient; and the present condition of that noble bridge, which does not in any place show the smallest change of shape, proves that he was not mistaken. Looking on this work as the first, or at least the second, specimen of masonic ingenuity that is to be seen in the world, we imagine that our readers will be pleased with a particular account of its most remarkable circumstances.

The span \( k \) (fig. 1) of the middle arch is 100 feet, and its height \( OV \) is 40, and the thickness \( KV \) of the crown is six feet seven inches. Its form is nearly elliptical; the part \( AVZ \) being an arch of a circle whose centre is \( C \), and radius 56 feet, and the two lateral portions \( A \) and \( B \) and \( Z \) and \( E \) being arches described with a radius of 35 feet nearly. The thickness of the pier at \( ab \) is 19 feet. The thickness of the arch increases from the crown \( V \) to \( Y \), where it is eight or nine feet. All the arch-stones have their joints directed to the centres of their curvature. The joints are all joggled, having a cubic foot of hard stone let half-way into each. By this contrivance the joints cannot slide, nor can any weight laid on the crown ever break the arch in that part if the piers do not yield; for a straight line from the middle of \( KV \) to the middle of the joint \( YI \) is contained within the solid masonry, and does not even come near the inner joints of the arch-stones; therefore the whole resists like one stone, and can be only broken by crushing it. The joint at \( Z \) is very nearly perpendicular to a line \( ZF \) drawn to the outer edge of the foundation of the pier. By this it was intended to take off all tendency of the pressure on the joint \( dZ \) to overset the pier; for if we suppose, according to the theory of equilibration, that this pressure is necessarily exerted perpendicularly to the joint, its direction passes through the fulcrum at \( F \); round which it is thought that the pier must turn in the act of oversetting. This precaution was adopted in order to make the arch quite independent of the adjoining arches; so that although any of them should fall, this arch should run no risk.

Still farther to secure the independence of the arch, the following construction was practised to unite it into one mass, which should rise all together. All below the line \( ab \) is built of large blocks of Portland stone, dovetailed with sound oak. Four places in each course are interrupted by equal blocks of a hard stone called Kentish rag, sunk half-way in each course. These act as juggles, breaking the courses, and preventing them from sliding laterally.

The portion \( aY \) of the arch is joggled like the upper part. The interior part is filled up with large blocks of Kentish rag, forming a kind of coursed rubble-work, the courses tending to the centres of the arch. The under corner of each arch-stone projects over the one below it. By this form it takes fast hold of the rubble-work behind it. Above this rubble there is constructed the inverted arch \( IeG \) of Portland stone. This arch shares the pressure of the two adjoining arches, along with the arch-stones in \( Ya \) and in \( Gb \). Thus all tend together to compress and keep down the rubble-work in the heart of this part of the pier. This is a very useful precaution; for it often happens, that when the centres of the arches are struck before the piers are built up to their intended heights, the thrust of the arches squeezes the rubble-work horizontally, after the mortar has set, but before it has dried and acquired its utmost hardness. Its bond is broken by this motion, and it is squeezed up, and never acquires its former firmness. This is effectually prevented by the pressure exerted by the back of the inverted arch.

Above this counter-arch is another mass of coursed rubble, and all is covered by a horizontal course of large blocks of Portland stone, abutting against the back of the arch-stone \( ZI \) and its corresponding one in the adjoining arch. This course connects the feet of the two arches, preserves the rubble-work from too great compression, and protects it from soaking water. This last circumstance is important; for if the water which falls on the road-way is not carried off in pipes, it soaks through the gravel or other rubbish, rests on the mortar, and keeps it continually wet and soft. It cannot escape through the joints of good masonry, and therefore fills up this part like a funnel.

Supposing the adjoining arch fallen, and all tumbled off that is not withheld by its situation, there will still remain in the pier a mass of about 3500 tons. The weight of the portion \( VY \) is about 2000 tons. The directions of the thrusts \( VY \) and \( YF \) are such, that it would require a load of 4500 tons on \( VY \) to overturn the pier round \( F \). This exceeds \( VY \) by 2500 tons—a weight incomparably greater than any that can ever be laid on it.

Such is the ingenious construction of Mr Mylne. It evidently proceeds on the principles recommended above—principles which had occurred to his experience and sagacious mind during the course of his extensive practice. We have seen attempts by other engineers to withstand the horizontal thrusts of the arch by means of counter-arches inserted in the same manner as here, but extending much farther over the main arch; but they did not appear to be well calculated for producing this effect. A counter-arch springing from any point between \( Y \) and \( V \) has no tendency to hinder that point from rising by the sinking of the crown; and such a counter-arch will not resist the precisely horizontal thrust so well as the straight course of Mr Mylne.

32. The great incorporation of architects who built the Origin of cathedrals of Europe departed entirely from the styles of the Gothic ancient Greece and Rome, and introduced another, in arches, which arcades made the principal part. Not finding in every place quarries from which blocks could be raised in abundance of sufficient size for forming the far-projecting cornices of the Greek orders, they relinquished those proportions, and adopted a style of ornament which required no such projections; and having substituted arches for the horizontal architrave or lintel, they were now able to erect buildings of vast extent with spacious openings, and all this with very small pieces of stone. The form which had been adopted for a Christian temple occasioned many intersections of vaultings, and multiplied the arches exceedingly. Constant practice gave opportunities of giving every possible variety of these intersections, and taught the art of balancing arch against arch in every variety of situation. An art so multifarious, and so much out of the road of ordinary thought, could not but become an object of fond study to the architects most eminent for ingenuity and invention. Becoming thus the dupes of

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1 We know from good authority that the counter-arch here spoken of, although originally intended, was never executed, because it was not thought necessary. The notion was, however, excellent, and it has, we believe, been actually executed in the Strand Bridge. We rather think the joggling was also abandoned, and, as far as we can judge, was not likely to be of any use. their own ingenuity, they were fond of displaying it even when not necessary. At last arches became their principal ornament, and a wall or ceiling was not thought dressed out as it should be till filled full of mock arches, crossing and abutting on each other in every direction. In this process in their ceilings they found that the projecting mouldings, which we now call the Gothic tracery, formed the chief supports of the roofs. The plane surfaces included between those ribs were commonly vaulted with very small stones, seldom exceeding six or eight inches in thickness. This tracery, therefore, was not a random ornament. Every rib had a position and direction that was not only proper, but even necessary. Habituuated to this scientific arrangement of the mouldings, they did not deviate from it when they ornamented a smooth surface with mock arches; and in none of the highly ornamented ancient buildings will we find any false positions.

33. This is by no means the case in many of the modern imitations of Gothic architecture, even by our best architects. Ignorant of the directing principle, or not attending to it, in their stucco-work they please the unskilled eye with pretty radiated figures; but in these we frequently see such abutments of mouldings as would infallibly break the arches, if these mouldings were really performing their ancient office, and supporting a vaulting of considerable extent. Nay, this began even before the Gothic style was finally abandoned. Several instances are to be found in the highly enriched vaultings of New College and Christ Church in Oxford, in St George's Chapel at Windsor, and Henry VII's Chapel in Westminster.

We call the middle ages rude and barbarous; but there was surely much knowledge in those who could execute such magnificent and difficult works. The working drafts which were necessary for such varieties of oblique intersections must have required considerable skill, and would at present occupy many very expensive volumes of Masons' Jewels, Carpenters' Manuals, and the like. All this knowledge was kept a profound secret by the corporation, and on its breaking up we had all to learn again.

34. There is no appearance, however, that those architects had studied the theory of equilibrated arches. They had adopted an arch which was very strong, and permitted considerable irregularities of pressure—we mean the pointed arch. The very deep mouldings with which it was ornamented made the arch-stones very long in proportion to the span of the arch. But they had studied the mutual thrust of arches on each other with great care; and they contrived to make every invention for this purpose become an ornament, so that the eye required it as a necessary part of the building. Thus we frequently see small buildings having buttresses at the sides. These are necessary in a large vaulted building, for withstanding the outward thrust of the vaulting; but they are useless when we have a flat ceiling within. Pinnacles on the heads of the buttresses are now considered as ornaments, but originally they were put there to increase the weight of the buttress: even the great tower in the centre of a cathedral, which now constitutes its greatest ornament, is a load almost indispensably necessary, for enabling the four principal columns to withstand the combined thrust of the aisles, of the nave, and transepts. In short, the more closely we examine the ornaments of this architecture, the more shall we perceive that they are essential parts, or derived from them, by imitation; and the more we consider the whole style of it, the more clearly do we see that it is all deduced from the relish for arcades, indulged in the extreme, and pushed to the limit of possibility of execution.

35. There is another species of arch which must not be overlooked, namely, the Dome or Cupola, with all its varieties, which include even the pyramidal steeple or spire.

It is evident that the erection of a dome is also a scientific art, proceeding on the principles of equilibration; and that these principles admit and require the same or similar modifications, in consequence of the cohesion and friction of the materials. At first sight, too, a dome appears a more difficult piece of work than a plain arch; but when we observe potters' kilns, and glasshouse domes, and cones of vast extent, erected by ordinary bricklayers, and with materials vastly inferior in size to what can be employed in common arches of equal extent, we must conclude that the circumstance of curvature in the horizontal direction, or the abutment of a circular base, gives some assistance to the artist. Of this we have complete demonstration in the case of the cone. We know that a vaulting in the form of a pent-roof could not be executed to any considerable extent, and would be extremely hazardous, even in the smallest dimensions; while a cone of the greatest magnitude can be raised with very small stones, provided only that we prevent the bottom from flying out, by a hoop, or any similar contrivance.

36. When we think a little of the matter, we see plainly, that if the horizontal section be perfectly round, and the joints be all directed to the axis, they all equally endeavour to slide inwards, while no reason can be offered why any individual stone should prevail. They are all wedges, and operate only as wedges. When we consider any single course, therefore, we see that it cannot fail in, even though it may be part of a curve which cannot stand as a common arch; nay, we see that a dome may be constructed having the convexity of the curve, by the revolution of which it is formed, turned towards the axis, so that the outline is concave. We shall afterwards find that this is a stronger dome by far than if the convexity were outwards, as in a common arch. We see also that a cone may be loaded on the top with the greatest weight, without the smallest danger of forcing it down, so long as the bottom course is firmly kept from bursting outwards. The stone lanthorn on the top of St Paul's cathedral in London weighs several hundred tons, and is carried by a brick cone of 18 inches thick, with perfect safety, as long as the bottom course is prevented from bursting outwards. The reason is evident: The pressure on the top is propagated along the cone in the direction of the slant side; and, so far from having any tendency to break it in any part, it tends rather to prevent its being broken by any irregular pressure from foreign causes.

37. For the same reasons the octagonal pyramids, which form the spires of Gothic architecture, are abundantly firm, although very thin. The sides of the spire of Salisbury cathedral are not eight inches thick after the octagon is fully formed. It is proper, however, to direct the joints to the axis of the pyramid, and to make the coursing joints perpendicular to the slant side, because the projecting mouldings which run along the angles are the abutments on which the whole panel depends. A considerable art is necessary for supporting those panels or sides of the octagon which spring from the angles of the square tower. This is done by beginning a very narrow pointed arch on the square tower at a great distance below the top; so that the legs of the arch being very long, a straight line may be drawn from the top of the keystone of the arch through the whole arch-stones of the legs. By this disposition the thrusts arising from the weight of these four panels are made to meet on the massive masonry in the middle of the sides of the tower, at a great distance below the springing of the spire. This part, being loaded with the great mass of perpendicular wall, is fully able to withstand the horizontal thrust from the legs of these arches. In many spires these thrusts are still further resisted by iron bars which cross the tower, and are hooked into pieces of brass firmly bedded in the masonry of the sides.

38. There is much nice balancing of this kind to be observed in the highly ornamented open spires; such as those of Brussels, Mecklin, Antwerp, &c. We have not many of this sort in Britain. In those of great magnitude, the judicious eye will discover, that parts, which a common spectator would consider as mere ornaments, are necessary for completing the balance of the whole. Tall pinnacles, nay even pillars carrying entablatures and pinacles, are to be seen standing on the middle of the slender leg of an arch. On examination we find that this is necessary, to prevent the arch from springing upwards in that place by the pressure at the crown. The steeple of the cathedral of Mecklin was the most elaborate piece of architecture in this taste in the world, and was really a wonder; but it was not calculated to withstand a bombardment, which destroyed it in 1578.

Such frequent examples of irregular and whimsical buildings of this kind show that great liberties may be taken with the principle of equilibration without risk, if we take care to secure the base from being thrust outwards. This may always be done by hoops, which can be concealed in the masonry; whereas in common arches these ties would be visible, and would offend the eye.

39. It is now time to attend to the principle of equilibrium as it operates in a simple circular dome, and to determine the thickness of the vaulting when the curve is given, or the curve when the thickness is given. Therefore, let $b \cdot A$ (Plate LIII. fig. 2) be the curve which produces the dome by revolving round the vertical axis AD. We shall suppose this curve to be drawn through the middle of all the arch-stones, and that the coursing or horizontal joints are everywhere perpendicular to the curve. We shall suppose (as is always the case) that the thickness KL, HI, &c. of the arch-stones is very small in comparison with the dimensions of the arch. If we consider any portion HA h of the dome, it is plain that it presses on the course, of which HL is an arch-stone, in a direction bC perpendicular to the joint HI, or in the direction of the next superior element $\beta b$ of the curve. As we proceed downwards, course after course, we see plainly that this direction must change, because the weight of each course is superadded to that of the portion above it, to complete the pressure on the course below. Through B draw the vertical line BCG, meeting $\beta b$, produced in C. We may take bC to express the pressure of all that is above it, propagated in this direction to the joint KL. We may also suppose the weight of the course HL united in b, and acting on the vertical. Let it be represented by bF. If we form the parallelogram bFGC, the diagonal bG will represent the direction and intensity of the whole pressure on the joint KL. Thus it appears that this pressure is continually changing its direction, and that the line, which will always coincide with it, must be a curve concave downward. If this be precisely the curve of the dome, it will be an equilibrated vaulting; but so far from being the strongest form, it is the weakest, and it is the limit to an infinity of others, which are all stronger than it. This will appear evident, if we suppose that bG does not coincide with the curve A bB, but passes without it. As we suppose the arch-stones to be exceedingly thin from inside to outside, it is plain that this dome cannot stand, and that the weight of the upper part will press it down, and spring the vaulting outwards at the joint KL. But let us suppose, on the other hand, that bG falls within the curvilinear element bB. This evidently tends to push the arch-stone inward towards the axis, and would cause it to slide in, since the joints are supposed perfectly smooth and slipping. But since this takes place equally in every stone of this course, they must all abut on each other in the vertical joints, squeezing them firmly together. Therefore, resolving the thrust bG into two, one of which is perpendicular to the joint KL, and the other parallel to it, we see that this last thrust is withstood by the vertical joints all around, and there remains only the thrust in the direction of the curve. Such a dome must therefore be firmer than an equilibrated dome, and cannot be so easily broken by overloading the upper part. When the curve is concave upwards, as in the lower part of the figure, the line bC always falls below bB, and the point C below B. When the curve is concave downwards, as in the upper part of the figure, bC passes above, or without bB. The curvature may be so abrupt, that even bG' shall pass without bB', and the point G' is above B'. It is also evident that the force which thus binds the stones of a horizontal course together, by pushing them towards the axis, will be greater in flat domes than in those that are more convex; that it will be still greater in a cone, and greater still in a curve whose convexity is turned inwards; for in this last case the line bG will deviate most remarkably from the curve. Such a dome will stand (having polished joints) if the curve springs from the base with any elevation, however small; nay, since the friction of two pieces of stone is not less than half of their mutual pressure, such a dome will stand although the tangent to the curve at the bottom should be horizontal, provided that the horizontal thrust be double the weight of the dome, which may easily be the case if it do not rise high.

40. Thus we see that the stability of a dome depends Stability of on very different principles from that of a common arch, a dome does and is in general much greater. It differs also in another important circumstance, viz. that it may be open in the middle; for the uppermost course, by tending equally from that in every part to slide in toward the axis, presses all to-gether in the vertical joints, and acts on the next course like the keystone of a common arch. Therefore an arch of equilibration, which is the weakest of all, may be open in the middle, and carry at top another building, such as a lanthorn, if its weight do not exceed that of the circular segment of the dome that is omitted. A greater load than this would indeed break the dome, by causing it to spring up in some of the lower courses; but this load may be increased if the curve is flatter than the curve of equilibration: and any load whatever, which will not crush the stones to powder, may be set on a truncate cone, or on a dome formed by a curve that is convex toward the axis; provided always that the foundation be effectually prevented from flying out, either by a hoop, or by a sufficient mass of solid pier on which it is set.

41. We have mentioned the many failures which happened to the dome of St Sophia in Constantinople. We imagine that the thrust of the great dome, bending the eastern arch outwards as soon as the pier began to yield, destroyed the half-dome which was leaning on it, and thus almost in an instant took away the eastern abutment. We think that this might have been prevented without any change in the injudicious plan, if the dome had been hooped with iron, as was practised by Michael Angelo in the vastly more ponderous dome of St Peter's at Rome, and by Sir Christopher Wren in the cone and the inner dome of St Paul's at London. The weight of the latter considerably exceeds 3000 tons, and they occasion a horizontal thrust which is nearly half this quantity, the elevation of the cone being about 60°. This being distributed round the circumference, occasions a strain on the hoop = \(\frac{7}{2 \times 22}\) of the thrust, or nearly 238 tons. A square inch of the worst iron, if well forged, will carry 24 tons with perfect safety; therefore a hoop of 7 inches broad and 1\(\frac{1}{2}\) inch thick will completely secure this circle from bursting outwards. It is, however, much more completely secured; for, besides a hoop at the base of very nearly these dimensions, there are hoops in different courses of the cone, which bind it into one mass, and cause it to press on the piers in a direction exactly vertical. The only thrusts which the piers sustain are those from the arches of the body of the church and the transepts. These are most judiciously directed to the entering angles of the building, and are there resisted with insuperable force by the whole lengths of the walls, and by four solid masses of masonry in the corners. Whoever considers with attention and judgment the plan of this cathedral, will see that the thrusts of these arches, and of the dome, are incomparably better balanced than in St Peter's church at Rome. But to return from this sort of digression.

42. We have seen that if \(bC\) be the thrust compounded of the thrust \(bC\), exerted by all the courses above HILK, and of the force \(bF\), or the weight of that course, be everywhere coincident with \(bB\), the element of the curve, we shall have an equilibrated dome: if it falls within it, we have a dome which will bear a greater load; and if it falls without it, the dome will break at the joint. We must endeavour to get analytical expressions of these conditions. Therefore draw the ordinates \(bd\), \(bd'\), \(BDB'\), \(C d C'\). Let the tangents at \(b\) and \(b'\) meet the axis in \(M\), and make \(MO\), \(MP\), each equal to \(be\), and complete the parallelogram MONP, and draw OQ perpendicular to the axis, and produce \(bF\), cutting the ordinates in \(E\) and \(e\). It is plain that MN is to MO as the weight of the arch HA to the thrust \(bC\) which it exerts on the joint KL (this thrust being propagated through the course HILK); and that MQ, or its equal \(be\), or \(bd\), may represent the weight of the half HA.

Let AD be called \(x\), and DB called \(y\). Then \(be = dx\), and \(eC = dy\) (because \(bC\) is in the direction of the element \(bB\)). It is also plain that if we make \(dy\) constant, BC is the second fluxion of \(x\), or \(BC = dx^2\), and \(be\) and \(bE\) may be considered as equal, and taken indiscriminately for \(dx\). We have also \(bC = \sqrt{dx^2 + dy^2}\). Let \(h\) be the depth or thickness HI of the arch-stones. Then \(h\sqrt{dx^2 + dy^2}\) will represent the trapezium HL; and since the circumference of each course increases in the proportion of the radius \(y\), \(hy\sqrt{dx^2 + dy^2}\) will express the whole course. If \(f\) be taken to represent the sum or aggregate of the quantities annexed to it, the formula will be analogous to the fluent of a fluxion, and \(f hy\sqrt{dx^2 + dy^2}\) will represent the whole mass, and also the weight of the vaulting, down to the joint HI. Therefore we have this proportion,

\[ \int hy\sqrt{dx^2 + dy^2} : hy\sqrt{dx^2 + dy^2} = b : bF, = b : CG, \]

\(= bd : CG, = dx : CG\). Therefore \(CG = \frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}}\).

If the curvature of the dome be precisely such as puts it in equilibrium, but without any mutual pressure in the vertical joints, this value of \(CG\) must be equal to CB or to \(dx\), the point G coinciding with B. This condition will be expressed by the equation

\[ \frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}} = dx, \]

or, more conveniently, by \(\frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}} = dx\). But this form gives only a tottering equilibrium, independent of the friction of the joints and the cohesion of the cement. An equilibrium, accompanied by some firm stability, produced by the mutual pressure of the vertical joints, may be expressed by the formula

\[ \frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}} > \frac{dx}{dt}, \]

or

\[ \frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}} = \frac{dx}{dt} + \frac{dt}{t}, \]

where \(t\) is some variable positive quantity, which increases when \(x\) increases. This last equation will also express the equilibrated dome, if \(t\) be a constant quantity, because in this case \(\frac{dt}{t} = 0\).

Since a firm stability requires that

\[ \frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}} > \frac{dx}{dt} \]

shall be greater than \(dx\), and CG must be greater than CB; hence we learn that figures of too great curvature, whose sides descend too rapidly, are improper. Also, since stability requires that we have

\[ \frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}} > \frac{dx}{dt} \]

greater than \(\int hy\sqrt{dx^2 + dy^2}\), we learn that the upper part of the dome must not be made very heavy. This, by diminishing the proportion of \(bF\) to \(bC\), diminishes the angle \(CBG\), and may set the point G above B, which will infallibly spring the dome in that place. We see here also, that the algebraic analysis expresses that peculiarity of dome-vaulting, viz. that the weight of the upper part may even be suppressed.

The fluent of the equation

\[ \frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}} = \frac{dx}{dt} + \frac{dt}{t} \]

is most easily found: it is \(L\int hy\sqrt{dx^2 + dy^2} = Ldx + Lt\), where \(L\) is the hyperbolic logarithm of the quantity annexed to it. If we consider \(dy\) as constant, and correct the fluent so as to make it nothing at the vertex, it may be expressed thus, \(L\int hy\sqrt{dx^2 + dy^2} = La = Ldx - Ldy + Lt\). This gives us \(L\int hy\sqrt{dx^2 + dy^2} = Ldx\),

and therefore \(\int hy\sqrt{dx^2 + dy^2} = \frac{dx}{dy}\).

This last equation will easily give us the depth of vaulting, or thickness \(h\) of the arch, when the curve is given. For its fluxion is

\[ \frac{hy\sqrt{dx^2 + dy^2}}{a} = \frac{dtdx + tdx}{dy}, \]

and \(h = \frac{adtdx + atdx}{ydy\sqrt{dx^2 + dy^2}}\), which is all expressed in known quantities; for we may put in place of \(t\) any power or function of \(x\) or of \(y\), and thus convert the expression into another, which will be applicable to all sorts of curves.

Instead of the second member \(\frac{dx}{dt} + \frac{dt}{t}\), we might employ \(p\frac{dx}{dt}\), where \(p\) is some number greater than unity. This will evidently give a dome having stability, because the original formula \(\frac{hy\sqrt{dx^2 + dy^2}}{\int hy\sqrt{dx^2 + dy^2}}\) will then be greater than \(dx\). This will give \(h = \frac{pdx}{ydy\sqrt{dx^2 + dy^2}}\). Each of these forms has its advantages when applied to particular cases. Each of them also gives \(h = \frac{adtx}{ydy\sqrt{dx^2 + dy^2}}\) when the curvature is such as is in precise equilibrium.

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1 The letters \(e\) and \(d\) are wanting in the plate; \(e\) ought to be at the intersection of \(bE\) and \(C e'\), and \(d\) at that of \(AD\) and \(Ce'\). And, lastly, if \( h \) be constant, that is, if the vaulting be of uniform thickness, we obtain the form of the curve, because then the relation of \( dx^2 \) to \( dx \) and to \( dy \) is given.

The chief use of this analysis is to discover what curves are improper for domes, or what portions of given curves may be employed with safety. Domes are generally built for ornament, and we see that there is great room for indulging our fancy in the choice. All curves which are concave outwards will give domes of great firmness; they are also beautiful. The Gothic dome, whose outline is an undulated curve, may be made abundantly firm, especially if the upper part be convex and the lower concave outwards.

The chief difficulty in the case of this analysis arises from the necessity of expressing the weight of the incumbent part, or \( \int hy \sqrt{dx^2 + dy^2} \). This requires the measurement of the conoidal surface, which, in most cases, can be had only by approximation by means of infinite series. We cannot expect that the generality of practical builders are familiar with this branch of mathematics, and therefore will not engage in it here; but content ourselves with giving such instances as can be understood by such as have that moderate mathematical knowledge which every man should possess who takes the name of engineer.

The surface of any circular portion of a sphere is very easily had, being equal to the circle described with a radius equal to the chord of half the arch. This radius is evidently \( \frac{\sqrt{dx^2 + dy^2}}{dy} \).

In order to discover what portion of a hemisphere may be employed (for it is evident that we cannot employ the whole) when the thickness of the vaulting is uniform, we may recur to the equation or formula

\[ \int hy \sqrt{dx^2 + dy^2} = \int \frac{a^2 dy}{\sqrt{a^2 - y^2}} \]

Let \( a \) be the radius of the hemisphere. We have \( dx = \frac{ady}{\sqrt{a^2 - y^2}} \), and \( dy = \frac{a^2 dy}{(a^2 - y^2)^{3/2}} \). Substituting these values in the formula, we obtain the equation

\[ y^2 \sqrt{a^2 - y^2} = \int \frac{a^2 dy}{\sqrt{a^2 - y^2}} \]

We easily obtain the fluent of the second member \( a^2 - a^2 \sqrt{a^2 - y^2} \), and \( y = a \sqrt{1 - \frac{y^2}{a^2}} \).

Therefore, if the radius of the sphere be 1, the half breadth of the dome must not exceed \( \sqrt{\frac{1}{2}} + \sqrt{\frac{3}{4}} \), or 0.786, and the height will be 0.618. The arch from the vertex is about 51° 49'. Much more of the hemisphere cannot stand, even though aided by the cement, and by the friction of the coursing joints. This last circumstance, by giving connection to the upper parts, causes the whole to press more vertically on the course below, and thus diminishes the outward thrust; but it at the same time diminishes the mutual abutment of the vertical joints, which is a great cause of firmness in the vaulting. A Gothic dome, of which the upper part is a portion of a sphere not exceeding 45° from the vertex, and the lower part is concave outwards, will be very strong, and not ungraceful.

But the public taste has long rejected this form, and seems rather to select more elevated domes than this portion of a sphere, because a dome, when seen from a small distance, always appears flatter than it really is. The dome of St Peter's is nearly an ellipsoid externally, of which the longer axis is perpendicular to the horizon. It is very ingeniously constructed. It springs from the base perpendicularly, and is very thick in this part. After rising about 50 feet, the vaulting separates into two thin vaultings, which gradually separate from each other.

These two shells are connected together by thin partitions, which are very artificially dovetailed in both, and thus form a covering which is extremely stiff, while it is very light. Its great stiffness was necessary for enabling the crown of the dome to carry the elegant stone lanthorn with safety. It is a wonderful performance and has not its equal in the world; but it is an enormous load in comparison with the dome of St Paul's, and this even independent of the difference of size. If they were of equal dimensions, it would be at least five times as heavy, and is not so firm by its gravity; but as it is connected in every part by iron bars (lodged in the solid masonry, and well secured from the weather by having lead melted all around them), it bids fair to last for ages if the foundations do not fail.

If a circle be described round a centre, placed anywhere in the transverse axis AC (Plate LII. fig. 11) of an ellipse, so as to touch the ellipse in the extremities B, b of an ordinate, it will touch it internally, and the circular arch B a b will be wholly within the elliptical arch BA b. Therefore, if an elliptical and a spherical vaulting spring from the same base, at the same angle with the horizon, the spherical vaulting will be within the elliptical, will be flatter and lighter, and therefore the weight of the next course below will bear a greater proportion to the thrust in the direction of the curve; therefore the spherical vaulting will have more stability. On the contrary, and for similar reasons, an oblate elliptical vaulting is preferable to a spherical vaulting springing with the same inclination to the horizon. (Fig. 13.)

44. Peradventure that what has been said on the subject convinces the reader that a vaulting perfectly equilibrated throughout is by no means the best form, provided that the base is secured from separating, we think it unnecessary to give the investigation of that form, which has a considerable intricacy, and shall content ourselves with merely giving its dimensions. The thickness is supposed uniform. The numbers in the first column of the table express the portion of the axis counted from the vertex, and those of the second column are the lengths of the ordinates.

| AD | DB | AD | DB | AD | DB | |----|----|----|----|----|----| | 0.4 | 100 | 610.4 | 1080 | 2990 | 1560 | | 3.4 | 200 | 744 | 1140 | 3442 | 1600 | | 11.4 | 300 | 904 | 1200 | 3972 | 1640 | | 25.6 | 400 | 1100 | 1260 | 4432 | 1670 | | 52.4 | 500 | 1336 | 1320 | 4952 | 1700 | | 91.4 | 600 | 1522 | 1360 | 5336 | 1720 | | 146.8 | 700 | 1738 | 1400 | 5756 | 1740 | | 223.4 | 800 | 1984 | 1440 | 6214 | 1760 | | 326.6 | 900 | 2270 | 1480 | 6714 | 1780 | | 465.4 | 1000 | 2602 | 1520 | 7260 | 1800 |

The curve delineated in fig. 15 is formed according to these dimensions, and appears destitute of gracefulness, because its curvature changes abruptly at a little distance from the vertex, so that it has some appearance of being made up of different curves pieced together. But if the middle be occupied by a lanthorn of equal or of smaller weight, this defect will cease, and the whole will be elegant, nearly resembling the exterior dome of St Paul's in London.

45. It is not a small advantage of dome-vaulting, that it is lighter than any that can cover the same area. If moreover, it be spherical, it will admit considerable varieties of figure by combining different spheres. Thus, a dome may begin from its base as a portion of a large hemisphere, and may be broken off at any horizontal course, and then a similar or a greater portion of a smaller sphere may spring from this course as a base. It also bears being intersected by cylindrical vaultings in every direction, and the intersections are exact circles, and always have a pleasing effect. It also springs most gracefully from the heads of small piers, or from the corners of rooms of any polygonal shape; and the arches formed by its intersections with the walls are always circular and graceful, forming very handsome spandrels in every position. For these reasons Sir Christopher Wren employed it in all his vaultings, and he has exhibited many beautiful varieties in the transepts and the aisles of St Paul's, which are highly worthy of the observation of architects. Nothing can be more graceful than the vaultings at the ends of the north and south transepts, especially as furnished off in the fine inside view published by Gwynn and Wale.

46. The connection of the parts arising from cement and from friction has a great effect on dome-vaulting. In the same way as in common arches and cylindrical vaulting, it enables an overload on one place to break the dome in a distant place. But the resistance to this effect is much greater in dome-vaulting, because it operates all round the overloaded part. Hence it happens that domes are much less shattered by partial violence, such as the falling of a bomb or the like. Large holes may be broken in them without much affecting the rest; but, on the other hand, it greatly diminishes the strength which should be derived from the mutual pressure in the vertical joints. Friction prevents the sliding in of the arch-stones, which produces this mutual pressure in the vertical joints, except in the very highest courses, and even there it greatly diminishes it. These causes make a great change in the form which gives the greatest strength; and as their laws of action are as yet but very imperfectly understood, it is perhaps impossible, in the present state of our knowledge, to determine this form with tolerable precision. We see plainly, however, that it allows a greater deviation from the best form than the other kind of vaulting, and domes may be made to rise perpendicular to the horizon at the base, although of no great thickness; a thing which must not be attempted in a plain arch. The immense addition of strength which may be derived from hooping largely compensates for all defects; and there are hardly any bounds to the extent to which a very thin dome-vaulting may be carried when it is hooped or framed in the direction of the horizontal courses. The roof of the Halle du Bled at Paris is but a foot thick, and its diameter is more than 200, yet it appears to have abundant strength. It is, on the whole, a noble specimen of architecture.

47. We must not conclude this article without taking notice of that magnificent and elegant arch erected in cast iron at Wearmouth, near Sunderland, in the county of Durham. The inventor and architect was Rowland Burdon, Esq.

This arch (of which a view is given at the article Bridge) is a segment of a circle whose diameter is about 444 feet. The span or chord of the arch is 286 feet, and its versed sine or spring is 34 feet. It springs at the elevation of 60 feet from the surface of the river at low water, so that vessels of 200 or perhaps 300 tons burden may pass under it in the middle of the stream, and even 50 feet on each side of it.

The sweep of the arch consists of a series of frames of cast iron, which abut on each other, in the same manner as the voussoirs of a stone arch. One of these frames or blocks (as we shall call them in future) is represented in Plate LIII. fig. 3, as seen in front. It is cast in one piece, and consists of three pieces or arms BC, BC, BC, the middle one of which is two feet long, the upper being somewhat more, and the lower somewhat less, because their extremities are bounded by the radius drawn from the centre of the arch. These arms are four inches square, and are connected by other pieces KL, of such length that the whole length of the block is five feet in the direction of the radius. Each arm has a flat groove on each side, which is expressed by the darker shading, three inches broad and three fourths of an inch deep. A section of this block, through the middle of KL, is represented by the light-shaded part BBB, in which the grooves are more distinctly perceived. These grooves are intended for receiving flat bars of malleable iron, which are employed for connecting the different blocks with each other. Fig. 4 represents two blocks united in this manner. For this purpose each arm has two square bolt-holes. The ends of the arms being nicely trimmed off, so that the three ends abut equally close on the ends of the next block; and the bars of hammered iron being also nicely fitted to their grooves, so as to fill them completely, and have their bolt-holes exactly corresponding to those in the blocks; they are put together in such a manner that the joints or meetings of the malleable bars may fall on the middle between the bolt-holes in the arms. Flat-headed bolts of wrought iron are then put through, and keys or forelocks are driven through the bolt-tails, and thus all is firmly wedged together, binding each arm between two bars of wrought iron. These bars are of such length as to connect several blocks.

In this manner a series of about 125 blocks are joined together, so as to form the precise curve that is intended. This series may be called a rib, and it stands in a vertical plane. The arch consists of six of these ribs, distant from each other five feet. These ribs are connected together so as to form an arch of 32 feet in breadth, in the following manner.

Fig. 5 represents one of the bridles or cross pieces which connect the different ribs, as it appears when viewed from below. It is a hollow pipe of cast iron four inches in diameter, and has at each end two projecting shoulders, pierced with a bolt-hole near their extremities, so that the distance between the bolt-holes in the shoulders of one end is equal to the distance between the holes in the arms of the blocks, or the holes in the wrought iron bars. In the middle of the upper and of the under side of each end may be observed a square prominence, more lightly shaded than the rest. These projections also advance a little beyond the flat of the shoulders, forming between them a shallow notch about an inch deep, which receives the iron of the arms, where they abut on each other, and thus give an additional firmness to the joint. The manner in which the arms are thus grasped by these notches in the bridles is more distinctly seen in fig. 4, at the letter H, in the middle of the upper rail.

The rib having been all trimmed and put together, so as to form the exact curve, the bolts are all taken out, and the horizontal bridles are then set on in their places, and the bolts are again put in and made fast by the forelocks. The bolts now pass through the shoulders of the bridles, through the wrought iron bars, and through the cast iron arm that is between them, and the forelocks bind all fast together. The manner in which this connection is completed is distinctly seen in fig. 4, which shows in perspective a double block in front, and a single block behind it. The abutting joints of the two front blocks are at the letters E, E, E; the holes in the shoulders of the horizontal cross pieces are at H.

48. This construction is beautifully simple, and very judicious. A vast addition of strength and of stiffness is procured by lodging the wrought iron bars in grooves formed in the cast iron rails; and for this purpose it is of great importance to make the wrought iron bars fill the grooves completely, and even to be so tight as to require the force of the forelocks to draw them home to the bottom of the grooves. There can be no doubt but that this arch is able to withstand an enormous pressure, as long as the abutments from which it springs do not yield. Of this there is hardly any risk, because they are masses of rock faced with about four or five yards (in some places only) of solid block masonry. The mutual thrusts of the frames are all in the direction of the rails, so that no part bears any transverse strain. We can hardly conceive any force that can overcome the strength of those arms by pressure or crushing them. The manner in which the frames are connected into one rib effectually secures the shifting joints from slipping; and the accuracy with which the whole can be executed secures us against any warping or deviation of a rib from the vertical plane.

But when we consider the prodigious span of this arch, and reflect that it is only five feet thick, it should seem that the most perfect equilibration is indispensably necessary. It is but like a film, and must be so supple, that an overload on any part must have a great tendency to bend it, and to cause it to rise in a distant part; and this effect is increased by the very firmness with which the whole sticks together. The overloaded part acts on a distant part, tending to break it with all the energy of a lever. This can be prevented only by means of the stiffness of the distant part. It is very true, the arch cannot break in the extrados except by tearing asunder the wrought iron bars which connect the blocks along the upper rail, and each of these requires more than a hundred tons to tear it asunder; yet an overload of five tons on any rib at its middle will produce this strain at twenty feet from the sides, supposing the sides held firm in their position. It were desirable, therefore, that something were done to stiffen the arch at the sides, by the manner of filling up the spandrels or space between the arch and the road-way. This is filled up in a manner that is extremely light and pleasing to the eye, namely, by large cast iron circles, which touch the extrados of the arch and reach the road-way. The road-way rests on them as on so many hoops, while they rest on the back of the arch, and also touch each other laterally. We cannot think that this contributes to the strength of the arch; for these hoops will be easily compressed at the points of contact, and, changing their shape, will oppose very little resistance. We think that this part of the arch might have been greatly stiffened and strengthened by connecting it with the road-way by trussed frames, in the same way that a judicious carpenter would have framed a roof. If a strong cast iron pillar had been made to rest on the arch at about 20 feet from the impost, and been placed in the direction of a radius, the top of this pillar might have been connected by a diagonal bar of wrought iron with the impost of the arch, and with the crown of the arch by another string or bar of the same materials. These two ties would cause the radial pillar to press strongly on the back of the arch, and they must be torn asunder before it could bend in that place in the smallest degree. Supposing them of the same dimensions as the bars in the rims, their position would give them near ten times the force for resisting the strain produced by an overload on the crown.

This beautiful arch contains only 260 tons of iron, of which about 55 are wrought iron. The superstructure is of wood, planked over a top. This floor is covered with coating of chalk and tar, on which are laid the materials of the carriage road, consisting of marl, limestone, and gravel, with foot-ways of flag-stones at the sides. The weight of the whole did not exceed 1000 tons, whereas the lightest stone arch which could have been erected would have weighed 15,000. It was turned on a very light but stiff scaffolding, most judiciously constructed for the preservation of its form, and for allowing an uninterrupted passage for the numerous ships and small craft which frequent the busy harbour of Sunderland. The mode of framing the arch was so simple and easy that it was put up in ten days, without an accident; and when all was finished, and the scaffolding removed, the arch did not sensibly change its form. The whole work was executed in three years, and cost about £26,000.

Appendix.

49. The excellence of the preceding article, written by the late Dr. John Robison, Professor of Natural Philosophy in the University of Edinburgh, for the Supplement to the third edition of this work, may be inferred from the fact, that almost every later writer on mechanics has spoken of it with approbation, and borrowed more or less from it. (See Gregory's Mechanics, book i. chap. 6; Hutton's Treats., vol. I., Bridges; Whewell's Mechanics, art. 72, &c.) There is however one part of the article, viz. the purely mathematical part, which must have been quite unintelligible as it originally appeared, because of its numerous typographical errors. These are here corrected, we believe for the first time. Even with this advantage, we fear it has rather a forbidding aspect to the student of modern mathematics, because of the very complicated diagram (see Plate LII. fig. 12) from which the differential equation of the equilibrated arch has been deduced. We shall therefore, as an appendix, treat of the equilibrated polygon and equilibrated arch upon the general principles of statics, and employ in this investigation only the most simple geometrical figures. The theory of the equilibrated arch cannot be delivered without employing the principles of the differential and integral calculus; but we shall endeavour to pass from the finite equation of the equilibrated polygon to the differential equation of the arch by the shortest and most direct road.

Equilibrated Polygon.

50. In Plate LIII. fig. 9, let ABCDEDCBA' be a polygon formed by beams or rods AB, BC, CD, DE, &c. of any length, movable about the points B, C, D, E, &c. as joints, and forming an equilibrium in a vertical plane by the mutual thrusts at the joints and by the weight of the beams, the extreme sides of the polygon being supposed supported or attached to fixed points. Let AB, BC, CD, be any three consecutive sides of the polygon; produce AB, DC, the extreme sides, until they meet in H. The beam BC is kept in its position by the thrusts of the adjoining beams AB, CD, in the directions BH, CH, and its own weight, which is equivalent to pressures or loads on the joints B, C, acting in a vertical direction. Let G be the point in BC, which is the centre of gravity of weights proportional to loads at B and C. By the principles of statics, the directions of these forces must pass through the same point; therefore G must be in a vertical line passing through H.

51. Draw BL, CK, perpendicular to the vertical HG. Let $\phi$, $\phi'$, $\phi''$, denote the angles which the lines AB, BC, CD, in their order, make with any horizontal line in the plane of the polygon; then

$$\phi = \text{angle HBL}, \phi' = \text{GBL} = \text{GCK}, \phi'' = \text{HCK};$$

also let $w$ denote the load on the joint B, and $w'$ the load on the joint C. By the nature of the centre of gravity,

\[ w : w' = CG : BG = CK : BL, \]

therefore

\[ w : w' = \frac{1}{BL} : \frac{1}{CK} = \frac{HG}{GL} : \frac{HG}{GL}. \]

Now

\[ \frac{HG}{BL} = \frac{HL}{GL} - \frac{GL}{BL} = \tan \phi - \tan \phi', \]

and

\[ \frac{HG}{CK} = \frac{GK}{HK} - \frac{HK}{CK} = \tan \phi' - \tan \phi''. \]

hence it appears that

\[ w : w' = \tan \phi - \tan \phi' : \tan \phi' - \tan \phi''. \]

If \( w'' \) denote the load on the next joint D, and \( \phi'' \) the angle which the line DE makes with the horizontal plane, it will appear in the same way that

\[ w : w'' = \tan \phi' - \tan \phi'' : \tan \phi'' - \tan \phi''' \]

and so on throughout the whole polygon; whence we have this important proposition:

The vertical pressures on any two joints of the polygon, whether adjoining or remote, are to one another as the differences of the tangents of the angles which the sides about the joints make with the plane of the horizon.

52. From this proposition we may infer, that if \( \phi, \phi' \) denote the angles which any two adjoining sides of the polygon make with the horizontal plane, and \( w \) the load on the joint at their intersection, then

\[ w = C (\tan \phi - \tan \phi') \]...........(A);

and in this formula, \( C \) denotes some constant quantity, which is the same for all the angles of the polygon. This is the general equation of the equilibrated polygon; and it shows that the loads at the joints depend entirely on the angles which the beams make with the horizontal plane, and are independent of the lengths of the beams themselves.

Equilibrated Arch.

53. We shall next investigate the differential equation of an equilibrated arch, deducing it from the finite equation of the equilibrated polygon.

Let us suppose an equilibrated polygon of a very great number of sides (fig. 10), and that a curve ABC passes through all the joints: the sides of the polygon will then be chords in the curve; and as the number of these may be conceived to be greater than any assigned number, and each shorter than any given line, they may be regarded as elements of the curve, and as constituting it.

54. Let us suppose the curve ABC formed in this manner to be the intrados of an arch of a bridge, and that the extrados is the line MDN, which may be curved or straight. Let AC be the span, or greatest horizontal width of the arch, and BH, which bisects AC at right angles, its rise or height; also let BD be the thickness of the arch of the crown: draw a straight line EDF perpendicular to DH, and draw AE, CF perpendicular to EF.

Let \( Pp \) be any indefinitely small part of the intrados ABC. Draw PRQ perpendicular to the horizontal line EF, meeting the extrados in R, and rp parallel to RP; also draw PK perpendicular to DH, meeting rp in q, and PT touching the intrados in p; and, referring the two curves to the same axes DE, let us put

\[ x = DQ = PK = a \sin \phi, \] \[ y = PQ = BD + BR = c + a (1 - \cos \phi), \] \[ dx = a \cos \phi d\phi, \] \[ dy = a \sin \phi d\phi. \]

Now, by formula (B),

\[ \frac{dy}{dx} = \frac{\sin \phi}{\cos \phi} = \tan \phi; \quad \frac{d^2y}{dx^2} = \frac{d\phi}{\cos^2 \phi}. \]

\[ \frac{d^2y}{dx^2} = \frac{1}{a \cos^3 \phi} = \sec^3 \phi. \]

Now, by formula (B),

\[ \frac{y - v}{c^2} = \frac{d^2y}{dx^2} = \sec^3 \phi; \]

therefore \( y - v = \frac{c^2}{a} \sec^3 \phi. \)

But when \( x = 0 \), then \( v = 0 \), \( y = c' \), \( \phi = 0 \), and \( \sec \phi = 1 \); therefore \( c' = \frac{c^2}{a} \) and \( c^2 = ac' \); hence we have

\[ v - y = c' \sec^3 \phi. \]

This shows that the vertical line between the intrados and extrados is always proportional to the cube of the secant of the angle which the radius OP makes with the perpendicular; a conclusion which agrees with section 25 of the preceding article.

57. The second problem, viz. having given the nature... of the extrados, to find that of the intrados, is the more important of the two, and more difficult. Its solution requires the integral calculus; but the difficulty is no greater than that of finding the area of a curve whose equation is given. This can always be accomplished, if not in finite terms, at least by infinite series. We shall now give a general solution of the problem.

Lagrange has shown (Théorie des Fonctions Analytiques, chap. viii.) that the integration of the equation

$$\frac{d^2y}{dx^2} - \frac{y}{c^2} = X,$$

where $X$ denotes an explicit function of the variable $x$, can always be accomplished when two particular integrals of this other equation, viz.

$$\frac{d^2y}{dx^2} - \frac{y}{c^2} = 0,$$

are known. Now $y = pe^{\frac{x}{c}}$ and $y = qe^{-\frac{x}{c}}$ are such integrals ($p$ and $q$ being arbitrary constants, and $e = 2.7182818$, the number whose Napierian logarithm is unity), as may be proved by differentiation; therefore, following Lagrange, to integrate the equation (B), viz.

$$\frac{d^2y}{dx^2} - \frac{y}{c^2} = -\frac{v}{c^2},$$

we assume

$$y = pe^{\frac{x}{c}} + qe^{-\frac{x}{c}},$$

for the complete integral equation; but now $p$ and $q$ are to be considered as functions of the variable $x$. To determine these, we differentiate, and get

$$\frac{dy}{dx} = \frac{1}{c} \left\{ pe^{\frac{x}{c}} - qe^{-\frac{x}{c}} \right\} + e^{\frac{x}{c}} dp + e^{-\frac{x}{c}} dq,$$

since $p$ and $q$ are indeterminate functions of $x$, we may assume that

$$e^{\frac{x}{c}} dp + e^{-\frac{x}{c}} dq = 0,$$

and then we have

$$\frac{dy}{dx} = \frac{1}{c} \left\{ pe^{\frac{x}{c}} - qe^{-\frac{x}{c}} \right\}.$$

Again, by differentiating, we find

$$\frac{d^2y}{dx^2} = \frac{1}{c^2} \left\{ pe^{\frac{x}{c}} + qe^{-\frac{x}{c}} \right\} + e^{\frac{x}{c}} dp + e^{-\frac{x}{c}} dq,$$

$$= \frac{y}{c^2} + \frac{1}{c} \left\{ e^{\frac{x}{c}} dp - e^{-\frac{x}{c}} dq \right\}.$$

This result, compared with equation (B), gives

$$\frac{1}{c} \left\{ e^{\frac{x}{c}} dp - e^{-\frac{x}{c}} dq \right\} = -\frac{v}{c^2}.$$

From this and the equation

$$e^{\frac{x}{c}} dp + e^{-\frac{x}{c}} dq = 0,$$

we obtain (putting $X = -\frac{v}{c^2}$)

$$\frac{dp}{dx} = \frac{c}{2} e^{-\frac{x}{c}} X, \quad \frac{dq}{dx} = -\frac{c}{2} e^{\frac{x}{c}} X,$$

$$p = \frac{c}{2} \int e^{-\frac{x}{c}} X dx + b; \quad q = -\frac{c}{2} \int e^{\frac{x}{c}} X dx + b.$$

Here $b$ and $b'$ are arbitrary constants, and the integrals are supposed to be taken so as to vanish when $x = 0$. The complete integral equation is now

$$y = be^{\frac{x}{c}} + be^{-\frac{x}{c}} + \frac{c}{2} \left\{ e^{\frac{x}{c}} \int e^{-\frac{x}{c}} X dx - e^{-\frac{x}{c}} \int e^{\frac{x}{c}} X dx \right\}.$$

To determine the arbitrary constants $b$ and $b'$, we must consider that a tangent to the intrados at the vertex is perpendicular to the vertical DH; therefore, when $x = 0$, then $\frac{dy}{dx} = 0$; but from the equation just found we get

$$\frac{dy}{dx} = \frac{1}{c} \left\{ be^{\frac{x}{c}} - be^{-\frac{x}{c}} \right\} + \frac{c}{2} \left\{ e^{\frac{x}{c}} \int e^{-\frac{x}{c}} X dx + e^{-\frac{x}{c}} \int e^{\frac{x}{c}} X dx \right\}.$$

Now, when $x = 0$, then $e^{\frac{x}{c}} = 1$, and $e^{-\frac{x}{c}} = 1$; and by hypothesis the integrals $\int e^{\frac{x}{c}} X dx, \int e^{-\frac{x}{c}} X dx$ in this particular case vanish; therefore, when $x = 0$, the last equation becomes $0 = b' - b$, so that $b' = b$. But again, in the general equation (C), when $x = 0$, then $y = c'$; therefore we have also $c' = 2b$ and $b = \frac{c'}{2}$. On the whole, the equation of the extrados is

$$y = \frac{c'}{2} \left\{ e^{\frac{x}{c}} + e^{-\frac{x}{c}} \right\} + \frac{c}{2} \left\{ e^{\frac{x}{c}} \int e^{-\frac{x}{c}} X dx - e^{-\frac{x}{c}} \int e^{\frac{x}{c}} X dx \right\};$$

the integrals being taken so as to vanish when $x = 0$. We have thus brought the solution to depend on the integration of the two differentials

$$e^{\frac{x}{c}} X dx, \quad e^{-\frac{x}{c}} X dx,$$

which in fact will only differ in their sign, because the branches of the extrados on opposite sides of the vertical are exactly alike, and therefore the substitution of $-x$ for $+x$ will not change the sign of $u$ nor of $X$. Now this integration can always be effected by known methods, therefore the second problem may be regarded as completely resolved.

57. Example. Let us suppose that the extrados is a horizontal straight line EF.

The line PT being supposed to touch the curve, let us as before put

$$c' = DB, \quad c'' = EA, \quad s = DE,$$

$$x = DQ, \quad y = PQ, \quad \phi = \text{angle TPK}.$$

In this case $v = 0$ and $X = 0$, and the equation of the curve is simply

$$y = \frac{c'}{2} \left\{ e^{\frac{x}{c}} + e^{-\frac{x}{c}} \right\}.$$

This case of the general problem has been resolved in sect. 25; the equation of the curve is, however, here given under a different form. We shall now deduce from it a formula for logarithmic calculation.

In all curves, tan. $\phi = \frac{dy}{dx}$; in the present case

$$\tan \phi = \frac{dy}{dx} = \frac{c'}{2c} \left\{ e^{\frac{x}{c}} - e^{-\frac{x}{c}} \right\};$$

Let $\psi$ be such an angle that

$$e^{\frac{x}{c}} = \tan (45^\circ + \frac{1}{2} \psi), \quad \text{then} \quad e^{-\frac{x}{c}} = \tan (45^\circ - \frac{1}{2} \psi).$$

Hence, by the arithmetic of sines (Algebra, sect. 244, (K) and sect. 240, (C) No. 1),

$$e^{\frac{x}{c}} + e^{-\frac{x}{c}} = \frac{1}{\cos(45^\circ + \frac{1}{2} \psi) \cos(45^\circ - \frac{1}{2} \psi)} = \frac{2}{\cos \psi} = 2 \sec \psi,$$

$$e^{\frac{x}{c}} - e^{-\frac{x}{c}} = \frac{\sin \psi}{\cos(45^\circ + \frac{1}{2} \psi) \cos(45^\circ - \frac{1}{2} \psi)} = \frac{2 \sin \psi}{\cos \psi} = 2 \tan \psi.$$ Also, by the theory of logarithms (Algebra, sect. xix.),

\[ \frac{x}{c} \log_e = \log_e \tan(45^\circ + \frac{1}{2} \psi) - \log_e \tan(45^\circ + \frac{1}{2} \psi) = \log_e \tan(45^\circ + \frac{1}{2} \psi) - 10. \]

From these expressions we obtain the relation of the three principal elements of the curve, viz. \( \varphi, x, y \), as follows:

\[ \tan \psi = \frac{c}{e} \tan \varphi \]

\[ x = \frac{c}{\log_e} \left\{ \log_e \tan(45^\circ + \frac{1}{2} \psi) - 10 \right\} \]

\[ y = \frac{c}{\cos \psi} = c' \sec \psi \]

But before these formulæ can be applied, the value of \( e \) must be known. To find this, let \( a \) denote the value of \( \psi \) when \( x = s \) and \( y = e' \); then equations (3) and (2) become

\[ c' = \frac{c'}{\cos a} \]

\[ s = \frac{c}{\log_e} \left\{ \log_e \tan(45^\circ + \frac{1}{2} a) - 10 \right\} \]

From these we obtain

\[ \cos a = \frac{c'}{e'} \]

\[ c = \frac{\log_e}{\log_e \tan(45^\circ + \frac{1}{2} a) - 10} \]

These formulæ determine \( e \), and this known, the values of \( x \) and \( y \) corresponding to any proposed value of \( \varphi \) may be readily found from formulæ (1) (2) and (3).

58. We may also determine \( y \) directly from \( x \) without \( \varphi \) by eliminating \( \frac{c}{e} \) by formulæ (2) and (5); we have then, to determine \( y \) from \( x \), these formulæ,

\[ \cos a = \frac{c'}{e'} \]

\[ \log_e \tan(45^\circ + \frac{1}{2} \psi) = 10 + \frac{x}{s} \left\{ \log_e \tan(45^\circ + \frac{1}{2} a) - 10 \right\} \]

\[ y = \frac{c'}{\cos \psi} \]

1. As an example, let us take the case of Blackfriars Bridge, for which a table of corresponding values of \( x \) and \( y \) has been given, sect. 26. Here the span is 100 feet, the height or rise forty feet, and the thickness at the crown six feet; and first, let it be required to find the ordinate \( y \), when \( x = 20 \) feet; we have now

\[ s = 50, \quad c = 6, \quad c' = 46, \quad x = 20, \quad \frac{x}{s} = \frac{20}{50} \]

Logarithmic Calculation.

\[ c = 6 \]

\[ c' = 46 \]

\[ \cos(\varphi = 82^\circ 30' 19") \]

\[ \log_e \tan(45^\circ + \frac{1}{2} a = 86^\circ 15' 0" 5") - 10 = 1-1837773 \]

To \( 10-0000000 \)

add \( \frac{3}{5} \times 1-1837773 = 0-7781512 \)

\[ \tan(45^\circ + \frac{1}{2} \psi = 71^\circ 25' 18") \]

\[ \cos(\psi = 52^\circ 50' 36") \]

\[ y = \frac{c'}{\cos \psi} = 9-9338 \text{ feet} \]

The greater part of the above calculation serves for all the values of \( x \), and need not be repeated in constructing a table of the ordinates.

2. Let it be required to find \( y \) when \( x = 32 \) feet.

To \( 10-0000000 \)

add \( \frac{3}{5} \times 1-1837773 = 0-7781512 \)

\[ \tan(45^\circ + \frac{1}{2} \psi = 80^\circ 5' 18") \]

\[ c = 6 \]

\[ \cos(\psi = 70^\circ 10' 36") \]

\[ y = 17-6933 \text{ feet} \]

In this way may all the numbers in the table of sect. 26 be computed with greater expedition than by the formula given there.

The value of \( \varphi \) to each value of \( x \) may be found from formulæ (5) and (1) of last article.

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ARCHÆUS, or Archeus. See Archeus.