MASONRY

Masonry. 1. MASONRY is the art of building with stones. The art of reducing stones to regular or determinate forms is sometimes called stone-cutting, but is usually considered a branch of masonry. Workers in marble are also called masons; but it is stone-masonry only which we intend to treat of in this article, as marble masonry is rather a manual than a scientific art.

Masonry defined.

Early history. 2. The art of building with stone is undoubtedly of great antiquity; and its early history is difficult to trace beyond the existing remains of ancient buildings, the oldest of which are objects of wonder, chiefly on account of the difficulty of moving, with ordinary powers, the immense stones of which they are formed. There is one thing remarkable in these stupendous efforts of human labour: its directors have often been happy in the choice of almost imperishable materials, for a lasting evidence of their command of power. The remains of this kind of gigantic masonry are found in various parts of the earth; some of the finest specimens are the ancient Egyptian buildings, which seem to have been intended to resist the power of men, as well as the slow operations of time.

Greek masonry. 3. The masonry of the ancient Greeks closely resembles that of the Egyptians. It is a more refined application of the same principles of construction, to a series of chaste and beautiful architectural forms, in which the ornamental part of the art has evidently attained to that state of perfection, which is rarely, if ever surpassed. The roof of the Octagon Temple of the Winds may be considered as the best example of their constructive skill, whilst it betrays their ignorance of the principles of the arch.

Roman masonry. 4. In Roman masonry we find less of ponderous strength, and of solid construction, than in the Egyptian and Greek, and rarely anything approaching to the accurate and highly finished labours of the latter, but considerably more artificial and economical knowledge. If the manner of forming arches and domes¹ was not actually invented by the Romans, at least the merit of applying them successfully, in the art of building, was undoubtedly theirs; and they also excelled in the composition of mortars and cements. Hence, they found it easy to construct large works at a moderate expense, which could not have been accomplished by the limited methods of building known to their predecessors. It gives us a high notion of the intrinsic value of the art of masonry, to examine its application by the Romans, whether it be in the celebrated Cloacae, Aqueducts, Bridges, or the Military Roads of that enterprising people. To them also we owe the beautiful idea of covering a temple with a dome.

Early English masonry. 5. After the decline of the Romans, the art of masonry, in Europe, gradually acquired its former importance, through its application to the construction of castles, towers, and other places of defence; and eventually, it gained a complete ascendancy over the other building arts, in the construction of cathedrals, monasteries, and such like edifices. In our own island it made an equal if not a greater degree of progress than it did upon the continent. The science of masonry² appears to have attained the most perfect state it

arrived at in those times, about the period when King's College Chapel, at Cambridge, was built, that is, about 1512. From that time, or soon after, the knowledge of construction declined; but the researches of men of science have, in modern times, more than replaced those lost principles which, there can be little doubt, the elder free-masons possessed. Unfortunately, such principles are, even at the present time, as inaccessible to a plain workman as the mysteries of the master-mason were to the apprentice and fellow-craft of former ages; unless it be in some rare instances, where the force of natural genius has risen superior to all difficulties, and a mere workman, like the "prentice of Roslin Castle," has outstripped the masters of technical science.

6. The most important principle of the free-masons, or, as they are usually called, the "Gothic builders," was that of reducing all the pressures of a vaulted roof to a few principal supports. These supports were either strong pillars, or buttresses, accordingly as the support was within the area, or formed a part of the external wall. The buttresses were made of considerable depth in the direction of the pressure, with a thin wall from buttress to buttress, for enclosing the building. If they had made the external walls of uniform thickness, according to the modern practice, a much greater quantity of material would have been required to balance the pressure of the vaulting. For similar reasons, the strength of their best vaulting consists in deep-moulded ribs; the spaces between these ribs being formed of thin light stones, supported from rib to rib. The principles of construction of the Gothic builders may be readily shewn by a model of wicker work, in the manner of Sir James Hall's truly elegant mode of explaining his ideas respecting the origin of Gothic vaulting.³ The earliest notice we have seen, in architectural works, of anything resembling the principles of construction just noticed, is given by Alberti, who, alluding to a method of building known to former architects, says, "The arches upon which the roof was placed were drawn quite down to the foundation with wonderful art, known but to few; so that the work upheld itself by being only set upon arches; for those arches having the solid earth for a chain, no wonder they stood firm without any other support."⁴

7. Masonry, with some other arts, having been drawn out of their ordinary course by the peculiar state of society in the middle ages, fell back to their common level, if not below it, at the Reformation; and the natural consequence of this change was the loss of the greater part of the knowledge which had been gained by the experience of several centuries. But even in the most depressed state of masonry, there were individuals in whom the love of that excellence which animated their predecessors, was not subdued by want of encouragement; and some scattered works were executed which are deserving of notice, if our limited plan would allow of it.

8. When Britain had happily become free from all internal disturbances, and there was little to occupy the time and attention of a rapidly increasing population, except the

¹ Perhaps the oldest arches, at present known, are those which Mr. Belzoni discovered in Egypt; they are executed in bricks of the same size, and of the same material, as those which the Egyptians used in the construction of their walls and pyramids. For further remarks upon the subject of Arches, see the Article Burges; and the Earl of Aberdeen's Inquiry respecting Grecian Architecture, page 191—211.

² What other term than science can be applied to that knowledge, which enables a mason to dispose large masses of stone-work over a considerable area, with only a few distant supports?

³ Essay on the Origin, History, and Principles of Gothic Architecture, London, 1813. Ware has collected most of the forms employed in his Tracts on Vaults, &c. London, 1822.

⁴ Architecture of Leo Baptista Alberti (Leon's translation), book i. chap. xii.

Masonry. improvement of their own condition in life, the chief fruit of their exertions for this purpose was, an unprecedented extension of the foreign and domestic trade of the country; wharfs, docks, harbours, and lighthouses were constructed; canals, locks, roads, and bridges, became the necessary appendages of this new state of things; and, accordingly, it was found desirable again to cultivate the art of masonry.

These important works also called forth a new profession, of which the celebrated Smeaton has been called the father. Smeaton's first work was the Eddystone Lighthouse, which, in originality of design, and soundness of construction, has not been equalled. Since its erection, such a succession of bold and useful works have been accomplished, that it would be difficult to enumerate them;¹ and it may be sufficient to remark, that the masonry of our own age and country, as it is exhibited in these works, is without a parallel in preceding times.

9. In the northern states of Europe, their best works are chiefly modelled after ours; and, with the exception of France, there is not in the southern states any considerable degree of encouragement given to any branch of masonry. It may nevertheless be remarked, that the principles of construction form a popular subject of study in Italy.

10. In France masonry has always been a popular art; partly, perhaps, from Paris being situated in the midst of a district which abounds in excellent building stone. The French government has constantly directed a considerable share of attention to the construction of roads, bridges, and military works; and, consequently, has afforded sufficient scope for its improvement. When, however, the larger works of the French masons are compared with those of our own countrymen, one very remarkable difference may be observed: the French works have more of the character of daring experiments, than that which ought to belong to the works of regular professors of an art; whilst the British works of the same kind have evidently been directed by men much better versed in practical construction than in the refinements of science. There is, perhaps, more of novelty in the French works than is to be found in ours; but it may be remarked, that this novelty of character is often obtained by a sacrifice of fitness, as in the catenarian dome of the Pantheon; or of strength, as in the bridges of Nogent, Nemilly, and others. The true criterion of excellence in a useful art seems to be, fitness for producing the desired end in the best possible manner.

I.—OF MATERIALS USED IN MASONRY.

11. The first object of attention, in a treatise on masonry, ought to be, the nature of the materials employed in it, because the greater part of the principles of an art always depend on the nature of the substances it is to be exercised upon.

Of Stones.

12. Building stone is a dense, coherent body, of considerable hardness and durability, but generally brittle. It possesses these qualities in various degrees, according to the nature of its chemical elements, or the state of aggregation of its parts. The structure of stones is either laminated or granulated, or of a mixed kind. The chemical constituents of building stones are silica, alumina, lime, magnesia, and metals, combined with acids, water, and sometimes with alkalis; some other chemical elements are found in building stones, but not often in sufficient quantity to affect the nature of the stones.

13. Laminated stones consist of thin plates, or layers,

cohering more or less strongly together; but when the layers are of considerable size, and cohere so slightly that they may be easily separated, the stones are said to be slaty. The layers are always nearly parallel to the quarry-beds of the stone, and they should always be horizontal, or as nearly so as possible, in a building, otherwise the action of the weather will cause them to separate, and fall off in flakes. In sandstones, the direction of the layers may often be discovered by their different shades of colour; and in others, by the position of minute scales of mica, which always lie parallel to the layers. In most stones the direction of the layers may be ascertained by the facility with which the stone yields to the tool in that direction; but a considerable degree of practice is necessary to acquire so nice a discrimination of resistance, and good workmen only attain it. Amongst laminated stones, those are the most durable in which the laminae are least distinct, and the texture uniform. When the laminae do not perfectly cohere, they are soon injured by frost, and they are wholly unfit for places alternately wet and dry.

14. Granular stones consist of distinct concretions resembling grains, either of the same or of different simple minerals cohering together. When the structure is uniform, and the grains or concretions are small, stones of this kind are always strong and durable, if the concretions themselves be so. Granular stones are sometimes open and porous, but when they are uniformly so, they seldom suffer materially by frost, because their uniform porosity allows the expansive force of congealing water to be distributed in every direction.

15. Stones of a compound structure, partly laminate and partly granular, have more or less of the characters of the two classes before described; for it may be observed in coarse-grained granite that the laminated structure of some of its parts renders it very susceptible of disintegration. All kinds of stone obtained from quarries are found divided by vertical or inclined seams, which are sometimes so close that they cannot be distinguished till the stones are wrought; but they often separate under the tool at such seams; and it is not safe to employ stone to resist any considerable transverse strain on account of the difficulty of knowing where those seams are.

16. In the present state of our knowledge of this important subject, we may attribute the failure of building stones to two causes; the one chemical, and the other mechanical, which we shall here distinguish by the terms decomposition and disintegration.

17. Decomposition consists in the chemical elements of a stone entering into new combinations with water, oxygen, or carbonic acid gas. Stones containing such elements as are readily acted upon by these external causes will be found most subject to decomposition; and the process will be, in many kinds, much hastened by a loose texture. Stones containing saline matter, as the felspar of some granites, are acted upon by water, particularly where the soluble salt is in considerable proportion; and in some stones the application of salt water soon destroys them. Dolomieu says, the houses at Malta are built with a fine-grained limestone, of a loose and porous texture, which speedily moulders away when it has been wetted with sea water.² Stones containing iron, which is not in a maximum state of oxidation, are often destroyed by the absorption of oxygen and carbonic acid; the presence of moisture accelerates their decomposition, and it is always still further hastened by increase of temperature. According to the observations of Kirwan, stones containing iron, in a low state of oxidation are of a black, a brown, or a bluish colour; and in some instances, when united with alumina and magnesia, they are of a grey,

¹ See the Articles BILL-ROCK LIGHTHOUSE, BRIDGE, BREAKWATER, and CALEDONIAN CANAL DOCKS.

² Kirwan's Geological Essays, p. 148, 149.

Masonry. or of a greenish grey; the former, as they become more oxygenized, change to purple, red, orange, and finally pale yellow; the latter become at first blue, then purple, red, &c.¹ But stones containing iron, combined with its maximum of oxygen, do not readily decompose, such are red porphyry, jaspers, &c. When stones contain manganese, lime, alumina, carbon, or bitumen, in particular states, they are subject to decomposition, from the affinities of one or other of these bodies; but nothing very decisive is, or perhaps can be, known respecting such changes, till some improvement be made in analytical chemistry, by which the state of combination of the constituents of minerals can be determined with more certainty.

Disintegration. 18. Disintegration is the separation of the parts of stones by mechanical action. The chief cause is the congelation of water in the minute pores and fissures of stones, which bursts them open, or separates small parts according as the structure is slaty or irregularly granulated. The south sides of buildings, in northern climates, are most subject to fail; because the surface is often thawed and filled with wet in the sunny part of the day, and frozen again at night. This repeated operation of freezing is also very injurious to sea walls, the piers of bridges, and other works exposed alternately to water and frost.² The decay and destruction of rocks being the effects of the same natural causes, the reader will find some further illustration of this subject in the article MINERALOGY.

Resistance to wear. 19. The resistance of stones to wear and tear is, for many purposes, a subject which it would be useful to investigate, since on this resistance depends also the labour of working them. From some experiments made by Rondelet, it appears that granite will bear eight times as much wear as veined white marble; and that the labour of sawing granite was about ten times greater than that of sawing veined white marble.³

Scotland abounds in quarries of excellent building sandstone; such, in particular, are the quarries at Culello, in Fife-shire, which furnished the stones for the monument erected at Yarmouth to the memory of Lord Nelson, and that at Edinburgh to the memory of Lord Melville. Nothing can exceed the beauty of the sandstone used in those noble structures; and besides beauty, and other valuable qualities, it has in a high degree that of being easily chiselled into the smoothest and finest forms.

Of Mortars and Cements.

Mortars and cements. 20. In the greater part of works executed in stone, it is necessary to use some kind of cementitious matter for connecting the parts together, to render them firm and compact. Works to be exposed to the action of water, immediately after being built, require this cementitious matter to be of such a nature that it will indurate under water. Hence it is, that we have occasion for two species of mortar; one which will set and harden under water, called a water mortar, or cement; and common mortar.

Nature of common mortar. 21. If a piece of limestone, or chalk, be slowly calcined, so as to expel the whole, or nearly the whole, of its carbonic acid, it loses about 44 per cent. of its weight; and on a small quantity of water being added, it swells, gives out heat, and falls into a finely divided powder, called slacked lime. The bulk of the powder is about double that of the limestone. If this powder be rapidly formed into a stiff paste with water, it sets or solidifies as a hydrate of lime, and ultimately hardens by the absorption of carbonic acid from the air. This constitutes common building mortar. Hydrate of lime consists of lime, 100 parts, and water 31 parts.

Masonry. 22. If any substance, reduced to powder, and containing much iron in a low state of oxidation, be mixed with slacked lime, the mixture will become harder and more coherent than when lime alone is employed; and it possesses the valuable property of acquiring an equal, or perhaps greater degree of hardness, if it be immersed in water as soon as it is formed into a paste. This constitutes water mortar. A paste of hydrate of lime alone softens and dissolves in water. Alumina, silica, and manganese, are endowed with the same property as oxide of iron, but in an inferior degree. It may be observed, in forming such combinations, that when the lime is in excess, it separates; and in favourable situations, either crystallizes or forms stalactites; indicating that there is a definite proportion according to which the materials should be combined to form the best cement.

23. If some kind of matter, in the state of particles or Use of grains, be mixed with the cementing material, and then said in the mixture be formed into a paste with water, the strength mortar, and hardness of the cement will greatly depend on the nature of the particles added to it. In order that the cement may be improved by such addition, it is necessary that the hardness of the particles should at least be equal to the greatest degree of hardness the cement can acquire; and also, that the affinity between these particles and the cementing matter should exceed the affinity between the parts of the cement itself. From this explanation it will readily appear, that any substance which is added to a cement for the purpose of increasing its bulk, its hardness, or its strength, ought to be in particles or grains; and that no soft, earthy, or pulverulent matter is fit for this purpose. It will be equally apparent, to those who have considered the nature of chemical combination, that the cementing part of the materials employed in the composition of mortars and cements, should be in the finest state of division that it is possible to reduce them to.

An attentive consideration of these principles will afford an easy solution of some of the most interesting questions to which this important subject gives rise; and correct some errors that have originated from a partial examination of the phenomena. We have now to examine the qualities of those materials which naturally occur in the earth; from these must be selected the kinds that can be obtained at the least expense for the place they are to be employed at, when their quality is suitable to the nature of the work.

24. The cementing materials are chiefly obtained from Limestone the different species of limestone, to which may be added puzolana, terras, iron ores, basalt, and other substances of a like character. The limestones may be divided, as regards cements and mortars, into three classes; common limestone, poor limestones (chaux maigre of the French), and cement stones.

25. Common limestones consist of carbonate of lime with very little of any other substance; they produce a white lime, which slacks freely when well burned; they dissolve in diluted muriatic acid with only a small portion of residue, and never contain more than a trace of iron. They differ much in external characters, as chalk, marble, common compact limestone, &c. These limestones do not form water cements without the addition of other kinds of cementing matter; and hence they are usually employed for common mortar. The hardest marble and the softest chalk make equally good lime when well burned; but chalk lime will slack when not perfectly burned, and, therefore, has seldom a sufficient quantity of fire; whereas stone lime must have sufficient to make it slack. It has also been observed, that stone lime does not re-absorb carbonic acid so rapidly as chalk lime.⁴

¹ Kirwan's Geological Essays, p. 145, 6.

² The decayed state of the piers of Westminster and Blackfriars' Bridges shews us how very little this important subject has been studied; as well as the necessity of studying it.

³ Traité de l'art de bâtir, tome i. p. 90.

⁴ Higgins on Mortars and Cements, p. 29.

Masonry. Lime, made from common limestones, sustains very little injury from being kept after it has been formed into a thin paste with water, provided the air be effectually excluded; indeed, Alberti mentions an instance of some which had been covered up in a ditch for a very long time, that was of an excellent quality.

Poor limestones. 26. Poor limestones consist of carbonate of lime united with silica, alumina, and metallic oxides. They in general produce a buff-coloured lime, but sometimes it is white. When the lime is white, there are no metallic oxides present. They contain a considerable portion of matter, which is insoluble in acids. They differ considerably in external characters; the blue lias, the Sussex clunch, and Sutton limestones, may be cited as instances. The lime of poor limestones does not slack freely; and it would always be desirable to reduce it to powder by grinding, in preference to slacking, because in slacking a part of its setting property is destroyed. When poor lime is slacked, it should be made into mortar, and used immediately. Moisture does such lime more injury than the re-absorption of carbonic acid; and mortar made of it ought not to be disturbed after it has begun to set or harden. Poor limestones are subject to vitrify in burning, unless the heat be increased gradually; but by a little management in this respect they will bear a heat sufficiently intense to convert them into good lime. The lime of poor limestones makes excellent common mortar, when the precautions we have pointed out have been attended to. It is much superior to mortar made from common limestones. It is not, however, sufficiently powerful for a water cement without the addition of puzzolana, terras, or other like substances.

27. Cement stones. We have taken the liberty of giving this name to the class of earthy bodies which afford a good water-cement without addition; and it appears to be essential for that purpose that there should be at least 50 per cent. of carbonate of lime in the stone. Those cements which answer best are made from stones containing about 60 per cent. The other substances usually contained in these stones are, silica, alumina, oxide of iron, sometimes oxide of manganese, carbonate of magnesia, with traces of other alkaline and earthy salts. Cement stones are in a great measure soluble, with effervescence, in diluted muriatic acid; they lose about one-third of their weight by calcination, after which they do not slack, but may be ground into a fine powder, of a brown colour, of different degrees of intensity according to the nature of the stone. Those now used to make cement are found in rolled pieces, or in nodules, with septa of carbonate of lime, on the sea-coast at Sheppy, Harwich, and near Whitby, and at Boulogne on the French coast. The nodules are plentifully dispersed throughout the London clay stratum, and the alum shale of Whitby; they are termed clay-balls, ludus helmontii, septaria, &c. Cement stones are also liable to vitrify in burning, either from being quickly exposed to a strong heat, or by making the heat too intense.

When cement stones have been calcined and ground, the powder thus obtained should be kept in a very dry place, as its setting property is soon destroyed by moisture. It absorbs moisture rapidly from the air; and, therefore, it should be as little exposed as possible; but when closely packed, and in a dry place, it may be kept a considerable time.

28. There are perhaps few mineral bodies which would answer as well as clay-balls for making cement, unless it be calcareous marls; but there are very few situations where the means of forming an artificial cement are wanting; the manner of proceeding must depend upon the nature of the substances employed. In general it will be necessary to mix slacked lime with the substance containing silica, alumina, and oxide of iron, and calcine them together, and afterwards grind them. The iron should be in the state of protoxide, to the amount of from 13 to 18 per cent. by

weight. The quantity of lime may be about 30 per cent. or 53 per cent. of carbonate of lime; the silica and alumina being in equal portions; and hence, in the raw materials, one hundred parts by weight may consist of

29. There are several mineral bodies which, combined with lime, form good cements. Some of these may be used in their natural state, but others require calcination. Puzzolana and terras are of the former kind; basalt, iron ores, and ferruginous clay, belong to the latter class.

Puzzolana is a volcanic production much used in making Puzzolana water cements. It was known to the Romans, and employed by them both in ordinary buildings and in water works. Its colour is reddish or reddish brown, and grey or greyish black; that of Naples is generally grey, that of Civita Vecchia more generally reddish brown. Its surface is rough and uneven, and it has a baked appearance; when broken and examined by a magnifying glass it appears to be a spongy substance, with innumerable cavities like a cinder, and not much harder. It comes to this country in pieces varying in size from the bulk of a nut to that of an egg. It is very brittle, and has an earthy smell; its specific gravity being from 2.570 to 2.785. It does not effervesce with acids; and is not diffusible in cold water. The iron it contains is not oxidated; and being finely divided and dispersed throughout its mass, offers a large surface which quickly decomposes the water with which it is mixed when made into mortar. The union of the oxygen of the water with the iron is the principal cause of the solidification of water mortar; and this union is greatly facilitated by the heat given out by the quicklime.

30. The substance called Terras or Trass, found near An-
dernach, on the Rhine, is endowed with similar properties. Its colour is light grey or brownish grey; its surface is rough and porous; and it is often found mixed with other matter. It is sometimes so hard that it yields with difficulty to the knife. According to Bergman, it contains more lime than puzzolana does, and effervesces slightly with acids. Smceton found puzzolana to be equal, if not superior, to terras in forming a water cement with Aberthaw lime; he also observed that terras is inferior for work which is to be alternately wet and dry.

31. If any stone be employed in which the iron is perfectly oxidized, it will be necessary to abstract some portion of the oxygen from the iron in the process of calcination, on the same principle as the reduction of iron from the ore is effected. (See IRON-MAKING.) It is also necessary to employ similar means to restore the cementing power of materials that have attracted moisture after being calcined.

II.—OF THE PRINCIPLES OF STABILITY AND STRENGTH IN MASONRY.

32. The strength and the stability of stone-work depends partly on its mass or weight, and partly on the resistance of the materials. And, since we cannot imagine incompressible fulcra, nor that the materials of masonry are infinitely hard and inflexible, as writers on elementary mechanics consider them to be, therefore, it is essential that the resistance of materials should be considered, and the effect of their weight allowed for in estimating the power of the straining force.

The resistance of stones being dependent on their state of aggregation, and not on the hardness or density of their

Masonry. elementary parts, their comparative strength cannot be judged of by these qualities; indeed, there are few kinds of materials of which the resistance is so uncertain as that of stone, and hence, it is not at all adapted for any support where its resistance depends on its cohesion only, unless it be very carefully examined, and abundant strength be allowed. (See § 15.) The resistance of stone to compression is less affected by its irregular nature, particularly as it is usually employed in blocks of inconsiderable height; and, in general, there is scarcely any reason to be sparing of a material which it is often more expensive to reduce than to employ in large blocks. When, however, works of great magnitude are to be constructed, the weight of the materials themselves forms the chief part of the straining force; and, consequently, in such cases it becomes desirable to form a tolerably accurate estimate of their power.

33. This power is limited by a property of bodies, that has not received that degree of attention which its importance would lead us to expect. We shall in this place make it the basis of an investigation of the power of materials to resist a force applied in any given direction, and shew its application to some of the cases where a mason is most likely to need the assistance of calculation.

When any material is strained beyond a certain extent, every time the strain is increased to the same degree, there is a permanent derangement of the structure of the material produced; and a frequent repetition will increase the derangement till the parts actually separate. (See the article CARPENTRY.) When a small base rests upon a considerable mass of matter, as a pier on the ground, the quantity of derangement will increase only till the mass be compressed to that degree which renders the increase insensible; but in many cases a number of years will elapse before the settlement becomes insensible.

34. The strain which produces permanent derangement in the structure of a material varies from one-fourth to two-fifths of that which would destroy its direct cohesion. In stone the lower value should be taken, on account of its being subject to so many defects; and, for the present, let this strain be denoted by $f$ lbs. upon a superficial foot.

35. Imagine ABCD to be a block, Plate CCCXLV. fig. 1, strained either in the direction EF or FE, by a force $W$; and let BDF be a line drawn in the same plane as the direction of the straining force, and perpendicular to the axis $ab$ of the block. Now, if we consider the resistance of the block to be collected at the centres of resistance $t$ and $c$; then $tF$ will represent a lever acted upon by three forces; that is, the resistance at $t$ and $c$, and the straining force at $F$. If the angle $FEg$, be denoted by $a$; then, the effect of the force $W$, reduced to a direction perpendicular to the lever, will be expressed by $\backslash cos.\; aW$. (1.)

Also, if $T$ be the resistance at $t$, and $C$ the resistance at $c$, we shall, in the case of equilibrium, have $C\; -\; T\; =\; \backslash cos.\; aW$. (2.)

And, by the property of the lever,

36. Without stopping to notice some maxims furnished by this equation, (see the article BRIDGE.) We will proceed to explain the notation used in the investigation which follows:

$l$ = the length AB.

$d$ = the depth BD, measured in the same plane as the direction of the strain.

$b$ = the breadth.

$z$ = the distance of the neutral point $e$ from the axis $ab$.

$y$ = the distance of the point $g$ from the axis $ab$.

$p(\backslash frac\{1\}\{2\}d\; -\; z)$, and $p(\backslash frac\{1\}\{2\}d\; +\; z)$ the respective distances of the Masonry centres of resistance $t$ and $c$ from the neutral point; and, consequently,

$pd$ = the distance, $cf$, between them. And, $g(\backslash frac\{1\}\{2\}d\; -\; z)$, and $g(\backslash frac\{1\}\{2\}d\; +\; z)$ the respective distances of the centres of gravity of the sections into which the neutral axis divides the block, counted from the neutral point.

The leverage, $cF$, expressed in this notation, will be $\backslash frac\{l\; \backslash sin.\; a\}\{\backslash cos.\; a\}\; +\; y\; -\; \backslash frac\{1\}\{2\}pd\; +\; (-p)z$; consequently, equation (4) becomes

37. Now, it has been shewn, by writers on the resistance of solids, that the resistance of any section, collected at its centre of pressure, is equal to its cohesive force multiplied by the distance of its centre of gravity from the neutral axis, and divided by the distance of the point of greatest strain from the neutral axis. (Emerson's Mechanics, prop. 77. 4to. ed.)

Therefore $\backslash frac\{C\}\{T\}\; =\; \backslash frac\{(\backslash frac\{1\}\{2\}d\; +\; z)^2\}\{(\backslash frac\{1\}\{2\}d\; -\; z)^2\}$; which being substituted in equation (5), we obtain

If the block be rectangular $p\; =\; \backslash frac\{2\}\{3\}$, and, therefore, the distance of the neutral point from the axis is

38. The value of $z$ for a rectangular section being determined, the magnitude of the straining force is easily found, so that it may not exceed the power of the material; for, by the properties of the lever,

39. In particular cases this formula becomes more simple; as, for example, when the distance of the point $g$ from the axis $ab$ is $0$, that is, when $y\; =\; 0$,

In a column, or pillar, the section of greatest strain will be at the middle of the length, as at BD in fig. 2; and the direction EF of the straining force is usually parallel to the axis; and then $\backslash sin.\; a\; =\; 0$, and $\backslash cos.\; a\; =\; 1$, and therefore,

When the direction of the straining force coincides with

Masonry. the axis, or when $y=0$, the strain on a column or pillar is expressed by the equation $W=fbd$ (13.)

These equations apply also to tensile forces.

When the strain becomes transverse, or when EF is perpendicular to the axis, as in fig. 3, then $\backslash cos.\; a=0$, and $\backslash sin.\; a=1$, hence $W=\backslash frac\{fbd^2\}\{6l\}$ (14.)

If the block be supported at the ends, and the load be applied in the middle of the length, as in fig. 4, the fracture will take place at BD; and W in equation (14) will be the pressure on either support, which is obviously half the load in the middle.

40. In any of these equations it is perfectly immaterial how the load be distributed, provided the line of direction be that which passes through the centre of gravity of the mass supported, and the weight be the whole weight of that mass. Or, if the strain be the combined effect of several pressures, then the direction must be that of the resultant of these pressures, as determined by the principles of mechanics. (See the article CARPENTRY.)

41. If a slab of equable thickness and width be supported along two of its sides, as at AC, AB in fig. 5, and it be strained by a force acting at D, in a direction perpendicular to its surface, and DE be made equal to DB, then the fracture will take place in the direction EB; for it may be

shown, by the principles of the maxima and minima of quantities, that the resistance, according to that line, is a minimum. And since, in that case, $EB=2FD$, we shall

have, by equation, (14), $W=\backslash frac\{fd^2\}\{3\}$ (15.)

A force uniformly diffused over the surface of the slab would fracture it in the direction CB, shown by a dotted line in the figure, and if $w$ be the load in pounds upon a square foot of the surface, then the proper values being substituted for the leverage and breadth in equation, (14),

The strength of a series of steps bearing upon one another, as in the perspective sketch, fig. 6, may be determined with sufficient accuracy by the last equation, supposing the depth to be the mean vertical depth of any one step; as, for example, taken at GH in fig. 7, the figure showing the ends of the steps.

42. The case to which equation (14) applies, affords the most convenient, as well as the most accurate, means of determining the value of $f$ for any material; and, supposing it to be one-fourth of the absolute cohesion (§ 34), the last column of the following table of experiments gives its value for various stones, mortars, &c. in the nearest simple numbers under the calculated value.

TABLE I.—Experiments on the Transverse Strength of Stones, &c. to the Case Equation (14.)

Numbers 18, 19, and 20, are from Barlow's Essay on the Strength of Timber, (p. 250), each being a mean of three trials. Number 23 is from Rondelet's Traité de l'Art de bâtir, (tome iii. p. 377), lowest result; the rest of the experiments were made by the writer of this article.

We have not here availed ourselves of the experiments of Gauthey (Rozier's Journal de Physique, tome iv.) on the transverse strength of stones; because those he fixed at one end appear to have been injured in fixing, and only a calculated result is given for the other specimens supported

at both ends. As to this, see the article on the STRENGTH OF MATERIALS.

43. Several experiments have also been made, with the intention of measuring the direct resistance to extension or compression; but theory indicates so nice an adjustment of the direction of the straining force as necessary in these experiments, that the reader may expect the results to differ as widely amongst themselves as they are found to differ from theoretical calculation.

The experiments, Nos. 3, 4, 5, 6, and 7, are extracted from Rondelet's L'Art de bâtir, (tome i. p. 312.) Nos. 1 and 2, are by Gauthey, (Rozier's Journal de Physique, tome iv. p. 414.)

44. In the resistance to actual fracture, from a compressive force, the joint effect of cohesion and friction is concerned, and, therefore, a much greater force is required to crush than to tear asunder the same quantity of material. The resistance to fracture might be investigated on principles analogous to those we intend to employ in determining the pressure of earth against retaining walls, &c. (See the article CARPENTRY.) But we conceive, that it is neither prudent, nor useful, nor necessary, to load the parts of a structure beyond that limit we have made the basis of our investigation.

(See § 34.) Rondelet has observed, that the load under which a stone began to split was nearly always two-thirds of that which crushed it; but that stone of some kinds began to split with half the load that crushed it, (L'Art de bâtir, iii. 86 et 101). The value of $f$ should, therefore, not exceed one-fourth of the force which splits stone, and supposing the splitting force to be always half the crushing one, we shall have $f\; =\; \backslash frac\{1\}\{8\}$ of the crushing force.

45. In this, as in the preceding tables, the reader will observe, that the results of all experiments are given in the original weights and measures; but that the value of $f$ and the weight of a cubic foot, are in English pounds avoirdupois, and for an English foot. The foreign weights and measures are distinguished by their foreign names.

The experiments, Nos. 1, 21, 22, and 36, were made by Gauthey. (Rozier, Journal de Physique, tome iv. p. 406.) Those numbered 3, 4, 5, 8, 9, 11, to 20, 30, 31, 32, 34, 37, 38, and 39, were made by Mr. George Rennie. (Philosophical Transactions for the year 1818.) The others were made by Rondelet. (Traité de l'Art de bâtir, tome i. and tome iii.) We have selected those which will be most useful, with others of a more interesting and curious nature; such are Rondelet's experiments on the effect of beating mortar, the strength and density of ancient mortar, and the resistance of stones used in ancient and in modern structures.

46. It was observed by M. Rondelet, in the course of his very numerous experiments, that it was not the heaviest stones which offered the greatest degree of resistance to compression, but those of a fine even grain and close texture, with a deep colour; that of granites, the most compact and perfectly crystallized were the strongest, (L'Art de bâtir, tom. i. 213, 215); and that, when all other qualities were the same, the strength was in proportion to some function of the specific gravity.

The writers who have contributed to our experimental knowledge of the strength of stones are not numerous. The chief are EMERSON, in his Mechanics, 4to. ed. p. 115; GAUTHÉY, in his Mémoire sur la Charge que peuvent porter les Pierres in ROZIER'S Journal de Physique, tome iv. 1774, and in his Construction des Ponts, tome i. p. 267; COULOMB, in his Mémoires présentés à l'Académie, 1773; RONDELET, in his Traité de l'Art de bâtir, tome i. et iii. (The latter volume contains the experiments made by PERMONET and SOUFFLOT); RENNIE, in the Philosophical Transactions for 1818, or Philosophical Magazine, vol. liii.; and TREDGOLD, in the Philosophical Magazine, vol. lvi. p. 290.

47. The last column in each of the three tables of experiments shows the greatest load that we suppose should be borne by a superficial foot of the different kinds of stone contained in those tables. We now propose to give the

results of some calculations respecting the extent to which stone has in practice been loaded. The foreign ones are reduced to our own weights and measures, and the whole stated in round numbers.

The pillars of the Gothic church of All-Saints at Angers, of the stone, No. 24, Table III. support on each superficial foot a pressure of 86,000 lbs.¹ The pillars of the dome of the Pantheon at Paris, the lower part of which are of Bagneux stone, (No. 2151, Table III.), support on each superficial foot 60,000 lbs.² The pillar in the centre of the chapter-house at Elgin, which is of red sandstone, supports on each superficial foot 40,000 lbs.³ The piers which support the dome of St. Paul's, in London, sustain a pressure on each superficial foot of 39,000 lbs.⁴ The piers which support the dome of St. Peter's at Rome, sustain a pressure on each superficial foot of 33,000 lbs.⁴ The pressure on the key-stone (No. 23, Table III.) of the Bridge of Neully has been estimated for each superficial foot at 18,000 lbs.⁴

In regard to these examples we have to remark, that the calculators of them have considered the pressures as uniformly distributed over the pressed surface; but this can only be true when the direction of the resultant of the straining force coincides with the axis of the pier or pillar; besides, stones cannot be wrought absolutely level, nor bedded in perfect contact. From these circumstances, the strength of piers, columns, pillars, and arch-stones, should be estimated by equation (12), and when the line of direction falls within the pier, always making $y\; =\; \backslash frac\{1\}\{2\}$ half the least dimension of the section, an allowance which will include the effect of the greatest possible inequality of action. We shall, in that case, have

If the pressure on the Bagneux stone in the piers of the dome of the Pantheon at Paris be estimated by this for-

¹ Gauthey, Rozier's Journal de Physique, tom. iv. p. 409, and Construction des Ponts, tom. i. p. 273.

² Rondelet, L'Art de bâtir, tom. iii. p. 74.

³ Telford, Edinburgh Encyclopedia, Art. BRIDGE, p. 505.

⁴ Gauthey, Construction des Ponts, tome i. p. 260.

Masonry. mula, it will be found that it is sufficient to split the stones, and this has actually happened.¹

Principles of arches, domes, &c. 48. The chief elements of the theory of arches have already been given in the Article BRIDGE, (sect. ii.) to which we refer the reader, at the same time expressing a hope that the excellent article referred to will be useful in correcting some absurd notions respecting catenary and other curves, which are too commonly entertained. The conical support of the lantern of St. Paul's is a fine example of an appropriate form, whilst the catenary dome of the French Pantheon exemplifies a scientific blunder of the first magnitude.²

The principles of domes, of groins, and of vaulting of every kind, are the same as those of arches, excepting that each kind has its peculiar manner of distributing the load on the different parts. See prop. M and N, art. BRIDGE.

Of the Pressure of Earth, Fluids, &c., against Walls.

Pressure of earth against walls. 49. When a high bank of earth, or a fluid, is to be sustained by a wall, as it is often necessary to do in forming bridges, locks, quays, reservoirs, docks, and military works, the construction is very expensive, however economical the means employed may be; hence it is desirable to devote some space to an object of which the importance is manifest.

Let EC, Plate CCCXLV. fig. 8, be the line according to which the earth would separate, if the wall were to yield in a small degree; then AEC will represent the section of the prism of earth, the pressure of which causes the wall to yield.

Put W = the weight of the prism AEC, when its length is unity.

R = the resistance of the wall, when its length is unity.

a = the angle ECa, which the plane of fracture makes with a vertical line.

c = the angle ACa, which the back of the wall makes with a vertical line.

F = the friction of the earth when the pressure is unity.

h = the vertical height of the wall aC in feet.

and S = the weight of a cubic foot of earth, water, or other matter to be supported.

If g be the centre of gravity of the prism of earth, the triangles rgg, CaE, being similar, the effort of the prism to slide, in the direction EC, reduced for the friction, will be

This effort is to be opposed by the resistance of the wall, which let us suppose to be collected at c, the centre of pressure, and, reducing it to the direction CE, the effect of friction being allowed for, it becomes

Hence, in the case of equilibrium,

50. But, in the case now considered, the radius being unity, $W\; =\; \backslash frac\{h^2\; S\}\{2\}\; (\backslash tan.\; a\; -\; \backslash tan.\; c)$. Therefore, $R\; =$

And, from the state of the variable quantities in this equation, it is obvious that it has a maximum value, which determines the angle of fracture. By the principles of maxima and minima, the maximum pressure takes place when

If the angle which the plane of repose (BRIDGE, Sect. III.) makes with a vertical plane be denoted by i, then

If the back of the wall be vertical, $\backslash tan.\; c\; =\; 0$, and then this equation reduces to the simple form, which Prony obtained, of $\backslash tan.\; a\; =\; \backslash tan.\; \backslash frac\{1\}\{2\}\; i$. (24.)

51. When we substitute in equation (21) the value of the $\backslash tan.\; a$, which has been found in equation (23), it becomes $R\; =\; \backslash frac\{h^2\; S\}\{2\; \backslash tan.\; i\}\; \backslash left\backslash \{\; \backslash tan.\; i\; +\; \backslash tan.\; c\; +\; \backslash frac\{2\}\{\backslash tan.\; i\}\; -\; 2\; \backslash left(\; \backslash tan.\; c\; \backslash tan.\; i\; +\; \backslash frac\{\backslash tan.\; c\; +\; 1\}\{\backslash tan.\; i\}\; +\; 1\; \backslash right)^\{\backslash frac\{1\}\{2\}\}\; \backslash right\backslash \}$ (25.)

And, when the back of the wall is vertical, it becomes $R\; =\; \backslash frac\{h^2\; S\}\{2\}\; (\backslash tan.\; \backslash frac\{1\}\{2\}\; i)$. (26.)

The $\backslash tan.\; i$ being the co-tangent of the angle of repose, if the matter to be supported be of so fluid a nature that it naturally assumes a sensibly level surface when at rest, the $\backslash tan.\; \backslash frac\{1\}\{2\}\; i$ becomes equal to unity, and consequently,

The same result may be obtained from the common principles of hydrostatics in the case of fluids.

Since the only variable quantity which enters into the calculation of the distance of the centre of pressure is the height h, whatever the nature of the supported material may be; therefore that distance counted from the base will always be $\backslash frac\{1\}\{2\}\; h$, as in the pressure of fluids. (See HYDROSTATICS.)

52. TABLE IV.—Table of Constant Quantities necessary for calculating the Pressure of some Materials.

¹ Gauthey, Construction des Ponts, tome i. p. 273.

² La charge considérable que cette voûte devait porter à son sommet, a déterminé à choisir pour la courbe de son cintre la chaînette. (Traité l'Art de bâtir, ii. 308.)

Masonry. In sand, clay, and earthy bodies, the natural slopes should be taken when the material is dry, and the clay and earth pulverised. When any of these bodies are in a moist state, the parts cohere, and the angle of repose is greater, though the friction be actually less. The preceding Table shews that the pressure of water is greater than that of any of the other kinds of matter, and from the nature of fluids it is evident, that if water be suffered to collect behind a retaining wall, calculated to sustain common earth only, it will most likely be overturned. Such accidents may be prevented by making proper drains.

53. The preceding analysis will apply, without sensible error, to the curved walls which have lately become fashionable. Fig. 10 is a section of one of these walls, as executed from a design by Rennie. The vertical height, AB, 21 feet; the wall of uniform thickness, with counterforts 15 feet apart; and the front of the walls described by a 69 feet radius, with the centre in the horizontal line DA produced. The wall is built of brick, and the uniform part is 4.5 feet thick. The radius is usually thrice the vertical height of the wall; when this proportion is adhered to, the angle $c$ will be ten degrees, for which the value of $R$ is calculated in the Table.

Resistance of Walls.

54. In the first place, we propose to investigate the resistance a wall offers to being overturned; and, in so doing, it appears desirable that the resistance of the mortar in the joints should be considered one of the elements of the strength of the wall. Good mortar adds much to the firmness of walls, and still more to their durability, and, all things considered, its first cost is less than that of bad; besides, the resistance of mortar to compression must be considered, for, in practice, we have no perfectly hard arrises to fulfil the conditions of common mechanical hypotheses.

Put $A$ = the area of the wall.

$w$ = the weight of a cubic foot of masonry.

$y$ = the horizontal distance, $ga$, between the vertical passing through the centre of gravity of the wall, and the point where the axis cuts the plane of fracture, the same notation being applied to the other quantities as in the foregoing equations.

Let $G$, fig 9, be the centre of gravity of the wall; and on the vertical $Gg$ set off $gI$, the height of the centre of pressure; also, let $IK$ represent $A\; \backslash times\; w$ = the weight of the wall, and $HI$ the force $R$ of the earth.

Then, completing the parallelogram, $EI$ will represent the direction and intensity of the straining force; consequently,

Which determines its direction, and its intensity is

But, we have found, equa. (10),

If the section of the wall be a parallelogram, then $A\; =\; hd$, and $-\backslash frac\{1\}\{2\}h\; \backslash tan.\; c\; =\; y$; these values of $A$ and $y$ being substituted in equa. (30), it becomes $-whd^2\; +\; fd^2\; +\; 3h^2wd\; \backslash tan.\; c\; =\; 2Rh$. $(31.)$

When the section of the wall is a rectangle $y=0$, therefore equa. (31) reduces to

This last equation is also correct for a wall of which the back is vertical, and the front sloping. We suppress the investigation, to afford the young student an opportunity of proving that the diminution of weight is exactly counterbalanced by the alteration of the distance of the centre of gravity from the axis.

The tendency of a wall to slide forward may be easily prevented, by giving an inclination to the joints.

55. To illustrate these rules we shall give two examples, and in these shew the construction of a table, which the reader may enlarge at his pleasure.

Example I. Let it be required to determine the thickness of a rectangular wall for supporting the front of a wharf 10 feet in height, the earth being a loose sand, and the wall to be built of brick.

The weight of a cubic foot of brick-work may be estimated at 100 lbs., and the resistance of mortar being valued at 5000 lbs. per superficial foot, the experimental value being 7900 lbs., Table III. Experiment 42, and the difference an allowance for any irregularity in building, consequently, $f=5000$; $w=100$; and by Table IV. $R=13.8h^2$; hence equa.

$\backslash sqrt\{\backslash frac\{h^2\}\{181\; -\; 3.62h\}\}$. When $h=10$ feet, then the thickness of the wall $d=2.644$ feet. If $h$ be made successively 10, 20, 30, 40, &c. feet, the numbers under the head of dry sand in the following Table will be obtained, observing that they are only calculated to the nearest tenth of a foot.

The proper thickness being found for supporting one kind of material, that for any other may be easily determined; as the thickness varies as the square root of $R$, equa. (33). Let the thickness for dry sand be $d$, then

$\backslash sqrt\{13.8\}\; :\; \backslash sqrt\{31.25\}\; ::\; d\; :\; 1.5d$ the thickness for supporting water. $\backslash sqrt\{13.8\}\; :\; \backslash sqrt\{17.85\}\; ::\; d\; :\; 1.14d$ the thickness for supporting moist sand. In this manner, by means of Table IV. the thicknesses for other kinds are easily calculated.

Example II. If a retaining wall be intended to support a sandy and loose kind of earth, to be constructed of brick, and to be inclined 10 degrees from the vertical, the thickness being uniform; it is required to determine that thickness for any given height.

hence $\backslash frac\{3\; \backslash tan.\; c\}\{2\}\; =\; .27$, and its square = .0729. Also $f=5000$,

For sandy earth $R=4.8h^2$, therefore $d\; =\; \backslash frac\{h^2\}\{50-h\}$

20, &c. feet, the numbers obtained will be the same within

Masonry. one-tenth of a foot, as those in the following Table, column fifth.

56. TABLE V.—A Table of the Thicknesses for Retaining Walls, Revetments, Dock-walls, &c.

Our investigation informs us that the mortar of high walls must be of a superior strength; indeed, we know that when its consolidation takes place, under considerable pressure, it is of much greater strength. According to what function of the pressure the strength increases, we have not experiments to determine, and we therefore point out the circumstance to the notice of experimental inquirers.

57. The preceding analysis being confined to the conditions under which the equilibrium could not be disturbed by the pressure, it would be quite unnecessary to consider the phenomena of actual fracture, if it were not for the proof which even these phenomena afford of the defects of the common mode of constructing these walls. The back of the wall is generally formed of inferior materials, hence the technical term face mortar and backing mortar; but, even with inferior mortar, the workmanship is so carelessly done, towards the back of the wall, that when it fractures a portion is left behind. A moment's attention to the direction of the pressure (see fig. 9) must shew the importance of using good mortar, and making good bond at the back of the wall; if any part be neglected it ought to be the middle, which is of least importance, provided the wall be well bound together by cross bond stones.

The strength of a wall to sustain earth will always be greatly increased by any roughness or irregularity in the back of the wall, such as projecting stones or bricks; in stone work it is easy to gain much stability by this means. The friction against a smooth wall must add much to its strength; we have not thought it necessary to include its effect in our calculations, but intend that, with counterforts, it should be considered as a set-off against accidental pressures, &c.¹

Counterforts are usually placed at about three times the thickness of the wall apart, and are made of the same width as the thickness of the wall. In fig. 11 is shewn a plan for building a wall to sustain the pressure of earth according to the form proposed by Vitruvius, (lib. vi. cap. ix.) And fig. 12 is a plan from Perrault's Notes. Various other plans have been proposed, the chief of which Colonel Pasley has collected in his Course of Fortification; but most of them have little to recommend them. It seems desirable that every kind of curved work should be avoided; and perhaps that plan which unites the most economy with the greatest stability is shewn by figs. 13 and 14, Plate CCCXLVI. The spaces A, A, A, are proposed to be filled with gravel or fragments of stone; the whole of the stone-work to be well bonded; and the front and back wall of that thickness which is best suited for bond, in the kind of material to be employed.

Bond of Walls and Cramps.

58. It is not sufficient to depend entirely upon the cementing power of mortar in the construction of walls; the stones themselves should also be bound together by their disposition. The art of disposing stones for this purpose is called bonding. Part of the longest stones should be employed to bind the wall in length, and the other part to bind it cross-ways; the former are called stretchers, the latter headers.

Figs. 15, 16, 17, 18, and 19, shew various methods of bonding walls; these are selected from Greek and Roman examples. The courses of stone are often irregular, as in fig. 17; and in some works we find both irregular courses and broken ones; that is, such as are intercepted by large blocks of stone. Broken courses should be avoided, because they occasion irregular settlements.

The bond of walls requires to be most carefully attended to in the construction of piers, angles, and, in general, every part exposed to great strain.

On this subject it may also be remarked, that crossing the joints properly is a more effectual means of bonding a wall than that of employing very long stones, unless they be very strong ones. For if a stone exceed about three times its thickness in length, it cannot be so equally bedded but that it is liable to break from unequal pressure; and the fracture commonly takes place opposite to a joint, and therefore destroys the bond of the wall. This defective mode of construction we have often had occasion to notice.

In works of hewn stones destined to support great pressure, or to bear the action of a heavy sea, it is necessary that the stones should be of great bulk, and connected in the firmest manner. Sometimes this is effected by forming the stones so as to lock them together. The Eddystone and Bell-Rock Light-houses are bound together at the base on this principle.

Where less strength is required, iron cramps are used, and sometimes pieces of hard stone are dove-tailed into the adjoining blocks. We think cramps of cast-iron might be employed with much advantage in all these cases.

59. The proper quantity of mortar to be employed in stone-work is another point to which it will be useful to direct the mason's attention. A stone cannot be very firmly bedded upon a very thin layer of mortar; and if the stone be of an absorbent nature, the mortar will dry too rapidly to acquire any tolerable degree of hardness, (Vitruvius, lib. ii. cap. viii.) however well it may have been prepared. On the other hand, if the bed of mortar be thicker than is necessary to bed the stone firmly, the work will be a long time in settling, and will never be perfectly stable.

When the internal part of a wall is built with fragments of stone, they should be closely packed together, so as to require as little mortar as possible. Walls are often bulged by the hydrostatic pressure of mortar, when it is too plentifully thrown into the interior, to save the labour of filling the spaces with stones.

The walls of houses are frequently built with hewn stone on the outside, and rubble stone on the inside. The settlement of these two kinds of stone-work during the setting of the mortar are so different, that the walls often separate; or where this separation is prevented by bond stones, the wall bulges outwards, and bears unequally on its base. These evils are best prevented by using as little mortar as possible in the joints of the interior part of the wall, and not raising the wall to a great height at one time.

Foundations.

60. The nature of the materials employed in masonry.

¹ See Philosophical Magazine, vol. li. p. 401, where the effect of such friction is considered.

having been considered, and also the methods of uniting them, we have, in the next place, to turn our attention to the nature of those foundations on which it is commonly required to raise permanent structures of such heavy matter.

In founding on dry ground, the nature of the foundation is ascertained without much difficulty. When it is found to be of firm hard rock, that will bear the action of the weather, no particular precautions are necessary; but in all other cases it is desirable to level the trenches to such a depth as will prevent them from being affected by the change of seasons. Frost, we believe, seldom penetrates so low as two feet below the surface, (see the article CLIMATE); but in clayey ground, the effect of shrinkage, by heat, is often sensible at four feet below the surface; for to that depth the cracks in summer often extend. Consequently, in clay, the depth of the foundations should never be less than four feet, and in heavy buildings, deeper in proportion to the weight they are to support.

In large works it is also necessary to examine the matter, inclination, and thickness of the under strata, particularly when the upper stratum is of inconsiderable thickness. For this purpose, the older writers on architecture, with much propriety, recommend that a well should be dug near the place, to ascertain these points. A knowledge of the inclination and nature of the strata will also be of use in planning drains, a subject of no small importance to the durability and comfort of a mansion.

In soft ground the base of the wall should be made wide, and it may be reduced to the proper thickness by small offsets or steps, as in fig. 20. On clay or dry sand the breadth at the bottom may be double the thickness of the wall. On compact gravel or chalk the breadth may be to the thickness as 3 is to 2.

If the ground be soft and wet, with a firm bottom within the reach of piles, then piles may be employed with advantage; but they are very likely to rot in a few years, where the ground is not wet; and, therefore, in the case of soft ground, not sufficiently wet to preserve piles from decay, we should recommend, in preference, a very wide base or footing, well bonded together with bars of cast-iron, disposed so that one part could not settle without causing the adjoining ones to go down at the same time; and the whole of the base should, for greater strength, be built in the best water cement.

It is a practice with some architects to employ timber beams and planking in such cases, and in consequence of its decay, in many instances it has been necessary to replace the timber with stone and brick at an enormous expense. It should be a maxim in construction never to employ timber in a permanent structure, where it is not either absolutely wet or perfectly dry.

When ground is very soft and wet, and the solid stratum is beyond the reach of piles, a solid mass may be formed to erect the superstructure upon, by means of a grating of timber, planked over. The brick or stone-work which is built upon the planking should be joined with a water cement.

When such ground is not absolutely wet, instead of planking we would employ a connected grating of cast-iron, with stone or brick-work built in water mortar.

In all these cases the greatest difficulty consists in preventing irregular settlements; and hence the advantage of employing wood or iron to bind the base together, and render it as far as possible an inflexible and solid mass.

In all edifices which press perpendicularly on their foundations, the centre of pressure should coincide, as nearly as may be, with the centre of gravity of the surface which sustains it. In wharf walls, terrace walls, abutments and piers of bridges, and the like, the resultant of the pressures should

fall in the centre of gravity of the surface which supports it. Foundations are most difficult to manage where the ground is irregular, particularly for highly finished buildings, which are so much disfigured by a small settlement. In such cases we would endeavour to procure an inflexible base by means of cast-iron beams. It is a good plan to form counter arches under the openings, provided these arches be carefully built; but where they are not well built, they yield so much as to be no better than common walling. Excavating and removing the earth from foundations is frequently a considerable part of the expense of large works; hence the peculiar species of management which will economize this branch of labour, has become an interesting subject of investigation. We intended to give an outline of the manner of treating it, but we find that it would extend this article far beyond its proper limits.

61. Founding in water may be done in various ways; but most of them are very expensive, presenting many difficulties in deep and rapid water. We shall confine our attention to this case only. The best method now in use consists in excluding the water from the space to be founded upon by means of a dam, called a coffer-dam, formed by rows of piles, with bricks or clay between the rows. When bricks are used, it is necessary to caulk the interstices between pile and pile. The space is kept clear of water by means of engines, and the foundation deepened and piled if necessary.

Considering the immense expense and risk of life which is encountered in excluding water of thirty or forty feet depth, we shall here propose, for suitable ground, a more economical and safe method, which is adapted for founding piers, abutments, sea-walls, &c. The space for the foundation being cleared, let the space it is to occupy be inclosed by a single row of piles, driven near to one another, but not so close as is necessary for a dam. The upper ends of these piles must be high enough for a stage to be formed upon them; which should be just above the height of floods or tides, as the case may be. From this stage the ground within the inclosure may be excavated, by means of a machine, formed so as to combine the principle of the field plough with that of the dredging-machine. (See DREDGING-MACHINE.) When the foundation has been cleared to a proper depth by this process, and levelled, the stone-work may be built in courses with a proper bed and joint of water cement. (See sect. i. art. 22.) A simple machine might easily be contrived for the purpose. If brick be employed instead of stone, it may be done by forming the bricks into blocks of three feet long and eighteen inches square, with cement, and using these blocks instead of stones. This method of building with blocks is already in use for constructing sewers in London. When the work is brought to that height which will enable the workmen to proceed in the ordinary manner, either then or afterwards, the piles may be cut off at low-water line, and a cap-sill being fixed upon their tops, they will remain, and serve as a protection to the work below water.

62. For many purposes it would not be necessary to excavate, nor yet to build in courses; for example, let us suppose it to be the pier of a light bridge, the row of piles being driven in an elliptical form, a strong chain should encircle it at one or two places, and the internal space filled with rough stones, thrown in with water cement, to fill the interstices between them. As the cement indurates, the whole mass would become one solid stone. This mode of construction is effected on the same principle as that which the French term "Les enrochements en béton,"¹ and we have the advantage of a much superior cement to any they have employed.

¹ Gauthey, Construction des Ponts, il. p. 276.

Of the beds or joining surfaces. 63. Before we proceed to explain the methods of forming stones to the particular shapes required for arches, vaults, &c., it may be remarked, that the young mason should be extremely careful to avoid making the beds of stones concave or hollow; for if this be done in any case where the stones have to bear much pressure, they will flush, or break off in flakes at the joints, and entirely disfigure the work. It is better that they should be slightly convex. In the construction of piers and columns, where perfectness of form is at least as much regarded as strength, this maxim should be carefully attended to. Nothing can be more offensive to the eye than a flushed joint, since it not only deforms, but also gives the idea of want of strength.

Methods of stone-cutting. 64. Stone-cutting may be equally well done by various methods; the most certain consists in forming as many plane surfaces to the stone as may be necessary, in such manner that these surfaces may include the intended form, with the least waste of stone, or in the most convenient way for applying the moulds. Upon the plane surfaces thus prepared, the proper moulds are to be applied, and the stone worked to them. It will generally happen that the bed of the stone will be one of the first plane surfaces, and the arrangement should always be made, so that there may be as little re-working as possible.¹

Describing arches. 65. When an arch is square to the face of the wall, the only difficulty is in drawing it to the proper curve. When the arch is circular, it may be described from a centre, unless the centre be very distant; and in that case a method proposed by Dr. T. Young² will be found extremely convenient for the mason's purposes. Fig. 21 represents the instrument. Three points in the curve being known, it is easily adjusted to the curve, and will also answer as a mould in many cases. AB is a straight bar of any convenient length; at each end, a small roller is fixed by means of two plates of brass; against these rollers the elastic bar CD slides as it adapts itself to a regular curvature, when moved by the screw E. The natural form of the elastic bar is shown by C'D', the depth in the middle, H, should be double the depth at either end, and the breadth uniform throughout the length. This bar, when of wood, should be of straight-grained ash, or lance-wood; the latter is best.

An elliptic arch may be described by continued motion, in the following manner. On a straight bar AB, fig. 22, if AC be made equal to the height of the arch, and CB equal to half the span, then if the end A be moved along a straight edge, ED, while the point B moves along another straight edge, FD, the point C will describe an ellipse.³ If the bar be made to move on rollers, an arch on a large scale may be easily and accurately described in this way, when a traumnal would become very unmanageable. For other methods see ELLIPTOGRAPH.

To find the direction of the joints, with a radius equal to half the span, from the point K, fig. 22, as a centre describe the arc GH, which determines the points G, H, called the foci. Let it now be required to draw the joint I, join IG, and IH, draw LI to bisect the angle GIH, and it is the joint at I. A parabolic arch may be drawn very easily on a large scale by means of tangents. Make AE, fig. 23, equal to the rise CA, and join ED and EB. Draw FG parallel to DB, and divide DF and EG (which are equal) each into the same number of equal parts, then join 11, 22, 33, &c., as is

shown in the figure; a curve drawn to touch these tangents is a parabola. Arches are most conveniently drawn on a large scale by means of parallel ordinates; and an extremely simple method of this kind, for a parabolic arch, has been described by Sir John Leslie. (Inquiry into the Nature of Heat, p. 503.) Let AB, fig. 24, be the span, and CD the height. Divide AB into twenty equal parts, and raise a perpendicular from each point of section. Let CD be 100 by a scale of equal parts, make the next ordinate on each side 99 parts, or $9\; \backslash times\; 11$, by the scale; the next pair of ordinates make 96 parts, or $8\; \backslash times\; 12$, and so on; those numbers being respectively as the rectangles of the segments into which AB is divided. To draw the joints of a parabolic arch, let I be a point, at which a joint is to be drawn, fig. 24; draw Id parallel to BA, and make DT equal to Dd; join IT, and make EI perpendicular to IT, which is the joint required.

66. The finest form for a Gothic arch is a cubic parabola, which is easily constructed from its equation. Observing that the vertex of the curve is at the springing of the arch, and making $x$ the abscissa, and $y$ its corresponding ordinate, by the nature of the curve $x\; =\; \backslash frac\{y^2\}\{a\}$. Now, if we make $y$ successively equal to 1, 2, 3, &c., feet, we shall have $\backslash frac\{1\}\{a\}\; =\; x$;

$\backslash frac\{8\}\{a\}\; =\; x$; $\backslash frac\{27\}\{a\}\; =\; x$; $\backslash frac\{64\}\{a\}\; =\; x$; $\backslash frac\{125\}\{a\}\; =\; x$; $\backslash frac\{216\}\{a\}\; =\; x$; $\backslash frac\{343\}\{a\}\; =\; x$;

$\backslash frac\{512\}\{a\}\; =\; x$; $\backslash frac\{729\}\{a\}\; =\; x$; &c. To find $a$, when $x$ is equal to half the span CD, fig. 25, Plate CCCXLVII, and $y$ the height AD; we have $a\; =\; \backslash frac\{AD^3\}\{CD\}$. If it be desirable that the ordinates should be $1/n$th part of a foot apart, then divide each by the $n^3$, which gives the dimensions in feet.

Example. In a Gothic building it is proposed to make an arch to an opening 10 feet wide, the height of which is to be 4 feet 6 inches above the springing line. There CD = 5 feet, and AD = 4.5 feet, therefore $a\; =\; \backslash frac\{AD^3\}\{CD\}\; =\; \backslash frac\{91.125\}\{5\}\; =\; 18.225$. And using this number for a divisor, the ordinates are easily found, by once setting a slide rule, to be,

And dividing the first, third, and fifth, each by 8 (considering $n=2$), gives the intermediate ordinates; $\backslash frac\{1\}\{8\},\; \backslash frac\{2\}\{8\},\; \backslash frac\{3\}\{8\}$. The advantage of this method consists in the facility of setting out the work on either a large or small scale. Every practical man is aware of the trouble of dividing a distance into equal parts, or of performing other geometrical operations on a platform or floor; but here, by an easy arithmetical operation, this is avoided. Draw the springing line CD, and the middle line AD, and let the line EC be drawn parallel

¹ Frézier may be consulted with advantage on this subject. Coupe des Pierres, tome ii. p. 14.

² Lectures on Natural Philosophy, vol. i. plate vi. fig. 83. This instrument might be usefully applied in ship-building.

³ See Edinburgh Review, vol. vi. p. 387. A most ingenious extension of the principles of describing curved lines has been invented by Mr. Joseph Jopling, and promises to be of much use in the arts, as well as a curious subject for mathematical speculation. The system is somewhat obscurely announced in a pamphlet, entitled The Septenary System of Generating Curves by Continued Motion. London, 1823.

to AD. Beginning at D, make a mark at every six inches on DA, and also on CE, beginning at C; then, through these divisions draw the parallel ordinates. Let the abscissas be measured off on these ordinates, from the line CE, by a rod divided into feet, tenths and hundredths of a foot. Put a nail in at each point found in the curve, and bend an uniform lath against the nails, and mark the curve.

67. Our next example is for the purpose of shewing the principles of constructing an arch for a bridge, when the span is considerable. In the article BRIDGE, prop. 5, the equation of the curve of equilibrium is found to be $y\; =\; \backslash frac\{\backslash frac\{1\}\{2\}ax^2\; +\; \backslash frac\{1\}\{2\}bx^4\}\{m\}$, for a disposition of the load which has place commonly in bridges. Making $x$ successively 10, 20, 30, &c. feet, we have

The curve of equilibrium being to pass at the middle of the depth of the arch-stones, CB, fig. 26, will be the height $=h$, and AB the semi-span $=S$; also let the depth of the arch and roadway at the crown, or $a$, be 7 feet; and suppose the quantity of matter be so regulated by hollow spans-drills that $b\; =\; \backslash frac\{10\}\{S^2\}$. Under these conditions we shall have

$m\; =\; S^2\; \backslash left(\; a\; +\; \backslash frac\{1\}\{6\}bS^2\; \backslash right)\; =\; \backslash frac\{S^2\}\{2h\}\; \backslash left(\; 7\; +\; \backslash frac\{10\}\{6\}\; \backslash right)$. If the semi-span be 72 feet, and the height 24 feet, then $m\; =\; 936$, and $b\; =\; .00193$. Calculating the ordinates from these data, we shall have

Construct the curve according to these ordinates, and divide it for the arch-stones. The joints should be perpendicular to this curve; but great accuracy is not necessary in this respect, provided the inclination from that perpendicular be considerably within the angle of repose. (See article BRIDGE, prop. 2.) The joints may be drawn thus, with any radius: from the next division on one side of the joint $a$ describe an arc, and from the next division on the other side, with the same radius describe another arc to intersect the former one, through the intersection and the division $a$ draw the direc-

tion of the joint. To find the depth of the key-stone, let the horizontal thrust $m\; =\; 936$ be multiplied by the mean specific weight of a cubic foot of the materials to form the bridge, and calculate the depth by equation (17). Suppose the mean to be 160 lbs., then the horizontal pressure will be 149,760 lbs., and $\backslash frac\{fbd\}\{4\}\; =\; 149,760$; which, considering $b$

$=$ unity, gives $d\; =\; \backslash frac\{599,040\}\{f\}$. For Craigleith stone, No. 15,

Table iii., the key-stone should be six feet deep. To find the depth of the arch-stones at any other part of the arch, set off $Be$ equal the depth at the key-stone, and draw $be$ parallel to a tangent to the curve of equilibrium at the point where the depth is to be determined, then $be$ is the depth at that point. The depth at a sufficient number of points being found as above, and set off equally on each side of the curve of equilibrium, the form of the intrados will be determined, which may be terminated by a circular arc at the springing; and it is not a little remarkable that the arch, thus described from principle, is a pointed arch.

68. When an arch cuts a plane wall in an oblique direction, there is a little more scope for the art of the stone-cutter. But previously to attempting to proceed further, we would recommend the young student to make himself master of the principles of projection, development, and solid angles. The first section of the article JOINERY is wholly restricted to these principles; all of which being equally applicable to both arts, it will be unnecessary to repeat them in this place. Let an elliptical arch be supposed to cut a plane wall obliquely, and the wall to be inclined, ABCD, fig. 27, is the plan of the arch; EF a section at right angles to its direction; IH a section at right angles to the line AB. Project the inclined face of the wall, as shewn at AO PB, by the method of projecting planes (JOINERY, sect. 1 and 7); and in doing this it will be found an advantage to produce the joints till they cut the base line EF, because the angles will be set out with greater accuracy from long lines. In the case where the wall is vertical, the section and projection of the face are not required. Next let the soffit be developed, on the supposition that the arch is a polygon of as many sides as there are arch-stones. (See JOINERY, sect. 1, art. 13.) KLMN shews the development of the soffit or moulds. The form of the bed of each stone is shewn by the planes $a,\; b,\; c$, and $d$, and is thus found, for the joint, 4, draw $he$ and $mf$ parallel to EF, and produce the joint 84 on the development to cut $he$ in $g$; set off $ge$ equal to $4i$, and draw $ef$ parallel to 48; then, a line drawn from $e$ through the point 4, will give the bevel at one end, and a line drawn from $f$ through 8 gives the bevel at the other end; its width is equal to $4k$ on the section. As there are no curved parts, except the soffit of the arch, the stones may be worked by means of bevels, without having moulds made for the soffits and beds.²

69. When a road crosses a canal in an oblique direction, the bridge is often made oblique. When the angle does not vary more than ten or twelve degrees from a right angle, the arch-stones may be formed as already described; but in cases of greater obliquity, a different principle of construction is necessary. These cases should, however, be avoided wherever it is possible; as, however solid the construction of an oblique bridge may be in reality, it has neither that apparent solidity nor fitness which ought to characterise an useful and pleasing object. An oblique arch may be constructed on the principle of its being a right arch

¹ Monge, in an elaborate article on the application of Descriptive Geometry to the use of Architects, has drawn a very erroneous conclusion respecting the joints of vaults and arches; for it is the direction of the pressure, and not the form of the soffit, which determines the best direction for a joint; but the views developed in the article BRIDGE were not known at the time Monge wrote. In other respects, the article of Monge is well worthy of the attention of the mason. (See Géométrie Descriptive, article 130, 4me ed. Paris, 1820.)

² Our method is analogous to that called Décis par Abrogi by the French writers. (See Frézier, tome ii. p. 133.) Other methods are given by Frézier, Simonin, Rondelet, Nicholson, &c. in the works referred to at the end of this article.

Masonry. of a larger span, as is shown in fig. 28. Let ABCD be the plan, and EFGH the corresponding points in the elevation, in this elevation the dotted lines show the parts which would not be seen. The joints of the arch are supposed to be divided upon the middle section, and therefore drawn to the mean centre K, which corresponds to the point I on the plan. Divide AD into any number of equal parts as at 1, 2, 3, &c. and transferring these points to the elevation; describe the arch belonging to each point, and also draw the parallel lines 11, 22, &c. on the plan. To find the mould for the arris of any joint, as a, draw ab parallel to the base line EF, and from a, as a centre, transfer the distances of the points where the arches cut the joint, to the line ab. Then let fall perpendiculars from the points in the line ab to the lines 11, 22, &c. in the plan, whence we find a, m, n, o, p, in the curve of the mould for the arris of the joint a. The mould for any other joint may be found in the same manner. The ends of the arch-stones will be square to the joints, and pde will be the mould for one end, and aodf the mould for the other end. It will be of some advantage in working the arch-stones to observe, that the arch-stone being in its place, the soffit should be everywhere perfectly straight in a direction parallel to the horizon.¹

70. If it be required to construct an arch in the wall of a circular building, as in fig. 29, where ABCD is the plan of the wall, the elevation EF should be drawn, and the joints in the same manner, as if the arch were in a plane wall. The curved surface of the soffit should be correctly developed by the process described in the article JOINERY, art. 13, and the moulds made of some flexible material; these soffit moulds are shown at a, b, c, &c. The mould for the joint 2 may be found by dividing the joint into any number m, n, &c. parts; and let a perpendicular fall from each point of division to the curved lines representing the faces of the wall on the plan, and from each point in which the curved lines are intersected by these perpendiculars, draw a line parallel to EF. Also, from 2, as a centre, transfer the divisions on the joint to the horizontal line f2, and from thence let perpendiculars fall, which will cut the lines that are parallel to EF in the points through which the curves of the mould must be drawn, as shown by the shaded part P on the plan. Any other joint may be described in the same manner.² In the figure, the section is drawn, because it shows somewhat more distinctly the size of the arch-stones; it is not necessary in finding the moulds, except the face of the wall be inclined, a case of very rare occurrence in practice. An arch in a circular wall always has the appearance of a want of strength on the convex side; and when the curvature is considerable, it becomes absolutely insecure. The method describing the raking mouldings, so as to mitre with horizontal ones, has been explained in the article JOINERY, and the same methods apply in masonry.

71. Respecting the general principles of stairs, we may also refer the reader to JOINERY, art. 39, 40, where the proportions of steps, &c. are shown; in masonry the kinds termed geometrical stairs are the only ones which offer any considerable difficulty in the execution. Each step of a geometrical stair is partly supported by wedging its end into the wall of the staircase, and it is further strengthened by resting upon the step below it. The outward end of a series of these steps is represented by fig. 7, Plate CCCXLV; the line abc shows the form of the joint between two adjoining steps; in the straight part of a flight of stairs, ab is made about an inch, and the part bc is made perpendicular

to the soffit of the stair, and of such a depth as may be required for the kind of stone. As this depth is determined by the mean depth necessary to render a stair safe, we shall here give an example of computing the mean depth of a step for Craigleith stone, by equa. 16 (art. 41). Put w = the greatest uniform load on a square foot, including the weight of the stone itself, = 300 lbs. the horizontal distance between C and D (fig. 6), = 10 feet, and the length of the step BD

That part of a step which is inserted into the wall of the staircase is made about eight or nine inches long for ordinary staircases, but ought to be longer when the steps are longer. Steps, and landings, and balconies, should be made to bear as evenly and firmly as possible upon their supports; and from a little consideration of the nature of the strains to be resisted in such operations, the mason may perhaps derive some instruction, since a mistaken view of the subject is likely to be attended with serious consequences. Let AB, fig. 30, Plate CCCXLVII, be a step fixed in a wall, CD being the face of the wall, and CA the part inserted in the wall. It will be obvious that the weight of the projecting part DB, of the step, with any load upon it, will tend to raise the fixed part at A, and to depress it at C. But it will require a less force at A to sustain the step than at any other point between A and D; and the nearer to D, the greater the strain will be, consequently a greater risk of failure. Hence the effectual resistance on the upper side should be at the extremity, A, of the step, and the support at C should be immediately at the face of the wall. We have often observed in stone stairs where steps are alternately in straight flights and winding ones, the soffit of the stair to be irregular, with sudden and abrupt changes of form where the winding steps began and terminated. These may always be avoided, by making a development of the ends of the steps, and forming the abrupt changes into easy curves, as a joiner does the hand-rail of a stair. (See JOINERY, fig. 43.)³

The earliest author on stone-cutting appears to have been Philibert De L'Orme, who, in the introduction to the fourth book of his work, remarks, that he had "never heard of anything that had been written on stone-cutting, either by ancient or modern architects." The labours of De L'Orme on this subject form the third and fourth books of his Treatise on Architecture, Paris, 1567, in folio. We shall close this article with the following list of some other authors on stone-cutting, in the order they were published:—Mathurin Jousse, Le Secret de l'Architecture, 1642, folio. Francois Derrand, L'Art des Traits et Coupe des Voûtes, Paris, 1643, folio. Abraham Bosse, Pratique du Trait à preux de M. Desargues, 1643. J. B. De la Rue, Traité de la Coupe des Pierres, folio, 1728; this was a republication of Derrand's work, with additions. Batty Langley, Ancient Masonry, London, 1733, folio. Frézier, Traité de la Coupe des Pierres, 3 tomes 4to, 1737–1739. Encyclopédie, article MAÇONNERIE. Encyclopédie Méthodique, MAÇONNERIE, 1785. Simonin, Traité Élémentaire de la Coupe des Pierres, 4to, 1792. P. Nicholson, Carpenters and Joiners' Assistant, 4to, London, 1797. J. Rondelet, Traité Théorique et Pratique de l'Art de bâtir, tom. ii. 4to, Paris, 1804. (G. G. G. G.)

¹ For further information respecting oblique or skew bridges, the reader may consult Gauthey, Construction des Ponts, tom. i. p. 390; Chapman, in Rees' Cyclopædia, art. OBLIQUE ARCH; and the article NAVIGATION, INLAND, in the Edinburgh Encyclopædia.

² Other methods are given by Frézier, Nicholson, &c.

³ The French methods of constructing stairs differ considerably from those of our own country; but it would extend this article too much to explain them; therefore, the reader may consult the works of Frézier, Rondelet, and Simonin, where he will find these methods detailed.