1902 Encyclopedia > Arch


ARCH, in building, a portion of mason-work disposed in the form of an arc or bow, and designed to carry the building over an open space. The simplest and oldest expedient for supporting a structure over a door-way is to use a single stone or lintel of sufficient length. On account of the difficulty of procuring stones of great size, this expedient can only be used for moderate apertures; nor can it be applied when there is to be a heavy superstructure, because the weight resting on the lintel would cause a compression of the upper, and a distension of the under side. Now, no kind of stone can bear any considerable distending strain, and thus stone-lintels are liable to fracture. The ancient Greek temples afford instances of the use of horizontal lintels of considerable size, but these architraves carry only the cornice of the building. The employment of a colonnade with flat architraves to support an upper story is contrary to sound principles, and, even in the case of ordinary houses, we see that the builder has been fain to relieve the pressure on the lintel by means of a concealed arch. In stone-work we must depend on compression alone.

Fig. 1.
When a lintel had been accidentally broken in two, we may suppose that the masons had set the ends of the halves upon the door-posts, and brought the broken ends
Fig. 2.
together. In this way there would be formed a support for the upper building much stronger than was the stone when entire; only there is a tendency to thrust the door-posts asunder, and means must be taken to resist this out-thrust. The transition from this arrangement to that of three or more wedge-shaped stones fitted together was easy, and thus the gradual development of the arch resulted.
So long as such structures are of small dimensions no great nicety is required in the adaptation of the parts, because the friction of the surfaces and the cohesion of the mortar are sufficient to compensate for any impropriety of arrangement. But when we proceed to construct arches of large span we are forced to consider carefully the nature and intensity of the various strains in order that provision may be made for resisting them.
Until the laws of the equilibrium of pressures were discovered, it was not possible to investigate these strains, and thus our knowledge of the principles of bridge-building is of very recent date; nor even yet can it be said to be perfected. The investigation is one of great difficulty, and mathematicians have sought to render it easier by introducing certain pre-supposed conditions; thus, in treatises on the theory of the arch, the structure is regarded as consisting of a course of arch-stones resting on abut-ments, and carrying a load which is supposed to press only downwards upon the arch-stones. Also cohesion and friction are put out of view, in other words, the investigation is conducted as if the stones could slide freely upon each other. Now, if the line of pressure of one stone against another cross their mutual surface perpendicularly, there is
no tendency to slide; and if this condition be adhered to throughout the whole structure, there must result complete stability, since the whole of the friction and the whole consistency of the cement contribute thereto. But if, in any case, the line of pressure should cross the mutual surface obliquely, the tendency to slide thereby occasioned must be resisted by the cohesion, and so the firmness of the structure would be impaired. Hence an investigation, conducted on the supposition of the non-existence of cohesion, must necessarily lead us to the best possible construction. But we can hardly say as much in favour of the hypothesis that the load presses only downwards upon the arch-stones. In order to place such a supposition in accordance with the facts of the case, we should have to dress the inner ends of the arch-stones with horizontal facets for the purpose of receiving and transmitting the downward pressure. But if, as is usually the case, the inner surfaces be oblique, they cannot transmit a vertical pressure unless in virtue of cohesion, and then this hypothesis of only downward pressure on the arch-stones is not in accordance with the fundamental principle of stability. In a thorough-investigation this hypothesis must be set aside, and the oblique pressure on the inner ends of the arch-stones must be taken into account. Since the depth of the arch-stones is small in comparison with the whole dimensions of the structure, and since the line of the pressure transmitted from one to another of them must always be within that depth, it is admissible to suppose, for the purpose of analysing the strains, that the arch-stones form an exceedingly thin course, and that their joints are everywhere normal to the curve of the arch. Eventually, however, the depth of the arch-stones must be carefully considered.
We may best obtain a clear view of the whole subject by first assuming that the load presses only downwards on the arch-stones, or that the inner ends of these are cut with horizontal facets.
Let Q'P'APQ (fig. 4) represent a portion of such an arch

Fig. 4.
placed equally on the two sides of the crown A, then the whole weight of the structure included between the two vertical lines P'H' and PH must be supported at P' and P, so that the downward pressure at the point P must be the weight of the building imposed over AP. This pressure downwards is accompanied by a tendency to separate the

supporting points P' and P. Now, as this tendency is horizontal its intensity cannot be changed by the load acting only downwards, and must remain the same throughout the structure, wherefore the actual pressure at P must be found by combining this fixed horizontal thrust with the downward pressure equal to the weight of the bridge from A to P. If, then, we draw ah horizontally to represent this constant thrust, and ap upwards to represent the weight of this portion of the arch, the line ph must, according to the law of the composition of pressures, indicate both in direction and in intensity the actual strain at the point P. This pressure must be perpendicular to the joint of the stones, and must there-fore be parallel to the straight line drawn to touch the curve at P.
Hence, if the form of the inside of the arch, or the intrados as it is called, be prescribed, we can easily discover the law of the pressures at its various parts ; thus, to find the strain at the point Q, we have only there to apply a tangent to the curve and to draw hq parallel thereto ; hq represents the oblique strain at Q ; aq represents the whole weight from the crown A to Q, and therefore pq is proportional to the weight imposed upon the position PQ of the arc.
Using the language of trigonometry, the horizontal thrust is to the oblique strain at any part of the curve as radius is to the secant of the angle of inclination to the horizon; also the same horizontal thrust is to the weight of the superstructure as radius is to the tangent of the same inclination. And thus, if the intrados be a known curve, such as a circle, an ellipse, or a parabola, we are able without much trouble to compute, on this hypothesis, the load to be placed over each part.
If we use the method of rectangular co-ordinates placing x along OH and z vertically downwards, so that Pir may be the increment of x, TTQ that of z, the tangent of the inclina-
tion at P is -g-, and therefore if h stand for the horizon-tal strain and w for the weight of the arch, we have—-
I /ox2 + 5zJ \
while the oblique strain is A / (—^\— )_ Also the change of weight from P to a proximate point Q is
5% ox2-
Let EST be the outline which the mason-work would have if placed compactly over the arch-stones, in which case RST is called the extrados, then the weight supported at P is proportional to the surface ARSP, and the incre-ment of the weight is proportional to PSTQ, hence if the weights and strains be measured in square units of the vertical section of the structure, and if y be put for PS, the thickness of the mason-work, we have—
8w = y8x, whence y = h
When the curve APQ is given, the relations of z and of its differentials to x are known, and thus the configuration of the extrados can be traced, and we are able to arrange the load so as to keep all the strains in equilibrium.
But when the form of the extrados is prescribed and that of the intrados is to be discovered, we encounter very great difficulties. Seeing that our hypothesis is not admissible in practice, it is hardly worth while to engage in this inquiry; it may suffice to take a single, and that the most interesting case.
If the whole space between the arch-stones and the road-way be filled up, the extrados becomes a straight line, and when this is horizontal we have y = z, so that the form of the arch must be such as to satisfy the condition—
= A

that is to say, z must be a function of x such as to be pro-portional to its own second derivative or differential coeffiient. Now this character is distinctive of the cate-narian functions, and therefore ultimately

e +e
where A is AO, the thickness at the crown of the arch, and e the basis of the Napierian system of logarithms. In this case, since Sw — z8x,—
w = AJJ
so that the form of the arch and also its weight may readily be computed by help of a table of catenarian functions.
Let us now consider the case when the ends of the arch-stones are dressed continuously, while the imposed load is formed of stones having vertical faces. The weight of the column PSTQ resting on the oblique face PQ is prevented from sliding by a resist-ance on the vertical surface QT, which resistance goes to partly oppose the horizontal strain transmitted by the preceding arch-stone; and thus the out-thrust of the arch, instead of being entirely resisted by the ultimate abutment, is spread over the whole depth of the structure. In this case the horizontal thrust against QT is to the weight of the column as Q7T. the increment of z, is to PTT, the increment of x; wherefore, putting H for the hori-zontal thrust at the crown of the arch, and h for that part of it which comes down to P, the decrement of h from P to Q is proportional to the rectangle under PS and Q7r, that is to say,
Sh = ySz.
Now, the whole decrement from the crown downwards is the sum or integral of all such partial decrements, and therefore the horizontal thrust transmitted to P is expressed by the symbol—
h = R-fySz,
while the whole, weight supported at P is the analogous integral
w = Jybx .
But the resultant of these two pressures must be perpendi-cular to the joint of the arch-stones, or parallel to the line of the curve; wherefore ultimately we obtain, as the con-dition of equilibrium in such a structure, the equation—
Sz(H -fy8z) = 8xfySx.
Since the vertical pressure at P is w, while the horizontal strain is h, the intensity of the oblique strain at P must be J{w2 + h?}. Now, in passing to the proxi-mate point Q, w becomes w + 8w, while h is reduced to
h - Sw^, so that the oblique strain at Q must be—
or, neglecting the second power of the infinitesimally small increment 8w, ,/| vP- + p* + 2w8w - 2h8w^ |, but

w — h^-, wherefore the strain at Q is J(w2 + h?), or exactly ox
the same as that at P. This result might have been obtained from the consideration that the thrust upon the surface PQ is perpendicular to the oblique strain, and can tend neither to augment nor to diminish it. Hence, as a characteristic of this arrangement, we have the law that the tension across the joints of the arch-stones is the same all along, and therefore is equal to H, the horizontal tension at the crown of the arch.
From this it at once follows that if r be the radius of curvature at the point P, y being the vertical thickness of the mason-work there, H = ry, so that if R be the radius of curvature at the crown of the arch, and A the thickness there, the horizontal thrust there, or the strain transmitted along the arch-stones, is H = RAH, being measured in square units of surface; hence also A : y :: r: R, or the thickness at any place, is inversely proportional to the radius of curvature there.
When the form of the intrados is given, its curvature at any point is known, and from that the thickness of the stone-work and the shape of the extrados can be found. The most useful case of the converse problem is, again, that in which the extrados is a horizontal straight line.
Let OH, figure 6, be the horizontal extrados, and A the crown of the arch ; make also AB such that its square may represent the horizontal thrust there; then, having joined OB and drawn BG perpendicular to it, and meeting the con-tinuation of OA in C, C is the centre of curvature for the crown of the arch. Or, if the radius of curvature and the thickness of the arch at the crown be pre-scribed, we may obtain the horizontal thrust by describing on CO a semicircle, cutting a horizontal line through A in the point B, then the horizontal thrust is equal to the weight of the quantity of the stone-work which would fill up the square on AB. The conditions of the problem require that the curve APQ be so shaped as that the radius of curvature at any point P shall be inversely proportional to the ordinate HP.
Resuming the general equation of condition—
Sz(H —JySz) = SxfySx,
and observing that in this case y = z, we have—
Sz(H —fz8z) = SxJzSx .
Now the integral fzSz is £z2, but as it must be reckoned only from A where z = A, the equation becomes
Sz(H + §A2 - Jz2) = SxfzSx.
The coefficient of Sz becomes less when z increases, and when ^z2 = H2 + -ijA2, this coefficient becomes zero, at which time 8x also becomes zero in proportion to 8z; that is to say, the direction of the curve becomes vertical. Wherefore, if we make OD = D such that D2 = A2 + 2H2, we shall obtain that depth at which the curve is upright, or at which the horizontal ordinate DQ is the greatest, and then the equation takes the form—
Sz(D2-z2) = 28xfz8x,
by help of which we should be able to find x in terms of z. The computation, however, is attended with considerable difficulty, and therefore it may be convenient to attempt a graphical solution. Since, for any vertical ordinate HP( = z), the horizontal thrust is £(D2 - z2), while the oblique strain is |(D2 - A2), the obliquity of the curve at P has for
D2 — z2
its cosine the value TTJ——„, wherefore the angle at which
D2 - A2' °
the curve crosses the horizontal line pP is known. Let then a multitude of such lines be drawn in the space between BA and DQ, and let the narrow spaces thus marked be crossed in succession from A downwards by fines at the proper inclination, and we shall obtain a representation of the curve, which will be nearer to the truth as the intervals are more numerous. The beginning of the curve at A may be made a short arc of a circle described from the centre C.
Since the minute differentials thus obtained are pro-portional to the sides of a triangle whose hypotenuse is D2 — A2, and one of whose sides is D2 — z2, we must have— D2-z2
Sx = v/{(D2-A2)2-(D2-z2)2}Sz ' and the integration of this would give the value of x. If we put <^> for the inclination of the curve at any point P, D2-z2 = (D2-A2) cos <£, .-. z = {D2 - (D2 - A2) cos </>}», and taking the differential,
& = i(D2 = A2) sin </>{D2-(D2-A2) cos 8<j>,
. H. cos <p. 5<p
_'_ N/{D2-2H cos?} '
where 2H is put for its equivalent D2 - A2. The integral of this expression may be obtained by developing the radical in terms arranged according to the powers of cos <j>, and then integrating each term separately. The result is a series of terms proceeding by the powers of cos <j>, the coefficient of each power being itself an interminate series; and the rate of convergence is so slow as to make the labour of the calculations very great. Such expressions belong to the class of elliptic functions, for which peculiar methods have been devised. Fortunately the actual calculation is not required in the practice of bridge-building, and therefore we shall only refer the reader to the above-named subject.
If the horizontal thrust and the thickness at the crown of the arch be prescribed, the radius of curvature there must be the same whichever of the two hypotheses be adopted ; now, if we sweep an arch from the centre C with the radius CA, the catenarian curve lies outside of it, while the curve which we have just been considering lies inside. Each of these is compatible with sound principles: the one if the inner ends of the arch-stones be dressed with horizontal facets, the other if the ends be dressed to a continuous curve; wherefore, between these two limits we may have a vast variety of forms, each of which may be made consistent with the laws of equilibrium by merely dressing the inner ends of the arch-stones at the appro-priate angles. Hence an entirely new field of inquiry, in which we may find the complete solution of the general problem:—
" The intrados and extrados of an arch being both prescribed, to arrange the parts consistently with the laws of equilibrium."
Let PQ represent the inner end of one of the arch-stones, the part Qg being vertical, and Tq being sloped at some angle which is to be found ; put t for the tangent of the inclina-tion of the joint P to the vertical, 6 for that of Pq to the horizontal line, then the horizontal

strain at P is —, while the corresponding strain at Q is
t + U
u> + 010

a w It
w it
^ or 6t = T- —
t iw t
Now, when the forms of the intrados and extrados are both given, the values of w, t, Sw, 8t, are thence deducible, so that the value of 6 may always be computed by help of differentiations only; excepting, indeed, that integrations may be needed for determining the value of w, which is the area included between the two curves.
In this very simple investigation we have the com-plete solution of the principal problem in bridge-building. The data needed for determining the shape of the inner end of the arch-stone are already in the hands of the architect, who must know, from his plans, the weight of each part and the inclination of each joint; so that, with a very small addition to the labour of his calculations, he is enabled to put the structure completely in equilibrium, even on the supposition of there being no cohesion and no friction; that is to say, he is enabled to obtain the greatest stability of which a structure having the prescribed outlines is susceptible. Even although he may not care to have the stones actually cut to the computed shape, and may regard their usual roughness and the cement as enough, he may judge, by help of the above formula, of the practi-cability of his design ; for if at any place the value of 68t come out with the wrong sign, that is, if w.&t be less than t.8w, the building is unstable, whereas if w.St be greater than t.Sw everywhere, the design, as far as these details go, is a safe one.
In every possible arrangement of the details, the hori-zontal thrust at the crown of the arch is transmitted to and resisted by the ultimate abutments. The only effect, in this respect, of varieties in the form of con-struction is to vary the manner of the distribution of that strain among the horizontal courses. Hence one great and essential element of security,—the first thing, indeed, to be seen to,—is that the ground at the ends of the pro-posed bridge be able to resist this out-thrust. Another, and not less important one is, that the arch-stones be able to withstand the strains upon them. In this respect much depends on the workmanship; it is all important that the stones touch throughout their whole surfaces: if these surfaces be uneven the stones must necessarily be subjected to transverse strains, and so be liable to fracture. The practice, too common among house-masons, of cheaply obtaining an external appearance of exactitude, by confining their attention to a chisel-breadth around the outside, is not permissible here, nor should any reliance be placed on the layer of mortar for making up the inequalities.
The limit to the span of an arch depends primarily on the quality of the material of the arch-stones. At the crown of the arch the horizontal thrust is the weight of as much of the masonry as fills a rectangle whose length is equal to R, the radius of curvature, and whose breadth is A, the effective thickness there; now this strain has to be borne by the arch-stones, whose depth we shall denote by d, and therefore these stones must be subjected, as it were, to the
direct pressure of a vertical column whose height is
This column must be much shorter than that which the stone is actually able to bear.
The ability of a substance to resist a crushing pressure is generally measured by the length of the column which it is able to support, without reference to the horizontal section; but it may be questioned whether this mode of estimation be a sound one, for it does seem natural to suppose that a block three inches square should bear a greater load than nine separate blocks each one inch square, seeing that the centre block in the entire stone is protected on all sides; and thus it is possible that we under-estimate the greatest practicable span of a stone arch. This difficult subject belongs to the doctrine of " Strength of Materials."
AECH, SKEWED. —In the earlier days of bridge-building the road was led so as to cross the river or ravine perpen-dicularly, but in modern engineering we cannot always afford to make the detour necessary for this purpose, and must have recourse to the shewed or oblique arch, having its plan rhomboidal, not rectangular.
If AB, CD, figure 8, represent the roadway, and EP,

Fig. 8.
GH, the boundaries of the abutment walls placed obliquely, we easily perceive that the thrust cannot be perpendicular to the abutments, for then it would go out on the side walls which have no means of resistance; the thrust can only be resisted in the direction of the road. Hence if the structure be divided into a multitude of slices by vertical planes parallel to the parapet, the strains belonging to each slice must be resisted within that slice, and each should form an arch capable of standing by itself. The abutment, therefore, cannot have a continuous surface as in tho common or right arch, but must be cut in steps to resist the oblique pressure ; wherefore also the ultimate founda-tion stones must present surfaces perpendicular to the road.
Attending for the moment to one only of these divisions, say to a thin slice contiguous to the side wall EG, let us study the manner in which the arch-stones in it must be shaped. At the crown I the pressure is horizontal in the plane EIG, and therefore the joint of the stones there must be perpendicular to AB, and so also must be its projection on the horizontal plane.
Proceeding along the line of the curve to the point R, we observe that the pressure there must be in the direction of a tangent to the curve, wherefore the surface of a joint at R must be perpendicular to that tangent, and the

exposed face of the stone must be right-angled. Now, the projection upon a horizontal surface of a right angle placed obliquely is not necessarily right; in this case it cannot be right, and therefore the course of a line of joints repre-sented in plan must bend away from being perpendicular to the side wall towards being parallel to the line of the abutment. Thus a continuous course of joints beginning at I must be shown in plan by some curved line such as IPp.
In many of the skewed bridges actually built, the out-line of the arch is divided into equal parts, as seen on the ends of the vault; the curved joint-lines l?p thus become portions of screws drawn on an oblique cylinder, and, although the arch-stone at the crown be rectangular, those on the slope cease to be so. The bearing surface is thus inclined to the direction of the pressure, and the tendency is to thrust out the arch-stones at the acute corners F and G. The fault is exactly the same as if, in ordinary building, the mason were to bed the stones off the level. The consequence is that skewed stone-bridges have not given satisfaction, the fault being attributed to the principle of the skew, whereas it should have been assigned to the unskilfulness of the design.
Let figure 9 be an elevation projected on a vertical plane parallel to AB, EIG, FSH, being the outlines of the ends

Fig. 9.
of the arch, and the sections taken at equal intervals along the crown line being also shown; then, since the projection of a right angle upon a plane parallel to one of its sides is always right, the joint at R, as seen on this elevation, must be perpendicular to the curve at R, and thus the curve IPp, representing one of the joint-courses, must cross each of the vertical sections perpendicularly. In this way each of the four-sided curvilinear spaces into which this elevation is divided must be right-angled at its four corners. This law is general, and enables us to determine the details of any proposed oblique arch.
If we draw, as in figure 9, the end elevation of the vault as intersected by numerous parallel planes, and lead a curved line crossing all these intersections perpendicu-larly, we obtain the end elevation of one of the joint-lines, and are able from it to prepare any other of its projections. The form and character of this end elevation IPp depends entirely on the nature of the curve EIG, but is the same whatever may be the angle of the skew. In order to examine its general character, let us take in the crown line two closely contiguous points I, K, and from these lead the joint-lines IP, KQ, of equal length, then the straight line PQ is equal and parallel to IK, on any of the projections.
If in the end elevation, figure 9, we continue the joint IP to meet the vertical section OQ in p, we may regard PQp as a small rectilineal triangle, right angled at p, while PQp is the inclination to the horizon. Now, PQ : Qp : : R : cos TQp, while PQ is equal to KI, the breadth of the arch-stone at the crown, wherefore the breadth of the course at the crown is to the breadth of the same course at any other place as radius is to the cosine of the inclination there. Hence it follows, as is shown in the end elevation, figure 10, that the arch-stones gradually diminish in breadth from still narrower, and an infinity of them would be needed to reach the abutment of a semicircular or semi-elliptic arch,

the crown downwards, being halved in breadth at an inclination of 60°. At a greater inclination they become

because the cosine of the inclination there is zero. In no properly-built skewed bridge can the arch-stones show equal divisions ; and it is impossible to continue the arch to the complete half circle or half ellipse.
Passing from the end elevation, figure 9, to the plan, figure 8, we observe that Qp on the plan is less than the actual Qp of the elevation in the ratio of the cosine of the inclination to radius, and, therefore, on the plan, the breadth at the crown is to the apparent breadth of the course at any other place as the square of the radius is to the square of the cosine of the inclination there; so that, at the inclina-tion of 60° the apparent breadth will be quarter of that at the crown.

Again, in figure 11, which is the side elevation of the
vault, or its projection
on a vertical plane per-
pendicular to the road,
the apparent distance
Qp is to the actual dis-
tance Qp of figure 9
as the sine of the in-
clination is to radius,
wherefore, the apparent
breadth Qp on this pro-
jection is proportional to g-
the product of the sine by the cosine of the inclination, that
is, to half the sine of twice the inclination. The width on
this projection is therefore greatest at an inclination of 4-5°,
being there just one-half of the actual breadth at the crown
of the arch.
This reasoning is founded on the supposition that the distance IK is excessively small, and the resulting con-clusions are strictly true only of an infinitely narrow course of arch-stones; they are, indeed, differential equations which must be integrated in order to be applied to actual practice. Thus we have seen that the curved line IP, figure 9, crosses the section NP perpendicularly at P, but then it does not continue in this direction for any perceptible distance. The draughtsman may attempt to trace it by making the sections very numerous, and by drawing perpen-diculars across the successive intervals; but however nume-rous he may make these sections, he can thus only effect an approximation to the true curve. We must integrate, that is, we must obtain the aggregate of an infinite number of infinitely small portions in order to reach an absolutely true result.
These conclusions hold good whatever may be the out-line of the arch. The most common, and therefore the most interesting case, is when the longitudinal section is circular, the cross section taken perpendicularly to the abutment being then an ellipse with its shorter diameter placed horizontally, the vault being an oblique cylinder. Figure 9 is actually drawn for the circular arch. If then O be the centre of the circular arc NP, the curve IP must

at P tend towards O, so that the draughtsman, while mak-ing the step across one of the intervals, has only to keep his straight edge up to the corresponding place of the centre. If we place the paper horizontally, fix a small heavy round body at P to the end of a thread OP, and then draw the end O of that string along the straight line HEF, P would always move towards the then position of the point O, and would trace out the curve of which we are in search. The projection, then, of the joint of an oblique circular arch upon a vertical plane parallel to the road, is always the curve known by the name of the Tractory. All tractories have the same shape, the size merely is regulated by the length of the thread OP, that is, by the radius of curvature of the circular arch. Hence, if the delineation of it have been accurately made in one case, the curve for another case may be obtained by mere enlargement or reduction; or, still better, in all cases it may be traced by help of a table of co-ordinates, such as that subjoined, which shows the dimensions of the tractory as represented in figure 12, in decimal parts of the radius of curvature of

o T
Fig. 12.
the arch. The computations have been made for equal motions of the point 0, corresponding, therefore, te equal dis-tances measured along the crown-line of the arch. The head-ings of the columns sufficiently explain their contents. By help of these the form of the tractory may easily be obtained, and with a piece of veneer or of thin metal cut to this shape, the architect may obtain all the details of the intended structure, first working out the said elevation, figure 9, and transferring the several points therefrom to the other projections.
If we put s for the angle of the skew, v for the distance IN measured along the crown of the vault, and i for the inclination at the point P, r being the radius of the arch, the distance IN or ¿0 of figure 10 is clearly v sin s, and as the result of the integration, we obtain—
V S™ S = Nap. log tan (45° + Jt) ,
by help of which equation we can readily determine i when v is known, or v when i is given. The table of Napierian logarithmic tangents being very scarce, it is convenient to convert these into denary or common logarithms. Putting, as is usual, M for the modulus of denary logarithms, that is, for '43429 44819, the above equation becomes— M
— . v. sin s — log tan (45° + Ji),
from which it is quite easy to tabulate the values of i cor-responding to equidifferent values of v, because the constant factor—
M. sin s

has to be only once computed; i, that is, the number of degrees in the arc NP being thus computed for each of the successive sections of the vault, we have only to divide a tape-line so as to show degrees and minutes of the actual circle in order to be able at once to mark the course of the
joints upon the centering of the arch; or better still, instead of the degrees, we may write upon the tape the successive values of NP, and then the commonest workman will be able to lay off the lines.

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o00033 o'10262 o00869 o02005 o03788 o06295 _09563 o13596 o18370 o23841 o29950 o36635 o43828 o51465 o59485 o57833 o76459 o85319 o94376 1-03597 1-12955 1-22426 1-31990 1-41633 1-51339 1-61097 1-70899 1-80737 1-90604 2-00495
The only other kind of skewed arch likely to possess any interest is the elliptic. In right arches the semi-ellipse is sometimes used on account of the grace of its form, but this reason for its adoption disappears in the case of the skew, because then we can only use a portion of the semi-ellipse. The end elevation of a joint in an elliptic skewed arch is a modified form of the tractory, and the general features of the arrangement are analogous to those of the circular arch.
The arch-stone of a common bridge is wedge-shaped, having two flat faces AacC, TibdD, inclined to suit the breadth of the course, but in the skewed bridge the corre-sponding faces are twisted, Cc not being parallel to Aa, and thus the dressing of them requires both skill and care. The dimensions of the stone and the inclinations of its four

Fig. 13.
edges may easily be computed when its intended position is known, and thus the degree of twist on each of its faces may be ascertained, and the lines may then be marked off on the ends of the stone.
The theory of the skewed arch was given for the first time in the Transactions of the Royal Scottish Society of Arts for 1838 ; from which it was copied into the Civil Engineer and Architect's Journal for July 1840, which see. (For the history and various forms of the arch see ARCHITECTURE.) (K S \

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