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i        \'i _                     _r / 








SKEW ARCHES.
ADVANTAGES AND DISADVANTAGES
OF
DIFFERENT METHODS
OF CONSTRUCTION.
BY
E. W. HYDE, C.E.
NEW YORK:
D. VAN NOSTRAND, PUBLISHER,
28 MURRAY STREET AND 27 WARREN STREET.
1875.








PREFACE.
THE author was led to make the investigations contained in this little treatise by
a desire to satisfy his own mind as to the
relative advantages of several different
methods which have been employed in
the construction of Skew Arches.
The two important points of comparison that naturally suggest themselves to
the investigator are:
1st. Relative security;
2d. Relative facility of construction.
A discussion and comparison of three
modes of construction, with special reference to these] points,k will be found in
the following pages, together with brief
descriptions of the manner of making the
necessary draughts, patterns, templets,
etc.
The paper first appeared in VAN NosTRAND'S ENGINEERING MAGAZINE, for
which it was written.
E. W.qHYDE.








SKEW ARCHES.
I PROPOSE in this paper to discuss to~
some extent three methods which have
been employed in the construction of
oblique or skew arches, and to make a
comparison of their relative security,,
facility of construction, etc.
The three methods will be designated as,
1st. The Helicoidal method.
2d. The Logarithmic method.
3d. The " Come de Vache" or Cow'shorn method.
The first two names are derived from,
the nature of the coursing and heading,,
joint surfaces and their intersections with
the soffit, and the third from the soffit,
itself, which is a warped surface that has,
been thus named. They will be considered in the order given above.




6
The following abbreviations will be
used throughout the paper:
C j c, for coursing joint curve, or intersection of coursing joint with soffit.
H j c, for heading joint curve.
C j s, for coursing joint surface.
H j s, for heading joint surface.
H P, for the horizontal plane of projection.
V P, for the vertical plane of projection.
P F, for the plane of the face of the
arch.
Ex. s, for the extradosal or outer surface of the arch.
THE HELICOIDAL METHOD.
In this method the C j s's and II j s's
are both   warped   helicoids, and  of
course their intersections with the soffit
helices.
Let C D D, C2 be the projection of the
soffit on the H P which coincides with the
springing plane of the arch, and D' V C,
is a semicircle whose radius will be designated by r.




7
The Ex. s will be taken as a conqentrie
cylinder projected in A B B1If: and B'
V' As and its radius will be designated
by r.
To construct the C and H j c's we will
first develop the soffit. Lay off O1 T=,r-3,1416 r; the points of the curve ) E F
C, may be found by the principles of de,
scriptive geometry or by means of the
equation of the curve, which we shall obtain. The latter method is much morm
accurate for construction upon a large
scale.
Let h=-D'D2, and a = angle CD2D' 
2r
obliquity of the arch;.. tang. a x-.
Also 0 = variable angle s QD'.    The
origin will be taken first at O1. We
have from the figure
Oi -=    = r 0.
y =    A,E    =- y...  h + y    r (1 -cos 0 )    = 1.-cos 0
h            2r            2
1 - cos e = h-+ 2y
A




8
and        cos 0     2=A
x          os -( - -)y)
whence       x - r cos   -     y.
Solving for y we have
h      z
(1)     Y       2 cos-,
which is the equation of D E F q, with,
the origin at 0O. If the origin be moved
to 0, the equation becomes
h      x
(2)1     y =      sihn. -
From either of these equations the
values of y for given values of x may be
easily obtained by the aid of a table of
natural sines and cosines.
Having constructed the curve D E F C1,
join D and C1 by a straight line. This
will be the development of a H j c. At
0, draw 0O S perpendicular to D C,. o0,
S is the development of J of a spire of the




9
helix which forms the C j c's, and corresponds to the curve 0 P R.
The angle TO2,S= TC102= tang.-' X,
(32                 72r2. *  = O0 T tang TO,S= -=   7r r tan a
Now divide D E F Ci into a convenient
odd number of equal-parts, so arranging
it as to cause one of the Cj c's as 6 D,
to pass through D,. The developments
of the C j c's are of course drawn parallel
to 02 S through the points of division of
the curve D E F C,. If it were not convenient to divide D E F C1 in such a manner that a line through D2 parallel to O0 S
would exactly pass through one of the
points of division, the direction of the
C j c's might be slightly changed, or the
divisions between Da 6 and C2 6 might be
made very slightly larger or smaller, as
the case required, tnin the divisions from
D to 6 and from C3 to 6'. The latter
method seems preferable, since it preserves the perpendicularity between the
C and H j c's, and the difference in the




10
size of the voussoirs would be so small as;
not to be noticeable.
All the courses below 6 and 68 run out'
into the abutment, and the impost must
be cut in steps, as shown in the figure,.
into which the voussoirs will fit.
A G H B1 is the development of the extrados, and the right line x y that of a
helix in which one of the H j s's intersects the Ex. s. 04 R1 is the development of the curve 0 L R., in which a
C j s intersects the Ex. s.
ann z   O,= - --
tr 2h      2
r   t r r  t r
-, -- -        tan a.
r' h      2 rn
The C j s's are generated by a right,
line moving on the axis 0 Q as one directrix, the helix 0 P R as a second, and
remaining always perpendicular to the
former. Hence the V P is a plane directer of the surface.




11
The details of the construction of a.
skew arch by this method are fully developed in "A Practical and Theoretical
Essay on Oblique Bridges," by 4Uo
Watson Buck, M. Inst. C. E. He, however, makes the C and H j s's hyperbolic
paraboloids instead of helicoids, as follows: The corners of the voussoirs are
normals to the soffit at the intersection of
the H and C j c's, and hence are elements
of the warped helicoids, which should
form the H and C j s's. The points
where two of the normals on the same
side of the voussoir pierce the soffit are
joined by a right line, and this line is
moved on the normals as directrices in
such a manner as to pass over equal distances, measured on the normals in equal
times, by which operation a hyperbolic
paraboloid is generated. The accompanying figure is an exaggerated representation of the effect of cutting the stones
in this manner. A B C D and E F G H
are the developed introdosal surfaces of
two voussoirs in successive courses. If
the courses were not required to break




12
joints, the stones would fit perfectly, but
as this is necessary to the stability of the
arch, that portion of a stone which is too!z
full will come opposite to the portion of
~the one in the next course which is likewise
too full, and similarly the hollow portion'too fuill, and similarly the hollow portion




13
of one opposite to the hollow portion of
the next.
The truth of these statements will
appear as follows:   The hyperbolio
paraboloid evidently cannot coincide with
the helicoid, as they are surfaces constructed according to a different law.
The normals to the two adjacent corners
of a voussoir are elements of both surfaces.
The normal midway between these
two is also an element common to
the two surfaces.   Hence it is eviident from the mode of their generation that the two surfaces intersect
each other along each of these three
lines. A section of the surfaces by a
plane perpendicular to the middle normal would give something like the accompanying figure,    Em-     the straight
line being the intersection with the hyperbolic paraboloid and the curve that with
the helicoid.
However, if the voussoirs are small
compared with the whole arch, as in Fig.
1, the difference between theparaboloidal




14
and helicoidal surfaces in the length of
one voussoir will be exceedingly small,
and the stones will fit with sufficient exactness for practical purposes. Nevertheless, the tendency is to cause the pressure to be unequally distributed, concentrating it at K, K1, K, and at A, C, E and
G. The difficulty of cutting the warped
faces of the voussoirs is considerably
diminished by this approximate method.
An investigation will now be made of
the security of an arch constructed according to the helicoidal method.
P In order that there may be no tendency
in the successive courses to slide upon
each other, it is evident that each C j s
must be at every point normal to the
direction of the pressure at that point.
We shall consider first the direction of
pressure as regards its parallelism to a certain vertical plane, without reference to
the angle it may make at any point with
the H P. This vertical plane is the place
of the face of the arch. It is probable
that the direction of pressure varies somewhat with reference to this plane in differ



15
ent portions of the arch, especially if the
crown settles to any extent after removal
of the centre. Still it must be approximi tely parallel to the P F, otherwise the
portions near B and A2 would be unsupported and would consequently fall.
For the purposes of the investigation
then the direction of pressure will be
assumed to be in a plane parallel to the
P F, and from the results obtained we
shll be able to see without difficulty
the effect upon the security of the arch
which would be produced if the direction
of'pressure were not parallel to the P F.
Proceeding then on this assumption, a
line drawn on the C j s of any voussoir
and lying in a plane perpendicular to the
P F and to the H P ought to be horizontal, but as this line would be the intersection of a warped helicoid by a plane not
containing an element of the surface, it
must be.a curve, and can only be horizontal at a maximum or a minimum
point, at infinity, or at so:ne singular point.
The intersection of the C j s of a voussoir
by a horizontal plane should give a line




16
perpendicular to the P F, but this line
would be also a curve, and could have the
required direction only at one or more
points. It becomes necessary then to discover the nature of these curves and the
direction of their tancgents at the point of
piercing the soffit.
In Fig. 1 (Frontispiece) the curves Q
P'4 a, Q P'3 b, etc., are the vertical projections of the curves cut by the vertical
planes, P4 8, P3 k, etc., from the helicoid
whose directrices are the axis 0 Q and thle
helix O P R. The curve Q P'c has a maximum point at P P' where it pierces the
soffit; all those above it have a maximuml point outside the soffit, and those
below it have one inside the same.
The curves 0 P4 r, 0 P3 q, etc., are cut
by horizontal planes through the points
P4 P', P3', etc. The curve through P is
not drawn, but if it were it would be tangent at P to the line P h.
It is plain from inspection of these curves
that the courses below P P' have a tendency to slide upon each other in a direction
from As towards A, which increases as the




17
abutment is approached. In Fig. 1 this
point P P' is on the tenth course from the
abutment, and there are three coursing
joints which would have a tendency to
slide throughout their whole length with
nothing to resist it except friction between the surfaces. The partial courses
would be prevented from sliding by the
steps cut in the impost. Above P P' the
tendency to slide would be in the opposite direction, but would be so small as
not to affect seriously the stability of the
arch. This will be evident from inspection of the curves O P1 p and O P. o.
We will now investigate these curves
analytically and determine the position
of the point P P'.
The distance T S, Fig. 1, is i the
height of one spire of the helix O P R, and
from eq. (3) TS=-    = r   7 r r tan a.
2h 
Call the height of one spire of the helix
Ah, then
Ahl _     --- =2 7rr 2 r tan ar.
h




18
Let O be the origin ofcoords, the axis of
the cylinder, O Q, the axis of z, O T the
axis of a, and the axis of y a vertical line
through 0. Then we shall have for the
helix 0 P R,
75
2        l, 2 7    hi7
2 z z     7r
whence 0        -- 
/i1  +  2
Also
2 t   7. 27rZ
=rcos =-r cos -, + 2 =-r sin-,
or substituting for t, its value
hA
(4)     x    -r sin —
which is the equation of O P R, the projection on X Z of one of the C j c's.
The vertical projection, or the projection
on X Y of this helix is
(5)        X2 + Y =2.'To obtain the equation of the C j s, we




19
must find the equations of an element,
and then eliminate the constant which
fixes its position. Let one equation of
the element be
(6)          z=z1.
To find the equation in terms of x and
y, substitute in the equation
Y - Y1i - y- Y
x - X1    x - X,
of a line through two points the proper
values of xi, x, yl and y,. We have since
all the elements cut the axis of'z,
x = o and y   - =~ o.
Substituting from (6) in (4)
h
2    -- r sin
snd substituting this value of xi in (5),
y,/ 2-/ — rs rs sins __ lr  1 -- sins h
7t rr nhk.h
hz,
r cos- 2,r r2




20
h z1
r cos 
y   _      rr        h z,
~'.-.. = -.. —COt 
x          h z        7rc  r
r sin
trr
or making z1 general by dropping the
subscript,
hz
(7)    y = -x cot,
which is the equation of a Cj s.
Now, intersect this surface by a vertical
plane perpendicular to the P F, whose
equation is
)2r
(8)    z   2^r(a - -),
in which a, is the intercept on X.
(9).. Y  -x cot (2 (a,-  )
which is the equation of the curves
Q P' c, etc.
In equation (9) if
X   o,  y = o;
if     (2 n - 1) z2 r
f al-  4  --




21
(2 2t- ]) 7
y -xcot          2 —--         0
n being an integer;
n 7rt2 r
if x   a- a     2
y -- X cot n 7t    c,
n as before being an integer. Hence the
curve has an infinite number of branches.
It is to be noticed tlhat equation (9) does not
contain A, the only constants being 7r, a,
and r, hence the form of the curves Q P' c,
Q P' d, etc., is entirely independent of
the obliquity of the arch. We will next
differentiate equation (9) for a maximum.
d= u(2( 1-x).2x
d     ~x  \  7 r    7 Ttrssin 2 (al-x))
V7 r
Let 2 (a - x)
Let --            e
7 r
(lo).    Y   -  co u -     2 
dx                 7 r sin' u
For a mnximum
2.r
cot u - -
7t r sin' t'




7 r sin u cos u
(11) * ymax --
By solving equation (11) the values of
x for which y is a maximum or minimum
can be obtained. It is only capable, however, of an approximate solution, but if
we obtain the x co-ordinate of the intersection of the locus of equation (9) with
the circle
(12)      x -+y2 = r2
in which r2 may have any value, and place
this value of x equal to that in equation
(11), we shall obtain the value of xymax
when the maximum point is at the intersection of the two curves... substituting from (9) in (12)
x -+- xi2 cot2 i = r,2
x2 (1 + cot2 ui) = x2 cosec2  == r2
(13).  x = r2 sin u.
by (11) and (13),
f r sin Ui cos u,
r2 sin i, = -       2
2 r2.' COS U= --
n r




23:2 (a,-  )      =  o-l(_ 2 r
r r                     7t r
7t r,  /2..  ^    a,  2 cos (-1 _  ).
Denoting by xmi the value of x for
which y is a maximum when'the locus
of (9) is subject to the condition of having a maximum point at its intersection
with x2+- y2 = r22, and substituting the
value of x just found]in the 2d member
of (13), we have.    a a-a,+ r C  (   2r2,  I
i=r'    7r             j
=r, sin cos-( 2 r- 
7n r
— r2 sin sin- 
~2 7,2
V2 r2 - 4 r22
r2   2 r2./V2 r2- 4 r,22
(14).. XmI r2' 
7r r




24
Let r2   r, this being the condition
that the maximum point shall be at the
point where the locus of (9) pierces the
soffit, and we have
(15)  xml- t-    2    4 - 2 4 = r cos r.
Whence
(16) xm=0.77118r = r cos 390 32' 23".
This value of r may also be found by
means of the curves OPlp, OPo, etc. Intersect the surface of equation (7) by the
horizontal plane
(17)          y   b.
hz
b =- x cotrt r
(18)  or  x= —b tan      -,
the equation of the curves OP1 p, etc.
Differentiating
dz     trr      hZ
(19)   d      -- cr' 
(19)   dx      b hd  c h  7r ir
7rz  r
I!fz= ~ -- in (18) and (19) we have




25
dz
x = c( and -    = o,
7r2 r
showing that the lines z =  ~ 2    are
asymptotes of the curves.
Intersect the circle x2 - y2 = r22 by y  b.  x —  4/r2 _ br 2
Substitute this value of x in (18) and:1  ir22- b2 = -b tan     2 r; whence
7C r'
(20) z1 =     tan   (:'r22     b
z is the z co-ordinate of the point in which
the locus of (18) pierces the cylinder, concentric with the soffit, whose radius is r2.
Substitute from (20) in (19), thus
d    =    rr2            _____ -    12
2d,        rcos2 tan   (            }
b h   cos cos      r2_




26
7r b( r~
(21)           h     )
If r2 = r, when the cylinder mentionedE
above becomes the soffit,
dzxi       b
(22)      d,    --  h'
which is the tangent of the angle between:
the tangent to the locus of (18) and the
axis of x at the point where the locus
pierces the soffit. If we make the condition that
dzi      2 r
(23)      d    -,   h
that is, that the tangent at this point shall
be perpendicular to the P F, we have
from (22) and (23)
7t bi  2r
h     h
(24).'. blr= —0.63662 r=r sin 39~ 32' 23"
which agrees with equation (16).
It is thus apparent that in an arch constructed by this method the portions below PP' or beyond the line r/y on the




27
development, should be omitted; i.e., thes
arch should be always segmental with a
span not exceeding
2 r cos r=2 r cos 39~ 32' 23"-1.54236r,
provided that there is to be no tendency
in the successive courses to slide on one
another. The amount of this tendency
to slide and its relation to the obliquity
of the arch will next be investigated.
Let us obtain an expression for the
angle < P4 u between the tangent line to.
O P4 r at P4 and the line P,4 perpendicular to the P F. Let < P4 u = O' then,
equation (22),
1 2r        -1    rb
0'    ta-1n     - tan     
h             h
(25.. tan'(ta 1 2r   -an1 7rb)
A h            h
2r     n b
h    ~h     h (2 r-  7r b)
2 7brb r  h   2 zr b r
1+ -
If h - o, tan O' = o,.'.' - o.
If h = oa tan O' = o,.-. 0' = o




28
-.' must be equal to 0, when tan 0'  o
because it can never equal, much less exceed 90~; for if it can equal 90~, then we
must have from (25)
h2A  2 It b r r=0.-. h. - 2 7r b r'
i. e., h must be imaginary. It follows
that there must be some value of h for
which tan 0' and therefore 0', is a maximum. By placing the first differential
co-efficient of the function with reference
to h= — we find this value to be
(26)     h = /"2 n b r.
By this equation h = 2 r cot a = 1.0445 r,
0' max    O' max
when b - r sin 10~,-. a - 62~ 25' for
0' max
this value of b.
Now the tendency to slide at any point,
as P4 P4', depends upon the angle between the normal to the C j s at that
point and the direction of the pressure,
This last we have assumed to be in a
plane parallel to the P F.,We will now




29
assume for the purposes of the investigation that it coincides at the point P,P,'
with the direction of the tangent at that
point to the ellipse cut from the soffit by
a plane through the point parallel to the
P F. Let pi= tangent of angle between
ground line and tangent line to curve
Q P4' a at the point P4'; then the tangent
of the angle between the tan line to this
curve in space at the point P4 P4' and
the H P will be =p, cos a.
Let p2 = tangent of angle between a
vertical line and the tangent line to the
circle D' V C, at the point P4' = angle
P4' Q C,, then the tangent of the angle
between a vertical and the tangent
line at P, P4' to the section of the soffit
parallel to the P F will be =p,:
cosec a.
Now to find the angle between the
normal at P, P4' to the C j s, and the tangent to the section parallel to the P F,
we have given two sides and the included angle of a spherical triangle, the ineluded angle being 0'.
Let N = angle between normal to




30
j s, at P, P4' and vertical = angle bete
\  _ _ _  _,___ ____
tween tangent plane to C j s at P, P4' an
H P, then we find the value of tan N to
be tall N =px cos a cosec 0'.
(27).sin N P1 cos a cosec 6'
1   pl-2p Cos a cosec0'
(28)   cos N
/l+p2-p cos2 a cosec2 06:and if t. = tan-'(p, cosec a), we have
P2 cosec a
(29)   sin t== — 1 - ~  2cosec2 a
1 +c p22 cosec2 oa
(30)    cos tq=
/1-+P2 consec2 a.




31
By spherical trigonometry
(31) cos 0 =  cos N cos te + sin N
sin t. cos 0'; whence, by substitution
(32)        cos  =
1 + p1i 2icot oa cot 0'
V'(l+-2_ cos0' a cosec2 0') (1+2-t2 cosec2 a)
Multiplying numerator and denominator
by sin a sin 0', this becomes
(33)
sin a sin d'-+ -p p2 cos a cos 0'.cos 5 —-,/(sin,2 o'+p'2 cos2 a) (p2-+Sin' a)
If in this equation a = o, then by equation (25), since h = 2 r cot or, 0' = o, and
hence cos 0  P'P     1,.. - == o.
P1P2
If c=90~-i.e., the obliquity=O-by
(25) 0'-0, and hence cos 0 =- - indeterminate. The reason for this is, that
the tangent lines to the curves  P,))
QP,'a, and OPr))P',7t by which we have
fixed the position of the tangent plane




32
coincide when a —-90~. The value towards which cos 0 approaches, however,.
is unity, as a approaches 90~, for from
the relations between the quantities it is
evident that when 0'=0, N must equal
t,, and hence by equation (31),
COs 0 =COs2 t- + sin' t- 1;.'.  =o.
It follows, therefore, that there must
be some value of a for which cos 0 is a
minimum, and therefore 0 a maximum;
for any given values of p1 and p. Owing to the complexity of the expression
for cos 0, this value would be very difficult to obtain by differentiation; but it
can be determined approximately by
calculating a series of values of cos 0 for
different values of a.  This has been
done, with the following results:
The point for which the calculations
are made is P, P', for which
x= - r cos 10%, y = b = r sin 10~, z 
9 h4 P2 = tan 10~, and p, = 0.47011 =tan 25~ 10' 43", the value of pi being:
found by equation (10).




33
o   o   o    C   0
E-q o   o CII
II  II  II  II   II
0   0   0t   1
0   0   0   C   0
o 0    o c0     o
c    Rc   cc  Cs
From this table it appears that C is a
-maximum when a has some value not far
from 200.
If the point considered be PP' for
which x -r cos r    cos'39 32' 23," and
y=    =r sin 390 32' 23",
y==b-rsin 39Q 32' 23",




34
we have 0'=o andp,=-=o.'. in equation (33) cos - = -, and fiom the same
considerations as when a =- 90~, we find
cos 0 = 1, whence 0 = o,
without reference to what the value of
a may be.
If, as is very probably the case, the
direction of pressure is not exactly parallel to the P F, but inclines somewhat
towards the plane of the right section of
the soffit, the effect will be to increase
the tendency to slide on the C j s's.
Near the springing plane, however, the
direction of pressure in a vertical plane
will not be exactly parallel to the tangent to the oblique section, but will
make -a less angle with the H P, which
will tend, especially when the obliquity
is considerable, to counteract the previously mentioned effect. This refers to
portions of the arch below P P' for which
y = r sin 39~ 32' 23"; above this, the
tendency to sliding being in the opposite
direction; if the line of pressure incline
towards the plane of right section, it will




35
decrease this tendency, and thus add to
the stability of the arch. It is evident
from inspection of the table of values
of 0 that a full centred arch should not
be built by this method unless with a
small obliquity, and then it would need
to be thoroughly supported by the
spandrels.
These results differ entirely from those
obtained in Chap.VII. of Buck's treatise
on oblique arches already referred to; in
fact, are in direct opposition to them. He
derives a formula for the value of r dependent upon the obliquity of the arch, so
that f decreases with a, or as the obliquity increases, and infers therefrom that
the safety of the arch increases as it becomes more oblique up to about 25~. I
should state that his r has a slightly different signification from mine. He undertakes to find the point at which the curves
Q P'c, Q P'd, etc., cut the intrados and
extrados at the same height above the H
P, and calls the angle included between
a radius drawn to this point of the intrados and the H P the angle r. It will be




36
evident from inspection of the drawing
that this would give to, r a somewhat
greitter value than I have obtained, but
still independent of a, since the only constants in equation (9) are a, 7r, and r.
The following table taken from his
book gives the values of r obtained by
Buck from his formula for certain values
of a, irom which he derives his inferences
with reference to the security of oblique
arches.
TABLE FROM BUCK'S ESSAY ON OBLIQUE BRIDGES.
When a = 65~, then r    27~ 17'
"   =55~, "'    r=25~ 13'
a    r45~, "   r   210 47'
a    350, "  r- 150 38'
"   — 2.05~40'   =-  00 0'
Immediately under this table he remarks, " It will be observed that the last
angle is given 25~ 40', at which the point
r descends to the level of the axis of the
cylinder, and the whole semicircle is safe."*
By looking at the table of values of 0 it
* The italics are mine.




37
will be seen that, for this value of a, 0
will be about 32~ or 33~ at 10~ above the
springing plane, r of course being 39~ 32'
23" instead of o, and hence the arch will
be anything but safe if a complete semi-cylinder.
One incorrect assumption made by Buck
in his derivation of the formula for the
value of r is as follows. He says: " It
may be shown that the tangent of the
angle which the tangent to the intradosal
spiral makes with the horizon diminishes
as cos r." He does not show it, however.
N      1
A'
pD^
C          --
P/      /
$^\
E/lP




38
Call the above mentioned angle ib, and
the angle between the tangent to the
helix at any point and axis of the cylinder
/3, then the true relation is
sin  = sin /3 cos r
sin 6 cos z
tan ib 
/n  - sin' fp cos' r.
In the figure suppose A P to be the
horizontal projection of an element d s of
the helix coinciding with the tangent line
D E. A' P' is its vertical projection, and
A1 P1 its revolved position about a horizontal line in a vertical plane through P.
The angle A1 P1 C1 =- b, and the angle
C A' P' — r... ds sin b-=Al C =A' C-A' P' cos r;
but A' P' == ds sin /;
sin -- =  sin /3 cos r.
At the springing plane the tangent to
the elliptical section parallel to the P F
becomes vertical, and the value of 0 is
(34) tan  = —-        cot a.
7 r    n




39
At the crown it is
(35) tan q = tan O'   = (2 —  ) cot 
— 2 cot2 a +  r
The following is a table similar to table
I, giving the values of 0 at the crownL
and springing plane, derived from equations (34) and (35).
TABLE II.
At the Crown.  At the Spring.
At the Crown,  lng Plane.
When a=60~ -~c,=    9~ 50' 0p-==18~ 20'
"  a=50~ -, —,=11 53'    sp=-28~ 7'
"  a-=40~ -4c,=12~ 49' Ap-=37~ 11'
"  a=300 -c,= 9~ 16'     p =470 48'
- = 00' is a maximum by equation (26)
when a = 38~ 35', for which value of a we
find 0, = 12~ 50'.
At the angle a = 25~ 40'-the last one
given in the table extracted from Buck's
work-the line which he finds to be horizontal makes an angle with the H P of a
little more than 29~ 51', that being the
angle between the tangent to the locus of
equation (9) at the point R) C2 and the
HP.




40
It will be noticed that the negative sign
of I0 in Table II. comes from the fact that
in equation (35) 7t is greater than 2. It indicates that the tendency to sliding is
in the opposite direction from what it
is below r.
LOGARITHMIC METHOD.
In this method the soffit is cylindrical,
as in the last, and the case considered
will be that in which the right section is
circular.
The heading joints are planes parallel
to the P F, which, therefore, cut the soffit
in ellipses, which are the h j c's. The
c j c's are drawn on the soffit in such a
manner as to cut each h j c at right
angles.
Now, the angle between any two lines
of the soffit remains unchanged after the
development; hence, the developed c j c's
must cut the developed h j c's at right
angles. Also, if two lines be perpendicular to each other, their projections on a
plaue par;allei to one of them will be per



41
pendicular; hence, if the arch be projected
upon a plane parallel to the P F, the
vertical projections of the c j c's will cut
the ellipses which are the vertical projections of the h j c's at right angles. These
principles will be used in obtaining the
equations of the curves.
Let A B B2 A )) B' V' A, be the projections of the Ex. S., and C D D, 2 ))
D' V C0 those of the soffit. 0 Q is the
axis of the cylinder, D CQ C8 D, is the
development of the soffit to be constructed from the projections, or by
means of ordinates found from equations
(1) or (2).
The developed h j c's are all parallel to
D E 02 F C,, and hence may be drawn
from a pattern constructed by means of
this curve. Divide D D, into such a
number of equal parts as will make the
voussoirs of convenient size, and draw
the h j c's through the points of division
by the pattern. The length of one of
these curves is equal to that of a semiellipse cut from the soffit by a P F, and
is calculated by the aid of the following




42
formula, obtained by integrating the
differential of the elliptical arc.
If S = length of semi-ellipse, a = semimajor axis and e  eccentricity, then
e2  3 e4  5 e6    )
(36) S =a(1-      -    -64  78  etc.
The middle h j c, n o, must be next
divided up into a convenient odd number of equal parts. The division is done
on the middle line, in order that the two
faces of the arch may be alike. We will
next find the equation of the curve
k O0 m of which the c j c's on the development of the soffit are portions.
We have already found the equation
of D E F C, equation (1), in considering
the helicoidal method. With the origin
at O1, it is
h     x
-       2 COS r
Differentiating
7 dy   h     x
(3)      dx - 2 r sin r
If k 0, m cut D E 0, F CQ at right angles,




43
we must have, at the point of intersection,
dy     dy'         dy'       dx
1+ ~ —-          or. do, ory
dy'
in which    7 is the differential co-efficient of the curve to be found. Substitute
dx
the value of -  from equation (37).
dy'          2 r
dx' -           x or dropping the
h sin -
primes, since the x and y in both members of the equation refer to the same
point,
2r       dx
r
sin.
*' Integrating
2r2
(38)    y  - -       tan 2   +c,
in which lo signifies Naperian logarithm.
frwelet whenbeing
If we let y =- o, when c = - 2-, this being
2




44
the condition that the curve shall pass
through 02, we have
2 r2 
0 -- -     le'(l)  C;..    o, and
(39) y   -      l tan 2
If we move the origin to O2 by placing
x' _t+  7 r r, equation (39) becomes
y - - t tan   2'      -
(22 r
2h -he tan(2r
or dropping primes and reducing
1 + tan 
(40) y                  tan —n -
\- tan




45
In (40) if
2 r'
= o, y =-       r   (1) = o;
r  ~2 r2.==-Ey7= —        4 —  (o) =- - ~;
Ztr        2r'
X      2' /Y  -    e(X)=+ci;
X   -$-,y= — 2        (o)   +   D.
the curve is asymptotic to the right
lines
=      7r -' i. e., to I X and D D'.
If the obliquity of the arch in fig. (2)
were in the opposite direction, the right
hand members of equations (1), (38),
(39), and (40) would all be positive.
To adapt equation (39) to convenient
computation, let x = n 7t r, which is
equivalent to dividing the distance nt r 
01 X into n equal parts, for which ordinates are to be calculated... Substituting. common logarithms for
Naperian,
2r   1       n n
(41) y=-    -. M     tan -,




46
in which l, signifies common logarithm,
and M is the modulus of the common
system. From this equation any number
of ordinates may be easily calculated for
the construction of the curve k 02 m.
The equation of the curve K 0 P P1
will next be found. This is the horizontal projection of the curve on the soffit of
which k O0 m is the development. Its
vertical projection is of course the semicircle D' V C2. It is plain that for any
value of y the x for the new equation
will bear a certain relation to the x of
equation (39), and hence may be derived
from it. Call the co-ordinates of K O P
x' and y', x and y being those of equation (39),
x' — -   r(l - cos-); whence z 
r cos(1 -— ), and y' =y, the origin
being at 01.
Substituting in equation (39) and omitting primes,
() 2 - r2  t  r cs —r
(42) y= --       tan  cos'()




47
In (42) change the origin to 0, and
we have
2 r'          x
(43) y -       tan (cos-^)
Now we have by trigonometry
I 2
X           r2 -r
os -   ta —  -    r tan-  ta
r          x          x —
r
also
tan i((tan- r2  2)-  ~ 2   2
(44) Y  h              - r  x 
= /r2  x2 ~r~+ x
Substituting in (43) we have
which is the equation of K O P with the
origin at 0, 0 Q being the axis of y.




48
In (44) if x  o, y  -- 1. (1) = o
h
" x --— r, y  -- - (o)- -oo
x;= -r, y=      4(o) =- oo.
Equation (39) solved for x is:
hy
(45)  x =- 2 r tan-l e- 
in which e is the base of the Naperian
system of logarithms. It is sometimes
desirable to consider y as the independent variable, in which case the equation
takes this form.
We will now give a table of values ot
1      nr n
M la tan -  corresponding to a series
of values of n, also the values of x in
equation (44) for which the ordinates are
equal to the corresponding ordinates of
the curve k OQ m.




49
TABLE II.;.<^    ~    Values of x in
c ~     Values of   Equation (44)
1          for whiclh yis
|o     I   an 7[                   R EMARKS.
la tan      equal to the  REMARKS.
~o  M'   2    y of Equat.e~. ~.tion (41).
0.01     - 4.2331     0.99951r   N.B. —The values of x in col0.02     - 3.4601     0.99803r    umn3aretobe
umn 3  are  to beo
0.03     - 3.0541     0.99556r   laid off from
0,04    -2.7659       0.99211r   the axis of the
cylinder   as
0.05    — 2.5421      0.98769r    c    de    as
the axis of y.
0.10    -1.8427       0.95106r   In equation
0.15    -1.4266       0.89101r    (41) the axis
oJ y is 01D,.
0.20    - 1.1240      0.80902r   To obtain the
0.25    - 0.8810      0.70711r    corresponding
values of x for
0.30    - 0.6742      0.58779r   the curve I 
the curve I O
0.35    - 0.4897      0.45399r    L M, substitute
0.40    - 0.3195      0.30902r   r, for r in the
3d column, the
0.45    - 0.1577      0.15643r    axis of y being
0.50    - 0.0000      0.OOOOOr    the same.
By   means of this table       the   curves
k 02 m and K O P may be easily and
accurately constructed, as well as the
curves I 07L     M   and   N  R   04  for the
Ek s.
For the curve I O L M, which is the




50
intersection of a cj s, with a cylinder
concentric with the soffit, we have if x be
the abscissa of K 0 P, x' the abscissa of
I 0 L M and r, the radius of the concentric cylinder,
x     r        rx'
-= orx.
X'    rl'      rt
This value in equation (44) gives
(46) Y  _h e(2  )  (primes omitted).
For the curve N R O0   which is the
development of I 0 L M, calling the
abscissa of k O0 m x, and that of N R 0O
-.', we have
x     r        rx'
2;=  *1. X  -
x    rl r* 
which, in equation (40), gives, omitting
primes,
(47) y   -- -h le      -
\1  tan 2r'
which is the equation of N R 0, with the
origin at O,.   The ordinates for this




51
curve are the same as those for the curve
k O0 m when we give the proper values
to x, that is, make x - n iT r1, and m easure it from the line B1 B, as the axis of
ordinates.
The curve S V' T is a projection of the
curve K 0 P P1 on a plane parallel to the
P F, of which p q is the horizontal trace.
In constructing an arch by this method
it would be desirable to project it on
such a plane, and hence this curve would
be needed. It cuts at right angles the
projection on its plane of any ellipse cut
from the soffit by a plane parallel to the
P F, and its equation is found by a'process similar to that by which equation
(39) was obtained. Let gp be the axis
of x and 0' V' the axis of y. The equation of the ellipses to which the curve is
to be normal at point of cutting is
(48)  b2 (x - a,)' + a'y' = ab
in which a, is the variable abscissa of the
centre. Differentiating
dy      b2 (x-a- 1)
d       --    al Y  or substituting value
of a1 from (48)




52
dy _ b Vb2   y2
dx       ay
dy     dy'
This in the formula 1      d+ dy x' 
dy'          ay
gives d' = - 6b2      y2; or dropping
primes, since x and y in the two members
refer to the same point,
dx =    -  b y-1 dy b _ y2
Whence, by integration
a          b- 4/b —y2   /b  - 
If we let x = o where y = b-i. e., place
the centre of the curve at O'-then c - o,
and
(49)
a b  le  _ b+*/ — bb2 y}
-a \    b      "2_-y'-     y2




53
This curve is symmetrical about both
axes of reference, and is asymptotic to
the axis of x.
To obtain (49) in a convenient form
for computation, place y =  n b, then
(50) x-                           -
From this equation have been calculated
the values of the parenthesis for a series
of values of n, as given in the following
table.
TABLE IV.
Values of n.        Values of- a
0.1               ~  2.000
0.2               i  1.313
0.3               ~  0.921
0.4               ~ 0.650
0.5               ~  0.450
0.6               ~  0.299
0.7               i 0.184
0.8               ~ 0.093
0.9                t 0.032
1.0                  0.000




54
The c j c's on the development of the
soffit may be constructed by means of a
pattern, one side of which is cut by the
curve k 02 m, and the other side is
straight and parallel to I X. By placing
this pattern upon the curve k 02 m, laying a straight-edge along the back, and
then sliding the pattern till it passes
through the different points of division
on the line n o, the proper portion of the
pattern for each curve will be found.
For construction upon a very large scale
-as, for instance, upon a platform the
true size of the arch-this method would
be impracticable. The point of the curve
k O2 mw which passes through any point
of division on the curve n o may then be
found by calculation. Find by measurement or computation the value ofx forthe
point of division-i. e., its distance from
the line D D2; take this distance as a
fractional part of V7 r-i. e., as a value of
n-and, substituting in equation (41),
find the corresponding value of y; this
will be the constant c of equation (38).
Subtracting this from the ordinates




55
found from Table III. will give the ordinates of the curve measured from a line
parallel to O0 X through the point of
division. It would also be desirable to
determine the exact point in which any
c j c cut the curve D2 C3. We can approximate very closely to this as follows:
Find the point as nearly as possible by
sketching in the curve from Table III.;
through the point thus found draw a
parallel to O0 X; find the distance from
this parallel to the one drawn through
the point of division on n o corresponding to the same c j c; add this distance
to the value of c found as above; take
this as the value of y in equation (45),
and compute from it the corresponding
value of x; this will be the true abscissa
of the curve on the line parallel to 01 X
through the point as first found, and of
course will give the intersection of the
c j c with the curve D, C, with great
accuracy.
The h j s's are, of course, planes parallel to the P F.   The c j s's are
a species of conoid generated by a




56
right line moving on the axis of the
cylinder as one directrix, a c j c as
another, and remaining always perpendicular to the former. Any plane perpendicular to the axis is therefore a plane
directer. The intersection of any c j s
with the P F will be a curve, several
points of which will be needed in a construction upon a large scale. These may
easily be found by drawing one or more
semicircles between D'V'C2 and B'V'A,,
finding, by Table III., the curves corresponding to I 0 L M, for the cylinders
of which these semicircles are the vertical
projections, and then erecting perpendiculars to the ground line from the
points of interlsection of these curves
with the P F (or planes parallel to it) to
meet the semicircles. Projected on a
plane parallel to the P F, these curves
of intersection of the c j s's with the P
F will be normal to the elliptical section
cut from the soffit by the P F at the
points in which they cut it, and hence
will be tangent to the projections on the
same plane of the c j c's. All the above
remarks apply also to the intersection of




57
a c j s with any h j s-i. e., any plane
parallel to the P F.
In order to cut the voussoirs patterns
must be made of the cylindrical and
plane faces of each. It will also facilitate the operation to construct each in
isometric projection, drawing on the
cylindrical face one or more elements
of the cylinder as guides in the cutting.
Cut first the two plane faces of the voussoir precisely parallel to each other; cut
roughly another plane face, making an
angle with these two approximately equal
to that between the P F and a tangent
plane to the soffit at some point of the
cylindrical face of the voussoir (i. e., to
the angle fi of equation (51), from which
equation its value may be found); lay
the stone in such a position that the first
two faces shall be vertical, and the third
uppermost; apply to the vertical faces
the patterns for the ends of the voussoir,
with the edges next the soffit uppermost,
their relative position being fixed by
measurement from the drawings; mark
out the ends by the patterns, and also
the points where one or more elements of




58
the cylinder pierces the h j s; sink
drafts in the upper face connecting these
points, and then cut the cylindrical face
by a templet cut to the radius of the
soffit, and applied at right angles to the
elements. Next apply the pattern of the
soffit, and draw by it
c.    -,d the lines a b, c d, for
P^']- ~~-J tthe edges of the
ff? -   bsa^'~-^l b stone. In the figure
a d is an element of
the cylinder, the stone being seen from
above, and being taken from a course
near the crown of the arch.
The warped faces can be cut by means
of a templet of the form shown in the
accompanying figure. The arc a b c )
a' b' c' is cut
to the radius
of the cylini l \q    / der, and the
arm a d )) a'd'
is firmly
fixed so as
to be normal,P to the are at
the point




59
a )) a'. The cross-bar ef is perpendicular
to a c, and its upper edge also perpendicular to the plane
e        ofa bc)a' b'c'
and a d ) a' d'.
g ))g'and g))g'
d   a         b/,  care braces.
Suppose the
fg  ii ~  stone were cut,
then, if the cur5        v ed edge of
the templet
were applied to the cylindrical face with
the edge of cross-bar e f coinciding with
an element of the cylinder, and the point
a )) a' at the edge of the voussoir, the
edge a d )) a' d' of the normal arm
would lie upon the warped face, and
coincide with an element of it. If a
number of drafts be sunk in the stone,
then, by means of this templet, the
warped faces can be easily cut. It only
remains to cut the Ex s of the voussoir.
If this surface is to be a cylinder concentric with the soffit, the intersections of
this cylinder with the c j s's can be at




60
once laid off on the stone by the aid
of the templet. Measure on a d ) a' d'
from a) a' the distance r,-r, between
soffit and Ex s; mark the point, and,
applying the templet as described
above, mark on the stone any number of
points in the required curve. If the
arch is to be built upon with stone, it
would be better to cut the Ex s in steps,
each stone having one or more horizontal
and vertical plane faces when in position
in the arch. This would not add to the
difficulty of cutting the stones, though it
would somewhat to that of making the
drawings.  This way of cutting the
stones is shown in the drawing of the
arch according to the "cow's horn"
method.
The diedral angle at the edge of a
voussoir must not be less than a certain
limit, otherwise the stone will be deficient in strength at this edge.  The
limit is usually taken at 60~; hence it
follows that a full centred arch should
not be constructed by this method with
a greater obliquity than 60~; but, as the




61
angle between a tangent plane to the
soffit and the P F increases as the element of contact is taken higher above
the springing plane, up to 90~ at the
crown, it is evident that a segmental
arch may be built with a greater obliquity than 60~, whose diedral angles
will be within the limits. To find the
chord  of the   segment (i. e., the
span),
Let a = C2 D, D' = obliquity of
arch.
Let p = angle between P F and tangent plane to soffit; i. e., its limiting
angle.
Let y = angle between the tangent
plane to soffit and the H P.
Then we have a right-angled spherical triangle, and by spherical trigonometry,           cos    cos a siny
cos /
or sin y -c  a'
2 cos a
(51).*. Span = 2 r sin y  2 cos 
K'  r'    ~~~~cos a




62
If we let f = 60~, then cos f6 =;
r.. Span=          r secant a.
cos a
r
Also cos a =s    from which we can
span
determine the limit of obliquity for a
given segment.
If the span = r, then cos a = 1. and
a = o.   Therefore we may give any
obliquity to the arch that we please without passing the limit. If a = 300, the
limiting span is
2r
Span =  2=3 1.1546 r.
The security of the arch constructed
after this method will next be considered.
In Fig. 2 the curves Q P' a and Q P1 b
are the vertical projections of the curves
cut from the c j s, whose directrices are
the axis of the arch, O Q, and the curve
K 0 P, by the planes P c and P1 O.
These curves will be shown to have maximum points at P P' and P1 P' where they
pierce the soffit. h K O L g is the horizontal projection of the curve cut from




63
the same c j s by the horizontal plane ef,
and it will be shown that its tangent at
the point of piercing the soffit is perpendicular to the P F. It will likewise be
shown that the tangent plane to the c j a
at any point of the c j c is perpendicular
to the tangent to the elliptical section
parallel to the P F at that point. Hence,
where it intersects the soffit every c j s is
exactly normal to that direction wlich
we have assumed to be the direction of
the pressure.
First to obtain the equation of a c j s.
Let the line 0 X be the axis of x, the line
Z O Q the axis of z; then let the axis of
y be a vertical line through 0. Then the
equations of the curve K 0 P P1 will be
(a)       X2 + y2 = r2
r2   r -  \
(b)    z=-     ( -- +  )
Equation (b) is derived from equation
(44) by substituting z for y and changing
the sign of the right hand member, because the axis 0 Q in equation (44) was
considered as positive from 0 toward Q,




.1                             O f i  -     ~0  
+ +
CD0                                           D    C 0  II  
+g  -                                CD
P'  s^-^>S                 0
0~-  --                                     0
0~      J     C~C     
b~~~, I~~~~~ C~~~~r~ 36,QI           CP~~~~~~~0~
Q g Cn co co 0




65
Since 0 Q is a directrix of the surface, we
shall have
(56)       x2 - y2a  o.. substituting from (53), (54), and (56)
in (55)
y    2r e, 1 4+ e
x    i + e"' r (1-e%,);
or reducing and making z1 general by
dropping the subscript,
ce       hz
y     2'    2 er2
(57)     x-1     ea        zX
1   e-  
which is the equation of a c j s.
Intersect this surface by the vertical
plane P c d, whose equation is
(58)     z=      (a,-x)
in which a,  0 c - intercept on X, and
2r
--   tan a.
h
Substituting value of z from (58) in
(57),
a,, -
y     2 e  r     2e
(59)  y_       2 (al-)   e
1-      ) -- l
1-e r




66
if we let u  --
r
This is the equation of the curves
Q P' a, Q P1' b, etc. In (59) ifx = c,
y = o      b;.. the line x= a, is an asymptote. Also if x = o, y = o, and if =
_t oo, y = o, and the axis of x is an
asymptote.
Differentiating (59)
d   2(1-e2u)(eu+xeu( d))+4 68Ud)
dx             (1 -/'e)2
du      1
But du = -r, therefore after reduction
r^1 ^r - x - (r + z)e )
(60) -y   2 e u  -r (1 - e+ )e
dy
For a maximum d-   o
* r -   — (r +   X) e = o.
<61)      e2 u - r --
r +- X;




67
This equation gives the values of x for
which y is a maximum.
Now eliminate y between equations
(59) and (a),
4 ea2   2.  2 +.(ie22)2= r; whence
21 (ii +e) =   r2; therefore  solving
for e2",
(61')     e2"   r -
~  r +-X 
This equation, which gives the x co-ordinate of the point of intersection of the
curves Q P' a, Q P' b, etc., with the
circle D' V C2, is identical with (61),
which gives x where y is a maximum;
therefore the curve whose equations are
2 eux
y -1 e2"
z=h (a,-x)
has its tangent line horizontal at the
point of piercing the soffit.




68
To find the inclination of the tangent
at the point where the curve pierces a
cylinder concentric with the soffit whose
radius is rl. If the co-ordinates of this
point be xl y, we have by substituting in
(61'), u for u, r1 for r, and xa for x,
e2   rl - x1.       1e  - X1
l  -e 1=  x.e   -^r,+
Substituting these values in (60),
dy  2j    r-r-x  - (r+xi)rl+: X
dYL 2  rvzxl T  "(1 r,+      Xi
whence by reduction,
(62) dy   (r-r1) Vri2 - XI   _   _y
(62)dx           -
)dx,  ~  r x^           r xI
if we let q - r = 6.
dyti
If we multiply -y by cos a, we obtain
the tangent of the angle between the tangent line to the curves c P ) Q P'a,
O, P )) Q Pl'b, etc., at the point of pierc



69
ing the Ex s and the H P. Call this
angle t, then
6Yo cos a
(63)   tan t-      y co.
2
Suppose a = 60~, -r   5 then
If xa=rt cos 10~, y,=r, sin 10~, and t=0 40' 20"' x,-=r, cos 20~, y-=r, sin 20~, and t==1Q 23' 20"
x' -=r, cos 30~, y-=r, sin 30~, and t=2~ 12' 15"
Intersect the locus of equation (57) by
the plane
y =b;.  -'  l —ei;.      be  +  2 x e -b- o;
solving this for e 2 as a quadratic,
ecz   - _  
(64)..    - 2 7/- 2 Ile
C   \             j




70
Differentiating and reducing,
(6 dz_     2            2r'
(65) ~dx~ IFc'    zF h  +
dzo     2r2      r
When x-=o,- do   f- F -T _ =  T  tan or.
b —-r, and xo  dzc,- 2r r =  tan a,dz tor,   a 2r  r_
To find the angle at which the locus of
(64) pierces a cylinder concentric with
the soffit, eliminate y between y = b and
-, + y' = r,.
$2   ~ +y2        -62..
x2 -- rl2 -b
Substitute this value of x in (65),
(66)
dz1_      2 r2         2rr   r
r      r        t   ad. ThrWe th'rr
If r ==  dxr    tan a. We see therefore that the locus of (64) pierces the soffit at the angle a.-i. e., its tangent is
there perpendicular to the P F-and that
it pierces any cylinder concentric with




71
the soffit at a constant angle for any
given value of a, no matter what value b
may have.
The surface is thus seen to differ very
essentially from the helicoid previously
considered, as regards tendency to sliding on the coursing joints, as is indeed
evident from a comparison of the two
drawings.
The tangent plane to the c j s at any
point of the c j c, as P P', must contain
the element of the surface through the
point and the tangent line at the point to
the curve cut out by a horizontal plane
through the point; therefore it must be
perpendicular to the P F. The tang. of
the angle between it and the H P will be
equal to
(67) tan'P Q C, cosec a = tan N
if N be the angle between the normal at
P P' and a vertical line; but this is also
the expression for the tangent of the angle
between a vertical line and the tangent
line at P P' to the elliptical section of the
soffit parallel to the P F; therefore the
assumed direction of pressure is normal




72
to the cj s at any point of the cj c.
This result might have been predicted
from the mode of construction of the cj c's.
The curve cut from a c j s by the plane
(68)      z =  2 ( -,)
2r
parallel to the P F is always convex towards the springing plane between the
soffit and Ex s. Its equation found by
substitution in (57) is
h2
(69) y      2          2era
if we place 43 (x - a) = v.
By differentiation and reduction we obtain
in the same way in which we found (62)
(70)  dy,     (4  4 3r    )
which is the tangent of the angle between
the tangent line to the locus of (69) at
point of piercing the cylinder whose ra



73
dius is r1 and the horizontal plane. To
obtain the angle between the tangent line
to the curve in space whose equations are
(68) and (69) and the H P, multiply (70)
2r
bysin a=- /^ 2+4 2
Call this angle tl, then
4 / 4r +h2 r,
(71)  taet t   1,  x ( 2.h2+  r -
XI P, x,.2 V.A + 4 =
y1 r sill' a -- r1 cos2 a'
xl      r sin a
If r, == r, which gives the point of piercing the soffit,
Yi            Y1r2-}-h'  y
(72) tan t, =-.    -      C osec'a,
XI    2r      xI
which   is  identical  with  equation
(67).
Ifrl
rIf  =-5 and a - 60~, then
when 4Y —tan 10~, t=-11~ 52' 50", t=11~ 30' 30"
" l-=tan 20~, t=23~ 28' 16", ts=22~ 47' 45'
0, =3 33' 0", =3         41' 24'  -t.tan 30~, It.34~ 33' 30", s-33~ 41' 24'?.",.




74
The points for which t1 and ts are found
being on the same radius are of course on
different curves, though these curves are
so near together that the difference between the angles t4 and t8 is very nearly
the same as the difference between the
slope of the tangents to a single curve at
the points of piercing the soffit and Ex s.
The function n = t1 - t, is found by
differentiation to be a maximum when
I
_Y          — =_:   = tan 44~ 32'
x1   V/ sin a + ~'_ cos.
r
where a and rl have the values assigned
r
above. The maximum value of the function in this case is t - ts = 55' 31".
It appears, therefore, that at no point
on the c j s, between the soffit and the
Ex s does the normal to the c j s vary to
any extent from the assumed direction of
pressure.
COW'S HORN METHOD.
IN this method the soffit is a warped
surface called the Corne de Vache, or
Cow's Horn, generated in the following




SS 7.  lplrrAl  n iU,1              \  i,,,    A:,,i..,..-          In T'        \ \  |;_\
T~~~~~ T,'. — = —..~.~,,,7    X      X.'i I/\,'' I.\,..,,,.     \._-;,, \_,.. —;___  \. x^ ^ \-'A7-''V!.
x"       -------—..',~       \Vm~. \ ~';':I:  \' i"' ^  —, ",,'" "' 
a\, * \  id\'''* *''  1! \!,  h  * \  * 0  }'''1 X -'  \,, 
\ 0'.,..,,'''.\'n,  $-ft x.'x,,:..-F:,'.,..'...'  -~..................-..............-........................-.. —i.-. — ^,, ^..!1.:,,;\,- ~~!x  \....-,\i\       _H,,,,. t._^l.,Jj~                           X    A,.~,,~  -' x-,,'
*~~~~~~~~'  i -: "','\'  ~       ~,\!\ \  il-" \.',', I'. —?   \ ^   \ "".. 
i:. —~~ ~ ~ ~ ~~'"  \'  \:''\ _.;'*' t! L. -: —-4  i,  \-  x _., \,'  ~ \.,'~~  ~   ~_ _: i_.........' 1'!l........................'.........'-!''1  "'iX:1,1":......-'^ - -  *'''*..!    \!'..  \:''V\.  / [, ~,!        ^ l,!;'i i!:', I  1",',':  \  \ - " i,  x',:.^ \^''^N>^\^;v <!>^'''A~~i^'*/'/'  ///   1.'**v  N'^i^  ^^''^ Y ^^"''-1'; h^^'^^^^''''''-:.......   /   k
-'\i     \'.,',\ \'.^/^:'  ~      \ *""I  i ^! p ^ e M':'^^ \,,.-,\,"31..-'  L.
j -/^ /!;;  I  |  F ^ -V'..^ }-"'"" "I.'*  ^   ^,,/'^.  "^;~~~~~~~~~~~~~~~~~~~\__...' -...~,,....... \   -,,   ~'~.......\..../',,.!   - -., ~   -.',,'~',  ~',~'tl'i'  \:~~~~~~~~~~~~~~~~~~~~~~~~~"A'~'~"
T-~~~~~~~'"   \"'..' A.....      ~  -'', /..,:.'.-..'?,   ~ -!.'  i' ",'- -''I..  " - ~ ~,,'''.-....  \,..'.',,'.. ~.   ~   -.'-4,.'' - — ~,~  ~   i   I'   ~,'   ~ -~~~~~~~~~~A'  \;.L. —-----—:,',q'i'...:....J-~ ~   ~.   "   ~. -''   "'   "'"',"'"'".'""'  ~' /....~."'-  ~'.......  /- ~.''.'.,~':'''   -~-/'"'~~~~~~~~~~~~~~~~~~%..  -.: - -   \ ~:,~' ~:u. ~.',~:.>>  -— ~.,.   ~........,,:/.:.,  ~,'.'-'.-..,...'.,..  ---.-', -- — i>,.,.. 7''":~ —--.:~ i~'  <  --— = — _.,. "   - -   ~ i1.4,








manner. A right line moves on three
directrices, which are: 1st, two equal
ellipses in parallel, vertical planes, having their transverse axes in the springing plane of the arch; and 2d, a right
line drawn in the springing plane perpendicular to the plane of the ellipses,
through the centre of the parallelogram
formed by joining the extremities of the
transverse axes of the ellipses. These
ellipses may, of course, as a particular
case, be circles. In Fig. 3, the plane of
one face of the arch is taken as the V P,
and the springing plane is the H P. In
referring to Fig. 3, the following notation
will be employed. Any letter with h
written above it as an exponent, means
the horizontal projection of a point, and
the same letter, with exponent v, is the
vertical projection of the same point, and
this point will be referred to as the point
A, B, etc.; i.e., the point whose projections are Ah AV, Bh BV, etc. A line drawn
through these two points would be,
therefore, the line A B. If one projection of a point is in the ground line, h or




76
I, as the case may be, is replaced by o,
and if the point itself is in the ground
line, it will be designated by the letter
alone without exponent.
In Fig. 3, the parallelogram Ah Bh C D
is the horizontal projection of the soffit.
Its centre O is the point through which
the rectilinear directrix of the soffit is
drawn perpendicular to the V P, since
this coincides with a P F.  The three
directrices of the cow's horn surface are
then the ellipses D S I C, and A K N B,
and the right line O Z lying in the H P.
The elements of the surface are to be
drawn so as to cut these three directrices.
The vertical projection of O Z is a point
in the ground line at 00, hence the vertical projections of the elements will be
lines radiating from this point. By dropping perpendiculars from  the  points
where the vertical projections of the elements meet the vertical projections of the
elliptical directrices to the horizontal
projections of the same, the horizontal
projections of the elements will be found,
as will be seen in the case of the elements




77
R S, P Q, etc.  The elements of the srface are the edges of the voussoirs-that is,
the cj c's, which, therefore, in this method
become right lines, while the h j c's, being
sections of the soffit parallel to the P F,
are curves of the 4th degree. It is, of
course, impossible to develop the soffit,
since the consecutive elements are not in
the same plane.
The arch must be divided up into
courses on the median section in order
that the two faces may be alike. To find
this, draw the vertical projections of a
number of elements, and bisect the portion of each included between the points
in which it cuts the vertical projections
of the elliptical directrices; through
these points of bisection the median
curve may be drawn. In Fig. 3, it is the
line L T, and is only drawn as far as the
crown of the arch. The length of this
median curve would have to be ascertained by construction upon a large scale,
and accurate measurement. It may then
be divided into a convenient odd number of equal parts, and the elements of




the surface, which are the cj c's drawn
through the points of division.
The h j s's in this method, are planes
parallel to the planes of the faces of the
arch, while the c j s's are hyperbolic
paraboloids the method of whose construction will next be shown. It will first be
shown that a hyperbolic paraboloid may
be drawn having an element in common
with any warped surface, and normal to
this surface at every point of the common element. It is proved in works on
descriptive geometry, that, if two warped
surfaces hive a common element, and
have common tangent planes at three
different points of this element, they are
tangent to each other throughout the
length of this element. Therefore, we
can always draw a hyperbolic paraboloid
tangent to a warped surface along an
element; for draw tangent planes at
three points of any element, and in these
planes, through the points of tangency,
draw right lines parallel to some given
plane; if a rectilinear generatrix be
moved on these lines, a hyperbolic para



79
boloid will be generated tangent to the
warped surface along the element. Now,
revolve this tangent surface about the
common element as an axis through an
angle of 90~; it will then be normal to
the other surface at every point of the
element. If the lines drawn in the tangent planes are perpendicular to the common element, after revolution through
90~, they will be perpendicular to the
tangent planes, and hence normals to the
warped surface. Hence it follows that
the directrices for a c j s may be three
normals to the soffit, drawn at any convenient points of the corresponding c j c,
or element. The points atwhich normals
are most easily constructed are those in
which the element cuts the three directrices, but as the intersection with the
rectilinear directrix will generally be beyond the limits of the drawing, some
other point must be used instead of this
one. A method will now be given by
which a tangent line can be easily and
simply constructed at any point of any
section of the cow's horn surface, by a




80
plane parallel to the elliptical directrices.
This tangent line being found, of course
the normal to the surface at the point of
i/i
tangency can be drawn at once. Let
A V B and C V D be vertical projections




81
of the elliptical directrices, P8 that of the
rectilinear directrix, P, X that of the
element through P1, which is a point of
some section of the soffit by a plane
parallel to the P F, and O F and E F
tangents to the curves A V B and C V D,
at the points where they are cut by P3 X.
E F and O F will meet on P8 F because
this is the axis radical of the two curves.
Let O E   == a, O F  b, P, O =- a and
O P4 = b3. Take O as the origin of coordinates, the line P8 X as the axis of
abscissas, and the line O Y as the axis of
ordinates.  Then the equation of the
tangent line E F will be
(72)      -+    -1;
that of the line P, Ps,
(73)      X     Y   1;
and that of the tangent line O F,
(74)         x = 0.
We will now find the equation of the
line cutting O X and P8 P5 in such a
OP,    P, P2.
manner that   P, E  -p p  Let 0 P=
P1 E     P2,-5




82
n X OE _ n a. To find the co-ordinates
of the point P, eliminate between (72)
and (73)
b
y5     b   -   x  = — x  b  -l
a           a1
a al (b - bl)
5 a b+ a bl, similarly
5    b1 (a + a,)
a1 b — a b,
For the point P, the co-ordinates are
x,? O      /4 y4- bl
Hence for P2 we shall have
n a al (b- b)
x2 --- n   --
al b -- a b,
y2=bl+  L(y5-b l) _b,(al bb+a b6 —na(bl —b))
a, b + a b,
and for P1, x1 = n a, y -= 0.
Substituting these values of x, y, x, y, in
the equation of a line through two points
(y-yi) (XI    ) — (a$ —XI) (y —y2)
we have after reduction,
_) y  (n a-x) (a, b+a b, —n a(b-b) )
(75)  n     n a (a + a;)
which is the required line.




83
Now in this equation we may give to
b, any value we please, positive or negative; suppose it to change gradually in
value from some positive quantity to some
negative, the line G P1 will change position accordingly, and at the instant in
which b, passes through zero it will be
tangent to the section through P,; for
the law of this curve of section is, that it
cuts off an nth part of the portion of
any radial line included between the two
curves A V B and C V D; hence at the
instant that b, = 0 the line G P1 coincides with an element of the curve. In the
figure G P1 is this limiting case; i. e., the
tangent at PI, and the line of equation
(75) would cut the axis of y very slightly nearer to F, but the two lines would so
nearly coincide for this position of P, PR,
that G P, is made to answer for both.
Now in equation (75) make b, =0 and we
have the equation of the tang. to the curve
of section.
7  b (n a- x) (a, + n a)'d
(Fo) th    inte c a (a +  1)Y -
For the intercept on Y let x = 0,




84
Y7  _ b (a, + n2 a)
YO7) *       a + aor putting it into the form of a proportion,
(78) a+ a: n a +a:: b: y.Hence to draw a tangent at any point of
a curve of section of a cow's horn surface
by a plane parallel to the curve directrices, draw the vertical projection of the
element through the point (V P supposed parallel to P F), and tangents to
the curved directrices at their points of
intersection with the element; lay off
E H and P4 I each equal to P, 0 =-a,;
draw H F to the point of intersection of
the tangents previously drawn, and I G
parallel to H F; through G draw G P,,
then will G P1 be the required tangent
line; for
O H=- O E +E H = a+ a,,
OI=OP +PI =na + a,,
and 0 F = b.-. by (78) O G =y,
We will now construct a hyperbolic
p araboloid normal to the soffit, and intersecting it in the element R S. Since
lines perpendicular to each other have




85
their projections on a plane parallel to
one of them perpendicular, the vertical
projections of the normals can be drawn
at once perpendicular to the tangents to
the vertical projections of the curve directrices and the median section at the
points R, U, and S. Their horizontal
projections will be perpendicular to the
horizontal traces of the tangent planes to
the soffit at R, U, and S. These tangent
planes will of course be the planes
through the tangent lines to the soffit
at R, U, and S, already drawn, and the
element of the surface R S. Portions
of their horizontal traces are af3, y6, and
e<, to which the horizontal projections of
the normals S a,,U b, and R c are perpendicular. These three normals are the
directrices of our c j s. To find an element of the 1st generation, pass a plane
through one directrix and find the points
where the other two pierce it; join these
points by a right line; this line will be
an element of the surface. Take, for
convenience, the plane which projects
a S on the vertical plane of projection,




86
then b and c will be the points in which
the other directrices pierce this plane;
therefore a c is an element of the first
generation. Any number of other elements may now be found by merely dividing up a c and S R, or a S and c R
proportionally, as in Fig. 3. In the figure the horizontal projections of several
elements of each generation are drawn,
but the vertical projections of those of
the first generation only.
Next the intersections of the c j s
with the P F's and with the Ex s must be
found.  The vertical projection of the
intersection of the c j s just constructed
with the V P is the line 7v Sv, of whicli
the portion drawn varies but little from
a right line. There -are a number of
forms in which the Ex s may be cut.
It may be a cylinder whose axis passes
through O and is parallel to E H; or a
co-axial cow's horn surface generated on
the extradosal ellipses H I, G and E K,
F; or the exterior surface of each course
may be cut like the course M1 Q1 by one
vertical plane through f and k, and one




87
inclined through d, f, g and k; or each
course may be cut in a series of steps,
as in the course P N, by a number of
horizontal and vertical planes. The last
method would be preferable for the voussoirs at the ends of each course, however the others were cut.
Except in the case where the Ex s is
cylindrical, no face of a voussoir can be
cut by the aid of a templet. Cut first
two plane faces on the stone precisely
parallel for the ends of the voussoir. If
the Ex s is to be cut in the manner of
P N or P1 N,, it would be best to cut
next the other plane faces of the voussoir of which patterns can be made
from the drawings.   Then apply the
patterns of the heads and mark the lines
on the stone, marking also the points
where one or more elements of the ruled
surfaces forming the soffit and c j s's
pierce the plane of the head of the voussoir. The warped faces can then be cut
by a straight edge.   The soffit face
should be cut first, and the elements
forming the edges of the voussoir mark



88
ed; then all the bounding lines of the
coursing joint faces will be given on the
stone, and draughts can be sunk by a
straight-edge in a direction perpendicular to the soffit edges of the stone by
which the c j s's may be cut.
The curves k~ I m and n o p are the
evolutes of the ellipses A K B and D I
C, and are convenient in drawing the
normals to these curves which are required.
The curved directrices of this arch we
take elliptical so as to correspond with
the curves cut from the soffit in the
other methods by planes parallel to the
PF.
In each of the drawings the direct
span is 30 ft., the oblique 34.64 ft.
a=-60~, and the number of courses is 49.
It is evident fiom the drawing that if
a perpendicular to the H P be erected
at the point 0, it will pierce the soffit in
the line K I, which is parallel to the
rectilinear directrix and lower than the
highest points of the elliptical directrices,
so that the crown of the arch curves




89
downward toward the middle, from whiich
peculiarity the surface derives its name,
" cow's horn."  Plainly, if this curvature
were so great as to cause the median section of the soffit to be convex toward the
springing plane at the crown, the arch
would be unsafe; indeed, could not stand
at all. We will investigate the conditions
under which this will be the case, and to
this end will obtain the equation of the surface. Let the line O X be the axis of X,
O Z the axis of Z, and let the axis of Y
be a vertical line through the origin 0.
The equations of the three directrices
will then be
(79) r( +         E    quati o  of
1   a2'2      Equations of
1               f       DIC;
(80) z= —      
(81) F(-e)-+         1       
a2'b1      Equations of
(83) x = 0, y    0, equation of OZ;
in which e = 0~ = 0~r, and
6 = OK = 01 -     distance between the faces of the arch.




90
The equation of a plane through the
axis of z is
(84)         y =- m x
in which m = tangent of angle between
plane and II P.
We will now find, by elimination, the
co-ordinates of the points in which (79)
and (81) pierce this plane, obtain the
equations of the element through these
points, and then eliminate the constant m.
From (79) and (84) we obtain after reduction, and placing ac m2 M - b+   k2,
b [- b  i a Vk2- ]e2n
Similarly fom  (81) and (84),
r2 bL  E   s,c 612  _2 In,2
Also from (80) and (82) we have
z- = -  6 and z,= +- +'
Substituting these values of x,, x,, z,, and
--- 1    22 -— z 
z~ in the equation     -      r   of a
X  X  X - Xi2




91
line through two points in X Z, we have
We2bb /    -b2    a b 4/k — E ml\
and by reduction
(85)    b   z -  a2  m2 x -  6 b2 x
= a b 6'a2m-2 + b2 — 2 m
which is one equation of the element
through (vt z1) and (xz z2), equation (84)
being another.
Squaring (85), introducing the value
of m from (84), and reducing, we obtain
622 (   y2 2   2 6x zE za2y2
-     + +  22  - E2    a2 + 2   J
62   + b+  ) 2   a    ab +  b  0;
or factoring,
(86) +                      - 2
+ 62      + a    2     o)
which is the equation of the cow's horn
surface. If E be taken negative, the oh



92
liquity of the arch will be in the opposite
direction from that of Fig. 3.
Equation (86) contains only even
powers of y, hence the surface is symmetrical with respect to the plane X Z.
If z =   6, it becomes
(x T  )4)2  y2
a2     - 
the equations of the curved directrices.
Ifx = 0, we have y i  b  1 -c, two
right lines parallel to the rectilinear directrix.
6 (x T: a)
If y-0, then z 
If z    some   constant =  n 6, we
have the intersection by a plane parallel
to the P F.
(87) { ~+-^b2__f       1 } (~.~-'32)
l 2/2'   2. ^2\
2 n2 X2   Y 2
nd if  0, this becomes
axnd if n s 0, this becomes




93
(88)(  -+ Y2 2     -2 (1   2    _ 0;
the equation of the median section.
In (88) let y = b    1 -  -   height
above X Z of lowest point of crown
of arch _ ordinate of median section
where x - 0.
*'i                 1i - -        0;
whence by reduction
(89)  -x    ~ +  2 ~ 2  a.
This equation gives the x co-ordinates
of the points in which a tangent to the
median section at the extremity of its
minor axis cuts the curve. In order that
the arch may stand, these points must be
imaginary.
In (89) when e > a Vf > 0.7071 a, x is real;
"     E=a /=  0.7071 a, x = 0;
E < a f- < 0.7071 a, a is imaginary.
The third of these cases is therefore
tile condition of stability.




94
The same result may be obtained by
dy
differentiating (88), placing d —  O, and
making the condition that there shall be
only one value of x for which y is a
maximum.
In equation (86) if e  0, we have
X2   y2
a cylinder whose axis is 0 Z.
If = a, by transposing and extracting
square root
\a +    -b-/ -   a   a
the equation of two cones tangent to each
other along the axis of z.
The equations of the c j s's can be
found without difficulty, but they contain
so many constants and are so complex
as to be of no practical utility. The
character of the arch as regards stability
and tendency to sliding on the coursing
joints, can be easily seen by examination
of Fig. 3 and comparing it with Figs. 1
and 2. It will be noticed that the vertical




95
projection of each element takes a direction between those of the normals to the
elliptical directrices at the points where
it cuts them; therefore at some point between these it must coincide with the
vertical projection of the normal to the
section at that point by the heading
plane through it. It follows that at this
point the element will be perpendicular
to the direction which has been assumed
to be that of the pressure, and from the
manner of its construction the c j s will
be also normal to this direction. At the
crown this point is midway between the
faces of the arch, and as we approach the
springing plane it moves toward the
points A and C. The curve X U 3 is cut
from the c j s, R S a c by a horizontal
plane through the point U, and the portion of it from U toward - which would
lie upon the coursing joint would evidently be nearly perpendicular to the
direction of pressure. The curve,u 7, t
is cut from the coursing joint surface
through the element M N, and the portion which is upon the voussoir is almost




96
exactly perpendicular to the P F. /11'i1 6
is cut from the surface of the lower face
of the same course M Q. 7'rp S is the
curve cut from- the c j s through the first
c j c on that side by a vertical plane perpendicular to the P F through B. It varies but slightly in the distance 7t 5 from
a right line. tr piS~ is a similar curve cut
by a plane through L. The first one or
two c j s's from the springing plane vary
so slightly from a plane in the portion included between the inner and outer surfaces of the arch that they might well
enough be made exactly plane when the
number of courses is large.
This method of constructing the arch
gives results therefore, as regards tendency to sliding in the coursing joints,
intermediate between those found in the
two methods previously considered, but
approaching far more nearly to those obtained in the logarithmic method; that
is, the c j s's are nearly normal to the direction of pressure.
An arch may also be constructed with
the cow's horn soffit and plane coursing




97
joints as follows: On any element, as M
N, take points midway between the heading planes, and at these points draw normals to the surface; through these normals and the element pass planes which
will form the cj s's. The coursing joint
will then be cut in a series of steps, and
a portion of the voussoirs will have a triangular vertical face midway between
the two ends. If the Ex s be also cut
in steps, the voussoirs will have all their
faces plane except the soffit, and all their
edges straight lines except the intersections of the heading planes with the soffit. The construction of the drawings
and cutting of the stones would thus be
comparatively easy.
What was said in treating of the logarithmic method with regard to the limit
of obliquity by reason of the edges of
the voussoirs becoming too sharp where
a is less than 60~, applies equally well to
this method of construction.  In this
case, however, as in that, segmental
arches can be built in which a < 60~.
Equation (51') could be used to ascertain




98
approximately the allowable span for a
given radius and obliquity when the
semi-axes of the elliptical directrices have
b. L;. t..'..
the ratio to each other -   cos. a; that
TV      a
i!,PY
I
j   It
\~!-' 
I S
\ - - - "    /




99
is, such that if the soffit were cylindrical,
its right section would be a circle.
If the curved directrices are circles, we
may obtain an approximate result as
follows, by supposing the soffit to be
cylindrical.
Let the angle P1VP'- = 0,
(      PED      y,
"     ABBO B    a,
the distance         C    Ph X=,
"           D PV = y,
and 6f  the angle between the P F and
the tangent plane to the soffit (supposed
cylindrical).
A Vv B~ will then be a right section of
the cylinder, and its equation will be,
if Vh P =r,
x2      y2
r2 sin2 c  r2
The equation of the tangent plane tc
the cylinder will then be
z x 
r.2 Si'21 a+ r2
r2 a    $
or        y 
Y1    y, s^'~ a
*   tan y.;y, and
7 y, s'^2 a 




100
XI
sin y -, _        == er
y  /x +) y2 sin a
cos f3    1
But by (51) sin y  cos     2 cos a
cos a   2 cos a
if we let / -= 60~.
*1           1,xl2 + yl2 sin4     2 cos a.4 x2, cos2 a-  x2 +- y2 sin4 a.
Dividing by W2,
L2
4 cos2 a = 1 +- y sin, oa.
Xi
We have tan.        Y1 — =  -- whence
7' =- -tana2 q cosec2 a.
Substituting
4 cos2 a-  1  tan2 q sir2 at
tan     /4 cos2    1 whe e
sin a
tan a
cos  _= ~/ --




1.01
(90) Oblique span  2 r cos O
2               tan a
-,r tan a =- 2 r.   
~/ 3            tan 600.
2
If a   30-, oblique span  - 3 
Ifa=45~, oblique span -  r-r==1.1547 r
In the cow's horn soffit this formula, as
well as (51'), will make the diedral angles at the edges of the voussoirs in the
first course slightly greater or less than
60~, according to their position in the
course. The exact solution of the problem' involves an equation of the 4th degree with four real roots.
COMPARISON OF THE THREE METHODS.
There is one advantage possessed by
the helicoidal method over each of the
others; viz., that it may be constructed
of brick. This is owing to the fact that
the successive c j c's are parallel, so that
the voussoirs, except those at the ends of
the courses, are all exactly alike, while ill




102
the other methods each stone is different
from the next one, though the two halves
of the arch on each side of the keystone are alike, so that any stone cut
for one side will fit also in the corresponding place on the other side. The
fact that the different voussoirs are alike
in the helicoidal method, of course lessens the labor of preparing the drawings,
and of making the necessary measurements. As regards the difficulty of cutting the stones, however, this method
does not seem to have any serious advantage over the others even by the approximate method of cutting which has been
mentioned, while if the coursing and
heading joint faces were cut with exactness, as helicoids, the difficulty would be
fully equal to if not greater than that by
the other methods.
It may be considered an advantage as
regards appearance that the quoin-stones
should be all alike, or rather those faces
of the quoin-stones which coincide with
the faces of the arch. This, of course, is
the case only with the helicoidal method.




103
It appears to me, however, that the gradual decrease in the size of these faces
from one side of the arch to the other
would not be displeasing to the eye,
when taken in connection with the direction of the c j c's which would make the
reason for the decrease obvious. The
real test, however, of the relative value
of the different methods would appear to
be that of security. When this test is
applied, the logarithmic and cow's horn
methods both excel by far the helicoidal.
It has been shown that in the last mentioned, when semi-circular, there is always a tendency to sliding on the coursing joints, both above and below a certain point; that is, the assumed direction
of pressure is nowhere normal to the
coursing joints except at a certain height
above the springing plane equal to r sin.
39~ 32' 23", and that near the springingplane this tendency to sliding increases
rapidly with the obliquity up to a == 20
(about); while in the logarithmic method along each c j c this tendency is zero;
that is, the assumed direction of pressure




104
is normal to the c j s at any point of the
c j c, and in the cow's horn the tendency
is small as compared with the helicoidal.
The logarithmic method, therefore,
seems to approximate to theoretical perfection as regards security, is followed
closely by the cow's horn, and at a great
distance by the helicoidal.
The cow's horn soffit admits of plane
coursing-joints, as has been shown, which
are not feasible in the others, and thus
possesses an advantage over them, if such
an approximate construction be desirable. If cheapness be an important item
to be considered, the last-mentioned
method would seem to present most advantages, as avoiding almost entirely the
luse of curved surfaces, and at the same
time reducing the sliding tendency to a
small amount.
If the main thing to be considered is
security, the logarithmic method must
stand first.