FI2 
eentre of the conterminate equilateral 
hyperbola C A its semi-axis, and C R 
will be equal to CB. Ry) 
By Corol. 4. If the angle B DT be 
made equal to, AC R, the right. line 
BD F will touch the catenaria in D. 
By Prop. 2. Corol. 1. If AT. be the 
‘conterminate equilateral hyperbola, and 
A&A P a parabola, whose parameter is 
equal to four times the axis of the hyper- 
bola; BF, the ordinate of the catenaria, 
wul be equal to the parabolic curve A P. 
fess BH, the ordinate of the hyperbola. 
By Corel. 2. The curve of the cate- 
naria AD is equal to BH, the corre- 
sponding. ordinate of the conterminate 
equilateral hyperbola. 
By Prop. 6. and Corol. 4, 2, 3,4, 5, 
and 6.. If VA be the evolute, VO 
will be the osculatory radius; and OZ 
2 tangent to the catenaira at the point Q. 
A€:CN:: NE:QOM, and the right 
ine NC will be equal to MV. ‘The 
evolving right line VA will be a third 
proportional. to the lines AC and CN. 
‘Fhe radius K A of a circle -equi-curved 
with the ehain, will be equal to the semi- 
axis AC of the conterminate hyperbola, 
and the chain A D, and the hyperbola 
A H wiil have the same degree of cur- 
wature at the vertex A. The curve VA 
tess AJ will bea third proportional to 
the right line AC, and the curve A L 
or the right line NE. The right line 
K Q wil be double AN. 
Prop. (%.. Corel. 1,2, 3. Tf 0 ALG 
be a logarithmic curve, whese subtan- 
gent W S is equalto AC; and if a point 
A& be taken, whose distance AC from 
the assymptote I W be equal to the sub- 
tangent; and from the points I W any 
how taken in the assymtote, equally dis- 
cant from the point C, and if ordinates 
-W GandIU be erected to the logarith- 
mie curve, to half the sum of which ID 
or W F be made equal; the points D 
and F will be in the catenaria, corre- 
sponding te the right line A C. 
If AC be unity, whose logarithm is 
equal toO. To find the logarithm-of C. A, © 
or of the ratio between C A and Ca. 
‘Fo the right linesC A and C A let the 
third proportional be C a; and !et half 
the sum of C AandCabeC 8. The or- 
dinate tothe eatenaria from B (that is 
BD) is the logarithm required. 
On the contrary, if from the logarithm 
given, C Lor C W, the correspondent num- 
ber 1 Uor W G be required,- or the ratio 
WG to C A, or 1Uto CA from’ W 
or 1, let fall a perpendicular meeting the 
‘4 
Of the Catenuria, Ke. 
(Sept. 1 2. 
catenaria. in D or F, and let C R be 
.made equal to ID or W F, that is, to 
CB. Then will A R be the semi-dift 
ference of the lines required 1 U W G; 
as I D or CR, is their semi-sum, ‘er 
CR+AR, and CR—A R, are the 
members W G or TU. 
ID the semi-sum of the ordinates 
IU WG of the logarithmic curve, ap- 
plied perpendicularly to 1 W at I gives 
the ordinate of this eatenaria; so the 
semi-difierence AR applied perpendi-. 
cularly to CA in B is the ordinate of 
the equilateral hyperbola B H described 
within the centre C and vertex A, and _ 
is equal to the catenaria A D. 
Now it appears, that this new mode of 
describing the eatenaria does produce 
the same curve, called the catenaria by 
Bernouilli, Leibnitz, and Gregory ; and 
that any geometrician, whether ac- 
quainted or not with algebra and fluxions, 
may verify the fact. What naportant 
results, in the other branches of mixed 
mathematics, may be deduced from the 
simplicity of this mode of construction, @ 
little time may probably shew. I need 
not make any apology for submitting 
the following observations on the theory 
(which Dr. Hutton has called an attempt 
towards perfection, and which he has 
acknowledged to be hastily composed, 
but which one solitary individual has ra- _ 
ther inconsiderately called ‘* ihe true . 
theory,”) by the celebrated author of thee 
articies, River, Roof, and Arch, in the 
Encyclopedia Britannica.—** But we 
beg leave to say, with great deference 
tothe eminent persons who have prose- 
cuted this theory, that their speculations 
have been of little service, aud are little 
attended to by the practitioner. We 
venture to aflirm, that a very great ma- 
jority of the facts, which occur in the 
failure of old arches, are irreconeileable 
to the theory. The way in which eirca- 
jar arches commonly fail, is by the sink- 
ing of the crown, and the rising of the 
flanks. It will be found, by calculation, 
that in most cases it ought to have been 
just the contrary. But the- clearest 
proof is, that arches very rarely fail, 
where their load differs most from that 
which this theory allows. We hope to 
be excused, therefore, by the mathema- 
ticians, for doubting the justness of this 
theory.” . 
Gf the theory of abutment-piers, per- 
“haps the gentleman, who, intuitively we 
presume, knows it to be ‘* the true 
theory,” through your Magazine will ex- 
plain 
