Ware Properties Arches

7/27/2019 Ware Properties Arches

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A

TREATISEOF THE

PROPERrrIES OF ARCHES,AND THEIR

ABUTJ\;fENT PIERS:

CONTAINING

Propositions for describinR g~om etricalIy the Catenaria, and theExtrauosscs of all Cur\'CS, so that their several Parts

a nd th eir l> ie rs m ay c 'lu ilib rate;

A L S O

CONCERNING BRIDGES, AND TIlE F LYiNGBUTTRESSES OF CATHEDRALS.

To w hic h a rc a dd ed , in Illu stra tio n,

SECTIONS OF TRINITY CHURCH.) ELY; KING'S ~OLLEGECHAPEL, CAM BRIDGE; W ESTM INSTER ABKEY; SALIS..

B URY, E LY, LIN CO LN , Y OR K, A ND l)E TER .BO RO UG H C ATH ED RA LS.

By SAM UEL W ARE, ARCHITECT.

. "True~ in deed, the great load ab ove of the w alls, and v aulting of the" nave, should seem to confine the pillars in their perpendicular .tation," that there should be no need of butment inward; but experience hath

" shew n the contrary."-Sir C . Wren'i Survey of Salisbury C athedral," Et e converso solre catenarire Sllnt fornices sive arcus legitim i: et"cu ju scllnq ue alteri1 .ls fig urre arcu s ideo su stin et.u r, q uod in ilJiu s crag ..

" sitie quredam catenaria inclusa !$it."-Phil. 'frans. 1O !};. D r. D avidG regory


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P R E F ACE.

THE inquiry into the theory of the equili-

bration of arches has hitherto been almof1:

exclufively confined to mathematicians; and

the moft celebrated of the prefent age ha vc

thought it worthy their attention.

The difference of opinion which cxifis on

this fubjeet may be attribqted to the cir-

cumfiance, that praCtical m en have not joined

in the inve11:igation.

M uch inconvenience has been experienced

by the architeCt and engineer from fome

jnconfifiencies in the treatifes which' have

been pubHfued on arches and piers and alfo


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.VI PREI.'ACE.

This invefiigation was commenced for the

author's own information, and for a particular

purpaCe; but he has been led to purfue the

{ubject further than he intended, and now

.offers the following e{fay to the public, with a

hope that [ainc of the defeCts init will be

a{crihed to the interruptions which other pur-

{n its ha vc nccc{t1rily occafioned .

111 tbe early progrcfs of this treatife, the

paper of Dr. David Gregory on the catenaria,

Phil. rrranf. 1697, induced a conviCtion, that if

a fin1plc mode of de[cribing that line geome-

tricaIIy could be diicovercd, it would be an im -, . ,

portant fiep towards the attainm(:nt of a cor-I

rea tlJcory oftheequilibrqtion of arches, andthci~ abutment pi~rs. This difficulty, after

an expenfe of much time,. the author has

furmounted (fce Prop ixjig 24 ) and a'


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PREFACE. \'11

buildings would be a p]ain and effeCtual mod~

of ~fi:ablifhing the truth of what is there ad-

vanced: Trinity C~urch, Ely, and King's

Col1ege Chapel, Cam bridge, were, therefore..

fixed upon for. this purpofc; and an oppor-

tunity was taken, which profcffional buuncfs

in the ncighbourhood afforded, to n)ake the.

dra\\'ings figurts ~6. and 27. The fcction in

Price's Ob{ervations on Salifbury Cathedral

lcd to the propofition on flying buttrefies.

An the {eetions herein publifhed (\vith the. ~xception of tbat"of Salifbury cathedral) have

been m ade on purpofe for this traCt, from accu-

rate admeafurements; and m uch tin1e and

cxpenfe have b~enbeftow ed i'nobtaining then).'fhc author thinks that, in pre{enting them to

the public, he iliall afford to the antiquary


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. ,.Vlll PREFACE.

ne{s of the columns and abutmel1ts in there

edifices.

Under the head of IntrodllClory Defini...tions and Rcmarks, the wordequipolIence

is adopted. as a genera) tcrm, applicable to

a]] arches which are ~nablcd to retain their,

aCtual ftate; whether from the equilibratiqnof their parts, or from ' the rnode in which

they arc conneCted. As the theory of the

equilibration of arches, and the properties

of the eaten aria, are deduced from the la\vsof motion, tho[e laws, which arc direCtly ap'"

p1icable to tbc[e fubje8:s, are given. in the firft

feClion; it being conceived, that the, con-

venience of a prompt reference to them, ifnccc{fary, v\'ould facilitate to the reader his

c.xam~nation of the fyfiem prop9fed to be


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PREF ACE. .IX

[pace: enclofcd by a number of chains of equal

links, conneCled together, but free to m ove

according to gravitation, fufpended from acircular hoop, bc:ars the fam e relation to a

dome and its ~butment wall, as the catc:naria

does to an arch and its abutm ent piers;

though he has not had Ieifure to afcertain thctruth of this hypothefis. I-Ie is not aware

that the .curve, which w()uld be form ed in

t his m~nner, has already been confidere.d by

m /athc:m aticians: fhould, ho\\'ever, "ny thinghave been publifhcd concerning it, he will ,be:

obliged to any gentlem anw h o th a Urefer him

to fuch a work. '

14, John Street, A delphi, London,.Jan. I, J8o


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CONTENTS.

P~e

IntroduCtory Definitions and Remarks. . . . . . . . .I

SECT. I.-Of the general Laws of Motion.. . .. . 15SECT. n.-Of ,Arches of Equilibration. .. 26

SECT. IlL-Of the Catenaria.. . . . . . . . . . . .37SECT. ly.-Of Abutment Piers of Equilibration j of

Bridges of many Arches; of the Flying Buttreff'cs

of Cathedrals. . . . . . . . . . . . . . . . . . . 4'~


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INTRODUCTORY,

DEFINlrrIONS AND RE~.iARKS.

-Pig. I. AN arch of equipollence is a curve,

whofe feveral parts are prevented fronl follow -ing their natural dircCtions towards the centreof the earth by mutual oppotition. TherefpeC tive aC tions of the parts are fo rcgu~ated,that through their" reciprocity they muG: re-

main in their aC1:ualfi:ate; or, if they begin togravitate at the fame time, each part muG:.move through the fame perpendicular fpac~ inthe fan1C~ime) and the chord A B will alwayskeep its parallelifm. It is, therefore., of no im-portance to the fiability or form of the arch ofwhat confifiency the foundations A, Bare; fothat they are equally com preffible, and defcend


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( ~ )

If th~ lines of direaio~1 of all the: parts c~n-tercd in onc point, that point muft I"cccive anequal prc{furc from each part.

An arch may he equipollcnt from either oft\VO caufes: the onc from the relation of the::w eights of the 1110veable parts, independent ofany affiftancc frorn co~tinuity, as in an arch of

equilibration; the other from conti~uity, whenthe parts do not equilibrate.

Equipollence in an arch does not' imply thatevery part i5 equally ahle to fu:lt:ain a given.weight, but that it is able to bear an equaladditional relative weight on 7ach part, orthat the continuity may be abl~ to balanc~ ordcftroy any inequality in libration : as in bothcafes the aCtions of the parts are t~e fame, the

word applies both to the arch of equi1ibrq,-tion, and to the arch cnabJcd to ftand byvirtue of themDde in which the parts are.conneCted.


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( 3 )fIzes, and the fame fpecific gravity, and faft..cn~d as in a catenaria; or of any fizes, and

any fpeciiic gravity, but {ecurely fai1:ened, as

in arches which are built rather from expcri-Lncc than principle.

lIcre it is to be obfcrved, that the tenacityof the parts, or the eohdion, is ahva}'s a re{ifi:-ing power; and fronl this circuil1irance an arch

111ay be doubly firong: it ~nay be equipollcntfrom both equilibration and faftening.

In our endeavours to fOrID a pcrfett arch,the confideration, that the more nearly \Veattain equilibration, the nearer \VC approachto perfeCtion, is of the utrnofi: ilnportanc~.

An arch of equilibration being able to fiandof it(clf ,,,ill be Inorc: able to fuHain an addi-ti0nal weight, and the \vho1c rcfif1:ance of the

cohefion of its parts will oppofe that additional,veight; whereas in an arch, which does notequilibrate, part of the coheiion \I\~iIlbe em-ployed in rcfiftio.g the inequality in the a~ions


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( { )plank cd grating or f~eepers) al10wing an extra

\

height for tbe l)fobable -linking. \Vhcd thefoundations are not uniforn1, an equal bed

nlufi: be obtalned by excavation or piling.As no \vcight bas any horizontal force, andthe refii1ance \vhich a force, aCting in a hori-zon tal d ircc9:ion, 111CCtSvith from a body lyingon a horizontal plane, is cauied by the cohe-

:lion bet\v~en that body and the plane; noweight, therefore, however fina11, can be fuf-pe.nded between two powers acting in a:ltraight line, the parts being perfectly free tornGve.

Fig. I. If. A werea )u b ~ i~ ~ _ ~ ~particle , andF A a lubricous horiz~ntal pla~e, no lubri-cous wcight could refift the force of A in thedireetion A F; two infinite powers mUll be

therefore obtained by the depth in which thefoundations aref u n kin the earth.

A ll bodies lying on level bed$, 'w hich receivea lateral im pul(c, fhould have as m uch coh~fion


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( .5 )It has hitherto been afTcrted, that if this line

w ere inverted, and com pofed of infinitely fm anrigid and pohi11ed ~)hercs, it would retain the

fame form: from this it has been concludc;d,,that an arch of equal voufioirs, forn1ing thiscurve, would be an arch of equilibration.

Thefal1acy of this opinion appears nlani..fc:ft, as the conncxion which equalizes theaction in onc c1fe is of no u[e in the other;therefore it could not be even equipollent.

I t is evident the connexion is neceifary toequalize the aCtions, fince bodiesin indined

dircCtions, whether fupported or fufpended,lo{e part of their gravity,. according tQtheir direCtions; and as the directions of thelinks in a chain differ, their aCtions mutt beu~cqua], unlefs {ome other force except gra-

vity be applied to equa1iz~ them. This error{eems to have ari[el1 fi'orn confounding theequipol1ence of a catcnaria with equilibra..tlon


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( 6 )

" a very th in arch, or fornix; thqt is, infinitely" fm all rigid and polifhcd fpheres ~ifp~{ed in an

\

" inverted curve of a catenaria will [ann an" arch: no part of vvhich will be thruit out-" ,yards or invvards by other parts, but, theCG l


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( 7 )

" For the force, which in the chain d~a'vs in-" wards, in an arch equal to the chain drives"outwards. An other circumfiances, con-"cerning th~ firength of walls to \vhich~'arches are appli~d, may be geonletrically"determined from this theory, which are

" the: chief things in the confi:ruttion of edi-

"fict::s. Inftead of gravity, if any other power"exert its force, atting in like manner on a

" fl~xible line, the fame curve will be pro-"duced. For e~ample: if the wind be fupw" pofed e:quable,'~and iliould blow, according" to right lines, parallel to a given line, the" line thus inflated by the wind wou~d be the" fame as the catcnaria. For finee all things" obtain in this other force, as we haVe fup-

" pofed in gravity, it is evident the fame line" muff be produced."The truth of the fentence in Italics, in the

abov~ quotation, is evident, though drawn


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( 8 )

'\"\"hich is given, to prove that the ca:t~na.riawould keep its figure unchanged in the onefituatioil as the other, would apply to an archof equilibration inverted, as its parts wouldhave the fanlc inclination to the horizon.

.W hen a chain forms acurvc, and its parts arefree to gravitate, they endeavour to aifU 111Che

longeft poffi ble 1ine; the curve formed byparts, which ar~ in:equilibration, and are freeto gravitate, is tbe 1horteft poffib]c line: inthe one cafe mutual diftention retains themin their fituation, in the other mutual com-preffion. If the aClions of the parts of an in~verted catenaria be equa1 by gravitation, asthey mutt be to retain their fituation, thenevery. joint in a chain is equally liable to

be broken by the gravitation of the parts;but the contrary is evident from expedence:therefore, the inverted curve of a catenaria,com po{ed of cqual* rigidp o 1 i f b e dfphcrc:s in a


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( 9 )

equal bodies, in different degrees of inclina-tion, arc unequal; but the aB:ions of the linksof a chain, free to gravitate:, are equal by mu-

tual oppo{ition, and are in different degrees ofinclination: therefore the ab{olutc gravities 'ofthe links mufi: be either proportionaJIy in-cr~afed by the oppofition according to theinclination to render their actions equal, orthe conncxion, as a refifting power, lnuft be afubfiitute for this increafe. Now bodies may10fe, but cannot acquire force by oppofitian,therefore it is the connexion which renders

the aCtions of the ]inks equal; and the in-verted curve of a catenaria, compafed of eqpalrigid poliilied fpheres in a plane pcrpendicularto tite hori~on, withop'(. conncxion, cannotkecp its figurc.

. Fig. .2. Every particle of thc arch Bc B,A C A; is in an oblique diJe8:ion,{Jh, a h,&c. exccpt tho{e in the dire8:ionc C, which


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( 10 )

in any part. 1'0 fubfiitute an aqdition to thethickncfs of the arch for this ligature is a de.fidera tu lTI in arch irc


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( I I )The larger the .parts are in an arch, or, in

other words, the few~r the joints, the lefsfubjet1: is it to variation, as the cohefion is

gr~ater: but the larger the parts, the marcliable is the material to fraCture.jt-igures 3. and 4. AB CD are two blocks

of itanc, taken from a quarry in which theyha vc been preffed in th~ direCtion from B to D

ever {inee their formation, {o that their partsmay be confidered as linear, and {lmilar to theparallel lines A Band C D. Stone, in thedireCtion from B to D, will bear almoft anyweight; but in th~ direB:ion from B toAwould be crufh~d or fraCtured by a 'compara-tively fmall one. It is vid~nt that the VOllf-fair E fhall be fironger than the vouffoir F, asthe fide a b partakes more of the frangible

direCtion A B of the fion~ thancd: thenearer; therefore, the fides of a voutToir af-fin1ilate to parallel lines, or the more vouf-(airs th~re are in an ~rch the firon~er will be


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( 12 )If no affifiance be required frqm cement,

fril9:ion, and f~li1:cning, but an equable at1:ionand rcfiftaflce in the parts be obtainedby

weight, th e fiz eof the vou{Io irs, t he in frang i-bi1ity~ the incom preffibility, if they be hom o-gencous, is of no importance. Celnent, fric-tion, and f~dtcning, and iulidi ty of the mate-ria1, do not at all dcfi:roy the equilibration of anarch; they, on the contrary, tcnd to refiit theuncertainty of preffure ron1 other caufes, andare not to be negleCted. A ,brick arch is com-pored of parallelograms, with the illtcdhccs

fi]]ed up with cement and nates, or otherhard {ubfiances, or Cel'llCnt alone: the diffi-culty in praCtice of l11aking thefe implc-ments homogeneous with the bricks occa-fions a {ett1ing .

Fig. 5. rrhe intrados a b is {uppo(ed toequilibrate: the mattcr in the interfiices hasthe fame effeCt as a horizontal force preffingi th di i f t k th b i k i thG


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( 13 )

them together, except where the joints acci-dentally meet at0; by which method they

diminiOl the fize of thG interfi:ice, and th~arch of courfe approaches nearer to hon10-gcneity, and the ikill of the worknlan be-com es lcfs im portant.

.F~r;.6. This arch is con1po(ed of fiones,

with parallel beds and perpendicular joints:it is evident no part of the arch can falluntil (ame one of the fi:onesb b flidc out-wards, or cru01.

1t is probable arche~ of this kind were an-terior ~o the arches whofe conftituent partshave oblique lines of cantaB:.

In this arch it will be {een, that the hori-zontal and perpendicular forces are refolvablc

into oblique forces cc; and if the anglescd c were taken off, the intrados would b~an arch.


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.A

rr It EAT I S E,

cfc. o/c,

- -

SEC1'. 1.

uF TIlE anNERAl. LAW S OF M OTION)APPLICABLE TO THE

S U.l!JE CT S 'T RE AT ED OF IN 'l 'HE I/OLLOW ING W ORK.

1~ROP03ITION I.Fig. 7. IF two forces urge a body B at th~

fame time in the direClions of, and propor-

tionally to, the fides -B A and B C of a paral-lelogram, it win move through the diagonalBD.. W hile B moves from B to C fuppofe theline B C to move parallel to itfelf into the

line AD; the body lTIU1tth~refore be alwaysin the. moving line BC. W hen.]3 C !hallarri ve at a b, let the" body be arrived atd;th fi b th th li B C d'th b d


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( 16 )

I. The forces in the direCtions BA, B C,\

and B D, arc refpeC'tivc1y proporti\pnal to theI

lines B _1\,B C, and 13 1).

~. 1-'hc two forces B A and B C nlay beCOll1pounded into the finglc force B D bydrawing the diagonal of the para1JelogramB I).

3. 'rhe fing1c force B D ll1ay be refolvedinto the two f(.>rccsB A and B C by de-fcribing a parallelogranJ, having B D as adiagonal, and B A andB C the direB:ions ofits fides.

4. If two forces, B A andR C, aB: inthe direCtions B A, BC, re{pcCl:ively, draw13e to the middle of the right line A C, andtwice B e is the force compounded out ofBAand 13 C, and Be its direCtion.

Fig. 8. As the angle A B C 1ha1Jdecrea{e,:n D will incrca{e; and when the angle hasvanifhed, the forces B A and B C will aCtt g th i th di Cti B D d th


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( 17 ):no compound forc~. If the forces be equalthey \viH balance or dcf1:roy each' otrlcr ;. 011the contrary, if they be l1neqn:ll, tb~ let; ofthe tVYO\ ' .1111 he dci1:ro)\ d, ~Ll1dbe rnovcd in

the dirc(:liuli of the grcntcr with a force equalto 1Jl~.~di


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( 18 )lite force A and HE, orP F to it$ oppouteforce B: therefore the three forces are refpec-tively as H F, F D, and D H~

I. Hence the forces A B Dare refpec-tively as the three fides of a triangle drawnp~rpendicuIar to their lines of diret1ion, asfilch a triangle will be umilar. *

~. The diagonal of a p"rallelogran1 muft

a1ways be the line of direetion of the rcfift-ing power; and the two fides interfcctingit mufi: be in the direaion of the aCtingpowers.

* Fig. A., Demonflration: If the line N D be at right.m gIes t'1 the fide K !vi (as w e fuppofe), w e have the tw otri-angles, M N D and ~1 K P, alike; as they have each a riglHangle, and the angle KM P comm on: the angle K will beequal to the angle N D M . By a fil11ibr reafoning it will~ppear, that the angle L is equal to the angle0 D M ; b u t

a s th e a n g te0 D l\~ is eq ual to the altern ate an gle E H D,it fo llowsthat the triang le KML is fim ilar to th e triang leEHD.

The power A m ay therefore be expreffed by KM , tne


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( 19 )3. If any number of forces aB: againfl

any point in one plane, and equilibrate, theymay be all r~duced to the aCtion of three, or

of two, equal and oppoGte ones.

PROPOSITION Ill.

Fig. I I. If onc body atts again{t another

body by any kind of force whatever, it exertsthat force in the direCtion of a line per-pendicular to the furface whereon it aCts.

Let the bouy B be aBed uFon hy the forceA B. The plane C D, and obG:acJe E, hinderthe body B from moving: divide the forceA 13 into the two forces C Band F B, theone parallel to the plane C D, the other per-pendicular; . then the plane' will receive the

perpendicular force F B, and. the obftacle Ethe parallel force C B. Ta~e a"way the ob-11:acle E,. and the force C B will ,move theb d l h l C D i h h


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( 20 )fore, having no effeCt, the remallllng forceF B "rill be the only one ,,,herebyth d b o d y13aEts again(t the p11ncCl), and that in th~

d ireCtion F 13, perpendicu lar t o i t .

PROPOSITION IV.

}7ig . 12 . 'The force whcre\vith a roIling

body dc[ccnds upon an inclined plane is, tothe force of its ab{olutc PTavitv, by 1Nhich...") .Jit ~ould dc(ccnd perpendicularly in a ti'ce{pace, as the hcight of the plane is to itslength.

Draw F E perpendicular to A B, (by Prop.II.Fig. 10.) '\E (being horizontal, and con-fequently at right angles to the direction lineof gravity,) will exprefs the abfolutc gravity of

C; . F E will Gxprcfs the force with "vhich itde[cends in thedireClion AB, being at rightangles thereto; andf\ F, th~ force with whichit pre[fcs the plane A B being at right angles


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(~ [

). .

SllppO[C the pL1nc A B to be paral1el to the:huri z()n, the cy!j ucler H \vilI be atrca uponany part of the plane where jt is laid; becaufcthe ii l pport or hi n dcrJl1cc frl) [11 t he plane isequal 10 the \\'holcahfulut~ gravity, and itsfurcc is dcitroyed, and there is no height torcp rcfcllt any.

.

Jf the plane i\ B be elcvated; fo as to be

perpendicular to the horizon, the cylinderC 'will dcfccnd with its whole force ofgra-vity, bccaufe the plane contributes nothingto its fllpport or hindcrance.

I . If t\VO bodies' defcend . by thc force

of gravity from the points D. and .B; oneby the inclined plane B A, and the otherby the line D A; and the fOrI~1er be equalto B A, and tbe lattcr to D A; they \-villrequire a fimilar force to fu{tain thcrnintheir direDion, or ihall de[cend with equalforce: or if there be t\VO Flanes(fig. B.)

of the f:une height, andtV~IObodies be laid


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C 2% )"

,

,% . Th~ gravity of a body is e~crea{tdthe nearer its direCtion is to a horizontalline: if the direCtion be a horizontal line,it has no gravity. *

3. I'he difficul~y of moving bodies on ahorizontal lineari(es from the refifiance ofcohefion; and that of continuing the motion,from the refiftance of friCtion.

4" In bodies on- inclined planes, what-evc:r be the length of the plane, jf the ver-tical lines or heights be equal, the forces ofthe bodies (their weights being in proportionto the length of the inclined planes,) thall b~equal.

PROPOSITION v.

Fig. 13. Let th~rc be any number of lines,A B, B C, C D, D E, &c. aB in the fame ver-tical plane, conneeted together, and m oveableabout the joints A~'B C D &c (the two ex


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( ~3 )trcrnc points, }, and G, being fixed). It isrequired to find the proportionat the w cightstu be laid upon the joints B, c.\ D, &c. in the

din.:C1ionof the radii from thofe points, fo thatthe \vhole may remain in equilibrio.From the {everal joints, hqving drawn the

lines B b, C c, D d, &c. repre{enting thedirection in which the weight.s arc:to a C 1 :

at . the joints B, C, D, &c. ereCt the equalperpendicular lines Bn, C p, D d, &c.. anddraw the horizontal linesn b, p c, &c.. in te r-fceling Bb, C c, & c. in h, c, &c.; then Bh,C c, D d, &c. win be proportional to theweights required,


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( 24

TA BL.E

.")

1.

Tab!e sl/'.'ll'ing {lIe 1"C.',jicctiIJC(lbso!it!1? w ~ i g h f $of bodie.s slidir.g(lO n'it inc /ii/{'d jJlw u'\' :ll tlit:d U T crc lIt d el~ l'c es(! ! incl inat ion, be-LIl;C CiI a //(I/'i-;O liilll ((lid jJi'ljJ('ndiculuT line, so that their Jorc.;,~

11'(/~' be Cl}Ilil!.

~ .'-~ ~.-'~ ~'~ . ~ .~ ~.~ ~--~ ~.,...",,~-~twur~.~~ : '

IJtg""~,. '\:'/'/:II!


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Degrees. Force. J:jq;recs. Force. pegrers. Force.

Horizontlll. 0

1 17 31 515 61 875~35 32 530 62 8833 52 33 545 63 801

70 34 559 64 89987 35 57~3 65 g06

104 36 588 66 91312.9 37 002 07 920139 38 616 68 92 7156 39 029 6g 934174 40 6-13 70 940191 41 05(j 71 945208 42 (j(ig 72 951

~.95 43 68~ 73 956f4.9 44,' 6g5 74 96 1259 45 707 75 g6(j276 46 71g, 76 970,'~92 47 731 77 974309 4;8 743 78 97 ~,325 49 ,, 755 79 98234.9 50 766" 80 9aS358, 5~ .777 81 g88375 52. 788 82 990391 53 799, 83,,' 'g9~407' )54; 8 0!)' : 84 994

( )

To.ble sh tw in g th e fo rce o f e t b o d yJ w h o s e w~ig ltt is 1,000, !>1 id ingd o w n inclincd planes in the different degreeJ cif inclinatiorl# b~-lwren a Iw rizpntal andp,erpcl1dicrtlar line.

~s

TABLE 11.


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( z6 )

SEC T. 11.

OF .A,ttCHES of EQ.yILIBRA. TI01l{.

IN the (oIIaw ing prapofitions the {cC l:ionsofthe arches and abutments are ured for theirfoliditie~; and the parts arc canfidered as ho-mogeneous lubricous reB:angular planes ofequal thicknefs which pre{ent their edge inthefecHon.

PROPOSITION VI.

Fig. 14-.,Given the length of onc planeV b, ,and its direCtion; and the dire CtiIl:ofother,,'planesed,: thee:x;tr.em itiesdd, & c. of

all thcplaI1cs'aretapgcl1ts.to -the {ern icircle.AVB,and;cpnl'pofe tg~ {clI1icircle.

, The'p]anesar,~ J 'e~~pgular,aJ190f ~q~al~hickl1~lS;c ()~ fe q 'u ~ n t l> !j:tlj~ i~ m a 1 fe s w iH b eas their


35/89

( :1.7 )1)r3w d (.' parallel, and equal to V band Ce

horizontaJ, cutting de in e, e: d~, de, fhall bethe 1cngths rrquircd.

Dt.'J!l(jJ!llratioll: The aCtions o~ the weights( 1,e tI , are equal; as cd, c d, arc: equal: and

A V B is an arch of eqHilibration, havinge e eas an extrados.

I. rf the planes were not at right angI~s to

the tangents at the pointsdd, the lines de, de,&c. could not exprc(s the maifes of the planes;as d e, de, \-vould differ in their w idths.

2. 'The plane de, being at right anglC$to the tangent, muff: be the joint effeCtof the two equal aCtions of the adjoiningplanes, and the .three forces JhalI equilibrate.

, . 3.Jf th~ arch had any folid abutmentof rock, 9r cal'th~ aef; the refifiance

"fromfA d e w ou1d be alone peceffary,and,efwouldbe the boundary on that' fide;or tl1eroc~ ' mightbecut hi theforrned "


36/89

( .28 )weight free to gravitate can be {ufj)cndedby t\VO finite forces acting in a horizontal1ine; froIn C towards 1\, and fron1 D to-

V\Tards B. An infinite horizontal refif1:anccmui1: intcr[cCt: the linee e iomewhcre; th(\t

~he arch may equi1ibrate: thc u>lid earth orrock is fuppofcd to be that rcfifiancc; there-fore in no cafe can the height of V, abovethe earth or rock, be equal to thc abfciifaof the arch, ",-hen'e e continued fhall . bethe form of the extrados.

. S. It will be ob[crved, that the planes a,re

confidered r~a:aI1gular paral1elogram s; conN OfequentIy, when placed ill the circumferenceofa circle, th~re,vould beiI1tcrft:ices betw ~eI1them,increa{ing in width from4 tow ards e,weretl1earchonIycompofcd of planes ilfuipg

out, a11.dhaying extre,mitics in the il1trados.Bt1t this isnotthe ca[c,C~cryinterfiicy,'s itQCCUrS ', eing'fille4~ i th '[l,lCce!IiQnsofreeta l1 -


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( :49 )6. It is held by {ome, that an arch which

fprings fron1 a horizontal bed, CA, exertsno thruft on the pier which fufiains it;

and by others) that: it does: the former ist hcorcticall y, the latter cxperim en tall y , trite.By Prop. iii.A~'. 11. if A C ,vcre a lubricousplane,. the arch A V vvould ilide off the pierat the level A C, being alfo lubricous, how...ever thick or :Ocnder it l11ight be. The archit(l~]f can, therefore, exert no thruH: on thepicr; but as the two furfaces always cohere,either from the weight or cement, or, from

both jointly, the arch A V, in endeavQuringto i1ide off thc pier in the horizontal dircc~tion, is refilled by that cohefion; and affeCtsthe pier.

.

Figures IS. IQ. ~7.18. 19. ~Q.~"7. The('unc principles~


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( 3 )8. The lower the arch at V E may

be in proportion to its {pan A B, \and thegreater the thickne{s at Vb may be, thenearer w jIl the ,extrados, im n1cdiately overA B, approach a horizontal line.

PROPOSITION VI.T.

Fig. Z I. Givcn the horizontal line A B,being the extrados of an arch, whofe partsarc lubricous, and ar~ fufpended betweentwo lubrico\1s abutments; the force of one

of the parts, Vb, being alfo given., Required the intrados, fo that the parts

,

fhaJl bcin equilibrio?'It is evident that C D wouid be the intra-

dosrcquired; and CJ1 ,e a, D B,&c. would be

the. dire~ion'oftheparts,\\'ere it ,poffihl~toobtainarefHt.ingabutment ,in a horizOQt~l'direcU ()l1:orrather. ,to fll,fp~I1d A BD C,


39/89

(31

)

A e and B e until they meet in E. From E,as a centre, defcribe the archd d, and takeany ~um ber of partsd, d; through whichdraw the lines a e, a c, equal and parallel toVb; and draw through d, d, c d f, c d f, radi-ating from E.

I. If c}: c f, &c. were the direCtions of theplanes, the moment liberty of motion ihould

be given them, they would turn ond as ac~ntre, and faH in the direCliona e.

~. If th~ extremities c c w ere he1dfirm ,they would either bend in the direClion.c d c, or all the parts below the lined d d,vollld break. off, beginning from.e ~t -the'abutment, * as thofe parts could receive norefiftance from the abutments, and the archm u 1 . 1 :fall. .. .. .. .

3. ]fthe extrados had beep.hh, anythingbelow b d, the 'archrn.uftfallin!t:antaneQQ f1y,as if couldp.otreceiy~ ~ny fupport frprn th~Ilbutmctltsbetween m d1n "

_ , __ , , ~ ' _- '- - '- - '- " "- '"..,.._~


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( .~~ )5. 1"'hc in~rados of an arch, 1vhofe ex-

trados fhall be horiznntaI,ll1uft be hori-zontal alfo, and vic~ vcr[a; Jnd the partsmuft :-.1iobe in a ycttical dircction, \-vhichis in1pofilblc: alid no ca"ch of equilibrationcan have either a horizontal extrados orintrados.~~

I>ROPOSITION VIII.

Fig. 22. Given thG extradQs Cb D, andintrados A V B, being a fen1icircle to whichthe c;r.trados is paralle1 .

.

Required the direClion of the parts, fo th~t.they l11ay be. in equilibrio ?

*W h~re the fofite~ 9f: ~rches, over the.peningsof

w indow s,& c. ".rc,"requiredto behori;?;ontal, the ~rches are

ofthis.dcrcri~ tioh,411though there,:lrchcS ;1P l~G ;1r to bear..greatw ~ight'iil1re~ litythey' only fB Haifltheirow n~ . T hey

"are. (i[1 bl~ick..\Vor~) only four illche~ thi


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( 33 ),

'

The force at V muft be Vb. Draw frdmany points, d, d, the lines c d, c d, paraUciand equal to Vb, and c e horizontal, cut...ting the extrados ine e; draw e d, e d, whichare the direCtion of the parts having equalaaions~

Referring to the laft propoution :of the ho-rizontal arch, it may be obferved, that a. frac-

ture would a1fo take place at this intra dos". ' ,the part b d A of every pJane being fubjeettobe broken. The plan'cde prefi'cs on the ad-Joining plane in the diretHond Il, an~ isre~fied in an oppofite direa:ion from the

abutrnent; and the, part,d h A can. only ,bepreffed by a force equal to that with whichit prc:ffes, theref


42/89

( 34 )reprefent their refpeB:Jve forces: and Cb D'cannot be an extrados of equilibration toA VB.

-

W hen the extrados and intrados, of acom pofition of ]ubricous' forces in equilibrioin the fame plane, are parallcJ, they mufibe right ]inc$; as thcdiretlion of each ofthe forces muft: form the fame angle with

a horizontal line:. no arch of equilibration,therefore, can have its intrados and extradosp~rallel.

FrQm this fcd:ion it will be' fcen, that thecx tradoffes of equiJib rat~on, Qf circular, eIlip ...tical, and cycloidal arches, a!re themoft con-venient form s for thecJ(:t'ra'do(fes of bridges:and thu~ natllre~irl this inaanc~, has Jlot beenfo mubh afyariance, with' herfelf, 'as byhe(

']aWsto ..prefcrib~fafQrtrt,t.qth~e~trad6s9 f a, perfe:~;a'rch".'~{'~hi~h',.'..;is..';Vvho lly..'..,' .in~J>plitabl'e',


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( 3S ).~vith the abfciffa of the curve, as they are'diirant fi'om the. vert~x, con{equently the ex-trados and ir.d',rados of the hyperbolic arch

cannot approach each other; nor can thofeof the parabolic keep always at the famedifiance: hut all curves~ compafed of Lama-geneous materials, mufi be thicker at th~ir'extremities than at the vertices, to be in

equilibration. . ,Thefe re[ults areiil oppofition to thofe of

Pr. Hutton; but the n1ethods, by the re[o-Iution of forc


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( $6 )pre1'fes its aCtion in the direaion of the pJana ~the fame Jaws apply to the voufloirs ~f anarch. The objeCt ~f i~quiry; in our endc:a-

vours to obtain ~n arch of equilibration, isthe abfoJute weight of e'ach vouiToir in itso~lique direClion;. whether arifing from thematerials folely of, which the arch is com-pofed, or partly from the weight incumbent:and it is always to be kept in, mind, thatw hen gravity is the: foIe a8:ing power in ~com pofition of lubriGous forces, that an add i-tionalforce aC ting vc:rtically ,and not dcfiroy-

ing' tbeequi]ibration, cannot alter the direc-tions of the parts; .and,confequcnt1y, thatth~>cqui1ibration of an arch mutt invariabiydepend upon equable' aCtions in the direCtionsofitsvouifoirs. ~t is, p~rb~psi not unnecef..faryagaintorep(:at,thatthc direCtions oftbe,'voufih irs 1hould_,b~.atrightangles to t4eta.n -gents of therefpe~ivepart$ofthe' il1tradQs of


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( 37 )

other branches of architeC1:ure, which are butlittle' underitood, may alfo derive affiftancefrom the cxtenuve mathematical knowledge

which its author poffeffcs.

SECT. lIT.

OF TH~ C4TE~ARIA.

'IN in'veftigating,the properties of the cate-naria, it is neceff'ary to confider the means b;ywhich we acquire a knowlcdge of its forml;1I1dother ~ircum {tances attending.that m 9dcof forroatiqn. ,

.Fig.~$.-I.'fllC~ lin~AB 'is (orlDed~frQma " chaincoropofed pfliIlk$' connea~titogether h~ving afreemodoninany'd'ir~~~


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( 38 )~h~y endeavour to affume the Jongefi: pO(qfib le line.

4. The lines formed by a chain are, eithe~firaight or curved.

S. 'Vhen the length of the ,chain DC i$equal to the ihorteft ,difianCe between the,cxtremitie>, and is free to move, the lin,c

'js vertical,.6.'VhenJ the lcngth of the ~hain excee~$

the :fhortcft diftancc betw een the extrem itiespf the chain forming the curvc, and the link$~reJr~e to rl1ove, the line,A. 13, formed by the~hain, is ~curve" and the Iongcfl: which thelinks C'1p. th~n a{fum~~ and is c'}ll~d tpe cate-

. . . . .pafl~." '

,

, ,7 . /W hona, c~ ~ini$ fr~ e .t.o:n1ove a,t ~very

conhexiorG andform s ,a curve, a fqrc~aG ts at~ach,~xtrcrni~yA ]l' ofthe,,'cllain' form ing,~he catedaria, apdth(:fe;:,only; and \\'hep itform s~( ftraight)ine~ ~Jorcea ~so~ly aton~


47/89

( 39 )can fi~fpend a perfeCtly flexible w eig4t in, afi:rAight line.

10. The caten~ria is multilateral, * the line

frOm a to d through the link being a fide:\v ben the links are ~qua~, and the liJJe isfiraight, the lines forming the catcnaria areeq ua1. W hen the links ,are equal, and thecatcnaria is curved, the line nearefi the centre

of the earth is th,e shorteft and horizontal,that link lapping moil: over the adjoininglinks; and the lines increafe ~n length as theyare' ' difiant from that line, and they. approach;nearer to vertical lines. If the forces were ataninnnite diftance from tl1oiliQ rteftJine~ tl.}etw o line~Q eareft ,the fo.rcesw ould be yertic~J,and the longeft. ' '

, I I.. Every fide is a tangenftQ th e C u rY e ~

the 'd ire~ ionQfeacb link i;~~tright'anglc~~qthe' tan,gent of the,curye, ortQ die fiqc m adeby. the link in the, curyc;.,', ,'.

"


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( ~o )red:ions form w ith a horizontal line~ is equalto J 80 degrees, the abfciffa of that catenaria.m JIft be infinitdy long.

13. "AI~ catcna:tias are {jmi]ar to oneccanother, f inc~ they are gene~ated by a like." con11:rucH on of like figures um ilarly pooted." vVhence two righ t lines, alike incIi~cd to the"horizon, drawn through the vertices of the

ccchains,will cut off fimilar figures and por-~, tions of the chains, which are proportio~al."to the right lines fo cutting t~em oft:

. 14., "If the chain A C B is fufpended at" the points A and E,which a~e at unequal~'heights,~4e part of the curve AC, E ,con~"tit1uestberarn~as if it haci been fufpel1ded"at the poiI1ts,A andB,which(ire equally"high;b~caufe .' it

.

is an one,wbether the

"PQIl} tE


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( 41 )(eB:at b, by the lilJts a h, a h, at right anglesto the tangtnts at thofe points. M akec:bequal to the linee b,and confiruCt th~ ihnilar

triangles c b d.Through d d draw th~ line requircQ.. *Catenarire may be of as many forms as the

gra vity and fhares of the links in a chain maybe .varied. . The catenaria form ed by um ilarand equal links will be::found from the circlt,,as in this propqfition.

. . .

. II()w~:verfio .p leth js U1Qdeof con firu8 ing the cate..

l1~riamay he, the reader is referred to Dr. DavidGregory 'spaper,w h iLh wilJJhow thetQought and labo~rnccc{fary','

."I, '.,'

','

'.'"', ' , '

",

for fuch an inquiry~,

A lthoQ gh th c apthorh;td earJyr ~ a f o nto doubt the un iverfal ()pinion, that the prqperties ot",thisline w en~ exclufivel y\vithinthercachofthe" higher ll1a.-thernatit;s , th i$ metbodisnevehhcleCs the rc r~ ltQfyar~ou~~

-.

':""."

.", .

'-:

..

',' .'-. '; '.! ", .,

..'.

~l1d repc~ted f\ttempt~~, " ..',


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( 42 J

SEC T. IV.

~r ABUTM ENT raRS o r' E QU IL IB RATI ON ; 01 !RIDGES 0.:M ANY ARCHES; AND OF THE FLYING BUTTR~S5ES

OFCATIlB URALS.

PROPOSITION X.

Fig. 25.* B,ELO'V the lineE A B is [olidimmovable rock: UpOl1which it is requiredto ereCt a fCJ11icirculararch,C V, & c. fiand~ing on abutment piers,. w hofoheight is equal

to C A. from the line EA B, and the thick-nefs of the arch at the vertex is equal to Va;fa that the arch and abutment piers fballequilibrate in.~very p~rt..

/

'Defcri9e.~he~~xtradosof~quilibratione e 4,tothe qrch C V,(l>Y Prop. vi.jig. 14). By

. A p ier is underfiood tobya b o ~ y which has Qnly .;1


51/89

( 43 )Prop. ix.fig. .24. draw the c~tenarian 11;a,cutting the line E A B in H. Draw the lineH F, which interfe(t, in I; from J, as a.centre, with the. radius I H, interfeCt theextrados of the arch of equilibration in K.DJaw K M and K H, and complete the p~",allelogp~.rn K H L M .

V M L H K is half the arch and abut-

ment pier required: A N is the depth theabutment pier muft be funk in the rock.M L N ~s a filling-.in to form th~ openingA ,V B. The direCtion of the parts, betweenM and V, converge to F; the remainder areparallel to K M . . It is neceffary to obferve,that the caten~ria itfelf mig~tbe the extradosof the arch; but then the diretlion of theparts In\l!t b~ atrigqt angles to t4e.tanpent~

of the.intrados of that catenaria..- ,;. .1_.

-';.

. ':

.,- - 4'

. " : ': " .-. '~,

.'. : ~

I. Although. the part,MCLN is of nQ~bfolriteufe il1. fl1ppqrtiI1g the arch~ or i~

f i i ih lj?l ( i f


52/89

( 44 )buildings, fhew n infigures 26. '7. 29. and3. that the joints of th~ mafonry ~n theabutment piers maybe horizontal, the \cohe-

!ion being a fubfiitute for the oblique direc-tion. *

By Prop. iv jig. J 2. 'I( 1\1, e In, a V, &c.

exprcis the f()rce with \vhich they ~n; prertdby a body iJiding down til::nJ,\NhGll :;111(.x.,

pre1Tes tbe ad ion of the body in a horlZc;~':1>:t1.dire8ion. l\ow the refiftence which i'...1\J1e m, &c. can make to the body is infinite, ad..:

~i~ting that its fubordinate parts,are :nCOf11-

preffible; therefore there can be no limit tothe weight which K M,e 11!,V a, &c. canfuftain, that weight a~ing at right an~]es totheir direB:ions: therefore, if the lubricity bedefiroyed, and the dir~a.ions of the forces re~

main the fame, the homogeneity of the ma-terial is' o f no importance., Hence arife twodifferent modes of obtaining the extrados of


53/89

( 45 )m itting lubricity as a datum) although arifingfrom different applications of the r~folution of

. forces: the one is, as before defcribed, by

confidering K M ,ern, &c. expreffive of theabioJute gravity; the other, by confid~ring

~ M , e m, &c. expreffive of the preffure atright angles theeto. in dther cafe them odes of confiru8 :io nmutt produce an equa-ble aCtion; but it is of theut malt import...ance that wc fhould confider the latter inprattice as the true and jufi met hod, and inall cafes to be reforted to. By altering the

homogeneity of the parts according to thef1rft m ethod, that is, by m aking {ome of thevouiToirs of lefs fpecific, gravity, they wouldconfequently be protraCted beyond the ex-trados a k; and the arch will, neverthelefS,

equilibrate, but the lower protraCted partsm utt nece1rarily be t1l1refifted, and be brokenby the fuperior. On the contrary, ifk 1n,


54/89

( 46 )one can never apply in pra8ice, as 'no ma-terials are lubricous; the other muff applyinvariably.

3. As the catenaria may be extended toinfinity, fo may the height of the arch andabutm ent piers., The greater the height ofthe arch, the nearer will the ]ine K H ap..proximate to a perpendicular; and if theheight be infinite,: C N continued fhall beone fide of the abuttTIcnt "pier, and a lineparallel at an infinite difi:ancc [rpm C fhaflbe the other. In no other cafe can a per-

pendicular line form the boundary line of anabutment pier, containing the lea!t poffiblematerials.

4. If any part of the abutment pier K HM Lwere under water, that part would lofe

fo J11uchof itswcight, as might be' equalto its bulk of water, admitting that the,vater preffed on the underfide.


55/89

( )7

b~ing immer[ed, or the fame effea willbe:produced.

I t i~ al(o proved, that a ~ody fpecificallylighter' than water may be made to fink,

or lie at the bottom of a veffel, by preventingthe water acting on the underGde : its fpecificgra vity will therefore be increa[cd, and thisdin1inution or augmentation of fpecific gra.yity \vilI be in proportion to the depth.

5., If .K 1-1, one fide of an abutment pier,he imm erfed in water from P to H, it willbe preifed by the ,vaterP R H, and its fpecificgravity will be increa[ed; becau(e, if it \vere

to rife, be1idcs the refi11ence of its own gra-vity, it would have to remove the waterp H. I-I.-(S~e alfo the fides Er, N i,jig. 28.)

6. .If the piers of the arch' were built uponfij}ts, or on, ground through which waterexudes or ~ulls up, it is evident the fpecificgravity w0111d pe decreafed by the 'waterpreffing upon the under furface; but this


56/89

( 48 )c1rches, it is advifable that each pi~r maybe confidcred as an abqtment pier;, that is,. ,that it may be able to refifi: the thruft of l1alfan arch. If this be the cafe, if all the arch~s'are of the fame fize, one centre will ferve forthe \\ hole bridge.

M r. 'Label y, in the ere8:ion of Weftm infierbridge, from following this principle, made

nine centres ferve for the fifteen arches ofthat bridge. He turned the three middlearches firft; he th~n ordered fix centreifa tthe arches on the W efi:minfter fide, whichafterwards {erved for the arches on the Surf'

fide.B. F ;g ltre s~6. and ~7. are fecHons of Tri-

nity Church, Ely, and King's College Chapel;Cambridge..

The dotted lines fuew how the abutmentpiers ought to have b~cn confiruB:ed accord..ing to the foregoing theory: to contain theleaft quantity of materials' and fupport the


57/89

( 49 )lc\'eral lines at right angles to the fevcl'altangents of the curve; in th~ fame manneras the catenaria n rl was found to the {cmi-

circle C V.

PROPOSITION XI.

F~fJ'28. Given a river, of which A B is

half the width: it is propofed to build abriuge over it of thre~ arches; the middlearch to rIfe the height AV, above its spring..ing A E, and the upper part of it to be aportion of a circle, whofe radius. is CV:its thicknefs at the vertex to be V F. Belowthe level G H is folid rock.'

Required th,e form of the three arches,the thickn~[s of the pier~, the depth the piers

fhould be funk in the rock, and the extrados'of ~he bridge; fo that the w holemay be incql.lilibrio?,

'Vf


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( S )ing A B in X; from X

JrE, touching Vf at J.of the m iddle arch.

Find by Prop. x.fig. 2S.the ex trados andabutment pier F'I f Z j {3 to the arch V E.W ith the radius X E defcribe the indefinitearch N h, touching the linez i, whofc centreR . is in 'the line A B.

E N is the thickne{s of the pier.By Prop. 'x.fig. 2S.nd the extrados and

abutm ent piers'm 1', r tt, to the arch Nh,having V~"' as the thicknefs at the vertc~,

and the level G H as a lateral rcfiftance: theline 'Y r nluft ncceffarilybe' a tangent to thearch f'E contil1\1cd. Drawr I Ki, and thepier i~ ,complc .te .,

From BJ on the line A B,' fet off thef.liftancc B T equal toR N; and from TJ as a~entre,. defcrjpeth~ indefin~te archBIf.. 'Find.

'

~he centre -t of t~e ~irclewhich1hal1 touch

defcri be the circ1eV fE is the form


59/89

( 51 )

the arch Vf at V, and the arch Bk. De-fcribe from it the archVk, touching the archB k in k; and by Prop. x.jzg. 25. ~nd the

extradoi of equilibration and abutment 'pierF W 0 M L to the arch Vxy k B, whofetl;icknefs at the vertex is V F, and havinga lateral refifiancc at the level G H. F \V is.the extrados of the bridge.

.

Fr01TIthe centre of the arch Vk draw theline e k. through T, interfeEting the arch ink.

. .

M ake' N h equal to Bk; and froln h, throughH , draw h P, in terfec1inge k in P. Frqm P,

as a centre, de1cribeh k,'touching the archesN' It and B k i11hand k.. .

N It k B is the fide arch required.. Thebridge is now com ple.te.

I.' By making E N an abutment pier,.either of the arches nlay be turned, and thecentre .firuck; and one centre will ferve forthe two arches. . .


60/89

( 52 )that t~G arches areindcpendyut of each other,and that the deHruC1:ionof onc arch will notb~ followed by that of the whole bridge;: and

,t,hat the: nliddlc arch Jnay be turncdfirfi, bywhich the u[ual fcttlclllcnts in the fidc archesmay be avoided.

~. The pier IDN is too "vide by7f1Jl,whenthe bridge is com p1ete, and every part hagtaken its proper bearing; generally it will befound, that the archv x y k will be too flatfor the thicknef~ V J? ~tthe vertex: the parte A 7f tn is contrived to obviate that. difficulty;

and it does not deftroy the equilibration ofany part of the bridgc~ and is found th us,Let fall the p~rpendicular lines7f ' U and m g,and

,', find the point in thehoriz;ontal lineY Zdrawn thrQugh 7l' 111,which fha1l be a

centre to the arch e '1r,tot1ching V Band theline '1fU,. andde[cribetbe ftrche '!r. Set ()ffmZ equal tY 'ff and'd~~c~ibc the indefil1itc


61/89

( S3 )

3. From the foregqing it will be feen howin1portant it is, that


62/89


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t ss )and is j u!1: adequate as a pier for the archBC; but the architeCt found that it was pru~dent to \.viden it.~(See the fets off randnl.)

The follo\ving quotation is from Sir Chrif-topher \\Tren's {urvey of that cathedral..

" Aln10ft all the cathedrals of the Gothic" forn1 are 'weak and dcfeClive in the poife ofle the vaults of the ifles: as for the vaults of

" the nave, they are on both fides equally((

{uprortedJ and propped up fron1 fprcading,'t by the bowes or flying buttre{[es, which, , 'r i C e from the out\\'ard wal1s of the HIes"It

But for the vaults of the iflcs, they are;H indeed fupported on the outfide by theH buttreiTcs, but inwardly they have no otherIt fiay but the pillars them {el res; w hich, a~

" they are ufualIy proportioned, if they flood" alone withGut the weight above, could not

I

H refia the 'fpreading of the H Iesqne m inute.lI~rruc, indeed, the grtat load above of the


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( 56 )(c any Gothic cathedral that I hav~, [cen, atH home or abro3;d, wherein I have \not ob-

" ferved the pillars to yield and bend inwards"from the weight of the vaults of the ifie." But this d e f e C t is n10il: confpicuous, upon,( the angular pillars of the crofs; for there,tI not only the vault wants buttn1cnt, but

" alfo the angular arches that relt upon that" pillar; and, therefore, both confpire toH thrufi it inwards, towards the centre of" the crofs."

3. From the foregoing it will be fcen, that

experience contradiCts. the theory which pro-pofes height as a fubftitute for abutments;as it is clear, however great the incumbentweight may bc) no advantage is gained, ex-cept from the increa(ed cohefion.

If the author's th~ory be true, the prin-. ,

~iples upon which Dr.' I-Jutton prop~fes toconfiruCt the abutment piers fo" that they


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( S7 )could not ftand a mom ent; and further,the: thickne[s of the pier in the: latter exampleought to be lefs than that in the firfi:.

In corol1ary the {econd of the fe8:ion ofpiers, in the fame: work, there: is the follow-ing conclufion: H So that in this cafe it" l11akes no difference of whatever height" the pier is to the fpringing of the arch;,t for though the drift of the arch be in-le .c.r~afed "\\yith the length of the lever, or

" height of the pier, the \vcight of the pier" itfelf, which aCts againft it, ii alfo incrc:afed

" in the fame proportion." . And again, inthe fubfequenf fcho1ium: {, In the fore-" going propofition I have conudered cir-lC cu1ar arches only, as it will m ake no(( difterepce of any confequence t? fuppofe" the arches of any other curve of the famelC fpan or pitch." .

To the former quotation the auth or


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( 58 )

that uze is governed by the tangents ofthe curve. *

The author would not have, prefumed tp

have obferved, as he has done, on the workabove mentioned, or to have contefted quef-tions of fo much difficulty with fo eminent amathematician as Dr. Hutton, had the fub-jet1: been fq1ely theoretical; but the differencobetween them is upon the application of thelaw s of m echanics to praC lice.

Pig. 3. is a {eCtion of Wefim infter abbey.The catenaria (J b c is drawn to ihew how

nearly the abutment piers correfpond withthe principles laid down in'the precedingpropofition. W hen there are double flyingbows) as on the fouth fide of the abbc:y, thethickncfs of the fecond. piers may be foundby drawing other catenarice over the firftbows, which will be interfeCted by thc fe-cond bows in afimilar manner to the firft


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( 59 )It will be fcen al{o, that on the contrary, ifthe thicknef~ of the piers be given, the fitua-.tion of the flying bows may be found. The

pillars A and B are evidently infidncient forthe arches of the :fidei 11 e s ,and have beenfound fo in the building.

PLATE 13, is a {caion of the ChapterHoufe at I-,incoln:~ PLAT~S 14. and 15. arefeB:ions of IJincoln cathedral. The architetl:employed in the ereCtion of there buildingshas not follovved implicitly the principles byw hich th~ architeCts were governed who~re5ed the other edifices of which feCl:ions arehere gi vcn. The fituations and forms of thefly ing bows of the cathedral are not the moftadvantageous, and the abutment piers arcfomew hat deficient in fubfiance; the walls of

the nave have beenconfequentIy thrutt out..wards, and the vaultings fiill continue that


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( 60 )aCtion upon them. The: huttrefTes to theChapter Houfe, which are larger than necef-fary had the [pringing of the bows b~e~

lower, arc thruft confidcrably out of a per-pendicular, and one 'of the bows mull be im-mediately rebuilt. The author fuggefted toan inteIJigent carpenter, who attended him,the height in the abutment from which the:

bow fhould. fpring.* .. The am bitious hopes of a.ttaining ftill fur-ther, knowledge in the fcience of building,evidently induced the artift who confiruCtcdthefe edifices to m ake the daring experim entsin vaulting, which are there fh~wn.' Upona reference to dates, it will be found, that

. .

Salifbury~athedral was ereCted {Qon afterthat at Lincoln:' this circumfiance, and the:

analogy between them, indicate, that to the. . T he author has 6bfervedin his exar~ inatiori of buH d-

. .

ing& ,but particularly in c


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( Gr )experience gained in. the conftru'C1ion of t"h~latter, the former is indebted for much of theadm iration it has occaG oned.

PLATE J6. is a feBion ofEI}' cathedral.PLATES 17. and lB . are fetlions of Peter...borough cathedral: the one is of the newbuilding at the east end, and appears, fromthe ilyle of architeCture, to have been ereCtcd

by the fame architetl: who ereBed King'sCo1Jege Chapel, Can1bridge. It is worthy ofremark, that the roof of Peterborough cathe-dral is 1imilar in principle to that defcribedto be in the fauxbourg Saint Honore atParis;"* which, although \vithout tie-beami,is ftatcd to have no ]atcral thruft on thewalls on which it refts.' It appears, how-ever, that the arGhited: here entertained a

different opinion, and builders in general willcloult. the pofition.

PLA'rB 19. is a fecHon of York cathedra1.


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( 6% )

ihould be stone, as the parts of the flyingbuttrelfes, the ftone couffinets, and the fize ofthe abutments, evidently 1hew: the vaultingsover the il1es are :tone.

THE END.


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