C YL

221

C YL

iin es to ln strument may also be applied to the drawing of'vhj f; h . an y inaccessible points, by means of the middle bar,ptn sa |ji " a y s bisects the angle, and therefore will be indis-y useful in the practice of perspective.

f'>rrn) a ^ O' rom the Greek kvkXoq, a circle, and eiloc,! >lotu, a "Sure described by rolling a circle upon a plane,c Urve p ? ai S h ‘ edge ; then the moving point will trace the‘hie fi tt ae< * a cycloid, or trochoid. The middle portion ofQuires 6 IS ver y appropriate for the arch of a bridge, which°f thj 3 ‘° have its roadway raised on the top, as the extradosQy Urve has a gentle convexity.

‘he earl ° PEAN ARCHITECTURE, that practised byC ° I ' 8|; t'Upr‘ C0 0n ' Sta Gi reece > more remarkable for its massivelon than any other feature, whence also its name.Asgi an Architecture.

con.

b,

*t 0 , JS TYLAR, a term applied to those erections whicha circular range of columns, without a central

uil<i,

Er rn e j k h*ER, (from the Greek kv\iv<W, to roll) a solidMallei I rn °ving a straight line in the periphery of a circle,G an °ther straight line, which passes through the' e cir e ] e w h>ch makes any given angle with the plane ofTl le ’ Un td the line come again into its first position.ind e g n ? Ce escribed by the moving line from the circle to

aight ] n ' te ex tension, is called a cylindric surface; theCa Ueii tl 1(J ne ' v hich passes through the centre of the circle isc yl>nd er 8 **® °f the cylinder, and the circle the base of the

^>nd er jg X ' sbe at right angles to the plane of the base, theu *‘ >8 c 'll ^ a right cylinder ; but if at oblique angles,, Euclid „ al e< ^ an oblique cylinder.

0 6 ®aes i t t ° nfi »es his definition only to a right cylinder, and

t °f it s -a a ao ^ ( i formed by revolving a rectangle round■It io - .

passingopposite

.. _ _cutting

cylindric surface. From the definition here

tin

jj 0u gh th ent / rom this definition, that all sectionsW 88 Parafi ? X ‘ 8 ’. or P ara bd to the axis, have their£i 1)8 and tv,’ V ^ Z- ’ those which are formed by the cuttingt0 > the f n c yl'ndric surface. From the definition herePa °^ v 'ous t ow ' n " consequences may be drawn, as beingt 0| a hel to a ° rec l u ire any formal demonstration. If a plane,bn t the n . er plane, drawn along the axis of a cylinder,Par i ‘be h^iphery of the base, the plane so posited willbas e l to ace ,°f the cylinder, and will meet it in a linec ylf the * e , axis ’ hut if such plane cut the plane of the*f>iesurf Cylb,ler w hh>n its periphery, it will cut thelast) Plane ^ * n two P ara H e l Hues, and the common section* it ,y> the Co an< ‘ the cylinder will be a parallelogram, and,a$i s ‘be cyjj a ' rn ° n section of a plane, parallel to the base,'tilf' Eft t u s riea l surface, is a circle with its centre in theHirn^at t|.„ ri0 ' v consider the property of a section which‘berg ‘‘Hi n •^ ane °1' the base of the cylinder, butto. cither pass along the axis, nor be parallel

of

Figure 1. —Let A m l b be a section of the cylinder throughthe axis, cutting the section proposed, and let the cuttingplane meet the plane of the base AFEBNoinRs: throughthe centre d of the base, draw the diameter A b, at right anglesto R s. Let the plane be drawn through A b and the axis d hof the cylinder, meeting the cutting plane in the line mhgli,and the surface of the cylinder, in am and bl. ThroughH and any other point, G, in M L draw qk and IP parallel toR S; through k q and d h, draw the plane k q o f, and throughi p draw the plane ipne, parallel to the plane kqof, cuttingA b at c ; and because K Q and I p are parallel to R S, theplanes kqof and ipne, are also parallel to r s ; thereforef o and e n, are respectively parallel to k q and i p; conse-quently, the figures kqof and ipne, are parallelograms,and therefore K Q is equal to F O, and i p equal to e n : becausee n is parallel to r s, it is at right angles to a b, and hence itis plain that IP is bisected by m i,. Now the triangle amt,has the side A t cut into several parts by the intermediatepoints d, c, b, and mt cut into other parts at the intermediatepoints h, G, L, by lines parallel to the side A m, and thereforethe parts ad, d c, Cb, are respectively in the same ratiowith the parts MH, hg, Gl; these being premised, we havetherefore

ac : ad :: mg:mh

Cb:ad::gl :mh

Therefore A c

X

C B ;

A D 2

: : M G

X

G L

: mh 2

but A c

X

CB =

E C 2

= Gl*

&

AD 2

= HK 2

consequently

Gl 2 :

H K 2

:: mg

X

G L

: mh !

therefore the section is an ellipsis.

But this demonstration, which is in principle that com-monly given, only shows the general property in respect ofthe axis, or at most in respect of only two conjugate dia-meters, of which one must be parallel to the base ; and as wehave never seen a general proof of this valuable propertyfor any two conjugate diameters drawn from simple princi-ples, without being under the disagreeable necessity of drudg-ing through nearly a whole treatise of conic sections, beforewe are able to arrive at the conclusion, we therefore subjointhe following demonstration.

Figure 2.—Let there be any cylinder, right or oblique,standing upon the base GEAPQD, and let C be the centre ofthe base ; through c draw G Q at pleasure, and a c d at rightangles to G Q. Through G Q and the axis C M draw the planeg Q s h, cutting the section proposed in the straight line H s,also through A d and the said axis draw the plane adkk,cutting the section in the straight line li n. In one of the dia-meters, A D, of the base, take any point, e, between a and C,and draw e p parallel to G Q, and the plane e p r f, parallel tothe plane g q s H, cutting the section in f r.

From the construction of the cylinder, it is evident that thelines k n, f r, h s, are each divided into parts, which are as theparts of the lines ad,ep,gq. Therefore as e p, g Q, and A D,are bisected, so will f r at L, h s at m, and k n at M.

aa ‘ be premises we have. A b

"y proportional lines . bd

How

b y multiplying the homologouXh e ‘p 01 tbe property of the circle,

How ° r ?’ ^ substitution, we havea g a in, by proportional lines ...

but'tf' 0 " tbe ter ms of the last analogyy the fourth annlncv. we have_

At >d by

A CA C

K LLN

K MKM

fourth analogy, we have.

UstT Com P arin S ^e antecedents and consequents of thetw ° analogies we obtain.

A B

X BD

: AC 2 : :

KL

X

L

N :

: km*

A B

X BD

= EB , 2 &

A C 2

r=

G

c*

E B 2

: gc 2 : :

KL

X

L

sf ;

: KM*

E B

: gc : :

FL

: ii M

E B®

: g c 2 : :

F L*

: HM*

E B*

: g c 2 : :

KL

X

L

n :

: KM 2

FL* : HU* : : KL X LH 1 KK 1

3 L