D OD

274

DOG

this proposition, a — x x a — x* and a; 5 + a x — a 1 ; and re-solving this quadratic equation, we s hall ha ve x? + ax -f- \ a?=a’>+^a 2 =%a' 1 ; whence* + and*= V j a? —la.

Let the radius a be = 1, and g o, or the_side of a regulardecagon inscribed in the circle, is = 1 Hence it

appears that the sine of 18° (or half the side of a decagoninscribed in the circle) is = J V •£ — i = i V 1 -25 — j =

1.118033 98, &c ^ _x_ _ 55901699) & c .— .25= .30901699, &c.

If the side of a dodecagon be 1, its area will be equal to

three times the tangent of 75° = 3 X 2 + ^3 = 11.1961524nearly; and, the areas of plane figures being as the squaresof their sides, 11.1961524 multiplied by the square of theside of any dodecagon, will give its area. Hutton’s Mensu-ration. , p. 114.

To inscribe a dodecagon in a given circle. —Carry theradius six times round the circumference, which will divideit into six equal parts, or form a hexagon, (see Hexagon) ;then bisect each of those parts, which will divide the wholeinto 12 parts, for the dodecagon.

DODECAHEDRON , (from ctacttca, twelve, and Upa, seat),one of the regular bodies comprehended under twelve equalsides, each of which is a pentagon ; or, a dodecahedron maybe conceived to consist of twelve quinquangular pyramids,whose vertices, or tops, meet in the centre of a sphere con-ceived to circumscribe the solid; consequently they have theirbases and altitudes equal.

To find the solidity of the dodecahedron. —Find that of oneof the pyramids, and multiply it by the number of bases,viz , 12 ; the product is the solidity of the whole body. Orits solidity is found by multiplying the base into one-thirdof its distance from the centre, twelve times; and to find thisdistance, take the distance of two parallel faces; the half isthe height.

The diameter of the sphere being given, the side of thedodecahedron is found by this theorem; the square of the dia-meter of the sphere is equal to the rectangle under the aggre-gate of the sides of a dodecahedron and hexahedron inscribedin the same, and triple the side of the dodecahedron. Thus,if the diameter of the sphere be 1, the side of the dodecahe-dron inscribed will be ( — V3) -7- 2 ; consequently, that

is to this as 2 to ( V 3 - — n/^-) and the square of that, to thesquare of this, as 6 : 3 — f5. Therefore the diameter ofthe sphere is incommensurable to the side of an inscribeddodecahedron, both in itself and its power.

If the side, or linear edge, of a dodecahedron be s, its sur-face will be

15s , < v /l+-f\/5 = 20.6457788 s’: and its solidity

5 s d 4/ -f- 21 5

-- 7.66311896 s\

40

If the radius of the sphere that circumscribes a dodecahe-dron be r, then is

— V3"

its side or linear edge =--- r >

3

its superficies = 10 r s */2 —f f5, and

20 r s V 3 + n/5

its solidity 8 30

The sides of a dodecahedron inscribed in a sphe re

bed > n

to the greater part of the side of a cube insert w" r0 pof'same sphere, and cut according to extreme and mean P pgtion. If a line be so cut, and the lesser segment bef,.

the side of a dodecahedron, the greater segment w 1 0 {tb eside of a cube inscribed in the same sphere. The si , g ofcube is equal to the right line which subtends • the the°s» mea pentagon, of the dodecahedron inscribed m 1sphere. . mIJ s

DODECASTYLE, an edifice having twelve coimfront. „ n tb e

DOG-LEGGED STAIRS, such as are solid betw^ th o

the

upper flights ; or those which have no well-hole1»

an°

.(Tress;

five

-, . ..

rail and balusters of both the progressive and retrogf-f t0flight, fall in the same vertical plane. The steps are 0 fstrings newels, and carriages ; and the ends of the stepthe inferior kind terminate only on the side of thewithout any housing. ^ 0 {

No. 1 . The plan—a the seat of the newel; 9 the ^the upper newel. . $$

The dotted lines represent the faces of the risers, » Dcontinued lines the nosings of the steps. #rt t c

o- . Ihe elevation. —a 11 the lower newel, the P . n ,erbeing turned ; ch the upper newel ; D E, F the 1 aand upper string-boards, framed into the newel; th 0joist framed into the trimmer 1; hi, no, q r > * 'ferfaces of the risers; mn, p q, s t, the heads, or ^boards ; rn, p, s, Sfc., nosings of the steps ; M °> Fupper and lower ramps. , 0 b e

To describe the ramps. — Suppose the upper one .rawn , pioduce the horizontal part, hm, of the ^dra w m N perpendicular to p h, and produce the s" b tpart, B o, to n : make n o equal to nm : draw o F ^ jt 0 ;angles to bn : from p, as a centre, describe the » id ®and describe another concentric circle to meet the u nd m l) eof the rail, and the sloping part b o ; and the rampcompleted. . a fre

R s, the story-rod ; this is a necessary article insteps ; for if a common measuring rule be used for eKC e&pose, the workman will be very liable to err either 1 " ^.jelior defect, and thus render the stairs extremely faulty > r i,e r 'cannot be the case if the story-rod be applied to every ^and if the successive risers be regulated thereby- ” ® i# 3 'are put up without the use of the story-rod, theerror is liable to multiply. . t

. construction of dog-legged staircases, the r ^

IS to take the dimensions of the stair and the he£* f0 t b»story, and lay down a plan and section upon * $ th 08full size, representing all the newels and (f*

the situation of the carriages, pitching-pieces, 1° » s tri n »and cross bearers, will be ascertained, as also of t»boards. ^ th e

The quantity of room allowed for the stairS ’:„e n ' b 5situation of the apertures and passages, will detei e -q uiir Ler there are to be quarter-paces, half-paces, a ]l t .winders, or two-quarter winders. In order to g ll9 ist

» - -fl.'/vht tO c

th 0

WiUUClS, UA tWU*l|UUHCl WliiUCIO* .X

variety possible, we shall suppose the nig

, 1V 9

tw o-quarter winders.

dhe strings, raik- -

must he fi X e d “ fp an,i newels, being framed toget „w,Ii show the sifi,.,.; ten ‘Poniry supports ; the*str ,nPut up next j n , on °f the pitching-pieces, which '

and fixing the oh’ Wed S in g one end firmlyPitch up the , t0 the string-board ; this being

°f the flye r8 . and thus finish the

3re seldom o-Iued °°~^gged staircases, the steps 0,1

Ba PPose them, theret^T in cases of returned n v’ e ore > to be of separate pieces P'