E MB

373

E MB

'"•(i*')* =„(!+£)*

si “ce d = 1—1

t 1

‘ave Z _

2 f ~

1 — d

^ pt( l^ =p ,(l + ^ =pt

/ d F_ d 3 5d* \

2) ~pt [l 2 2 2 3 .4 2 4 .8 2 5 .16/

the periphery; but the true periphery = p t

11_ d q ,n q q £,73.

\ ? — 2?4J — &c - Now this series a s rees

^ the former in the first and second terms, and differs inonly — * the rule is therefore an approximation.

Th ^

let e s e ? u ^ e now delivered is still too long for practical uses ;therefore try the following:

5ti(J J''® 1 —Multiply the half sum of the two axes by 3.1416,jf e product will be the periphery, nearly.

—The same still as the preceding, viz., 50

50

40

: )90

45

3.1416

45

15 7080125 664

141.3720 the periphery.

Th‘

tow' 3 rule is exceedingly easy, and sufficiently near for allPot v Ca Purposes, where the eccentricity of the ellipsis isthg tr S r eat. It gives the periphery nearly as much belowH ere Ut ^ as the preceding rule is above ; and, consequently,both ® rea t accuracy is required, if the result be found bys ho ly ®?®thods, the half sum will be exceedingly near. ToThn * S ky an example :

■R resul t by the first rule 142.241971e result by the last rule 141.3720

2 )283.613971

tl ^ ll e half sum of both 141.806983 which is very

'Hit v, e '■ruth, as it agrees in five places of figures with theliLjj series.

lg ^'-i SOGrRAPH, an instrument usually constructed of

Sll( W/ 0 <? ^ escr ibing a semi-ellipsis at one movement of theJljV Ellipsis, Problem 1, Method 3.

^lin.. ^PSOIT") n Qnliil opftnp.rnt.pH hv rf

SOLD, a solid, generated by revolving a Semi-te * r eund either of its axes. This solid is understood by° e the same as spheroid. See Spheroid.

>pi 0v I 1Q Arch, a portion of the curve of an ellipsis,''Hla as an arch. This curve has some advantages over4t the , arcs > in bridge-building, as it leaves a greater space'Hnti launches for the passage of vessels, and, conse-C ° l ' s tru^ f 3aves a considerable quantity of materials in the

>is :

^ 0MPASSES - See Ellipsograph.lii 4 , p Tlc Conoid, the same as Ellipsoid.

Plip S j f Tlc ; Winding Stairs, a winding stair , having anits plan. See Stairs and Winding Stair.

Hstr u A1S ’KMENT, a large body, mound, or bank of earth,cte d or thrown up in different ways, according to cir-

5 c

cumstances. Embankments are of various kinds, accordingto the purposes for which they are designed, as RailwayEmbankments, which carry a line of railway over valleysand low ground at the elevation required for the level of therails: Canal Embankments, for confining the water of acanal or reservoir, or upon which a canal or aqueduct isformed ; and Embankments constructed with the view ofguarding, protecting, and defending lands on the bordersof the sea, rivers, and lakes, from being inundated andinjured by them, and for reclaiming lands from the sea.

We shall treat first of the latter description of embank-ments. These are of different kinds and forms, accordingto the nature of the situations and the. materials of whichthey are constituted. In embanking against the sea andlarge rivers, where the slopes next them are naturally gentleand easy, they are mostly of the earthy description,- beingwell put together, and covered on the surface with turf cutfrom the tough sward of the land in the neighbourhood; butin cases where the banks, borders, and shores, are steep andbold, they are usually of a more hard and solid nature : as otstone, brick, gravel, sand, shells, and other similar substances,laid closely in some sort of tenacious material, such as clayor mortar, and other matters of the same quality. Timber isalso frequently employed in their construction in a varietyof forms.

In works of this sort, very much depends upon the formin which they are constructed, and the nature and manage-ment of the materials made use of. In respect to the first, itmay be remarked, that banks of these kinds are commonlyconstructed with too narrow bases for the heights which aregiven them ; from which circumstance, the sides whichare opposed to the effects of the water become too steep andupright ; consequently, in cases of high tides or floods, theyare utterly incapable of resisting their weight, which hasequally a lateral and downright pressure. Besides this, thereis another disadvantage attending this method of formingthem, which is, that the floods, as well as the tides, in ebbingand flowing, have a more continued action on one part thanwould be the case, if the slopes were more gentle and gradual:consequently, they have a much greater tendency to breakdown and destroy the superficial parts of the banks. Withsome variations in the forms, most of the embankments inthis country are, however, made in this way. They maysucceed in some particular instances ; but in general it isfound, that breaches are frequently taking place in them, fromthe effects of the sea or floods, which are not capable of beingfilled up or repaired, without considerable difficulty andtrouble; and which, if suffered to continue even for a shortspace of time, endanger the whole embankment.

The common form of embankment is shown at Figure 1,and the improved form pointed out at Figure 2.

The angles or slopes of these sorts of works are made verydifferent in various cases ; but that shown in the above figure,seems in general well calculated for the purpose of resistingthe impression of heavy tides, or the waters of floods. Thegreater breadth they have in proportion to their height, themore effectual they must be in resisting the power ofthe waters which come upon them. In regulating theheights of embankments, it is necessary to ascertain thegreatest depth of water at the highest tides or floods ;making the summits of them about two feet higher than thepoints to which they rise at such times. By some, a lessheight than this above the highest mark of the tides or floodshas, however, been considered sufficient; but it is alwaysproper to be on the safe side, as the consequences of an over-flow are very serious.

In forming embankments with stones, or other similar