^ifcle • ' "

^ e h v 111 ^ le same plane, so that in each circle, the distanceHiqj! een the point of contact at the commencement of the

, an d the point of contact at any instant while infe re ° n > * s ec l ua l one to the other. Hence if the circum-H 0V j Ces ar e equal, all parts of the circumference of the

the ® Clrc le will have been in contact with all the parts ofIf !, CUm ^ erence of the quiescent circle,the,. . S ener ating circle proceed along the convexity ofif a ^ eri phery, it is called an upper, or exterior epicycloid;

q^S the concavity, a lower, or interior epicycloid.ci rcl e P ar t of the quiescent circle which the generating

Th,

m °ves along, is called the base.

j)oi n j"l l e ngth of any part of the curve, which any given^ le rev °l v * n g circle has described from the time oftli e v St ^ e * n g in contact with the quiescent circle, is, to doubleof tj i ers f ( I ®ine of half the arc, as the sum of the diametersfiw? Clr cles, to the semi-diameter of the quiescent circle ;Moq„ e ! the circumference of the moving circle be carriedtl le ® the convex side of the quiescent circle; but if uponSaid ncave side, as the difference of the diameters to theD^i-diameter.

of a ji‘ tialley gives us a general proposition for the measuring

tpicv ?. cloi( is and epicycloids : thus, the area of a cycloid orvCioici ... - ...

either primary, contracted, or prolate, is to the

/, 7 V/Uiiui jjiuiatt/) is tu buc

Metier • generating circle, and also to the areas of the partsSe gia e 6l * * n these curves, to the areas of the analogousthe c n * s °f the circle, as the sum of double the velocity ofVfc l<JCit re ’ an< I the velocity of the circular motion, to the®«e pi Z °I the circular motion. The demonstration hereof,

. Trans. No. 218.

>hg p arf -as of epicycloids may be determined by the follow-Portion : As the radius of the circle of the base is to’>nes that of the radius, together with twice that of thect 0r "'o circle, so is the circular segment to the epicycloidale t>icy ( _!i°. I '. , b e whole generating circle to the whole area of theAs i

'i^oid.

a ^3 tr\

'Put a .. l “e tangents, it is known from the time of Descartes ,Vi, ,n e drawn from any point to that of the base, which^alar ^ le circle > whilst this point is described, is perpen-jVj a to the curve, and consequently to the tangent.

^ t'e Vn l >e ' rt; * us > discussing this subject, conceived a polygone 1Un}. Ve upon another, the sides of which are respectivelyHi c |’ - 0rie the angles described a curve, the periphery ofcif c 3 formed of arcs of circles, and the area is composedjV p r ^ ar sectors, and right-lined triangles. He determined° V^on of the area, and of the periphery of this figure!°lyt, 0 n tbe generating polygon. He also supposed those9(1 ep; c 3 to become circles, the figure described to become°id, and the above-mentioned proportion, modified

1 die ^ *° tb is supposition, gave him the area and peripheryIt d ®Picyeloid. Mem. de VAcad. 1727.

^ViU. 6 - 8 , n °t appear that any writer published an account of

h 6 first'? 8 ’ ^ e f° re the celebrated Sir Isaac Newton , who, inp'aplg b °°k of his Principia , proposed a general, and a veryi "ou'h^^d °f rectifying these curves. After him,'p-Uis d/ 1 ’, 1 during his residence at Paris , showed how, by

N.

I ar ^' e lutregal and differential calculus, to determinean d rectification. The invention of epicycloids is,1 r ° Sa Dl ascribe d to M. Reaumur, the celebrated Danish ,74. "!® r > during his residence at Paris , about the yearues e curves appeared to him to be such as bestJ1S !.®. teeth of wheels, constructed so as to diminish theirr- 0r Hi. 5 t ' on > and to render the action of the power morel| ' h, ' 8p ’ Uence he was led to consider them, and to this

thev

si*

However, M. de la Hire

was

!,(■ -vs „ ~y have been applied,aiso. Mention of Reaumur, and seems to claim the meritutfttrical and mechanical invention. But M.Leibnitz,5 a

who resided at Paris in 1674, and the two following years,says, that the invention of epicycloids, and their applicationto mechanics, was the work of this Danish mathematician, andthat he was esteemed the author of It.

EPISTYLE, (from cm, upon, vtAoe, column ) in ancientarchitecture, a term used by the Greeks for what we callarchitrave, viz., a massive stone, or a piece of wood, laidimmediately over the capitals of columns, from one to theother. 1

The epistyle is the first or lowest principal division of theentablature.

EPISTYLAR ARCUATION, a term applied to thatmethod of building in which arches are thrown from columnto column, instead of horizontal architraves.

EPITIIHEDAS, (Greek eiri riOrj/ji) a word used by theGreeks, to express the simas, or cymatium, or crowningmoulding of the entablature. By some the term is restrictedto the upper member of a raking cornice.

EQUAL ANGLES, are those whose containing lines aremeasured by equal portions of equal arcs, described from themeeting of the two containing lines.

Equal Arcs. See Arcs.

Equal Circles, are those whose diameters are equal.

Equal Curvatures, are such as have the same, or equalradii of curvature. See Curvature and Curve.

Equal Figures, are those whose areas are equal, whetherthe figures be similar or not.

Equal Solids, are those which comprehend, or containas much as the other, or whose solidities or capacities areequal.

EQUIANGULAR FIGURES, such as have equal angles.

EQUIDISTANT STRAIGHT LINES, are a series ofparallels with equal intervals.

Equidistant Solids, are those whose intervals are termi-nated by parallel planes, at equal distances on the corres-ponding sides, comparing that of any two adjoining solids,with that of any other two adjoining solids.

EQUILATERAL (from cequus, equal, and latus, a side)having equal sides.

Equilateral Figure, that which has all its sides equal.

Equilateral Hyperbola, that in which the asymptotesare at right angles to each other.

EQUILIBRIUM, in mechanics, the equality of forces inopposite directions, so that they mutually balance each other ;equipoise.

ERGASTULUM, among the ancients, a house of correc-tion, or workhouse, where slaves, by the private authority oftheir masters, were confined and kept at hard labour for someoffence. It was likewise called sophronisterium.

ESCAPE, a concave quadrantal moulding used to jointwo parallel members of different projections, as the shaft ofa column with the fillet at its foot and junction with thecapital.

ESCUTCHEON, a shield charged with armorial bearings.Decorations of this kind were much used in the later periods ofGothic architecture, carved on bosses, dripstones, spandrels,&c. The term is also applied to the metal plate on doorssurrounding the key-hole, and to that from which the handleis suspended. Beautiful specimens, of excellent design andworkmanship, are to be found in old mediaeval buildings.

ESCURIAL, the palace or residence of the kings of Spain .The word originally signified a little village in Spain , situatedin the kingdom of New Castile, twenty-two miles to theN. W. of Madrid . Here king Philip built a stately monas-tery, of the order of St. Jerome, held by the Spaniards as oneof the wonders of the world. It was begun in 1357, andfinished in about 22 years, at the expense of 6,000,000 of