ARCH

, Arc, Arcus, in Geometry, a part of any curve line; as, of a circle, or ellipsis, or the like.

It is by means of circular arcs, or arches, that all angles are measured; the arc being deseribed fiom the angular point as a centre. For this purpose, every circle is supposed to be divided into 360 degrees, or equal parts; and an arch, or the angle it subtends and measures, is estimated according to the number of those degrees it contains: thus, an are, or angle, is said to be of 30 or 80, or 100 degrees.—Circular arcs are also of great use in finding of fluents. |

Concentric Arcs, are such as have the same centre.

Equal Arcs, are such arcs, of the same circle, or of equal circles, as contain the same number of degrees. These have also equal chords, sines, tangents, &c.

Similar Arcs, of unequal circles, &c, are such as contain the same number of de- grees, or that are the like part or parts of their respective whole circles. Hence, in concentric circles, any two radii cut off, or intercept, similar arcs MN and OP.—Similar arcs are proportional to the radii LM, LO, or to the whole circumferences.—Similar arcs of other like curves, are also like parts of the wholes, or determined by like parts alike posited.

Of the Length of Circular Arcs. The lengths of circular arcs, as found and expressed in various ways, may be seen in my large Treatise on Mensuration, pa. 118, & seq. 2d edition: some of which are as follow. The radius of a circle being 1; and of any arc a, if the tangent be t, the sine s, the cosine c, and the versed sine v: then the arc a will be truly expressed by several series, as follow, viz, the arc ; where d denotes the number of degrees in the given arc. Also nearly; where C is the chord of the arc, and c the chord of half the arc; whatever the radius is.

To investigate the length of the arc of any curve. Put x=the absciss, y=the ordinate, of the arc z, of any eurve whatever. Put ; then, by means of the equation of the curve, find the value of x^. in terms of y^., or of y^. in terms of x^., and substitute that value instead of it in the above expression ; hence, taking the fluents, they will give the length of the arc z, in terms of x or y.

, in Astronomy. Of this, there are various kinds. Thus, the latitude, elevation of the pole, and the declination, are measured by an arch of the meridian; and the longitude, by an arch of a parallel circle, &c.

Diurnal Arch of the sun, is part of a circle parallel to the equator, described by the sun in his course from his rising to the setting. And his Nocturnal Arch is of the same kind; excepting that it is described from setting to rising.

Arch of Progression, or Direction, is an arch of the ecliptic, which a planet seems to pass over, when its motion is direct, or according to the order of the signs.

Arch of Retrogradation, is an arch of the ecliptic, described while a planet is retrograde, or moves contrary to the order of the signs.

Arch between the Centres, in eclipses, is an arch passing from the centre of the earth's shadow, perpendicular to the moon's orbit, meeting her centre at the middle of an eclipse.—If the aggregate of this arch and the apparent semi-diameter of the moon, be equal to the semi-diameter of the shadow, the eclipse will be total for an instant, or without any duration; and if that sum be less than the radius of the shadow, the eclipse will be total, with some duration; but if greater, the eclipse will be only partial.

Arch of Vision, is that which measures the sun's depth below the horizon, when a star, before hid by his rays, begins to appear again.—The quantity of this arch is not always the same, but varies with the latitude, declination, right ascension, or descension, and distance, of any planet or star. Ricciol. Almag. v. 1, pa. 42. However, the following numbers will serve nearly for the stars and planets.

, in Architecture, is a concave structure, raised or turned upon a mould, called the centering, in form of the arch of a curve, and serving as the inward support of some superstructure. Sir Henry Wotton says, An arch is nothing but a narrow or contracted vault; and a vault is a dilated arch.

Arches are used in large intercolumnations of spacious buildings; in porticoes, both within and without temples; in public halls, as ceilings, the courts of palaces, cloisters, theatres, and amphitheatres. They are also used to cover the cellars in the foundations of houses, and powder magazines; also as buttresses and counter-forts, to support large walls laid deep in the earth; for triumphal arches, gates, windows, &c; and, above all, for the foundations of bridges and aqueducts. And they are supported by piers, butments, imposts, &c.

Arches are of several kinds, and are commonly denominated from the figure or curve of them; as circular, elliptical, cycloidal, catenarian, &c, according as their curve is in the form of a circle, ellipse, cycloid, catenary, &c.

There are also other denominations of circular arches, according to the different parts of a circle, or manner of placing them. Thus,

Semicircular Arches, which are those that make an exact semicircle, having their centre in the middle of the span or chord of the arch; called also by the French builders, perfect arches, and arches en plein centre. The arches of Westminster Bridge are semicircular.

Scheme Arches, or skene, are those which are less than semicircles, and are consequently flatter arches; | containing 120, or 90, or 60, degrees, &c. They are also called imperfect and diminisbed arches.

Arches of the third and fourth point, or Gothic arches; or, as the Italians call them, di terzo and quarto acuto, because they always meet in an acute angle at top. These consist of two excentric circular arches, meeting in an angle above, and are drawn from the division of the chord into three or four or more parts at pleasure. Of this kind are many of the arches in churches and other old Gothic buildings.

Elliptical Arches, usually consist of semi-ellipses; and were formerly much used instead of mantle-trees in chimnies; and are now much used, from their bold and beautiful appearance, for many purposes, and particularly for the arches of a bridge, like that at Black-Friars, both for their strength, beauty, convenience, and cheapness.

Straight Arches, are those which have their upper and under edges parallel straight lines, instead of curves. These are chiefly used over doors and windows; and have their ends and joints all pointing towards one common centre.

Arch is particularly used for the space between the two piers of a bridge, intended for the passage of the water, boats, &c.

Arch of equilibration, is that which is in equilibrium in all its parts, having no tendency to break in one part more than in another, and which is therefore safer and slronger than any other figure. Every particular figure of the extrados, or upper side of the wall above an arch, requires a peculiar curve for the under side of the arch itself, to form an arch of equilibration, so that the incumbent pressure on every part may be proportional to the strength or resistance there. When the arch is equally thick throughout, a case that can hardly ever happen, then the catenarian curve is the arch of equilibration; but in no other case: and therefore it is a great mistake in some authors to suppose that this curve is the best figure for arches in all cases; when in reality it is commonly the worst. This subject is fully treated in my Principles of Bridges, pr. 5, where the proper intrados is investigated for every extrados, so as to form an arch of equilibration in all cases whatever. It there appears that, when the upper side of the wall is a straight horizontal line, as in the annexed figure, the equation of the curve is thus expressed, where x=DP, y=PC, r=DQ, h=AQ, and a= DK. And hence, when a, h, r, are any given numbers, a table is formed for the corresponding values of x and y, by which the curve is constructed for any particular occasion. Thus supposing a or DK=6, h or AQ= 50, and r or DQ = 40; then the corresponding values of KI and IC, or horizontal and vertical lines, will be as in this table.

The doctrine and use of arches are neatly delivered by Sir Henry Wotton, though he is not always mathematically accurate in the principles. He says; First, All matter, unless impeded, tends to the centre of the earth in a perpendicular line. Secondly; All solid materials, as bricks, stones, &c, in their ordinary rectangular form, if laid in numbers, one by the side of another, in a level row, and their extreme ones sustained between two supporters; those in the middle will necessarily sink, even by their own gravity, much more if forced down by any superincumbent weight. To make them stand, therefore, either their figure or their position must be altered.—Thirdly; Stones, or other materials, being figured cuneatim, or wedge-like, broader above than below, and laid in a level row, with their two extremes supported as in the last article, and pointing all to the same centre; none of them can sink, till the supporters or butments give way, because they want room in that situation to descend perpendicularly. But this is a weak structure; because the supporters are subject to too much impulsion, especially where the line is long; for which reason the form of straight arches is seldom used, excepting over doors and windows, where the line is short and the side walls strong. In order to fortify the work, therefore, we must change not only the figure of the materials, but also their position.—Fourthly; If the materials be shaped wedgewise, and be disposed in form of an arch, and pointing to some centre; in this case, neither the pieces of the said arch can sink downwards, for want of room to descend perpendicularly; nor can the supporters or butments suffer much violence, as in the preceding flat form: for the convexity will always make the incumbent rather rest upon the supporters, than thrust or | push them outwards. His reasoning, however, afterwards, on the effect of circular and other arches, is not accurate, as he attends only to the side pressure, without considering the effect of different vertical pressures.

The chief properties of arches of different curves, may be seen in the 2d sect. of my Principles of Bridges, above quoted. It there appears that none, except the mechanical curve of the arch of equilibration, can admit of a horizontal line at top: that this arch is of a form both graceful and convenient, as it may be made higher or lower at pleasure, with the same span or opening: that all other arches require extrados that are curved, more or less, either upwards or downwards: of these, the elliptical arch approaches the nearest to that of equilibration for equality of strength and convenience; and it is also the best form for most bridges, as it can be made of any height to the same span, its hanches being at the same time sufficiently elevated above the water, even when it is very flat at top: elliptical arches also look bolder and lighter, are more uniformly strong, and much cheaper than most others, as they require less materials and labour. Of the other curves, the cycloidal arch is next in quality to the elliptical one, for those properties, and, lastly, the circle. As to the others, the parabola, hyperbola, and catenary, they are quite inadmissible in bridges that consist of several arches; but may, in some cases, be employed for a bridge of one single arch which may be intended to rise very high, as in such cases as they are not much loaded at the hanches.

Arch Mural. See Mural arch.