a=v2v.{1+ &c. } aphelion, and the lower apsis the | as the radii of the corresponding perihelion; or, according to the circles. ancient astronomy, the apogee and The length of circular arcs may perigee. The diameter which joins be expressed in terms of the sine, these two points is called the line cosine, tangent, &c. m the followof the apsides, and is supposed to ing manner: pass through the centre of the or Let a represent any arc to radius bit of the planet, and the centre 1.; s its sine, c its cosine, t ils of the sun or earth; in the modern tangent, is the secant, and v its astronomy, this line makes the versed sine; then will longest or transverse axis of the a=t-3+3+{t-1+1+}(9_&c. elliptical orbit of the planet; and 's 193 195 187 in its line is counted the eccentri &c. с 303 505 7c city of the orbit. 9c9 According to the above defini 1.93 1.3.85 Q=S+ tion, the lines of the greatest and + + 2.3 2.4.5 leasi distance are supposed to lie 1.3.5.57 in the same straight line; which + &c. is not always precisely the case, 9.4.6.7 as the two frequently make an 1 302 + + angle with each other; and what 22,3 23.4.5 this angle differs from 180', is call. 1.3.5v3 ed the motion of the line of the 24.46.7' apsides; and when this is less than 180°, the motion is said to be con a=sin. a, sec. Q, sec. a, sec. trary to the order of the signs; fa, &c. 3.1416 and when it is greater than 180°, d= .01745329, &c. the motion is said to be according 180 to the order of the signs. Xd; where d represents AQUARIUS, in Astronomy, the the number of degrees, 11th sign of the zodiac, beginning &c. contained in the from Aries; its character is m given arc. 60-C AQUEDUCT, a conduit oj' water, -, nearly. Where Cand in Architecture and Hydraulics, is 3 a construction of stone and timber care the chords of the built on uneven ground, to pre arc and half arc. serve the level of water, and to As since similar arcs are to each conduct it through canals from one other as their radii, it is obvious, place to another. that having the length of any arc ARCH, or ARC, in Geometry, a given to radius 1, the length of a part of a curve line: as of a circle, similar arc may be found for any ellipse, &c. other radius, by multiplying the Circular Arc, is any part of length of the first arc by the given the arc of a circle, and by which radius. "Or, since .01745329 is the the magnitudes of angles are com- length of an arc of 1°, to radius 1; pared ; an angle being said to con- the length of any arc, of wlrich the tain so many degrees, minutes, &c. measure is given, will be found by as are contained in the arc which multiplying the number of degrees subtends it. by .01745329: and that product again Co-centric Arcs, are such as by the given radius. have the same centre. ARCH in Astronomy, has various Equal Arcs, are such arcs of denominations, according to the the same circle, or of equal circles, circle of which it is a part. which have the same measure, or Diurnal Arch of the Sun, is part the same number of degrees, mi- of a circle described by the sun in nutes, &c. his course between rising and set. Similar Arcs, are those which ting. His nocturnal arch is that have the same measure, but belong. I described between setting and ris. ing to different circles. The lengths ing. of similar arcs are to each other The latitude and elevation of the pole are measured by arcs of the nate, x = the abscissa, and a=the ineridian, and the longitude by an thickness at the keystone; hence, arch of a parallel circle. when a, h, and r, are any given Arce of Progression, or Direc. numbers, a table is formed for the tion, is an arch of the ecliptic, corresponding values of u and y, which a planet seems to pass over, by means of which the carve is when its motion is direct, or accordo constructed for any particular ocing to the order of the signs. casion. Arch of Retrogradation, is an And in a similar manner, if the arch of the ecliptic described when curve of the intrados, and the depth a planet is retrograde, or moves of the key-stone be given, lhe equacontrary to the order of the signs. tion of the extractos may be com Arch of Position, or Angle oisputed. Thus, for example, in the Position, is the same with the horary case where the intrados is a circle, angle. the equation, readily deduced from ÅRcg of Vision, is the sun's depth the above construction, is below the horizon at which a pla y V (a? + b2-y). net or star, betore hid in his rays, begins to appear. This arch is differ ✓(72_62) ent for different planets; being for where a=the height; and (putting Mercury 10°, Venus 5o, Mars 111°, n=half the span, and m=the height Jupiter 10°, 'Saturn 11°; a siar or of the keystone) /(a + my-n?=b; the 1st magnitude 12°, 2d magnitude the distance of the point from the 13°, &c. This angle is not, how- centre of the arch=x, and its ever, constant in all cases for the height above the centre = y. It sanie planet, but varies a little will be found that a line at right with the latitude and declination, angles to the perpendicular, and &c. With respect to Venus, it is meeting it at the point where sometimes reduced to 0, as she is at times visible when the sun is v(a + m)? — n? falls, will be an some degrees above the horizon. assymptote to the extrados. Hence ARCH of Equilibrium, in the the extrados, in the case of a cir cular arch, is curve of the Theory of Bridges, is that which is fourth in equilibrio in all its parts, and order, very much retherefore equally strong through. sembling the conchoid of Nico oui, having no tendency to break medes, and that it coincides very in one part more than another. nearly with the curve, in which a It is not of any determinate curve, bridge. This holds good, of whal. road is usually carried over a but varies according to the figure of the extrados; every different ever portion of the circle the arch extrados requiring a particular in is supposed to consist. trados, so that the thickness in small circle of the sphere parallel ARCTIC Circle, in Astronomy, a every part may be proportional to the pressure. If the arch were from the arctic or northern pole. to the equator, and distant 23° 28/ equally thick throughout, the cate Arctic Pole, the northern pole nary curve would be the arch of of the world. equilibration; but as this can seldom or never happen, it is a mis- ficial measure or surface of any AREA, in Geometry, is the supertaken idea to suppose this curve figure. The areas of similar plane the best in all cases. It therefore agures are to each other as the appears, that when the upper side of the wall is a straight horizontal square of their like sides, or other lineal dimensions. Jine, the equation of the curve is thus expressed : AREOMETER, an instrument for a+x+ V (2 ax + x2) measuring the density or gravity log. of Auids. It is now commonly made of glass; consisting of a round y=hx hollow ball, which terminates in a atr tv (2 ar+r2) log. long slender neck, hermetically sealed at top; there being first as where h = half the span, r = the much mercury put into it, as will neight of the arch, y = the ordi- serve to balance or keep it swim a a ming in an erect position. They of the different nations among stem or neck is divided into de- whom it bas been used. Its origin, grees or parts, which are num- like that of all sciences, is involve bered, to show the specific gravity ed in obscurity ; nor is it fully by the depth of its descent. ascertained to whom we are in ARGUMENT, in Astronomy, is debted for the improved system in general a quantity upon which now in use. another quantity or equation de Arithmetic is divided into various pends; or it is an arch, whereby kinds, according to the nature of we seek another unknown arch, the numbers that form the subject bearing soine proportion to the of it; but the most simple form, tirst: hence and that which is the root or founl. ARGUMENT of Inclination, ordation of all the others, is the arithARGUMENT of Latitude, of any metic of abstract or simple numplanet, is an arch of a planet's bers. A nuniber is called abstract orbit, intercepted between the as- when it merely answers to the cending node and the place of the question, “ how many ?" and has planet from the sun, numbered no allusion to the value of the said according to the succession of the many, and no reference to things signs. of any kind, but which admits of Menstrual ARGUMENT of Lati- having the name and value of any tude, is the distance of the inoon's kind of things whatever applied true place from the sun's true to it. In this sense, one is the place; by which is found the smallest and simplest number that quantity of the real obscuration in can be mentioned; and the other eclipses. numbers proceed by constant ad. Annual ARGUMENT of the Moon's ditions of one, without limit. This Apogee, or simply Annual Argu- being the case, the first thing rement, is the distance of the sun's quisite for the formation of an arith. place from the place of the moon's netical language, is the invention apogee; that is, the arc of the of a limited number of original ecliptic comprised between those names, which shall comprehend a two places, great and almost endless variety of ARGUMENT of the Parallax, de numbers. For this purpose, what notes the effect it produces on an is usually called a "scale of num. observation, and which serves for bers,” becomes necessary, and the determining the true quantity of limits of this scale are the powers the horizontal parallax. of any number, m which is taken ARGUMENT of the Equation of as the modulus or root. Thus mio, the Centre, is the anomaly, or dismi, m?, m3, m4, will form five ranks tance, from the apogee or aphelion; or places in the scale, any number because this equation is calculated whatever being taken for m; and in an elliptic orbit for every de- the scale will receive its name acgree of anomaly, and varies accord-coriling to the nuruber so laken : ing to the variation of the ano- thus, maly. If m= 2, the scale is binary. ARIES, or the Ram, in Astro If m = 3, the scale is ternary. nomy, the first of the old 12 signs It m=: 7, the scale is senary. of the zodiac: it is marked r, If m=10, the scale is denary. in imitation of a ram's head. The li m= 12, the scale is duodesun enters this sign generally about nary. the 201h of March. When the lower or any of the ARITHMETIC, that part of ma- intermediate places is blank, it is thematical science which treats of supplied by a character (0) having the nature and properties of num. no separate value, and another bers, the representing of them by power of the modulus is given by proper symbois, and the application adding this character. of them to the business of calcula Whatever may be the value of tion. This science has undergone m, mno is always =1. Hence, the various improvements, and has par- tirst tive places in each of the uiken of the genius and language above scales, expressed in coin 52 mon numbers, or according to the tain nomber of times, would prodenary scale, will be, duce the given number, the proDenary 1, 2, 4, 8, 16 cess is called division. These four Ternary 1, 3, 9, 27, 81 processes are called the four funSenary 1, 7, 49, 343, 2401 damental operations, and the me. Denary 1, 10, 100, 1000, 10000 thods of performing them the four Duodenaryl, 12, 144, 1728, 20736 fundamental rules of arithinetic; and each of the latter is the conIf, however, any of the scale verse of the corresponding one of were adopted, all these express the former. Subtraction is the sions would be the same ; and converse of addition, and division each would require one character the converse of multiplication. less than the number of ones in m, For the method of performing thus, each of these, see the articles Denary Addition, Subtraction, &c. Ternary 1, 2, ARITHMETICAL, any thing reSenary 1, 2, 3, 4, 5, 6, lating to arithmetic. Denary 1, 2, 3, 4, 5, 6, 7, 8, 9, ARITHMETICAL complement of Duvdenary1, 2, 3, 4, 5, 6, 7, 8, 9,4:2, a Logarithm. See Anti-LOGARITHM and COMPLEMENT. In which last p stands for 10, and ARITHMETICAL Mean, is the x for 11. For the inethod of chang. middle term of three quantities in ing a number from one scale to an arithmetical progression, or half other, see Notation. the sum of any two proposed num. Arithinetic receives various de- bers. nominauons, according to the na. ARITHMETICAL Progression ture of the numbers treated of: and Proportion. See PROGRESSION thus, and PROPORTION. Simple or Integral arithmetic is ARITHMETICAL Ratio, is the the method of calculating by sim- difference between any two adja. ple or abstract numbers. cent terms in arithmetical progres. Fractional arithmetic is the mession. thod of calculating by numbers ARITHMETICAL Scales. See which are less than the number 1, NOTATION. and is of two kinds, vulgar and ARITHMETICAL Triangle. See decimal. For these, as well as for TRIANGLE. the arithmetic of exponents, logar. ARMILLARY Sphere, a name ithms, sines, &c. see those articles. given to an artificial sphere, which Whatever may be the nature of represents the several circles of the numbers which form the sub. the system of the world, put wge. ject of arithmetical computation, ther in their natural order. There are but two changes, of which The armillary sphere revolves any number is susceptible, namely, upon an axis, within a horizon, being made greater, and being which is divided into degrees, and made less; and, generally speak- moveable every way upon a brass ing, each of these may be per- supporter. The other parts are formed in two ditlerent ways. the equinoctial, zodiac, meridian, When a number is increased by the two tropics, and the two polar adding to it any number or num-circles. bers whatever, the process is called ARTIFICIAL Numbers, a term addition, and when it is increased sometimes used for logarithms, lo by repeating itself any number of garithmic sines, tangents, &c. times, the process is called multi ASCENDING, in Astronomy, is plication. When a number is di-, understood of those stars, &c. minished by taking away from its which are rising above the hori. any number or numbers whatever, zon, in any parallel of the equator. the process is called subtraction; ASCENDİNG Latitude, is the and' when it is diminished by latitude of a plavet when going taking away one or more numbers towards the north pole. equal to that which is left, or ASCENDING Node, is that point when (which is the same thing) of a planets orbit where it passes the number ieft, it repeated a cer. the ecliptic to proceed worth ward. ASCENSION, in Astronomy, is To the sine of the right asceneither sight or oblique. sion. Right AscENSION of the Sun or The sun's right ascension in time of a Star, is that degree of the is useful to the pracucal astronoequinoctial (counted from the be. mer in regular observations, who ginning of Aries,) which rises with adjusts his clock by sidereal time. the sun or star in a right sphere. It serves also for converting appaOr, it is that degree and minute of|rent into siderial time ; as, for in. the equinoctial, counted as before, stance, that of an eclipse of Jupi. which comes to the meridian with | ter's satellites, in order to know the sun or star. at what time it may be expected The reason of thus referring it to Lo happen by the clock. For this the meridian is, because that is purpose, the sun's right ascension always at right angles to the equi- tor ihe preceding noon, together noctial, whereas the horizon is with the increase of right ascen. only so in a right or direct sphere. sion from noon, must be adved 10 The right ascension stands oppos- the apparent time of a known star ed to the right descension, and passing the meridian: then subcorresponds to the longitude of iract the sun's right ascension in places on the earth. Two fixed time at noon, from the star's right stars, that have the same right as ascension in time, the remainder cension, that is, which are at the is the apparent time of the stara same distance from the first point passing the meridian nearly: from of Aries, or which is still the same, which the proportional part of are in the same meridian, rise at the daily increase of the same the same time in a right sphere, or right ascension from his apparent with respect to persons living un- time from noon being subtracted, der the equator. If they be not in leaves the correct time of the star's the same meridian, the difference passing the meridian. The sun's between their rising, or coming to right ascension in time is also use. the meridian, is the precise differ- ful for computing the time of the ence of their right ascension. But moon, or a planet's, passing the in an oblique sphere, where the meridian, horizon cuts all the meridians ob The Arch of Right Ascension, is liquely, different points of the that portion of the equator, inler. meridian never rise or set toge- cepted between the beginning of ther; so that two stars, on the Aries and the point of the equator, same meridian, never rise or set at which is the meridian: or it is the the same time; and the more ob- number of degrees contained in it. lique the sphere is, the greater is Oblique ASCENSION, is an arch of the interval of lime between them. the equator, intercepted between To find the right Ascension of the the first point of Aries, and that Sun or a Star, point of the equator which rises, Say, for the sun, together with the star, &c. in an As radius oblique sphere. The oblique asIs to the cosine of the sun'scension is counted from west to greatest declination, or obli- east; and is greater or less, ac- cording to the different obliquities Arch of Oblique ASCENSION, is To the tangent of the right the arch on which the oblique as cension is measured. ascension. If the obliquity of the ecliptic, ASCENSIONAL Difference, is and the sun's declination were the difference between the right given, the proportion for the right and oblique ascension. ascension would be: To find the ascensional Difference As radius of the Sun, having the Sun's DeIs to the cotangent of the ob. clination and the Latitude of the liquity of the ecliptic, Place. So is the tangent of the sun's Say, As radius declination Is to the tangent of the latitude, |