To find the Aberration of a Star in | 2. The aberration in declination Latitude und Longitude. at any other time, will be equal to 1. The greatest aberration in la. the greatest aberration multiplied titude, is equal to 201 multiplied by the sine of the difference, beby the sine of the star's latitude. tween the sun's place at the given 2. The aberration in latitude for time, and its place when the aberany time is equal to 2011 multiplied ration is nothing. by the sine of the star's latitude, 3. The sine of the latitude of a and the sine of elongation for the star : radius = the tangent of the same time. angle of position at the star : the The aberration is subtractive be- tangent of difference of longitude fore opposition, and additive after. between the sun and star. 3. The greatest aberration in 4. The greatest aberration in longitude is equal to 2011 divided by right ascension is equal to 20" mul. the cosine of the star's latitude; tiplied by the cosine of the angle and the aberration for any time is of position, and divided by the equal to that quotient multiplied sine of the difference in longitude by the cosine of the elongation of between the sun and star, when the the star. aberration in right ascension is noThis aberration is subtractive in thing. the first and last quadrants of the 4. The aberration in right ascenargument, and additive in the se- sion at any other time, is equal to cond and fourth quadrants. the greatest aberration multiplied EXAMPLE 1. To find the greatest by the sine of the difference beaberration of Urse Minoris, whose tween the sun's place at the given latitude is 750 13/. time, and his place when the aberHere the sine 75° 131 = .9669; con- ration is nothing. Also the sine of sequently 2011 X 9669 19.31, the the latitude of the star : the radius greatest aberration in latitude. Al the co-tangent of the angle of posi. 60 cosine 75° 131 = -2551; and there. tion at the star : the tangent of the 204 difference of longitude between fore =78.4", the greatest uber- the sun and star. 2551 ABERRATION of the Planets ration in longitude. is their geocentric motion, or the 2. To find the aberration of the space through which they appear same star in latitude and longitude, to move, as seen from the earth when the earth is 30° from syzy during the time of the light's passe gies. Here sine of 30°=5; and, there ing from the planet to the earth. It is evident that this aberration fore, 19.31" X 5= 9.67", the aber will be greatest in the longitude, ration in latitude. If the earth be and very small in latitude, be30° beyond conjunction, or before cause the planets deviate very litopposition, the latitude is dimi- tle from the plane of the ecliptic, nished; but if it be 30° before con- so that this aberration is almost junction, or after opposition, the insensible and disregarded ; the latitude is increased. Again, co- greatest in Mercury being only sine 30o = -866; consequently 78.4" about 4th, and much less in the * *886 = 67,891, the aberration in other planets. As to the aberralongitude. If the earth be 30° from tion in right ascension and declinaconjunction, the longitude is dition, it must depend upon the place minished; but if it be 30° from op- of the planet in the zodiac. The position, it is increased. aberration in longitude being equal Declination and Right Ascension. greater or less according to this To find the Aberration of a Star in to the geocentric motion will be 1. The greatest aberration in de-motion : it will be greatest in the clination is 20" multiplied by the superior planets, Mars, Jupiter, sine of the angle of position at the Saturn, and Uranus, when they star, and divided by the sine of are in opposition to the sun; but in the difference of longitude between the inferior planets, Mercury and the sun and star, when the aberra- Venus, the aberration is greatest at tion in declination is nothing. the time of their superior conjunc 4311 2911 tion. The maxima of aberration in all plano convex lenses, havfor the several planets, when their ing their couvex surfaces exposed distance from the sun is the least, to the parallel rays, the longitudiare as follow, vie. pal aberration of the extreme ray 1 Moon is equal to if of the thickness of the lens, 2 Mercury 591 0 In all double convex lenses of 3 Venus equal spheres, the aberration of 4 Mars 3711 the extreme ray is equal to ij of 5 Jupiter the thickness of the lens. 6 Saturn 27110 In a double convex lens, the radius of whose spheres are as 6 to 7 Georgian, or Uranus 25'10 1, if the more convex surface be And between these numbers and exposed to the parallel rays, the nothing the aberration of the pla- aberration from the figure is less nets, in longitude, varies accord-than that of any other spherical ing to their situation. That of the lens, being no more than 1 of its sun, however, is invariable, being thickness. constantly 201; and this may alter his declination, ABSCISS, ABSCISÆ, ABSCISSA, is by a quantity which varies from 0 to 8", being of a curve, comprised between any part of the diameter or axis greatest at the equinoxes, and vanishing in the solstices. any fixed point, where all the abABERRATION, in Optics, is that called the ordinate, which is ter scisses begin, and another line error, or deviation of the rays of minated in the curve. Commonly light wben inflected by a lens, or the abscisses are considered as speculum, whereby they are pre- commencing at the vertex of the vented from meeting or uniting in curve; but this is not necessary, the same point, called the geome as they may have their origin in trical focus. It is either lateral or any other point; but, generally, longitudinal. The lateral aberra when condition is specitied, tion is measured by a perpendicu.. they are understood as commenc, lar to the axis of the speculum; ing at the vertex. The absciss and produced from the focus, to meet ihe refracted ray. The longitudi- corresponding ordinate, considered nal aberration is the distance of together, are called co-ordinates, the focus froin the point in which of the curve is defined. and by means of these the equation the same ray intersects the axis. If the focal distance of any lenses ABSOLUTE Equation, in Astrobe given, if their aperture be small, nomy, is the sum of the optic and and if the incident ray homogene eccentric equations. ous and parallel, the longitudinal The apparent inequality of a plaaberrations will be as the squares, net's motion, arising from its not and the lateral aberrations as the being equally distant from the cubes of the linear apertures. earth at all times, is called its There are two species of aberra. optic equation; and this would tion, distinguished according to subsist it the planet's real motion their different causes ; the were uniform. The eccentric ine. arises from the figures of the spe- quality is caused by the planet's culum, or lens, producing a geo-motion not being uniform. For the metrical dispersion of the rays, illustration of this, conceive the when these are perfectly equal in sun to move, or appear to move, in all respects: the other arising from the circumference of a circle, in the unequal refrangibility of the the centre of which the earth is rays of light themselves. placed. Then it is manifest, that In all plano convex lenses, hav- if the sun move uniformly in this ing their plane surfaces exposed circle, he must appear to move to parallel rays, ihe longitudinal uniformly to a spectator on the aberration of the extreme ray, or earth; and, in this case, there would that most remote from the axis, is be no optic or eccentric equation. equal to 4 times the thickness of But suppose the earth to be placed the lens. out of the centre of the circle ; no one and then, though the sun's molion, proach to it, the more is their mo. should be really uniform, it would tion accelerated. Another class not appear to be so when seen held, that the earth emitted a sort from the earth; and in this case of attractive effluvia, innumerable there would be an optic equation, threads of which continually asbut not an eccentric one. Again, cend and descend, proceeding, like let us imagine the sun's orbit mot radii, from a common centre, and to be circular, but elliptical, and diverging the more the further the earth to be in its focus, then they so; so that the nearer a it is evident that the sun cannot heavy body is to the centre, the appear to have a uniform motion more of these magnetic threads it in such ellipse, and, therefore, his receives ; and hence the more its motion wili be subject to two motion is accelerated. equations, viz. the Oplic and Ec But leaving all such visionary centric Equations, the sum of theories, and only admitting the which is the Absolute Equation. existence of some such force as ABSOLUTE Term, or Number, gravity, inherent in all bodies, in Algebra, is that which is com- without regard to what may be pletely known, and to which all the cause of it, the whole mystery the other parts of the equation is of acceleration will be cleared up, made equal. and the theory of it established on ABSTRACT Mathematics, or the most obvious principles. Pure Mathematics, is that which Suppose a body let fall from any treats of the properties of magni. height,* and that the primary cause tude, figure, or quantity, absolule. of its beginning to descend is the Jy and generally considered, with power called gravity ; then, when out restriction 10 auy species in once the descent is cứnınienced, particular, such as Arithmetic and motion becomes, in some measure, Geometry. It is thus distinguished natural to the body; so that, if left from Mixed Mathematics, in which to itself, it would persevere in it simple and abstract quantities, for ever; as we see in a stone cast primatively considered in Pure from the hand, which continues Mathematics, are applied to sensi- to move after it is left by the cause ble objects, as in Astronomy, Me. that first gave it motion ; and which chanics, Optics, &c. motion would continue for ever, ACCELERATION is principally was it not destroyed by resistance used in Physics, to denote the in- and gravity, which cause it to fall creasing rapidity of bodies in fall to the earth. But beside this ten ing towards the centre of the earth, dency, which of itself is sufficient by a force called gravity, whether to continue the same degree of a property of matter, or an effect motion, in finitum, there is a conof the earth's notions. stant accession of subsequent ef. That natural bodies are accele- forts of the same principle, which rated in their descent, is evident from various considerations, both * Sir Richard Phillips, in his Esa priori and posteriori. Thus we says, maintains that there is no actually find, that the greater such torce inherent in matter as height a body descends from, the the attraction of gravitation, and more rapidly it descends, the that the cause of a body's descend. greater impression it makes, and ing to the earth, as well as all the the more intense is the blow wich other phenomena usually ascribed which it strikes the obstacle upon to the action of this force, are the which it impinges. natural and necessary results of the Some have attributed this acce- two motions of the earth. (See ar. leration to the pressure of the air ; ticles Attraction and Motion.)-This others to an inherent principle in explanation of the true cause of the matter, by which all bodies tend phenomena does not, however, alto the centre of the earth as their ier the law of acceleration, or, inproper seat or element, where they deed, any law of the earth or the would be at rest; and hence, say planetary system; though it varies they, the nearer that bodies ap- l our reasoning. began the motion, and which con, one second, then will lv represent tinues to act on the body already the velocity at the end of t sein niotion, in the same manner as conds. if it were at rest. Here, then, be And now if we represent by ing two causes of motion, and both the velocity of a body when it first acting in the same direction, the begins to fall, or its velocity ac. inotion they jointly produce inust quired in the first instant of its necessarily be greater than that of descent, then will the terms of the one of them; and the same cause series of increase acting still on the body, the descent of it must, of course, represent ihe successive velocities 0,2 0,3 0,4 0, &c. , be continually accelerated, at each successive instant; and, For, supposing gravity, or the earth's motion, to act uniformly on the time in uniform motions is since the velocity v, multiplied by all bodies, at equal distances from equal to the space, and since we the earth's centre; and that the time in which a heavy body falls form during any indefinite small may consider the motions as uni. to the earth to be divided into instant of time, the above may also equal parts, indefinitely small. Let be taken to represent the spaces this force incline the body towards described at each successive in. the earth's centre, while it moves stant; and hence the sum of them in the first indefinitely small space will be the whole space described of time of its descent; if after this in the time t. Now the number of the force be supposed to cease, the rernis in this series being t, and t body would proceed unitormly towards the centre of the earth, with be represented by v, and thus the being the final velocity, it may a velocity equal to that which resuni of the series will be expressed sults froin the force at the first im. by (4+v;. But since o reprepression, But since the action of the force sents the first velocily of the de. is supposed still to continue, in the scending body, it is indefinitely second moment of time the body small, and may be considered as will receive a new impulse down- nothing with regard to v. wards, equal to what it received therefore, be cancelled out of the in the first; and thus its velocity expression, and consequently the will be double of what it was in whole space will be represented the first moment: in the third, it by tv; or, taking s to represent will be treble; in the fourth, quad the space, we shall have s= s tv, ruple; and so on continually : for the space described by the inipression made in one mo a body uniformly accelerated in ment, is not at all altered by what any time, is equal to half the space is made in another. The whole that would be described by the are, as it were, united into one same body, in the same time, with sun. Hence the velocity will be a uniform velocity equal to that proportional to the time in which last acquired. it is acquired. It has been found by experiment Thus, if a body, by means of this that a body falling freely in the constant force, acquire a velocity latitude of London, passes through v, in one second of time, it will, in a space equal to 101, feet in the two seconds, acquire a velocity 2 v; first second of time; hence we have, in three seconds, it will acquire by representing this space by g,g= a 2v; and so on : and all bodies, tv, or g=v; because t=1, whatever be their quantity of mat- whence v=2g, is the velocity acter, will acquire, by the force of quired at the end of the first segravity, the same velocity in the cond of time, and, therefore, from same time; that is, supposing no what has been above demonstrat. resistance from the atmosphere, or ed, v=28 t, will represent the acany other obstacle in a perfect quired velocity at the end of any time t. Hence, if t be made to repre Also s=fto; substituting, there. sent the time a body has been fail-fore, for v, we have s=dt (2gt), ing, and v the velocity acquired in or s=t2g; and since g is constant It may, that is, – vacuum. V: it follows that the spaces describ When the accelerating forces ed by falling bodies are to each are different, but constant, the other as the squares of the times spaces will be as the products of of descent, the spaces themselves the forces into the square of the being accurately expressed by the times ; and the times will be as forinula sorg, where g repre. the square roots of the spaces di. sents 101 teet, the space a vody rectly, and of the forces inversely. falls through in the first second of Por when the force is given or time. constant, the velocity (V) is as the Hence we deduce the following time (T); and when the forces are the general laws of motions uniformly different, but constant, and accelerated, viz. time is given, the velocity (U) will But when 1. The velocities acquired are be as the force (F). constantly proportional to the neither the force nor the time is times. given, the velocity (v) will be as 2. That the spaces are propor. the product of the force into the tional to the squares of the time ; time, that is as (F x T). so that if a body describe Hence, v=FXT: 9xt any given space in a given time, it will conseq. Fx12: fxt=VXT:vxt= describe four times that space in a Sis s double time, nine times that space in a treble time, and so on. And therefore, T2: F: if universally, it the times be in S arithmetical proportion, as 1, 2, 3, T: *, &c. t, the spaces described will or, t=V7V f be as 1, 4, 9, 16, &c. t2. Thus a body, which talls by gravity From the properties above dethrough 1012 feet in the first semonstrated, we obtain the follow. cond, will fall through 64} feet in note the space passed over in the ing practical theorems : Let g de, two seconds, and so on. And since first second of time, by a body the velocities acquired in falling urged by any uniform force, deare as the limes, the spaces will be noted by 1; and let t denote the the square of the velocities: time or number of seconds in which and both ihe times and velocities the body passes over any other will be as the square roots of the space s, and v the velocity acquir. spaces. ed at the end of that time: then 3. The spaces described by a fall. we shall have v= 2 gt, and s=gt; ing budy in a series of equal mo and from these two equations we ments, or intervals of time, will obtain the following general forbe as the odd numbers 1, 3, 5, 7, 9, mulæ : &c. which are the differences of 25 the squares or whole spaces; that 1. is, a body which falls through 2g 16, feet in the first second, will 2 s fall through 3 x 16.s in the second, 2. 2gt = 5 X 16+ in the third, and so on. These properties may be other 3. wise represented, thus :- Let S, V, s=gt?= 48 2 be put for the space and velocity of a falling body in any time T; 4. and s, v, the same for the time t; .8= 2 t 4 $ then we shall have Let us now illustrate these for- mulæ by a few examples. EXAM. I. How far will a body : VSS : vs fall in 1211 in the latitude of Lon. T:t = don; and what will be its last veS:$= locity? 1672, and t = 12; there. T:t fore s= ge=167 X 144=2310 feet, Tot = SU V v the space required. as . 2ugs . V V2: |