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most simple and the most profound. Then we have Bridge, Wood, Ludlam, Simson, and a hundred others In Elementary Geometry we have Euclid; Playfair's last edition is decidedly the best. Again, in the practical parts, we have Hutton's Course, Crocker's Surveying, Keith's Mensuration, Moore and Mackay's Navigation, and an endless chain. On Trigonometry, we have Wood, Wodehouse, Keith, and many others. On Fluxions and the differential Calculus, we have Simson, Vince, Stone, La Croix, Boucharlat, and many others. In Mechanical Philosophy, we have Blair's Grammar, Young's Lectures, Bridge's Mechanics, Wood and Vince's Course, Playfair's Outlines, part of the same, by Leslie, and a Course by Millington. In Astronomy, we have Vince, Squire, the Wonders of the Heavens, and a variety of others. In short, if we have relaxed in our mathematical and scientific studies, it is not for want of books; for, though they be of lighter fabric, and fewer in proportion to the whole number of books than at some former periods, they are still numerous; and if we do suffer, it is not through want of books, but want of readers.

Every lover and student of the Sciences will duly estimate the value of a portable Dictionary of the Mathematical and Philosophical Sciences. Other Dictionaries of these subjects are, by their high price, placed beyond the reach of the general mass of purchasers, while by their fullness they tend to supersede elementary works, without supplying their places. The great use of a Dictionary is to aid study by convenient reference to particular points of difficulty, and to assist enquiry by an alphabetical arrangement of subjects. London, Sept. 10, 1823.

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ABACUS. A table used before of light, and the earth's motion in

the introduction of the modern or its orbit. This apparent motion is figurate arithmetic, for facilitat- so minute, that it could never have ing the busmess of calculation. been discovered by observations, Originally it appears to have been unless they had been made with nothing more than a smooth piece extreme care and accuracy. Dr. of board, covered with sand, and Bradley, astronomer royal, was served indifferently for arithmeti- led to it accidentally by the result cal computations, or geometrical | of some careful observations, diagrams. The word calculate is which he made with a view of dederived from the calculi, or small termining the annual parallax of pebbles, which were used along the fixed stars. If light be supwith the abacus. These were dis- posed to have a progressive motributed in rows, each row having | tion, the position of the telescope, a different value, in the same man- through which any celestial object ner as the ranks or places of figures is viewed, must be different from have in the modern scale of num- that which it must have been, if bers. As many rows were requir-light were instantaneous; and, ed as there were ranks or places in the largest number which entered into the calculation; and one counter less than the root of the scale of arithmetic was required for each row. For instance, it the root of the scale had been 10, nine counters would have been required in each; and to express any particular number, as many would have been required in each row as there were ones in the corresponding word or figure. Thus, 365 would have been expressed by 5 in the right-hand row, 6 in the second, and 3 in the third. It is easy to see how, by the help of such an instrument, the common operations of arithmetic could be performed.

therefore, the place measured in the heavens will be different from the true place.

Clairaut explains the aberration, by supposing drops of rain to fall rapidly after each other from a cloud, under which a person moves with a very narrow tube; in which case it is evident that the tube must have a certain inclination, in order to admit a drop which enters at the top, to fall freely through the axis of the tube, without touching the sides of it; and this inclination must be greater or less, according to the velocityof the drops in respect to that of the tube. In this case, the angle made by the direction of the tube, and that of the falling drops, is the aberraABERRATION, an apparent mo- tion, arising from the combination tion of the celestial bodies, occa- of these two motions. sioned by the progressive motion

To find the Aberration of a Star in

Latitude and Longitude.

1. The greatest aberration in latitude, is equal to 20" multiplied by the sine of the star's latitude.

2. The aberration in latitude for any time is equal to 20" multiplied by the sine of the star's latitude, and the sine of elongation for the same time.

The aberration is subtractive before opposition, and additive after. 3. The greatest aberration in longitude is equal to 20" divided by the cosine of the star's latitude; and the aberration for any time is equal to that quotient multiplied by the cosine of the elongation of the star.

This aberration is subtractive in the first and last quadrants of the argument, and additive in the second and fourth quadrants.

EXAMPLE 1. To find the greatest aberration of y Ursa Minoris, whose latitude is 75° 13.

Here the sine 75° 13/9669; consequently 20′′ × 9669 = 19.34", the greatest aberration in latitude. Al60 cosine 75° 13/= 2551; and there

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ration in longitude.

2. To find the aberration of the

same star in latitude and longitude, when the earth is 30° from syzy gies.

Here sine of 30°·5; and, therefore, 19.31" X 59'67", the aber ration in latitude. If the earth be 30° beyond conjunction, or before opposition, the latitude is diminished; but if it be 30° before conjunction, or after opposition, the latitude is increased. Again, cosine 30°866; consequently 78.4" 86667.89, the aberration in longitude. If the earth be 30° from conjunction, the longitude is diminished; but if it be 30° from opposition, it is increased.

To find the Aberration of a Star in Declination and Right Ascension. 1. The greatest aberration in declination is 20 multiplied by the sine of the angle of position at the star, and divided by the sine of the difference of longitude between the sun and star, when the aberration in declination is nothing.

2. The aberration in declination at any other time, will be equal to the greatest aberration multiplied by the sine of the difference, between the sun's place at the given time, and its place when the aberration is nothing.

3. The sine of the latitude of a star radius the tangent of the angle of position at the star : the tangent of difference of longitude between the sun and star.

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4. The greatest aberration right ascension is equal to 20 multiplied by the cosine of the angle of position, and divided by the sine of the difference in longitude between the sun and star, when the aberration in right ascension is nothing.

4. The aberration in right ascension at any other time, is equal to the greatest aberration multiplied by the sine of the difference between the sun's place at the given time, and his place when the aberration is nothing. Also the sine of the latitude of the star: the radius the co-tangent of the angle of position at the star: the tangent of the difference of longitude between the sun and star.

ABERRATION of the Planets is their geocentric motion, or the space through which they appear to move, as seen from the earth during the time of the light's passing from the planet to the earth.

It is evident that this aberration

will be greatest in the longitude, and very small in latitude, because the planets deviate very little from the plane of the ecliptic, so that this aberration is almost insensible and disregarded; the greatest in Mercury being only about 43", and much less in the other planets. As to the aberration in right ascension and declination, it must depend upon the place of the planet in the zodiac. The aberration in longitude being equal to the geocentric motion will be greater or less according to this motion: it will be greatest in the superior planets, Mars, Jupiter, Saturn, and Uranus, when they are in opposition to the sun; but in the inferior planets, Mercury and Venus, the aberration is greatest at the time of their superior conjunc

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7 Georgian, or Uranus 25" 0 And between these numbers and nothing the aberration of the planets, in longitude, varies accord ing to their situation. That of the sun, however, is invariable, being constantly 20"; and this may alter his declination, by a quantity which varies from 0 to 8", being greatest at the equinoxes, and vanishing in the solstices.

ABERRATION, in Optics, is that error, or deviation of the rays of light when inflected by a lens, or speculum, whereby they are prevented from meeting or uniting in the same point, called the geometrical focus. It is either lateral or longitudinal. The lateral aberration is measured by a perpendicular to the axis of the speculum, produced from the focus, to meet the refracted ray. The longitudinal aberration is the distance of the focus from the point in which the same ray intersects the axis. If the focal distance of any lenses be given, if their aperture be small, and if the incident ray homogeneous and parallel, the longitudinal aberrations will be as the squares, and the lateral aberrations as the cubes of the linear apertures.

There are two species of aberration, distinguished according to their different causes; the one arises from the figures of the speculum, or lens, producing a geometrical dispersion of the rays, when these are perfectly equal in all respects the other arising from the unequal refrangibility of the rays of light themselves.

In all plano convex lenses, having their convex surfaces exposed to the parallel rays, the longitudinal aberration of the extreme ray is equal to 1 of the thickness of the lens.

In all double convex lenses of equal spheres, the aberration of the extreme ray is equal to 13 of the thickness of the lens.

In a double convex lens, the radius of whose spheres are as 6 to 1, if the more convex surface be exposed to the parallel rays, the aberration from the figure is less than that of any other spherical lens, being no more than 15 of its thickness.

as

any part of the diameter or axis ABSCISS, ABSCISE, ABSCISSA, is of a curve, comprised between any fixed point, where all the abcalled the ordinate, which is terscisses begin, and another line minated in the curve. Commonly the abscisses are considered commencing at the vertex of the as they may have their origin in curve; but this is not necessary, any other point; but, generally, they are understood as commenc no condition is specified, ing at the vertex. The absciss and corresponding ordinate, considered together, are called co-ordinates, and by means of these the equation of the curve is defined.

when

ABSOLUTE Equation, in Astronomy, is the sum of the optic and eccentric equations.

The apparent inequality of a planet's motion, arising from its not being equally distant from the earth at all times, is called its optic equation; and this would subsist if the planet's real motion were uniform. The eccentric inequality is caused by the planet's motion not being uniform. For the illustration of this, conceive the sun to move, or appear to move, in the circumference of a circle, in the centre of which the earth is placed. Then it is manifest, that

In all plano convex lenses, hav-if the sun move uniformly in this ing their plane surfaces exposed to parallel rays, the longitudinal aberration of the extreme ray, or that most remote from the axis, is equal to 4 times the thickness of the lens.

circle, he must appear to move uniformly to a spectator on the earth; and, in this case, there would be no optic or eccentric equation. But suppose the earth to be placed out of the centre of the circle;

and then, though the sun's motion should be really uniform, it would not appear to be so when seen from the earth; and in this case there would be an optic equation, but not an eccentric one. Again, let us imagine the sun's orbit not to be circular, but elliptical, and the earth to be in its focus, then it is evident that the sun cannot appear to have a uniform motion in such ellipse, and, therefore, his motion wili be subject to two equations, viz. the Optic and Eccentric Equations, the sum of which is the Absolute Equation.

ABSOLUTE Term, or Number, in Algebra, is that which is completely known, and to which all the other parts of the equation is made equal.

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But leaving all such visionary theories, and only admitting the existence of some such force as gravity, inherent in all bodies, without regard to what may be the cause of it, the whole mystery of acceleration will be cleared up, and the theory of it established on the most obvious principles.

ABSTRACT Mathematics, or 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, absolute. of its beginning to descend is the Jy and generally considered, with-power called gravity; then, when out restriction to auy species in once the descent is commenced, 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 simple and abstract quantities, primatively considered in Pure Mathematics, are applied to sensible objects, as in Astronomy, Mechanics, Optics, &c.

to itself, it would persevere in it for ever; as we see in a stone cast from the hand, which continues to move after it is left by the cause that first gave it motion; and which 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 of the earth's motions.

That natural bodies are accelerated in their descent, is evident from various considerations, both a priori and posteriori. Thus we actually find, that the greater height a body descends from, the more rapidly it descends, the greater impression it makes, and the more intense is the blow with which it strikes the obstacle upon which it impinges.

motion, in finitum, there is a constant accession of subsequent ef forts of the same principle, which

* Sir Richard Phillips, in his Essays, maintains that there is no such force inherent in matter as the attraction of gravitation, and that the cause of a body's descend. ing to the earth, as well as all the other phenomena usually ascribed to the action of this force, are the natural and necessary results of the Some have attributed this acce- two motions of the earth. (See arleration 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 ter 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-lour reasoning.

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