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maining books of the elements both of Geometry and of Trigonometry are easily mastered; and then may come the higher equations. After this, the elements of the fluxionary or differential calculus may be studied; and then the doctrine of curves; and the more abstruse parts of the calculus. And, when this has been done, the student is ready for the mixed sciences; and, with such preparation, his progress can hardly fail in being both sure and rapid. Before, however, he proceeds too far in the mechanical sciences, it is advisable that he study the elements of Chemistry, just for the purpose of enabling him to decide what phenomena are mechanical and what are not. Such a course of study as we have been sketching, would, if rightly gone about, not occupy more time than most boys waste upon a few idle, or at least superficial, accomplishments, and it would fill society with men of another description than are now found in them. To contribute to such a result is one of the objects of this Dictionary; and we shall close this preface by mentioning the names of a few of the books which would be found useful as auxiliaries, or which would furnish those details which cannot be looked for in a work so small and so condensed as this:—For a general synopsis, perhaps no book is better than Nicholson; but Bezout, and some others of the French authors, are more systematic. Of Arithmetic, the books are many; but in English few of them are profound. Barlow's Theory of Numbers is among the best. The practical ones are without number, and many of them without variety. Joyce, Hutton, Bonnycastle, Walkingham, Vyse, and far from the worst, Dr. Hamilton, may be noted. In Algebra, Euler is at once the 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 the introduction of the modern or figurate arithmetic, for facilitating the business of calculation. Originally it appears to have been nothing more than a smooth piece of board, covered with sand, and served indifferently for arithmetical computations, or geometrical diagrams. The word calculate is derived from the calculi, or small pebbles, which were used along with the abacus. These were distributed in rows, each row having a different value, in the same manner as the ranks or places of figures have in the modern scale of numbers. As many rows were required 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, 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 commuon operations of arithmetic could be performed. ABERRATION, an apparent motion of the celestial bodies, occasioned by the progressive motion


of light, and the earth's motion in its orbit. This apparent motion is so minute, that it could never have been discovered by observations, unless they had been made with extreme care and accuracy. Dr. Bradley, astronomer royal, was led to it accidentally by the result of some careful observations, which he made with a view of determining the annual parallax of the fixed stars. If light be supposed to have a progressive motion, the position of the telescope, through which any celestial object is viewed, must be different from that which it must have been, if light were instantaneous; and, therefore, the place ineasured in the heavens will be dillerent from the true place. Clairaut explains the aberration, by supposing drops of rain to fall rapidiy 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 velocity of 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 aberration, arising from the combination of these two motions.



To find the Aberration of a Star in Latitude and Longitude. 1. The greatest aberration in latitude, is equal to 2011 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 20mgitude 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 93. Ursae Minoris, whose latitude is 75° 13'. Here the sine 75°13' = '9669; consequently, 201 x 9669 = 19.34, the greatest aberration in latitude. Also 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 5 = 9'671, the aberration 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, X ’866= 67.897, the aberration in longitude. If the earth be 30° from conjunction, the longitude is di. minished ; but if it be 30° from op

osition, it is increased.

o find the Aberration of a Star in

Declimation 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.

4. The greatest aberration in right ascension is equal to 2011 mul. tiplied 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 posi. tion 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 4}'', and much less in the other planets. As to the aberration in right ascension and declimation, 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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And between these numbers and nothing the aberration of the planets, in longitude, varies according to their situation. That of the sun, however, is invariable, being constantly 2011; and this may alter his declination, by a quantity which varies from 0 to 81, 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 ihe 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.

m all plano convex lenses, hav

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,

1n all plano convex lenses, having their convex surfaces exposed to the parallel rays, the longitudimal aberration of the extreme ray is equal to 14 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 spinerical lens, being no more than # of its thickness.

ABSCISS, Abscis/E, Abscissa, is any part of the diameter or axis of a curve, comprised between any fixed point, where all the abscisses begin, and another line called the ordinate, which is terminated in the curve. Commonly the abscisses are considered as commencing at the vertex of the curve; but this is not necessary, as they may have their origin in any other point; but, generally, when no condition is specified, they are understood as commencing 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.

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