mon numbers, or according to the tain number of times, would prodenary scale, will be, Denary 1, 2, 4, 8, 16 Ternary 1, 3, 9, 27, 81 Senary 1, 7, 49, 343, 2401 Denary 1, 10, 100, 1000, 10000 Duodenary1, 12, 144, 1728, 20736 If, however, any of the scale were adopted, all these expressions would be the same; and each would require one character less than the number of ones in m, thus, Denary 1, In which last Arithmetic receives various denominations, according to the nature of the numbers treated of: thus, Simple or Integral arithmetic is the method of calculating by simple or abstract numbers. Fractional arithmetic is the method of calculating by numbers which are less than the number 1, and is of two kinds, vulgar and decimal. For these, as well as for the arithmetic of exponents, logarithms, sines, &c. see those articles. Whatever may be the nature of the numbers which form the sub. ject of arithmetical computation, there are but two changes, of which any number is susceptible, namely, being made greater, and being made less; and, generally speaking, each of these may be performed in two different ways. When a number is increased by adding to it any number or numbers whatever, the process is called addition, and when it is increased by repeating itself any number of times, the process is called multiplication. When a number is diminished by taking away from it any number or numbers whatever, the process is called subtraction; and when it is diminished by taking away one or more numbers equal to that which is left, or when (which is the same thing,) the number left, if repeated a cer duce the given number, the process is called division. These four processes are called the four fundamental operations, and the methods of performing them the four fundamental rules of arithmetic; and each of the latter is the converse of the corresponding one of the former. Subtraction is the converse of addition, and division the converse of multiplication. For the method of performing each of these, see the articles Addition, Subtraction, &c. ARITHMETICAL, any thing relating to arithmetic. ARITHMETICAL complement of and COMPLEMENT. a Logarithm. See Anti-LOGARITHM ARITHMETICAL Mean, is the middle term of three quantities in arithmetical progression, or half the sum of any two proposed numbers. Progression ARITHMETICAL and Proportion. See PROGRESSION and PROPORTION. ARITHMETICAL Ratio, is the difference between any two adjacent terms in arithmetical progression. See ARITHMETICAL Scales. NOTATION. ARITHMETICAL Triangle. See TRIANGLE. name ARMILLARY Sphere, a given to an artificial sphere, which represents the several circles of the system of the world, put toge ther in their natural order. The armillary sphere revolves upon an axis, within a horizon, which is divided into degrees, and moveable every way upon a brass supporter. The other parts are the equinoctial, zodiac, ineridian, the two tropics, and the two polar circles. ARTIFICIAL Numbers, a term sometimes used for logarithms, lo garithmic sines, tangents, &c. ASCENDING, in Astronomy, is understood of those stars, &c. which are rising above the hori zon, in any parallel of the equator. ASCENDING Latitude, is the latitude of a planet when going towards the north pole. ASCENDING Node, is that point of a planet's orbit where it passes the ecliptic to proceed northward. ASCENSION, in Astronomy, is either right or oblique. Right ASCENSION of the Sun or of a Star, is that degree of the equinoctial (counted from the beginning of Aries,) which rises with the sun or star in a right sphere. Or, it is that degree and minute of the equinoctial, counted as before, which comes to the meridian with the sun or star. The sun's right ascension in time is useful to the practical astronomer in regular observations, who adjusts his clock by sidereal time. It serves also for converting apparent into siderial time; as, for instance, that of an eclipse of Jupiter's satellites, in order to know at what time it may be expected to happen by the clock. For this purpose, the sun's right ascension for the preceding noon, together with the increase of right ascension from noon, must be added to the apparent time of a known star passing the meridian: then subtract the sun's right ascension in time at noon, from the star's right ascension in time, the remainder is the apparent time of the stars passing the meridian nearly: from which the proportional part of the daily increase of the same right ascension from his apparent The reason of thus referring it to the meridian is, because that is always at right angles to the equinoctial, whereas the horizon is only so in a right or direct sphere. The right ascension stands opposed to the right descension, and corresponds to the longitude of places on the earth. Two fixed stars, that have the same right as eension, that is, which are at the same distance from the first point of Aries, or which is still the same, are in the same meridian, rise at the same time in a right sphere, or with respect to persons living un-time from noon being subtracted, der the equator. If they be not in the same meridian, the difference between their rising, or coming to the meridian, is the precise difference of their right ascension. But in an oblique sphere, where the horizon cuts all the meridians obliquely, different points of the meridian never rise or set together; so that two stars, on the same meridian, never rise or set at the same time; and the more ob-number of degrees contained in it. lique the sphere is, the greater is the interval of time between them. To find the right Ascension of the Sun or a Star. Say, for the sun, leaves the correct time of the star's passing the meridian. The sun's right ascension in time is also useful for computing the time of the moon, or a planet's, passing the meridian. The Arch of Right ASCENSION, is that portion of the equator, intercepted between the beginning of Aries and the point of the equator, which is the meridian: or it is the Oblique ASCENSION, is an arch of the equator, intercepted between the first point of Aries, and that point of the equator which rises, together with the star, &c. in an oblique sphere. The oblique ascension is counted from west to obli-east; and is greater or less, according to the different obliquities of the sphere. Is to the cosine of the sun's To the tangent of the right ascension. If the obliquity of the ecliptic, and the sun's declination were given, the proportion for the right ascension would be: As radius Is to the cotangent of the ob- Arch of Oblique ASCENSION, is the arch on which the oblique as cension is measured. the difference between the right ASCENSIONAL Difference, is and oblique ascension. To find the ascensional Difference of the Sun, having the Sun's Declination and the Latitude of the Place. So is the tangent of the sun's ASTROGNOSIA, signifies a knowledge of the fixed stars, their names, ranks, situations, &c. ASTRODICTICUM, an astronomical instrument invented by Mr. To the sine of the ascensional Weighel, by means of which sevedifference. ral persons may view the same When the latitude and declina-star at the same time. tion have the same name, the difference between the right as cension, and the ascensional difference, is the oblique ascension; and their sum is the oblique de scension; but when they are of contrary names, the sun is the oblique ascension, and the difference is the oblique descension. ASTROLABE, the name of an ancient astronomical instrument, very much resembling our armillary sphere. It is likewise the name of an instrument formerly much used at sea for ascertaining the altitude of the sun, stars, &c. which consisted of a brass ring about fifteen inches ASCII, are those inhabitants of the globe, who, at certain times of the year, have no shadow; such are all those who inhabit the tor-in diameter, graduated into derid zone. ASTEROIDS, in Astronomy, a name given by Dr. Herschel to the four new planets discovered by the foreign astronomers Piazzi, Olbers, and Harding. ASTERISM, in Astronomy, an ancient term, siguifying the same as CONSTELLATION. grees and minutes, and fitted with an index moveable about its centre, and carrying two sights; the whole being attached to a small brass ring for suspending the instrument at the time of observation. Modern astronomers use the term Astrolabe, to denote a stereogra phic projection of the sphere, either upon the plane of the equator, the eye being supposed to be in the pole of the world; or upon the plane of the meridian, when the eye is supposed in the point of the intersection of the equinoctial and horizon. ASTRONOMICAL, any thing relating to astronomy. ASTRONOMY, a mixed mathematical science, which treats of the heavenly bodies, their mo tions, periods, eclipses, magnitudes, &c. and of the causes on which they depend. That part of the science which relates to the motions, magnitudes, and periods of revolution, is called Pure or Plain Astronomy: and that which investigates the causes and laws by which these motions are regu lated, is called Physical Astronomy. History of Astronomy.—The early history of this science, like that of all others of ancient date, is too much disfigured by fabulous and allegorical representations, to admit of any regular or satisfactory elucidation. It is probable, however, that some knowledge of this kind must have been nearly coeval with the formation of society. ASTRAL, depending or belonging Many traces of it have been to the stars as ASTRAL year, &c. I found amongst various nations, ASTRÆA, in Astronomy, a name given by some authors to the sign Virgo. which show that several of the employ the time the sun moves 1o most remarkable celestial pheno- in the ecliptic. Their sidereal mena must have been observed, year consists of 365d, 6h, 12m, 308; and a knowledge of them dissemi-and the tropical, of 365d, 5h, 50m, nated, at a very remote period. 35s. They assign inequalities to But in what age or country the the motions of the planets, answerscience first originated, or by ing very well to the annual paralwhom it was gradually methodised lax, and the equation of the cenand improved, is extremely un- tre. Most authors, however, fix the certain. M. Bailly, in his elaborate his-origin of astronomy and astrology tory of ancient and modern astro- either in Chaldea, or in Egypt; nomy, endeavours to trace the ori- and, accordingly, among the an gin of this science among the Chal- cients, we find the word Chaldean deans, Egyptians, Persians, Indi- often used for astronomer, or asans, and Chinese, to a very early trologer. Both of these nations period. And thence, he maintains, pretended to a very high antiquity, that it was cultivated in Egypt and and claimed the honour of produ Chaldea 2800 years before Christ; cing the first cultivators of this in Persia, 3209; in India, 3101; and science. The Chaldeans boasted in China, 2952 years before that of their temple, or Tower of Belus, æra. He also apprehends, that and of Zoroaster, whom they placed astronomy had been studied even 5000 years before the destruction long before this distant period, of Troy: while the Egyptians spoke and that we are only to date its with equal pride of their colleges revival from this time. of priests, where astronomy was taught; and of the monument of Osymandyas, in which, it is said, there was a golden circle of 365 cubits in circumference, and one cubit thick, divided into 365 equal parts, answering to the days of the year, &c. In investigating the antiquity and progress of astronomy among the Indians, M. Bailly examines and compares four different sets of astronomical tables of the Indian philosophers; namely, that of the Siamese, explained by M. Cassini in 1689; that brought from India From Chaldea and Egypt, the by M. le Gentil, of the Academy science of astronomy passed into of Sciences; and two other manu- Phenicia, and was by that people script tables, found among the applied to the purposes of navigapapers of the late M. de Lisle; all tion, whence they became masters of which he found to accord toge- of the sea, and of almost all the ther, and all referring to the meri- commerce in the world. The dian of Benares. It appears that Greeks, it is probable, derived the fundamental epoch of the In- their astronomical knowledge chief. dian astronomy, is a remarkablely from the Egyptians and Pheniciconjunction of the sun and moon, ans, by means of several of their which took place at the distance countrymen who visited those naof 3102 years before Christ: and tions for the purpose of learning M. Bailly informs us, that by our the different sciences. most accurate astronomical tables, such a conjunction did really hap pen at that time. He farther observes, that at present the Indians calculate eclipses by the mean motions of the sun and moon commencing at a period 5000 years dis Several of the constellations are mentioned by Hesiod and Homer, who lived 870 years before Christ. The knowledge of the Greeks in this science, was greatly improved by Thales the Milesian, and others, who travelled into Egypt. Thales was born about 640 years before The solar year of the Brahmins Christ; and was the first among of Tervalore is divided into twelve the Greeks who observed stars, the unequal months, each being equal solstices, the eclipses of the sun to the time the sun occupies in and moon, and predicted an eclipse moving through a sign; and in of the sun. The science was fartheir calculations for a day, theyther cultivated and extended by tant. his successors, Anaximander, Anax- and motions of the heavenly boimenes, and Anaxagoras; but most dies. especially by Pythagoras, who having resided a long time in Egypt, &c. brought thence the learning of the Egyptians, taught the same in Greece and Italy, and founded the sect of the Pythagoreans. He taught that the sun was in the centre of the universe; that the earth was round, and people had antipodes; that the moon reflected the rays of the sur, and was inhabited like the earth; that comets were a kind of wandering stars, disappearing in the farther parts of their orbits; that the white colour of the milky way was owing to the united brightness of a great multitude of small stars; and he supposed that the distances of the moon and planets from the earth, were in certain harmonic proportions to one another. Philolaus, a Pythagorean, who flourished about 450 years before Christ, asserted the annual motion of the earth about the sun; and not long after, the diurnal motion of the earth, on its own axis, was taught by Hicetus a Syracusian. About the same time, Meton and Euctemon flourished at Athens, where they observed the summer solstice 432 years before Christ; and observed the risings and settings of the stars, and to what seasons they answered. Meton also invented the cycle of nineteen years, which still bears his name. Hipparchus, who flourished about 140 years before Christ, was the first who applied himself to the study of every part of this science; and, as we are informed by Ptolemy, made great improvements in it: he discovered that the orbits of the planets are eccentric, that the moon moved slower in the apogee than in her perigee, and that there was a motion of anticipation of the moon's nodes; he constructed tables of the motions of the sun and moon, collected accouuts of such eclipses, &c. as had been made by the Egyptians and Chaldeans, and calculated all that were to happen for 600 years to come: he discovered that the fixed stars changed their places, having a slow motion of their own from west to east: he corrected the Calippic period, and pointed out some errors in the method of Erastosthenes for measuring the circumference of the earth; he computed the sun's distance more accurately than any of his prede cessors: but his chief work is a catalogue which he made of the fixed stars, to the number of 1022, with their longitudes and latitudes, and apparent magnitudes; which, with most of his other observations, are preserved by Ptolemy in his Almagest. But little progress was made in parchus to that of Ptolemy, who astronomy from the time of Hipwas born at Pelusium in Egypt, in the first century of the Christian era, and who made the greatest part of his observations at the ce lebrated school of Alexandria in that country. Profiting by the ob Eratosthenes, who was born at Cyrene in the year 271 before Christ, measured the circumference of the earth; and being invited from Athens to Alexandria, by Ptolemy Euergetes, and made keeper of the royal library there, he set up for that prince those ar-servations of Hipparchus, and other millary spheres, which Hipparchus and Ptolemy afterwards employed so successfully in observing the heavens. He also determined the ancient astronomers, he formed a system of his own, which, though erroneous, was followed for many ages by all nations: he compiled distance between the tropics to be the Almagest, which contained the of the whole meridian circle, observations and collections of which makes the obliquity of the Hipparchus, and others of his preecliptic, in his time, to be 23° 51. decessors in astronomy; a perThe celebrated Archimedes also formance which will ever be valucultivated astronomy, as well as able to the professors of that scigeometry and mechanics; and con-ence. This work was preserved structed a kind of planetarium, or from the conflagration of the Alex orrery, to represent the phenomena andrian Library, and translated |