Example. Multiply 5 a −3 y + c 3 a 15 aa-6 ay + 3 ac DIVISION. General rule for the signs.-If the signs of the divisor and dividend be like, the sign of the quotient is; but if they be unlike, the sign of the quotient is The examples of division admit of three cases. Case I. When the divisor is simple, and a factor of every term of the dividend. Rule. Divide the coefficient of each term of the dividend by the coefficient of the divisor, and expunge out of each term the letter, or letters, in the divisor; the result is the quotient. Example. Divide 16abc by 4ac. 16 abc 4 ac From the method of notation, the quotient may be expressed thus, ; but the same quotient is more simply expressed by the rule just given, in this manner, 4b; the fours in 16 being four, and the like quantities ac, ac, being expunged. Case II. When the divisor is simple, but not a factor of the dividend. Rule. The quotient is expressed by a fraction; of which the numerator is the dividend, and the denominator, the divisor. Thus, the quotient of 3ab1, divided by 2mbc is the frac tion. 3 ab 2 mbc Case III. When the divisor is compound. Rule 1.-The terms of the dividend are to be arranged according to the powers of some one of its letters; and those of the divisor according to the powers of the same letter. 2. The first term of the dividend is to be divided by the first term of the divisor, observing the general rule for the signs; and this quotient, being set down for a part of the quotient wanted, is to be multiplied by the whole divisor, and the product subtracted from the dividend. If nothing remain, the division is finished; but if there be a remainder, it is to be taken for a new dividend. 3. The first term of the new dividend is next to be divided by the first term of the dividend, as before; and the quotient joined to the part already found, with its proper sign. The whole divisor is also to be multiplied by this part of the quotient, and the product subtracted from the new dividend; and thus the operation is to be carried on, till there be no remainder, or till it appear that there will always be a remainder. Example. 2a+3b) 8a+ 2 ab-15 b1 (4a-5b -10 ab-15 b* -10 ab-15 b* An equation declares the equality of two quantities; and is represented, as has been said before, by the sign =; as, 100s. = 51. A simple equation is that which contains only one power of the unknown quantity; thus, b+c=x, is a simple equation. Example 1.-If x+y=16, and x-y=12, to find the value of x and y. = 2 because a + y = 16. . Example 2.-What are the two numbers whose sum 65, and difference 9, make x and y the unknown numbers? How are the fundamental operations in algebra performed? What is addition in algebra, and how is it performed? The first case, second case, the third case? What is the rule for subtraction in algebra? What is the rule for the multiplication of simple quantities? What is the rule for multiplying compound quantities? What is the general rule for the signs to prepare for division in algebra? What is the first case and its rule? What is VOL. I. M the second case and its rule? What is the third case, and ASTRONOMY is that science which treats of the heavenly bodies; of their number, forms, motoins, and the laws by which those motions are governed. The term Astronomy, is derived from two Greek words, signifying star and law. It is a sublime and useful science. It may be regarded as the triumph of philosophy and the human mind. It has conferred upon mankind the most valuable benefits, as it certainly has been the grand conductor and friend of navigation. Astronomy is, without doubt, of very high antiquity; but its early history is too much disfigured by fabulous representations, to afford any firm ground of dependance. The Jewish historian, Josephus, asserts, that before the deluge, Seth and his posterity had made considerable advances in this science, and had engraved its principles upon two pillars, one of which existed in his time. The greater part of authors, however, agree in attributing the origin of astronomy to Egypt, or Chaldea. The Egyptians boasted of their colleges of priests, by whom astronomy was taught; and of the monument of Osyman dias, in which, it is asserted, was a golden circle 365 cubits in circumference, and one cubit thick. The upper face was divided into 365 equal parts, answering to the days of the year; and on every division were engraved the name of the day, and the heliacal rising of the several stars for that day; with conjectures, from their rising, concerning the weather. The position of the pyramids, those wonderful structures, seems to confirm the claim of the Egyptians to a very early knowledge of astronomy; for their faces are directed, with great precision, towards the four cardinal points of the compass; which circumstance indicates that they were acquainted with some method of drawing a correct meridional line, which is no easy operation. The Chaldeans had their temple of Belus, which was a very lofty tower, probably answering the purpose of an observatory. From the testimony of ancient writers, it appears that they taught the spherical form of the earth; that they knew the causes of eclipses of the moon; could foretel them as well as the appearance of comets; and that they attempted to measure the magnitude of the earth and the sun. From Chaldea, it seems that this science passed into Phenicia, and thence to Greece; where it was greatly improved by Thales, the Miletian, who travelled into Egypt, and brought back with him the first principles of the science. After him, Pythagoras greatly advanced the knowledge of astronomy; for he taught that the sun is the centre of our system, that the earth is round, that the moon is visible by reflecting the sun's rays, that |