A new, easy, and very concise Rule, for 2. Put down three times the quotient figure for a reserved number, which multiply by the quotient figure, for a trial divisor, and find how often it is contained in the dividend; neglecting units and tens, for the next quotient figure, which annex to the reserved number, and multiply the result by it, placing the product under the trial divisor, two figures farther to the right, and add for the true divisor, which multiply by the quotient figure, and subtract, as in division; bring down the next period, and try as before, by the last divisor, for the next figure of the root. 3. Over the product of the reserved number, place the square of the annexed figure; also to the reserved | number, with its annexed figure, add double that annexed figure for a new reserved number, to which annex the new quotient figure; multiply and set down as before, and add the product to the two lines over it, together with the above-mentioned square, for the next true divisor:-repeat the operations as far as necessary according to the foregoing directions, bringing down periods of ciphers for decimals where requisite. Note.-When the divisor is not contained once in the dividend, a cipher is annexed to the reserved number, two to the divisor, and a period of three figures or ciphers to the remainder or dividend; also, in trying for the quotient figure, some allowance is to be made for the increase of the trial divisor, to form that of the true one, as in the common rule. This is especially to be noticed in trying for the second and third figures, since in these the proportionate increase is greatest; thus, the third divisor is to be considered as something more than the second, with the line above it added. EXAMPLE 1. Let it be required to extract the Cube Root of 122615327232. The different parts of the operation | in this example will be easily understood, by comparing them with the rule. The superposited square numbers have a mark over them, to shew they do not belong to the divisors after which they are placed. When the cube root of a number is required to many places of decimals, having formed half of them, the rest may be obtained by the rule of contracted division of decimals. Thus, in example 2d, having found the first ten figures of the root, try for the next figure, which here is 8; put down in its proper place the last figure of the product of this quotient figure, and the reserved number, (which here is 3,) and add it and the two lines above 119104 73923904 it, from the place under which it stands, for the divisor; and from the remainder cut off one figure less than those omitted in the old divisor, and work by the method of contracted division. The figures employed in the following example, are about 480, and are all that are necessary to be put down in the whole work, which gives the answer true to the last figure. The rule by approximation, already noticed, is the best of those which have come into general use; but even this rule, in order to obtain the answer true, to the same extent as in our example, will require nearly 2000 figures, or at least four times as many as are here introduced. Ex. 2. Let it be required to extract the Cube Root of 2, true to 20 places of figures. 2.000(1.2599210498948731647 1 1000 728 272000 46875000 4383021000 100242201000 4999808512000 If one figure more than the half of those required in the root be found by the rule, the increase or addition to the last divisor will be unnecessary, as in the following example. Ex. 3. Let it be required to extract the Cube Root of 22, true to 10 places of figures. 22.000(2.802039330 These examples will be sufficient to elucidate the Rule, and to shew its advantages. The truth of the Rule may be easily shewn by the resolution of a binomial cube, from which it will appear, that as the work proceeds, the greatest even cube is always taken from the periods of the given number as far as the work has been extended. Buffier's Singular Wager. TO THE EDITOR OF THE IMPERIAL SIR, THAT there is a something in us which we call Liberty, or Free-will, is as Thus, if two figures of the root be certain, as that we possess an immatefound, let the value of the first be de- rial principle which we call our Soul; noted by 10 a, and that of the second and though there are persons who by b, then it will appear that a3 is taken espouse a contrary opinion, according from the first period, according to the to whom, man is nothing but a curious first article of the rule; or, which machine, and of course no more acamounts to the same, 103 a3 is taken countable for his actions than a windfrom the two first periods. Again, the mill; yet the conduct of these men trial divisor (units and tens being sup- is ever found to be at variance with plied) is 3.102 a2, and the product the principles they profess: they are arising from the reserved number is pleased or offended with the actions of (3.10 a+b)× b=3.10 ab+b2 ; which others, and praise or blame them, just being added to the trial divisor, as the as those do who believe themselves rule directs, becomes 3.102 a2+3.10 and the rest of their species to be free ab+b2; and this, multiplied by b, agents. From this circumstance it is gives 3.102 a2b+3.10 ab2 + b3, which, justly inferred, that a consciousness of according to the rule, is taken from our freedom is as deeply rooted in our the remaining part of the two first pe- nature, as the love of being itself; and riods; hence, in the whole has been if it be possible entirely to erase it, it subtracted, 103 a3 + 3.102 a2 b + 3.10 can only be done by unnatural vioab2+b3, or the cube of 10 a+b, that lence. The arguments against the freeis, the cube of the two first figures of dom of human actions, are of the same the root: and this is the greatest cube complexion as the arguments of cercontained in the two periods b, being tain philosophers, against the existtaken the greatest number the opera- ence of matter and motion, and should tion will allow. Now, suppose another be treated in the same way; that is, figure found; then to the former re- opposed by an appeal to matter of fact. served number, with its annexed If we refer to our inward consciousfigure, viz. 3.10 a+b, 2 b is added, ness, we shall find, that we are but making 3.10 a+b for a new reserved little less certain that we are free, than number, which is three times the two that we think; nor do I doubt, that if first figures of the root; also having we could but fathom the bottom of added according to the rule, as part their minds, who advocate the docof the next divisor, the former divisor, trine of "fixed fate," we should find a 3.103 a2+3.10 ab+b2, the former pro- latent, though to themselves unnoticed, duct by the reserved number, 3.10 ab+consciousness that man is free; and b2, and also b2, we have 3.102a2+6.10 ab+3 b2, or 3.10 a+b × 10 a+b, which is the new reserved number, mul tiplied by the two first figures of the root. Hence the new reserved num- which, in certain circumstances, would discover itself even to themselves. An actual expedient, something like the following, would probably verify the correctness of my views. "You say that I am not free," says Buffier, addressing himself to a Necessitarian," and that it does not depend on the mere determination of my will and choice whether I shall move my hand or not. If that be the case, it must necessarily be decreed, that within a quarter of an hour hence, I thrice successively: I cannot, thereeither shall or shall not raise my hand fore, alter this necessary determination. This being supposed, in case I lay a wager on one side rather than the other, I can be a winner only one side; that is, either by laying that I am not a friend to gambling in any of its modes; but as the above is a very singular wager, I send it for your inspection: if approved, its insertion will oblige yours, respectfully, HISTORY OF ASTRONOMY. S. (Continued from col. 345.) Astronomy of the Greeks.-We must not expect to find any thing relative to Astronomy amonst the Greeks, of equal antiquity with what has been related of the Chaldeans and the Egyptians; but the lively imagination of this people, in order to exalt themselves, led them to convert almost all their great men into astronomers. When the Egyptian and Phoenician colonies arrived in Greece, they carried with them into that country the arts and sciences of their native land. So early as the 13th and 14th centuries before the Christian æra, the position of the stars, with regard to the circles of the sphere, was established with great exactness; a strong proof that the sphere described by Eudoxus was the production of a more perfect system of Astronomy, and that the Greeks merely changed the names of the constellations, in honour of the adventurous Argonauts. In this fabulous period of Grecian history, it is impossible to ascertain the state of astronomical science, or even to name the individuals who contributed to its progress. Hesiod and Homer, the most ancient writers among the Greeks, and who, according to Sir Isaac Newton, lived 870 years before the Christian æra, both mention several of the constellations. Hesiod, in particular, directs the farmer to regulate the time of sowing and harvest, by the rising and setting of the Pleiades; and informs us that Arcturus rose, in his time, as the Sun set, 60 days after the | winter solstice. Homer informs us, that the Pleiades, Orion, and Arcturus, were used in navigation. Thales, the Milesian, was one of the most ancient, as well as the most celebrated, astronomers of Greece. He is supposed to have been the first who travelled into Egypt in search of knowledge, and who brought from thence the first principles of that science, in which the Greeks afterwards made such surprising advances. We are told by Diogenes Laertius, that Thales determined the height of the pyramids, when in Egypt, by measuring the length of their shadows, when the Sun was 45 degrees in altitude, and when, of course, the lengths of the shadows of objects are equal to their perpendicular heights. According to Sir Isaac Newton, Thales lived about the 41st Olympiad, or 615 years before the Christian æra. That the conjecture of Sir Isaac is nearly correct, may be inferred from the time of that famous eclipse, predicted by Thales, which happened at the time the two armies, under Alyattes, king of Lydia, and Cyaxares, the Mede, were engaged. This being regarded by each party as an evil omen, inclined both to make peace. Diogenes Laertius observes, that Thales was the first who taught in Greece, that the true length of the solar year is 365 days; and Plutarch assures us, that he divided the earth into five zones by the polar circles, and tropics; that he was acquainted with, and described the obliquity of the ecliptic, and shewed that the equinoctial is cut at right angles by the meridians, which | human mind. He rose above vulgar pass through and intersect each other in the poles. Stanly says, that he held the Sun's diameter to be one 720th part of his annual orbit; which, if true, is a degree of precision much to be admired, in the rude knowledge of these early times. He is also said, by some, to have observed the exact time of the solstices, and from thence to have deduced the true length of the solar year; to have observed eclipses of the Sun and Moon; and to have taught that the Moon had no light of her own, but that she borrowed it from the Sun. Anaximander, the disciple of Thales, followed his master, in the career of discovery; and he seems to have been the first of the ancients, who ventured to explore the heavens with the eye of a philosopher. He is said to have invented or introduced the use of the gnomon into Greece; to have observed the obliquity of the ecliptic; and to have taught that the Earth was the centre of the universe, that it was spherical, and that the Sun was not less than it. He is said to have made the first globe; and to have set up a sun-dial at Lacedæmon, which is the first we read of amongst the Greeks; the knowledge of which, together with that of the pole and gnomon, some think, was brought from Babylon by Phercydes, who was contemporary with him. This Phercydes, together with Thales and many other ancient philosophers, held that water is the first principle of all natural bodies; and as this doctrine cannot be derived from any very obvious principles, some are of opinion that it was traditional, and that it has some relation to the earth's having been created originally in a fluid state. Anaximander died in the second year of the 58th Olympiad, or the 547th year before Christ. Anaxagoras was the third in succession from Thales in the Ionic school. He is said to have predicted the eclipse of the Sun, which, according to Thucydides, happened in the first year of the Peloponnesian war; and to have taught that the Moon was habitable, having plains, hills, and waters, as our Earth has. Contemporary with Anaxagoras, was Pythagoras, the Samian, whose sublime genius has scarcely been ever equalled: he was endowed with those rare talents, which seldom adorn the No. 5.-VOL. I. opinions and prejudices, and gave new light, not only to astronomy and the mathematics, but to every other branch of philosophy. He was instructed in the Greek learning in his youth, and afterwards travelled in Phoenicia and Egypt, being_recommended to king Amasis by Polycrates, governor of Samos, and by that means was admitted to familiar conversation with the Priests, and, for many years, to a participation of Egyptian learning. Pythagoras taught that the universe was composed of four elements; that it was round, and had the Sun in the centre of this mundane system: that the Earth was round also, like a globe, had antipodes, and that the Moon was enlightened by the rays of the Sun. He was of opinion that the Stars were worlds, containing earth, air, and ether; that the Moon was inhabited like our Earth; and that the Comets were a kind of wandering stars, which disappeared as they ascended towards the superior parts of their orbits, but appeared again at their return, after long intervals of time. (To be continued.) |