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of which had been hitherto sought for by other methods in vain.

The following are the names of some of those who since the commencement of the eighteenth century, have excelled in this and other branches, viz. Mad. Agnesi, D'Alembert, Atwood, De Billy, James, John, and Daniel Bernoulli, Bezout, Borda, Birch, Batten, Browne, Le Bas, Bossut, Barlow, Bonnycastle, Bridge, Cousin, Courtivron, Cotes, Colson, Clairaut, Cramer, Condorcet, Craig, C. and G. Cooke, Christie, Demoivre, Dalby, Dealtry, Dodson, Euler, Emerson, Fontaine, Facio, Fagnanus, Frend, Farish, La Grange, Guisnée, Glenie, Olinthus Gregory, L'Hôpital, La Hire, Hayes, Hornbuckle, Hermann, Hutton, Hustler, Hellings, Jacquier, Jones, Kirkby, Kelly, De Lagni, Landen, Littledale, Manfredi, Monmort, Maclaurin, Montucla, Maseres, Milner, Nicole, D'Omerique, Ozanam, Pemberton, Prestet, Pingré, Peacock, Riccati, Reyneau, Robertson, Rigaud, Sterling, Saunderson, Le Sieur, Saurin, Simson, T. Simpson, Sowerby, B. Taylor, M. Taylor, Turner, Viviani, Varignon, Vince, Waring Wolfius, Watson, Woodhouse, Wood, &c. Astronomy has been cultivated during the same space by many learned men, among which the following are some of the principal, viz. Adams, Bradley, Bouguer, Bailli, Bulkley, the Cassinis, La Caille, Ferguson, Halley, Harding, Herschel, Juan, Koenig, Keill, La Lande, Long, Lax, Maupertuis, Mayer, Maskelyne, Olbers, Pound, Smith, Wolloston, &c. Some of these, with many others, excelled in various branches of the Mathematics, besides those we have ascribed to them; to particularize their inventions, improvements, and excellencies, with just discrimination, would far exceed our prescribed limits; and fully to understand them, recourse must be had to a great variety of modern treatises on every branch of mathematical science.

Thus we have endeavoured to shew, in a brief and general manner, the nature, great importance, and use of mathematical knowledge; and to point out a few of the leading facts in its history. The reader need not be informed, that by the improvements and discoveries in science, which have taken place during the three last centuries, and the application of mathematical and physical knowledge to Civil Polity, the Arts, Commerce, and Agriculture, the present generation enjoys advantages superior beyond comparison to those possessed by former

ages. Our own country was, but a few centuries ago, an overgrown wilderness, a prey to the wildest superstition, and scarcely supplied the bare necessaries of life to its scanty and savage inhabitants. Now, the conveniences and luxuries of every kingdom in the world are poured in, and added to the produce of our own, constituting a rich abundance for the supply of every want, and the gratification of almost every wish; it follows then, that our obligations to Providence are proportionably greater than those of former ages. If the ox knows his owner, and the ass his master's crib, let us not, more stupid and ungrateful than they, while we live in the enjoyment of infinitely superior benefits, be less dutiful in our attachment, nor overlook the kind hand which supplies them. Possessing more ample means than our ancestors possessed, it is incumbent on us to improve our advantages, by the strict and faithful observance of every religious, moral, and social duty.








Numerorum notitia cuicunque primis saltem literis eruditio necessaria est. QUINTILIAN.

ARITHMETIC, or the science of Numbers, is justly

considered as the basis of all the other mathematical sciences; and therefore a sufficient acquaintance with its principles and elementary rules ought to be acquired before any of the other branches are attempted.


Arithmetic holds a distinguished rank among the mathematical sciences; it even surpasses them all in usefulness its universal application to the common concerns of life renders it a part of knowledge not merely desirable, but necessary to every one who wishes to be serviceable to society, to manage his own private affairs well, and to guard against fraud and imposition. Nothing satisfactory can be offered respecting the origin and invention of Arithmetic; like almost every The word is derived from the Greek agıðμos, number, and μετρίων to

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other useful art, its beginning must have been extremely rude and simple, the fruit of pure necessity, and it must have originated in the first ages of the world, when men began to form societies; for it is not easy to conceive how social intercourse could have been maintained, differences and disputes adjusted, bargains made, and trafficking carried on, without the necessary aid of compu


Shortly after the dispersion of mankind, the sciences were carried by the descendants of Shem into Chaldæa and the East; in these countries Arithmetic was cultivated probably at an earlier period than in any other. The Phoenicians, who were descended from Canaan, the son of Ham, and settled on the eastern coast of the Mediterranean sea, were the people who first of any addicted themselves to commerce, to which they made navigation subservient; and as they must have practised Arithmetic to a great extent in their numerous mercantile transactions, succeeding nations have ascribed to them the invention.

Josephus informs us, that Abraham, having acquired

The Phoenicians inhabited the sea-coast, extending, according to Ptolemy, from the river Eleutherus on the north, to Pelusium on the south. They are called in the sacred writings Canaanites, and are remarkable for the series of awful calamities and judgments which a long and uninterrupted course of the most abandoned profligacy had brought upon them. They were, during the captivity of the Israelites in Egypt, (from 1635 to 1491, A. C.) considered as a great and powerful people. Their mercantile spirit and excessive riches are mentioned by the prophets Isaiah and Ezekiel, both of whom denounce the impending judgments of the Almighty on their pride and obduracy. Profane authors speak of their great industry, and represent them as the inventors of letters, arithmetic, commerce, navigation, and almost every thing that is useful.

Flavius Josephus was born at Jerusalem A. D. 37, and died A. D. 93; he was equally great as an historian and an orator, as is witnessed by his "History of the Antiquities and Wars of the Jews." His "Discourse on the Martyrdom of the Maccabees" is a masterpiece of eloquence; he is ealled by St. Jerome, "The Livy of the Greeks,"

a knowledge of Arithmetic in the East, was the first who instructed the Egyptians in the art. By the Egyptian priests Arithmetic was cultivated with ardour, and constituted no inconsiderable part of their theology and philosophy. The Grecian philosophers, who travelled into the East in quest of knowledge, transmitted this science from Egypt into Greece, where it must (in common with the other sciences) have received considerable improvements; among which the invention of the Multiplication Table is ascribed to Pythagoras, and a method of determining the Prime Numbers to Eratosthe



Joseph. Antiq. b. i. c. 8. Abraham was a native of Ur in Chaldæa, from whence he was driven by a famine into Egypt. If the account given by Josephus be true, we are sure that Arithmetic must have been known and practised by the Chaldæans about the time of their first settling in that country.

d" The combinations of numbers constituted one of the principal objects of his researches; and all antiquity testifies that he carried them to the highest degree."..... "He attached several mysterious virtues to numbers, and swore by nothing but the number four, which was to him the number of numbers. In the number three likewise he discovered various marvellous properties; and said, that a man perfectly skilled in Arithmetic possessed the sovereign good." It is supposed by some, that these expressions, and others of a like tendency ascribed to the ancient philosophers, are not to be understood literally, but that they have a figurative and hidden meaning unknown to us.

e Prime Numbers are such as cannot be divided by any number greater than unity without a remainder; the rest are called composite. The ingenious method alluded to above was called, "The Sieve of Eratosthenes :" for some account of it, see Dr. Horsley's paper in the Philosophical Transactions, vol. 62. p. 327. Eratosthenes was a native of Cyrene, a city of Lybia; he was the second person entrusted with the care of the Alexandrian library: grammar, poetry, philosophy, and mathematics, were the subjects that engaged his affections, especially the latter. He measured the obliquity of the ecliptic, making it only about 20 degrees; he also measured a degree of the meridian, and thence determined with tolerable accuracy the circumference of the earth. The invention of the armillary sphere is ascribed to him; and his consummate skill acquired him the names of The second Plato-The Cosmographer and Geometer of the World, &c. He starved himself to death, A. C. 194, in the 82d year of his age. Of his compositions a few fragments only remain.

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