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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 computation. Shortly after the dispersion of mankind, the sciences were carried by the descendants of Shem into Chaldaea 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 everything 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 Eratosthenes".

• Joseph. Antiq. b. i. c. 8. Abraham was a native of Ur in Chaldaea, 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 Chaldaeans about the time of their first settling in that country.

* “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.

* 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.

The Hebrews, Greeks, and Romans represented numbers by the letters of the alphabet peculiar to each nation. The most simple method of notation among the Greeks was, by making their 24 letters represent each a number in order, from 1 to 24; this method may be seen by referring to Homer's Iliad, or Xenophon's Cyropaedia, where it is employed in numbering the books : higher numbers were represented by small letters pointed underneath, by the capitals, and by inclosing the capitals with the Greek II, &c.

From Greece Arithmetic passed to the Romans, who do not seem to have made any improvement in the science; they merely adapted the letters of their alphabet to the numbers received from their masters. Their method, which is still employed in denoting dates, chapters, and sections in books, ought to be understood by every one, and is as follows. I stands for one, V five, X ten, L fifty, C one hundred, D five hundred, M. one thousands; these seven letters, differently placed or marked, were made to express all numbers. As often as any character is repeated, so many times its value is repeated; thus, II represents two, III three, XX twenty, XXX thirty, CC two hundred, MM two thousand. A less character placed on the left of a greater diminishes its value; thus, IV denotes four, IX nine, XL forty, XC ninety. A less character to the right of a greater increases its value; thus, VI denotes sir, VII seven, XI eleven, LX sixty,

* The derivation of these numerals is thus given by some : I, demoting initium, the beginning, was considered as the only fit representative of the first number, or one. V., (the ancient U,) being the fifth vowel, was with propriety put for five. X, being made up of two V's, represented two fives, or tem. C, centum, or one hundred. M, mille, or one thousand. L, being the half of the old C, which was square, was put for half a hundred, or fifty. D, dimidium mille, or half a thousand, five hundred. The D was frequently written IO, and the M, CIO ; hence these latter marks are sometimes put jor 500 and 1000 respectively.

CXX one hundred and twenty, DX five hundred and ten, DCC seven hundred, M, DCCCC, XC, IX one thousand nine hundred and ninety-nine. In some ancient books, records, and inscriptions, and on antique coins and medals, we meet with the C inverted; thus, IO denotes five hundred: every O added increases it tenfold; thus, IOO denotes five thousand ; CIO stands for one thousand, and a C and O added at the ends increase its value tenfold; thus, CCIOO denotes ten thousand, CCCIOOO one hundred thousand, CCCC10000 one million; a line over any number increases its value a thousand-fold; thus, WITI denotes eight thousand, X ten thousand, LXXX eighty thousand, CC two hundred thousand, MMM three million, &c. We have not the means of tracing the progressive improvements of Arithmetic among the ancients; judging from their works, (which however are not always to be depended on “,) there is reason to suppose that the science advanced. Beside Addition, Subtraction, Multiplication, and Division, the ancients possessed methods of extracting the Square and Cube Roots; they were acquainted with the theory of Proportions; Arithmetical and Geometrical Progression; and in general with the combinations of numbers, the reduction of ratios to their simplest form, &c. The ancient methods of notation were, however, but ill adapted to the practical operations of Arithmetic; and hence it is that the art, with respect to its practical part, must have made but slow progress. The destruction of

* Although the greater part of heathen antiquity has descended to us through the hands of the Greeks, yet their evidence must be received with caution, particularly that of the Helladians; they were a bigotted people, highly prejudiced in their own favour. There surely was never any nation so incurious and indifferent about the truth. Bryant's Analysis, vol. i. p. 143. 155,

the famous Alexandrian library, A. D. 642, has left us no particular treatise on the subject; we have, however, some of the most plain and useful properties of numbers in the seventh, eighth, ninth, and tenth books of Euclid's Elements, A.C. 280, and in the Arenarius of Archimedes, A. C. 220; there is likewise the Commentary of Eutocius on Archimedes' Treatise of the Circle, some fragments of Pappus", A. D. 400. The writings of Nicomachus, A. D. 100, which were published at Paris in 1538, and the treatise of Boethius', written at Rome in the sixth century, give us no very favourable idea of the ancient Arithmetic, which seems to have consisted principally of dry and tedious distinctions and divisions of numbers; so that on the whole, the acquisition of any considerable degree of knowledge in this most useful branch must have been attended with almost insurmountable difficulties". * Pappus was an eminent mathematician and philosopher of Alexandria; he lived in the fourth century after Christ; the greater part of his valuable writings are lost. His Mathematical Collections, in eight books, except the first, and part of the second, are still extant; parts of these have been published by the following authors, viz. Commandine, in a Latin translation with a commentary, 1558; Mersenne, 1664; Meibomius, 1655; Wallis, 1688; David Gregory, 1703; and Dr. Halley, 1706; also Dr. Hutton has given a brief analysis of these books in his Mathematical Dictionary, p. 187, 182. vol. ii. * Boethius was a celebrated Roman; he was put to death, A.D. 525, by Theodoric, king of the Ostrogoths, on suspicion of a conspiracy. During his confinement he wrote that excellent work De consolatione Philosophia. The best editions of his works are that of Hagenau, 4to, 1491, and that of Leyden, cum notis variorum, 1671. * Aldhelm, bishop of Shireburn, and one of the most learned men of the age, who flourished in the time of the Saxon Heptarchy, A. D. 700, complains bitterly of the difficulties he met with in learning Arithmetic, as almost surpassing the powers of the human mind. He thus writes to his friend Hedda, bishop of Winchester. “What shall I say of Arithmetic, whose long and intricate calculations are sufficient to overwhelm the mind, and throw it into despair 2 All the labour of my former studies, by which I made myself a complete master of several sciences, was trifling in com

parison of what this cost me.” Anglia Sacra, t. ii. p. 6, 7, quoted by Dr. Henry.

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