one of the most ancient musical writers, was born at Tarentum, a city in Magna Graecia, now Calabria. He was the son of a musician, and it appears that he lived about the time of Alexander the Great and his successors. His Harmonies ARITHMETIC

Is a science which explains the properties of numbers, and shows the method or art of computing by them.

History of Arithmetic.

At what time this science was first introduced into the world, we can by no means determine. That some part of it, however, was coeval with the human race is absolutely certain. We cannot conceive how any man endowed with reason can be without some knowledge of numbers. We are indeed told of nations in America who have no word in their language to express a greater number than three; and this they call "pataararorincooaoa:" but that such nations should have no idea of a greater number than this, is absolutely incredible. Perhaps they may compute by threes, as we compute by tens; and this may have occasioned the notion that they have no greater number than three.

But though we cannot suppose any nation, or indeed any single person, ever to have been without some knowledge of the difference between greater and smaller numbers, it is possible that mankind may have subsisted for a considerable time without bringing this science to any perfection, or computing by any regular scale, as 10, 60, &c. That this, however, was very early introduced into the world, even before the flood, we may gather from the following expression in Enoch's prophecy, as mentioned by the apostle Jude:

"Behold, the Lord cometh with ten thousands of his saints." This shows, that even at that time men had ideas of numbers as high as we have at this day, and computed them also in the same manner, namely, by tens. The directions also given to Noah concerning the dimensions of the ark, leave us no room to doubt that he had a knowledge of numbers, and of measures likewise. When Rebekah was sent away to Isaac, Abraham's son, her relations wished she might be the mother of thousands of millions; and if they were totally unacquainted with the rule of multiplication, it is difficult to see how such a wish could have been formed.

It is probable, therefore, that the four fundamental rules of Arithmetic have always been known to some nation or other. No doubt, as some nations, like the Europeans formerly, and the Africans and Americans now, have been immersed in the most abject and deplorable state of ignorance, they might remain for some time unacquainted with numbers, except such as they had immediate occasion for; and, when they came afterwards to improve, either from their own industry, or hints given by others, might fancy that they themselves, or those from whom they got the hints, had invented what was known long before. The Greeks were the first European nation among whom arithmetic arrived at any degree of perfection. M. Goguet is of opinion, that they first used pebbles in their calculations: a proof of which he imagines is, that the word Ψηφιστειν, which comes from Ψηφις, a little stone, or flint, among other things, signifies to calculate. The same, he thinks, is probable of the Romans; and derives the word calculation from the use of little stones (calculi) in their first arithmetical operations.

If this method, however, was at all made use of, it must have been but for a short time, since we find the Greeks very early made use of the letters of the alphabet to represent their numbers. The 23 letters of their alphabet taken according to their order, at first denoted the numbers 1, 2, 3, 4, 5, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 100, 200, 300, 400, 500, 600, 700, and 800; to which they added the three following, 5, 5, D, to represent 6, 90, and 900. The difficulty of performing arithmetical operations by such marks as these may easily be imagined, and is very conspicuous from Archimedes's treatise concerning the dimensions of a circle.

The Romans followed a like method; and besides characters for each rank of classes, they introduced others for five, fifty, and five hundred. Their method is still used for distinguishing the chapters of books, and some other purposes. Their numeral letters and values are the following:

I V X L C D M One, five, ten, fifty, one hundred, five hundred, one thousand.

Any number, however great, may be represented by repeating and combining these according to the following rules:

1st, When the same letter is repeated twice, or oftener, its value is represented as often. Thus II signifies two; XXX thirty; CC two hundred.

2d, When a numeral letter of lesser value is placed after one of greater, their values are added: thus XI signifies