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 paxarrarvorinouroac: 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 ἀριθμός, which comes from ἀριθμέω, 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 24 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, r, s, Θ, 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.

2nd. When a numeral letter of lesser value is placed after one of greater, their values are added: thus XI signifies signifies eleven, LXV sixty-five, MDCXXVIII one thousand five hundred and twenty-eight.

3d. When a numeral letter of lesser value is placed before one of greater, the value of the lesser is taken from that of the greater: thus IV signifies four, XL forty, XC ninety, CD four hundred.

Sometimes IO is used instead of D for 500, and the value is increased ten times by annexing O to the right hand.

Thus IO signifies 500. Also CIO is used for 1000 IIOO 5000 CCIOO for 10000 IIOOO 50000 CCCIOO for 100000

Sometimes thousands are represented by drawing a line over the top of the numeral, V being used for five thousand, L for fifty thousand, CC, two hundred thousand.

About the year of Christ 200, a new kind of arithmetic, called sexagesimal, was invented, as is supposed, by Claudius Ptolemaeus. The design of it was to remedy the difficulties of the common method, especially with regard to fractions. In this kind of arithmetic, every unit was supposed to be divided into 60 parts, and each of these into 60 others, and so on; hence any number of such parts were called sexagesimal fractions; and to make the computation in whole numbers more easy, he made the progression in these also sexagesimal. Thus from one to 59 were marked in the common way: then 60 was called a sexagesima prima, or first sexagesimal integer, and had one single dash over it; so 60 was expressed thus I'; and so on to 59 times 60, or 3540, which was thus expressed LIX'. He now proceeded to 60 times 60, which he called a sexagesima secunda, and was thus expressed I''. In like manner, twice 60 times 60, or 7200, was expressed by II''; and so on till he came to 60 times 3600, which was a third sexagesimal, and expressed thus, I'''. If any number less than 60 was joined with these sexagesimals, it was added in its proper characters without any dash; thus I'XV represented 60 and 15, or 75; I'VXXV is four times 60 and 25, or 265; X'II'XV, is ten times 3600, twice 60 and 15, or 36, 135, &c. Sexagesimal fractions were marked by putting the dash at the foot, or on the left hand of the letter: thus I,, or 'I denoted 60; I,, or "I, 7200, &c.

The most perfect method of notation, which we now use, came into Europe from the Arabs, by the way of Spain. The Arabs, however, do not pretend to be the inventors of them, but acknowledge that they received them from the Indians. Some there are indeed, who contend that neither the Arabs nor the Indians were the inventors, but that they were found out by the Greeks. But this is by no means probable; as Maximus Planudes, who lived towards the close of the 13th century, is the first Greek who makes use of them: and he is plainly not the inventor; for Dr Wallis mentions an inscription on a chimney in the parsonage house of Helendon in Northamptonshire, where the date is expressed by M° 133, instead of 1133. Mr Luffkin furnishes a still earlier instance of their use, in the window of a house, part of which is a Roman wall, near the market place in Colchester; where between two carved lions stands an escutcheon with the figures 1090. Dr Wallis is of opinion that these characters must have been used in England at least as long ago as the year 1090, if not in ordinary affairs, at least in mathematical ones, and in astronomical tables. How these characters came to be originally invented by the Indians we are entirely ignorant.

The introduction of the Arabian characters in notation did not immediately put an end to the sexagesimal arithmetic. As this had been used in all the astronomical tables, it was for their sakes retained for a considerable time. The sexagesimal integers went first out, but the fractions continued till the invention of decimals.

The oldest treatises extant, upon the theory of arithmetic, are the seventh, eighth, and ninth books of Euclid's Elements, where he treats of proportion and of prime and composite numbers; both of which have received improvements since his time, especially the former. The next, of whom we know anything, is Nicomachus the Pythagorean, who wrote a treatise of the theory of arithmetic, consisting chiefly of the divisions and divisions of numbers into classes, as plain, solid, triangular, quadrangular, and the rest of the figurate numbers as they are called, numbers odd and even, &c., with some of the more general properties of the several kinds. This author is, by some, said to have lived before the time of Euclid; by others, not long after. His arithmetic was published at Paris in 1538. The next remarkable writer on this subject is Boethius, who lived at Rome in the time of Theodoric the Goth. He is supposed to have copied most of his works from Nicomachus.

From this time no remarkable writer on arithmetic appeared till about the year 1200, when Jordanus of Namur wrote a treatise on this subject, which was published and demonstrated by Joannes Faber Stapulensis in the 15th century, soon after the invention of printing. The same author also wrote upon the new art of computation by the Arabic figures, and called this book Algorithmus Demonstratus. Dr Wallis says, this manuscript is in the Savilian library at Oxford, but it hath never yet been printed. As learning advanced in Europe, so did the knowledge of numbers; and the writers on arithmetic soon became innumerable. About the year 1464, Regiomontanus, in his triangular tables, divided the radius into 10,000 parts, instead of 60,000; and thus tacitly expelled the sexagesimal arithmetic. Part of it, however, still remains in the division of time, as of an hour into 60 minutes, a minute into 60 seconds, &c. Ramus in his arithmetic, written about the year 1550, and published by Lazarus Schonerus in 1586, used decimal periods in carrying on the square and cube roots to fractions. The same had been done before by our countrymen Buckley and Record; but the first who published an express treatise on decimals was Simon Stevinus, about the year 1582. As to the circulating decimals, Dr Wallis is the first who took much notice of them. He is also the author of the arithmetic of infinities, which has been very usefully applied to geometry. The greatest improvement, however, which the art of computation ever received, is the invention of logarithms. The honour of this invention is unquestionably due to Baron Napier of Merchiston in Scotland, about the end of the 16th or beginning of Chap. I.