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The Hindoos at least communicated a knowledge of it to the Arabians, and, as we are not able to trace it beyond the former people, they must have the honor of its invention. They do not, however, claim this honor, but refer it to the Divinity, declaring that the invention of nine figures, with device of place, is to be ascribed to the beneficent Creator of the universe.

But though the invention of modern Arithmetic is to be ascribed to the Hindoos, the honor of introducing it into Europe belongs unquestionably to the Arabians. It was they who took the torch from the East and passed it along to the West. The precise period, however, at which this was done, it is not easy to determine. It is evident, that our numeral characters and our method of computing by them were in use among the Arabians about the beginning of the eighth century, when they invaded Spain, and it is probable that a knowledge of them was soon afterwards communicated to the inhabitants of Spain, and gradually to those of the other European countries.

It is said, that the celebrated Gerbert, afterward Pope Sylvester II., returning to France from Spain, where he had been to acquire a knowledge of the Arabic or Indian notation, about the year 970, introduced it among the French.

About the middle of the eleventh century it is supposed to have been introduced into England by John of Basingstoke, Archdeacon of Leicester.

The Arabic characters, having been first used by astronomers, became circulated over Europe in their almanacs; but do not seem to have secured general adoption in Europe earlier than the twelfth or thirteenth century.

The science of Arithmetic, like all other sciences, was very limited and imperfect at the beginning, and the successive steps by which it has reached its present extension and perfection have been taken at long intervals and among different nations. It has been developed by the necessities of business, by the strong love of certain minds for mathematical science and numerical calculation, and by the call for its higher offices by other sciences, especially that of Astronomy. In its progress, we find that the Arabians discovered the method of proof by casting out the 9's, and that the Italians early adopted the practice of separating numbers into periods of six figures, for the purpose of enumeration. To facilitate the process of multiplication, this latter people also introduced, probably from the writings of Boethius, the Multiplication Table of Pythagoras.

The invention of the Decimal Fraction was a great step in the advancement of arithmetical science, and the honor of it has generally been given to John Muller, commonly called Regiomontanus, about the year 1464. It appears, however, that Stevinus, in 1582, wrote the first express treatise on the subject. The credit of first using the decimal point, by which the invention became permanently available, is given by Dr. Peacock to Napier, the inventor of Logarithms; but De Morgan says, that it was used by Richard Witt as early as 1613, while it is not shown that Napier used it before 1617. Circulating Decimals received but little attention till the time of Dr. Wallis, the author of the Arithmetic of Infinites. Dr. Wallis died at Oxford, in 1703.

The greatest improvement which the art of computation ever received was the invention of Logarithms, the honor of which is unquestionably due to Baron Napier, of Scotland, about the end of the sixteenth or the commencement of the seventeenth century.

The oldest treatises on Arithmetic now known are the 7th, 8th, 9th, and 10th books of Euclid's Elements, in which he treats of proportion and of prime and composite numbers. These books are not contained in the common editions of the great geometer, but are found in the edition by Dr. Barrow, the predecessor of Sir Isaac Newton in the mathematical chair at Cambridge. Euclid flourished about 300 B. C.

A century later, Eratosthenes invented a method, which is known as his “sieve,” for separating prime numbers from others.

The next writer on Arithmetic mentioned in history is Nico machus, the Pythagorean, who wrote a treatise relating chiefly to the distinctions and divisions of numbers into classes, as plain, solid, triangular, &c. He is supposed to have lived near the Christian era.

About the middle of the fourth century lived Diophantus, a celebrated mathematician, who, besides being the first known author on the subject of Algebra, composed thirteen books on Arithmetic, six of which are still extant.

The next writer of note is Boethius, the Roman, who, however, copied most of his work from Nicomachus. He lived at the beginning of the sixth century, and is the author of the well-known work on the Consolation of Philosophy.

The next writer of eminence on the subject is Jordanus, of Namur, who wrote a treatise about the year 1200, which was published by Joannes Faber Stapulensis in the fifteenth century, soon after the invention of printing.

The author of the first printed treatise on Arithmetic was Pacioli, or, as he is more frequently called, Lucas de Burgo, an Italian monk, who in 1484 published his great work entitled Summa de Arithmetica, &c., in which our present numerals appear under very nearly their modern form.

In 1522, Bishop Tonstall published a work on the Art of Computa tion, in the Dedication of which he says, that he was induced to study Arithmetic to protect himself from the frauds of money-changers and stewards, who took advantage of the ignorance of their employers. In his preparation for this work, he professes to have read all the books which had been published on this subject, adding, also, that there was hardly any nation which did not possess such books.

About the year 1540, Robert Record, Doctor in Physic, printed the first edition of his famous Arithmetic, which was afterward augmented by John Dee, and subsequently by John Mellis, and which did much to advance the science and practice of Arithmetic in England in its early stages. This work, which is now quite a curiosity, effectually destroys the claim to originality in some things of which authors much more modern have obtained the credit. In it we find the celebrated case of a will, which we have in the Miscellaneous Questions of Webber and Kinne, and which, altered in language and the time of making the testament, is the 2nd Miscellaneous Question in the present work. This question is, by his own confession, older than Record, and is said to have been famous since the days of Lucas de Burgo. In Record it occurs under the “ Rule of Fellowship.” Record was the author of the first treatise on Algebra in the English language.

In 1556, a complete work on Practical Arithmetic was published by Nicolas Tartaglia, an Italian, and one of the most eminent mathematicians of his time.

From the time of Record and Tartaglia, works on Arithmetic have been too numerous to mention in an ordinary history of the science. De Morgan, in his recent work (Arithmetical Books), has given the names of a large number, with brief observations upon them, and to this the inquisitive student is referred for further information in regard both to writers and books on this subject since the invention of Printing. It is remarkable that De Morgan knew next to nothing of any American works on Arithmetic. He mentions the “ American Accountant” by William Milns, New York, 1797, and gives the name of Pike (probably Nicholas Pike) among the names of which he had heard in connection with the subject. Of the compilation of Webber and the original work of Walsh, he seems to have been entirely ignorant.

The various signs or symbols, which are now so generally used to abridge arithmetical as well as algebraical operations, were introduced gradually, as necessity or convenience taught their importance. The earliest writer on Algebra after the invention of printing was Lucas de Burgo, above mentioned, and he uses p for plus and m for minus, and indicates the powers by the first two letters, in which he is followed by several of his successors.

After this, Steifel, a German, who in 1544 published a work entitled Arithmetica Integra, added

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considerably to the use of signs, and, according to Dr. Hutton, was the first who employed + and to denote addition and subtraction. To denote the root of a quantity he also used our present sign, originally r, the initial of the word radix, root. The sign to denote equality, was introduced by Record, the above-named English mathematician, and for this reason, as he says, that “ noe 2 thynges can be moar equalle,” namely, than two parallel lines. It is a curious circumstance that this same symbol was first used to denote subtraction. It was also employed in this sense by Albert Girarde, who lived a little later than Record. Girarde dispensed with the vinculum employed by Steifel, as in 3 + 4, and substituted the parenthesis (3+4), now so generally adopted. The first use of the St. Andrew's cross, X, to signify multiplication, is attributed to William Oughtred, an Englishman, who in 1631 published a work entitled Clavis Mathematicæ, or Key of Mathematics.

It was intended to notice several other works, ancient and modern, but the length to which this sketch has already extended forbids it.

We had thought of alluding to the ancient philosophic Arithmetic, and the elevated ideas which many of the early philosophers had of the science and properties of numbers. But a word must here suffice. Arithmetic, according to the followers of Plato, was not to be studied " with gross and vulgar views, but in such a manner as might enable men to attain to the contemplation of numbers ; not for the purpose of dealing with merchants and tavern-keepers, but for the improvement of the mind, considering it as the path which leads to the knowledge of truth and reality.” These transcendentalists considered perfect numbers, compared with those which are deficient or superabundant, as the images of the virtues, which, they allege, are equally remote from excess and defect, constituting a mean between them; as in the case of true courage, which, they say, lies midway between audacity and cowardice, and of liberality, which is a mean between profusion and avarice. In other respects, also, they regard this analogy as remarkable ; perfect numbers, like the virtues, are “ few in number and generated in a constant order; while superabundant and deficient numbers like vices, infinite in number, disposable in no regular series, and generated according to no certain and invariable law."

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NOTE TO TEACHERS. For the convenience of those who require a less extended course, beveral entire Articles, and some examples, have been marked o), to be omitted at the option of the teacher.

ARITHMETIC.

DEFINITIONS.

ARTICLE 1, QUANTITY is anything that can be increased, diminished, or measured ; as time, weight, lines, surfaces, and solids.

2. A unit is a single thing or quantity regarded as a whole.

3. An abstract unit is one that has no reference to any particular thing or quantity.

4. A concrete unit is one that has reference to some particular thing or quantity.

5. A number is an expression of quantity, representing either a unit or a collection of units.

6. An abstract number is a number whose unit is abstract; as, one, six, nine.

7. A concrete or denominate number is a number whose unit is concrete; as, one dollar, six pounds, nine men.

8. A simple number is a unit, or a collection of units, either abstract, or concrete of a single kind or denomination ; as, 1, 15, 1 book, 13 dollars.

9. The unit of measure of any quantity is one of the same kind with that by which the quantity is measured or compared; as, in the abstract number, six, the abstract unit is that of measure or comparison ; and in six pounds, the concrete unit, one pound, is that of measure or comparison.

10. ARITHMETIC is the science of numbers and the art of computing by them. It treats of the properties and relations

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