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, for the purpose of enumeration. To facilitate the process of multiplication, this latter people also introduced, probably from the writings of Boethius, the long neglected 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 Regiomontanus, about the year 1464. It appears, however, more properly to belong to Stevinus, who in 1582 wrote an 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.

The next writer on Arithmetic mentioned in history is Nicomachus, 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.

The next writer of note is Boethius, the Roman, who, however, copied most of his work from Nicomachus. He lived at the begin ning 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 Computation, 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 of 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 11th 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. He had also seen the Memoir of Zerah Colburn. Of the compilation of Webber and the original works of Walsh and Warren Colburn, 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 was followed by several of his successors. After this, Steifel, a German, who in 1544 published a work entitled Arithmetica Integra, added 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 Mathematica, 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 must not, however, omit to mention two American works, which have done much for the cause of practical Arithmetic in this country. These are the large work of Nicholas Pike, first published about 1787, and the little unpretending “First Lessons" in Arithmetic, by War ren Colburn. From the former of these many later authors have borrowed much that is useful, and the latter has exerted an influence on the method of studying Arithmetic greater, perhaps, than any other modern production. No better elementary work than that of Colburn has ever, it is believed, appeared in any language.

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 are, like vices, infinite in number, disposable in no regular series, and generated according to no certain and invariable law."

We conclude this brief sketch in the earnest hope that the noble science of numbers may ere long find some devoted friend who shall collect, arrange, and bring within the reach of ordinary students, much more fully than we have done, the scattered details of its longneglected history.

ARITHMETICAL SIGNS.

Sign of equality; as 12 inches1 foot signifies that 12 inches are equal to one foot.

+Sign of addition; as 8+6= 14 signifies that 8 added to 6 is equal to 14.

Sign of subtraction; as 8-6=2, that is, 8 less 6 is equal to 2.

× Sign of multiplication; as 7 x642, that is, 7 multiplied by 6 is equal to 42.

Sign of division; as 42÷6=7, that is, 42 divided by 6 is equal to 7.

12 Numbers placed in this manner imply that the upper number is to be divided by the lower one.

:: Signs of proportion; thus, 2: 4 :: 6: 12, that is, 2 has the same ratio to 4 that 6 has to 12; and such numbers are called proportionals.

12—3+4=13. Numbers placed in this manner show that 3

is to be taken from 12, and 4 added to the remainder. The line at the top is called a vinculum, and connects all the numbers over which it is drawn.

92 implies that 9 is to be raised to the second power; that is, multiplied by itself.

83 implies that 8 is to be multiplied into its square, or to be raised to the third power.

This sign prefixed to any number shows that the square root is to be extracted.

This sign prefixed to a number shows that the cube root is to be extracted.

Sometimes roots are designated by fractional indices, thus ; 9* denotes the square root of 9; 27* denotes the cube

root of 27.

() [] Parentheses and brackets are often used instead of a vinculum. Thus, (7-3) x 560 +3.

An edition of this work, without answers, is published for the accommodation of those teachers who prefer that the pupil should not have access to them.

A KEY, containing solutions and explanations, is also published for the convenience of teachers.

ARITHMETIC.

SECTION I.

ARITHMETIC is the science of numbers, and the art of computing by them.

The operations of Arithmetic are performed principally by Addition, Subtraction, Multiplication, and Division.

NUMERATION.

NUMERATION teaches to express the value of numbers, either by words or characters.

Numbers in Arithmetic are expressed by the ten following characters, which are called numeral figures; viz. 1 (one), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (seven), 8 (eight), 9 (nine), 0 (cipher, or nothing).

The first nine of these figures are called significant, as distinguished from the cipher, which is of itself insignificant.

Besides this value of the numerical figures, they have another value, dependent on the place which they occupy, when connected together. This is illustrated by the following table and its explanation.