Denominate Fractions . . . 185 Addition of Decimals . . . 200 Subtraction of Decimals . . 202 Multiplication of Decimals . 203 Division of Decimals . . . 206 Reduction of Repetends . . 219 Transformation of Repetends 222 Conjoined Proportion . . . 248 Discount and Present Worth 2S6 Fire and Marine Insurance . 335 Reduction of Currencies . . 34! Arbitration of Exchanges. . 358 Alligation Alternate . . . 364 Extraction of the Square Root 373 Extraction of the Cube Root 378 Arithmetical Progression . . 395 Geometrical Progression . . 400 Multiplication of Duodecimals 421 Division of Duodecimals . . 423 Mensuration of Surfaces . . 425 INTRODUCTION. HISTORY OF ARITHMETIC. It is difficult to determine who was the inventor of Arithmetic, or in what age or among what people it originated. In ordinary history, we find the origin of the science attributed by some to the Greeks, by some to the Chaldeans, by some to the Phoenicians, by Josephus to Abraham, and by many to the Egyptians. -The opinion, however, which modern investigations have rendered most probable, is, that Arithmetic, properly so called, is of Indian origin, — that is, that the science received its first definite form, and became the germ of modern Arithmetic, in the regions of the East. It is evident, from the nature of the case, that some knowledge of numbers and of the art of calculation was necessary to men in the earliest periods of society, since without this they could not have performed the simplest business transactions, even such as are incidental to an almost savage state. The question, therefore, as to the invention of Arithmetic, deserves to be considered only as it respects the origin Of the science as we now have it, and which, as all scholars admit, has reached a surprising degree of perfection. In this sense the honor of the invention must be awarded to the Hindoos. The history of the various methods of Notation, or the different means by which numbers have been expressed by signs or characters, is one of much interest to the advanced and curious scholar; but the brevity of this sketch allows us barely to touch upon it here. Among the ancient nations which possessed the art of writing, it was a natural and common device to employ letters to denote what we express by our numeral figures. Accordingly we find, that, with the Hebrews and Greeks, the first letter of their respective alphabets was used lor 1, the second for 2, and so on to the number 10, — the latter, however, inserting one new character to denote the number 6, and evidently in order that their notation might coincide with that of the Hebrews, the sixth letter of the Hebrew alphabet having no corresponding one in the Greek. The Romans, as is well known, employed the letters of their alphabet as numerals. Thus I denotes 1; V, 5; X, 10; L, 50; C, 100; D, 500; and M, 1,000. The intermediate numbers were expressed by a repetition of these letters in various combinations; as, II for 2; VI for 6; XV for 15; IV for 4; IX for 9, &e. They frequently expressed any number of thousands by the letter or letters denoting so many units, with a line drawn above; thus, V, 5,000; VI, 6,000; X, 10,000 ; X, 50,000; C, 100,000; M, 1,000,000. In the classification of numbers, as well as in the manner of expressing them, there has been a great diversity of practice. While we adopt the decimal scale and reckon by tens, the aborigines of Mexico, according to Humboldt, and some of the early nations of Europe, adopted the vicenary, reckoning by twenties; some of the Indian tribes, and several of the African tribes, use the quinary, reckoning by fives; and the Chinese for more than 4,000 years have used the binary, reckoning by twos. The adoption of one or another of these scales has been so general, that they have been regarded as natural, and accounted for by referring them to a common and natural cause. The reason for assuming the binary scale probably lay in the use of the two hands, which were employed as counters in computing; that for employing the quinary, in a similar use of the five fingers on either hand; while the decimal and vicenary scales had respect, the former to the ten fingers on the two hands, and the latter to the ten fingers combined with the ten toes on the naked feet, which were as familiar to the sight of a rude, uncivilized people as their fingers. — It is an interesting circumstance, that in the common name of our numeral figures, digits (digiti) or fingers, we preserve a memento of the reason why ten characters and our present decimal scale of numeration were originally adopted to express all numbers, even of the highest order. It is now almost universally admitted, that our present numeral characters, and the method of estimating their value in a tenfold ratio from right to left, have decided advantages over all other systems, both of notation and numeration, that have ever been adopted. There have been those, as Leibnitz and De Lagni, who have advocated the binary scale ; a few, with Claudius Ptolemy, have claimed advantages for the sexagenary scale, or that by sixty; and there are those who think that a duodecimal scale, and the use of twelve numeral figures instead of ten, would afford increased facility for rapid and extensive calculations; but most mathematicians are satisfied with the present number of numerals and the scale of numeration, which have attained an adoption all but universal. It was long supposed, that for our modern Arithmetic the world is indebted to the Arabians. But this, as we have seen, is not the case. 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 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. 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 Nieomachus. 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 |