39. Miscellaneous Problems in Interest . . . 181 - 182 45. Insurance and Policies . . . . . . 48. Practice . . . . . . . . . 196 – 198 49. Equation of Payments . . . . . 199 – 201 50. Custom-House Business . . . . . . 201 - 205 . 53. Compound Proportion, or Double Rule of Three . . 217 - 221 55. Partnership, or Company Business 57. General Average . . . . . . 227 - 229 60. Involution ; Evolution, or the Extraction of Roots ; 61. Extraction of the Square Root 62. Extraction of the Cube Root . 63. Arithmetical Progression . or Soriani 257 - 261 64. Geometrical Series, or Series by Qu 67. Annuities at Compound Interest 69. Alligation . . . . . . . . 275 – 279 70. Permutations and Combinations Mensuration of Solids and Superficies 316 - 327 77. Tonnage of Vessels . . . . . . 83. Astronomical Problems . . . . . . 345 – 347 84. Miscellaneous Questions . . . . . 347 - 354 INTRODUCTION. HISTORY OF ARITHMETIC. The question, who was the inventor of Arithmetic, or in what age or among what people did it originate, has received different answers. 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 Phenicians, by Josephus to Abraham, and by many to the Egyptians. The opinion, however, rendered most probable, if not absolutely certain, by modern investigations is, that Arithmetic, properly so called, is of Indian origin, - that is, that the science received its first definite form and became the regular 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. And 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 for 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, 1000. 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, &c. They fre quently expressed any number of thousands by the letter or letters de. noting so many units, with a line drawn above ; thus, V, 5,000; VI, 6,000 ; X, 10,000 ; L, 50,000 ; 7, 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, other nations have adopted the vicenary, reckoning by twenties ; others the quinary, reckoning by fives ; and others 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 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 has attained an adoption all but universal. It was long supposed, that for our modern Arithmetic the world was indebted to the Arabians. But this, as we have seen, was 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 common use among the Arabians about the middle of the tenth century, 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. Their general adoption in Europe would not seem to have been 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, 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 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 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 radir, root. The sign = to denote |