the science is not designed to be unnecessarily steep and rugged, the author does not desire to relieve the learner of all occasion for effort, nor make him feel that the "Hill of Science " is no hill at all, but only a fiction of former ages. The author's idea is, that, in order to become a thorough and accomplished arithmetician, one must study, and the National Arithmetic proposes no substitute for mental exertion. Still, it is not designed to be difficult beyond the necessities of the case, and no pupil, who is faithful to himself, will, it is thought, find reason to complain that enough is not done by way of suitable illustration to facilitate his progress. It is the opinion of some teachers, that no rules should be furnished the pupil to aid him in performing arithmetical questions, but that every pupil should form his own rules by the process of induction. But the author's experience has led him to a different conclusion, nor does he think that the insertion of proper rules, in a work like the present, interferes in the least with the necessity of study, or a thorough knowledge of the different numerical processes. The National Arithmetic is intended to be complete in itself; but the smaller works of the author will prepare the pupil for an easy entrance upon the study of it. The learner can omit the more difficult parts of the present work until he reviews it, if thought advisable by the teacher. A few rules, which are omitted in some works on Arithmetic at the present day, the author has thought best to retain, such as Practice, Progression, Position, Permutation, &c. For, though these rules may not in themselves be of great practical utility, yet, as they are well adapted to improve the reasoning powers, and give interest to the higher departments of arithmetical science, it is deemed desirable to place them within reach of the student. In closing these prefatory remarks, the author would earnestly recommend that the pupil be required to give a minute and thorough analysis of every question he performs, at least until he has proved himself familiar with all the principles involved in the rule under consideration, and also the manner of their application. He would further recommend a frequent and thorough review of the parts of the work which the pupil has gone over, the exercise having respect mainly to the principles involved in the preceding rules and examples. Bradford Seminary, September 1, 1847. CONTENTS. Questions to be performed by Analysis 17. Addition of Vulgar Fractions 18. Subtraction of Vulgar Fractions 19. Multiplication of Vulgar Fractions 20. Division of Vulgar Fractions 25. Multiplication of Decimals 30. Infinite or Circulating Decimals Reduction of Circulating Decimals 31. Addition of Circulating Decimals 32. Subtraction of Circulating Decimals 33. Multiplication of Circulating Decimals 34. Division of Circulating Decimals. 35. Mental Operations in Fractions, &c. 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 Phonicians, 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; C, 100,000; M, 1,000,000. The In the classification of numbers, as well as in the manner of ex pressing 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. 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. |