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AS perfection is seldom, if ever, attained in matters of science, there is still a chance for improvement. And, as reason, under the hand of refinement, acquires new powers, so almost every science for a century past has acquired new beauties, except that of Arithmetick, which has remained almost stationary. Even in our standard works, we find the Rules completely arbitrary, without a single reason given, why they should produce their results. Thus, a science which may properly be considered as the greatest work of human wisdom, has been left shrouded in mystery. Those who have written on the subject, have left the principles of the rules unexplained, as if they were beyond the reach of human investigation, or beneath the dignity of scientifick research. The object of the author, in this treatise, is to disperse the gloom, and represent the science in its true light, freed from all its obscurity. In this work, the principles of each rule will be found illustrated by plain demonstrations; a thing that is novel in the science of Arithmetick, yet, in the opinion of the author, it is the true and only mode by which the principles of the science can be communicated or studied to advantage. It is hardly possible for the student to retain arbitrary rules: WORDS are soon forgotten, while timu is scarcely sufficient to efface PRINCIPLES from the memory.
After the Examples given under each rule, it was thought best to annex a list of questions and answers for the benefit of teachers and students, illustrating more fully the principles of the rules and the operations of the work. These are calculated to brighten the faculties of the student, and leave a lasting impression on the mind. This method of instructing by questions and answers, is.considered by many to be so important, that some of the best institutions in the Union are now conducted solely on this plan.
The author has dwelt on the Simple Rules beyond what is usual, in order to make the student thoroughly understand the principles of numbers; the relation which these rules bear to each other, and their practical application in business. He believes it will generally be al. lowed by instructors, that most authors are so deficient in these rules, that the student passes to the more intricate parts, while he is yet ignorant of the first rudiments of the science; the consequence is, his ambition is dampened; he can see no beauty in a science in which obsculrity is back of him, and impenetrable darkness before him.
Federal money being next in simplicity to whole numbers, is introduced immediately after each of the Simple Rules. This arrangemeni will be of great service to him who has but little time or opportunity 10 devote to the science, since an accurate knowledge of these rules
and that Currency, will enable him to transact correctly most of the business of life. The teacher will discover that in this work, the labour of the student is greatly abridged in acquiring a knowledge of the Reduction of Currencies; for, on inspection, it will be found that many authors have given nearly or quite THIRTY RULES to perform what is here embraced in two. Most authors have given several different rules for casting INTEREST, which produce different results; neither giving the proper authority of our courts, nor deciding which rule should be followed by the student in practical business. The consequence has been, that even business men follow as many different modes of casting interest, as there are different rules our books, the results of which are widely different; but in this work the author has observed a strict adherence to legal interest; quoting the decisions of our courts. As Commission or Factorage, Brokerage, Ensurance, and likewise Buying and Selling Stocks, are depending upon the same principles with Simple Interest, they are placed under the same running title, with demonstrations as they severally occur, showing that the principles involved are the same.
The Rule of Three has, in this treatise, been so illustrated by reducing its principles back to Multiplication and Division,
as to render it almost as simple as those rules. Since the Double Rule of Three, Practice, Single and Double Fellowship, Tare and Tret, Barter, Loss and Gain, Alligation, Discount and Annuities, belong strictly to the Single Rule of Three, they are therefore placed under the running title of that rule, and shown to depend upon the same principles, so that the student, when he understands the one, may properly be said to understand the whole; because he will at once perceive that he is acquainted
with all the principles on which they depend. In short, all the rules throughout the work will be found demonstrated, not merely by examples worked out, dignified with the name of demonstrations, but by plain reasoning on numbers.
THE AUTHOR. Watertoron, N. Y. July 4, 1831.
Equation of Payments,
Is the art or science which treats of the nature and properties of numbers. Unity or unit is that by which every thing is called one, or the beginning of a number. An integer, or whole number, is some entire quantity, as one, ten, fifteen, twenty, &c.; so called in opposition to fractions, which are broken numbers or parts of integers; as, one-half, two-thirds, or three-fourths.
We have two methods of expressing all numbers; the Arabick and the Roman. The Arabick method is by ten characters or figures, nine of which are significant of value, the tenth is insignificant, or of no value.
Notation and Numeration of Numbers. The characters employed in the Arabick method, are expressed and written as follows :: Unit, unity or one, 1
And a cipher, 0 These figures are also called digits, from the Latin word digitus, a finger. The first nine figures are called significant, because each expresses value of its own; the cipher is called insignificant, because it expresses no value of itself, yet it alters the value of those at the left hand; thus, the number, 9, expresses nine, join a 0, it becomes ninety, 90. All num
expressed by the repetition and different arrangement of these figures.
NOTE.—It is about a thousand years since the Arabick method of notation was introduced into Europe by the Arabs, when they established themselves in the southern provinces of Spain. Although they introduced the Arabick numeral figures and the principles of notation in Europe, yet they were not the original inventors: they derived their knowledge from In