and that Currency, will enable him to transact correctly most of the business of life. The teacher will discover that in this work, the la bour 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 conse quence has been, that even business men follow as many different modes of casting interest, as there are different rules in 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 redu eing 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 ac quainted 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. Watertown, N. Y. July 4, 1831. Simple Subtraction, Federal Subtraction of Decimal or Federal Money, Simple Multiplication, Multiplication of Decimal or Federal Money, Simple Division, Division of Decimal or Federal Money, Reduction, Compound Addition, Decimal Fractions, Addition of Decimals, Subtraction of Decimals, Multiplication of Decimals, Reduction of Currencies, SIMPLE INTEREST, Commission or Factorage, Ensurance, Buying and Selling Stocks, 161 Practice, Compound Interest, SINGLE RULE OF THREE, Single Fellowship, Double Fellowship, 161 164 176 179 180 ARITHMETICK, 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: ས Unit, unity or one, 1 Two, Three, Four, 6849 Six, 2 Seven, 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 numbers may be 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 India, & Hundreds. of Trillions. Tens. of Billions. of Millions. Units. of Thousands. Hundreds. Tens. of Units. Units, 9,5 7 6, 3 6 4, 8 6 5, 4 9 6, 37 4 Figures expressing the same significant value, when standing alone, increase in a tenfold proportion when they are joined; thus in the number 66, the left hand figure expresses ten times more than the one at the right; but when the left hand figure expresses greater sig nificant value than the one at the right, the proportion is greater that |