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SEVERAL NEW METHODS OF OPERATION,
NEW SYSTEM OF PROPORTION;
THEORENCAL EXPLANATIONS OF ALL THE PRINCIPAL RULES.
A TREATISE ON MENSURATION,
BRIEF PRACTICAL SYSTEM
BÝ GEORGE WILLSON.
PUBLISHED AND BOLD, WHOLESALE AND RETAIL, BY
ALBANY; AND BENNET & BRIGHT, UTICA
ENTERED, according to the Act of Congress, in the year 1836, by GEORGE WILLSON, in the Clerk's office of the District Court of the Northern District of New York,
In constructing an Arithmetic for the use of schools, two errors are to be avoided; the total exclusion of explanation, and the redundancy of it :-the laying down of naked arbitrary rules, as if the scholar were incapable of comprehending the reason of them; and that minuteness and excess of explanation, which leaves nothing for his own discovery and investigation.
Explanations should, in the judgment of the Author, be applied to the principle of the rules, rather than to the solution of particular questions; and nothing should be done for the learner, v:hich he is capable of doing for himself. Every one recollects the satisfaction, with which, as he advanced in his mathematical course, he fu ind himself able to take the progressive steps by fumf, without the aid of his teacher or his fellowstudents ; and every one is conscious of the surer grasp, he yet retains, on the acquisitions he made by the unassisted exercise of his own faculties. To the mind as to the limbs, leading strings may be of service in taking the first steps; but, if never laid aside, there can never be confidence in the ability to go alone.
In the preparation of this treatisc, the Author has aimed especially at brevity, clcarness and precision ; and he has added only so much of theoretical explanațion, as will enable the scholar to understand the principle and reason of the several rules, and methods of operation.
A new method of statement in Proportion, and several new and convenient contractions, under different rules, will be found in this book.
The reader is referred to Decimals, to the theoretical explanations of Proportion, the Square and Cube Roots, and to Mensuration, as containing original matter, not to be met with in other systems.
A brief practical system of Book-keeping is subjoined, which will be found sufficient for all the purposes of ordinary business; and indeed, for mercantile establishments, those only excepted, whose extensive operations require the use of the more complicated system of double entry.
As the best preparation for the use of this book, or of
any other treatise of the same grade, the Author would recommend a thorough acquaintance with ColBURN'S MENTAL ARITHMETIC. The effect of this invaluable little work, in disciplining the minds of younger pupils to habits of clear and accurate thinking, and the aptitude and rapidity it gives in numerical calculations, are known to all who have used the book.
A word in conclusion, on the method of teaching Arithmetic. No school should be without a blackboardo The scholars should be formed into classes: every member of which, should in rotation, be required to work a question on the black-board, in the presence of his class, and to explain each step of the process; and the reason of it. If taken through the book in this way, it is hardly possible, that a pupil should fail of attaining a competent knowledge of the science,
ARITHMETIC treats of the properties, relations, and combination of numbers.
Its prircipal branches aro, Notation or Numeration, Addition, Subtraction, Multiplication, and Division.
Numeration teaches to write and read numbers.
METHOD OF NOTATION. The characters by which numbers are expressed, were derived fro:n Arabia. They are the following: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. By variously combining these, all possible numbers are expressed. The simple characters, however, carry us no higher in numbering than nine units. To denote an adäitional unit, the first figure is repeated, but in a higher place, having the cipher (which by itself has no value) on the right : the 1 then becomes a unit of the second order, and has ten times its simple value. The addition is continued up to nineteen, by placing the simple figures at the right of this unit of the second order, in the room of the cipher : thus, 11, 12, 13, 14, 15, 16, 17, 18, 19; which may be read, one ten and one, one ten and two, &c. When all the simple characters are repeated in this combination once, the addition is made by increasing the unit of the second order one, (making it two tens,) and repeating the cipher : thus, 20. Again the simple characters are added successively in the same manner as before, and we have 21, 22, 23, &c. When the whole are repeated, the second order is again increased, and becomes 30. After the nine digits have all in succession been used in the second crder, we are carried in numbering up to 99, or, nine tens and nine; then the additional number is expressed, by removing the original unit one place farther to the left, where it becomes a unit of the third order, and has ten times the value which it had in the second, and one hundred times its simple value : it is then named hundreds