A New, Copious and Complete System of Arithmetic

ADVICE TO STUDENTS.

The author having taught a number of years, found it necessary to impress students of Arithmetic and Mathematics, as they progressed, with the necessity of committing all the useful rules and definitions to memory; and also, to endeavour to comprehend the meaning, use, and application of the examples arising out of those rules and definitions. This enables them to command the necessary helps, at a moment's warning, without book or any thing else. It looks awkward in a man who pretends to be a scholar, to run to his books for every thing he may want in the course of business. It is true, that where the use of Mathematical tables is required in the investigation of Geometrical and other problems, books are to be adverted to. It is no where expected, that tables of logarithms, tables of latitudes and departures, and many others, can be committed to memory; but wherever it is practicable, it should be done. Useful rules and theorems should be deeply impressed upon the mind. Every useful definition should be clearly understood by the student, as he progresses; then, what would otherwise be mysterious, will ap pear as clear as the sun at noon day.

EXPLANATION OF CHARACTERS.

Where any of these marks are found in the body of the Arithmetical Instructor, look to the bottom of the page for their corresponding marks, and you will find La further explanation of what is spoken of above.

When this sign is found between two numbers, or quantities, it shews that they are added, and is therefore the sign of addition, called plus.

When this sign is found between two numbers, or quantities, it shews that they are subtracted, and is therefore the sign of subtraction, called minus.

When this sign is found between two numbers, or X quantities, it shews that they are multiplied, and is therefore the sign of multiplication.

16

4

or

This is the sign of equality, and reads equal, and shews that where it is placed between two numbers, or quantities, that these numbers and quantities are equal.

This is the sign of division; and shews, when placed between two numbers, or quantities, that the first is divided by the last.

When numbers, or quantities, are placed fraction-wise, as those in the margin, it shews that the upper number, or quantity, is divided by the lower one.

3)12(4 This shews, that 12 divided by 3 is equal to 4. When dots are placed between numbers, or quantities, as in the margin, they are called signs of proportion, and read, viz. as 2 is to 4, so is 8 to 16, or, as a is to b, so is e to d.

2:48: 16

a b c d

This expression is called an equation, and means

bc d that b, multiplied by c, and divided by a, gives a quotient equal to d.

This expression is also an equation, which, in Simple Interest, means that the principal, ptr+p=a time, and rate per cent, plus the principal, is equal to the amount,

This equation, in Compound Interest, means that prta the principal multiplied by the ratio, raised to the power of the time, is equal to the amount. 2x1, aX1 Shews that 2 multiplied by 1, and a mutipli21 a1ed by 1, give their first powers.

2x2, axa S Shews that 2 multiplied by 2, and a multiplied =22 aby a, give their second powers, or squares. 2×2×2=23 Shews the third powers of 2 and a, or a×a×a=a3 { their cubes.

And, in general, 2" and a" shew any powers of 2 and a. (1+7) and (a+b)2, or (2+7)" and (a+b)", shew that the first pair, inclosed in parenthesis, are squared, and the second pair raised to any power.

✓2 or 2 Va or va ✅a

When such characters are placed before numbers, or quantities, it shews that their square roots are extracted, and is therefore the sign of evolution.

(2+7) These expressions shew that the square roots S (a+b) { of 2+7, and a+b, are extracted.

2 or 2 These expressions shew that the cube roots Va or a of 2 and a are extracted.

n

And generally,
V2 or
√2 2

When the exponent of the root is expressed by n, it means that the nth roots of 2 Va or a Land a are extracted; that is, any roots, &c.

n

These figures, with the dash drawn across them, shew that they are cancelled; that is, that equals are stricken from both dividend and divisor.

This character, placed between two numbers, or quantities, shews that the one on the left is less than that on the right.

This, the reverse of the above.

Where this ambiguous sign is placed between two quantities, it signifies that sometimes both are to be added, and sometimes the one subtracted from the Lother.

This character, in music, denotes a flat.. #This, a sharp.