EVOLUTION, OR THE EXTRACTION OF THE SQUARE ROOT.-A square number is the product resulting from the multiplication of any number by itself. Thus, 25 is the square of 5. 5×5=25. When a number is squared, it is made at the same time multiplier and multiplicand. It is thus twice factor of the product, for which reason, this product or square is denominated the second power, of that number. When a square number consists only of one, or two figures, its root in whole numbers is either,

1 2 3 4 5 6 7 8 9 whose squares are 1 4 9 16 25 36 49 64 81

If the number be not exactly one of these squares, there will be a fraction in the root. The square roots of numbers which are not exact squares, are called surd, or irrational numbers.

EXTRACTION OF THE CUBE ROOT.. A cube number is formed by multiplying a number by itself, and then multiplying again, by that same number, the product of the first multiplication.

A cube is, therefore, the product of a square, multiplied again by the root, from which it originated. Thus 64 is the cube of 4; that is, 4x4 16 the square, multiplied again by 4,

16 × 464. To extract the cube root, is, therefore, to find out by particular rules and operations, the cube root of any number.

PROGRESSION. Arithmetical Progression is a series of terms, in which each term exceeds, by the same quantity, the term by which it is immediately preceded or followed, as

2, 4, 6, 8, 10 &c. or 10, 8, 6, 4, 2, &c.

Geometrical Progression is a series of terms, each of which contains the same number of times, that term, by which it is immediately preceded, or is contained in it, the same number of times.

In the first case, the progression is increasing, as, 2, 4, 8, 16, 32, 64, &c.

In the second case, it is decreasing, as 162, 54, 18, 6, 2, &c.

The number, by which the series is continually increased or diminished, is called the ratio.

PERMUTATION AND COMBINATION. Permutation of quantities is the changing or varying their order.

The combination of quantities, is the showing how often a less number of things can be taken out of a greater number and combined together, with regard to their places, or the order in which they stand.

DUODECIMALS, or cross multiplication, is a rule used by workmen and artificers, for computing the contents of their work.

Dimensions are usually taken in feet, inches and quarters, any parts smaller than these being neglected, as of no consequence. This takes place both in multiplylng them together, or in casting up their contents.

QUESTIONS.

What is reduction? What is arithmetical proportion? What is geometrical progression? What is the rule of three direct; Inverse; Double, or rule of five? What is practice? What is tare and tret? What is interest? What is commission or brokerage; and buying and selling stocks? What is insurance? What are discount and rebate? What are profit and loss, and fellowship? Alligation? Exchange? Commission? Position and the rule of false? What is involution? A power? An index? What is evolution, or the extraction of the square root? What is the extraction of the cube root? What is progression? What are permutation and commutation? What is the rule of duodecimals, or cross multiplication.

CHAP. XIV.

FLUXIONS AND LOGARITHMS.

FLUXIONS, or the method of fluxions, or as it is called on the continent the Differential and Integral Calculi, is a branch of mathematical analysis. It was invented near the end of the seventeenth century, and Sir Isaac Newton and Mr. Leibnitz, two of the most illustrious philosophers of that age, both claimed the discovery.

In the application of Algebra to the theory of curve lines, some of the quantities which are the subjects of consideration, may be conceived of, as having always the same magnitude; such

as the parameter of parabola, and the axes of an ellipse or hyperbola; while others are indefinite as to their magnitudes, and may have any number of particular values; such are the coordinates at any point in a curved line. This difference in the nature of the quantities which are compared together has equally place in various other theories, both in pure and mixed mathematics.

This naturally suggests the division of all quantities into two kinds, namely such as are constant, and such as are variable. A constant quantity is that which retains always the same magnitude, however other quantities with which it is connected may be supposed to change.

A variable quantity is that which is indefinite in respect of magnitude, or which may be supposed to change its value.

Thus in the arithmetic of sines, the radius is a constant quantity, while the co-sine, sine, tangent, of an arch, as also the arch itself, are variable quantities. Constant quantities are usually denoted by the first letters of the alphabet, and variable quantities by the last.

Any expression of calculation, containing a variable quantity, together with other constant quantities, is called a function of that variable quantity.

If a variable quantity be supposed to change its value, then a corresponding change will take place in the value of any function of that quantity.

LOGARITHMS.

An admirable contrivance for shortening cal

culations, was invented by John Napier, a Scotch philosopher. It consists in a set of numbers called Logarithms, or indices of numbers, which were so adapted to the numbers, to be multiplied or divided, that these being arranged in the form of a table, each opposite to a number termed its logarithm, the product of any two numbers in the table was found by the addition of their logarithms; and the quotient arising from the division of one number by another, was found by the subtraction of the logarithm of the divisor from that of the dividend.

Similar simplifications were introduced into the more laborious operations of involution and evolution.

Let two series of numbers be formed, the one constituting a geometrical progression, the other an arithmetical progression. Let the first term of the geometrical progression be unity, or, 1, and that of the arithmetical progression be 0. Thus : Geom. Pro. Arith. Pro.