The numbers which form the series, are called the terms of the series. The first and last terms in the series, are called the extremes; and the other terms the means. The number by which the terms of the series are continually increased or diminished, is called the common difference.

PRINCIPLES ASSUMED.

1. When four numbers form a progressional series, the sum of the two extremes is equal to the sum of the two means; and of any three quantities, in such a series, double the mean is equal to the sum of the extremes.

2. In any equidifferent series, the sum of the two extremes is equal to the sum of any two means that are equally distant from the extremes, and equal to double the middle term, when there is an uneven number of terms.

3. The difference between the extremes of an equidifferent series, is equal to the common difference multiplied by the number of terms less 1.

4. The sum of all the terms in any equidifferent series, is equal to the sum of the extremes, multiplied by half the number of terms.

The reason of these Principles, and the following Problems or cases, it is presumed, is sufficiently obvious without being demonstrated.

For brevity and perspicuity, let a=first term, l=last term, n=number of terms, d=common difference, s=sum of all the terms, and let the different cases be represented as in the following Table. By this means, a summary of the whole doctrine of equidifferent series may be presented at a single view.

In Progression, if three parts be given, the other two may readily be found. By an explanation of the first problem or case in the Table, the rest will be readily understood. In the first case of the Table, read-The first term, last term, and number of terms given, to find the common difference; or, sum of all the terms.

RULE. Divide the difference of the extremes by the number of terms less 1, the quotient will be the difference. Multiply the sum of the extremes by the number of terms, half the product will be the sum of all the terms.

II. PROGRESSION GEOMETRICAL,

OR, CONTINUAL PROPORTIONALS.

Any series of numbers, the terms of which gradually increase or decrease by a constant multiplication or division, are said to be Continual Proportionals.

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

The first and last terms of a series are called extremes, and the other terms means.

PRINCIPLES ASSUMED.

1. If four quantities be in continual proportion, the product of the two means will be equal to that of the two extremes, when continued, or if discontinued; and, of three quantities, the square of the mean is equal to the product of the

two extremes.

2. If four quantities are such, that the product of two of them is equal to the product of the other two, then are those quantities proportional.

3. If four quantities are proportional, the rectangle of the means, divided by either extreme, will give the other

extreme.

4. The products of the corresponding terms in continual proportions, are also proportional.

5. If three numbers be in continued proportion, the square of the first will be to that of the second, as the first number to the third.

6. In any continual proportion, the product of the two extremes, and that of every other two terms, equally distant from them, are equal.

7. The sum of any number of quantities, in continued proportion, is equal to the difference of the rectangle of the second and last terms, and the square of the first, divided by the difference of the first and second terms.

As the last term, or any term near the last, is very tedious to be found by continual multiplication, it will be very necessary, in order to ascertain it, to have a series of numbers in arithmetical proportion, called indices, or exponents, beginning either with a cipher, or a unit, whose common difference is

one.

When the first term of the series and the ratio are equal, the indices must begin with a unit; and, in this case, the pro

duct of any two terms is equal to that term signified by the sum of their indices.

Thus, 1, 2, 3, 4, 5, 6, &c., indices, or arithmetical series. And 6+6=12, index.

Then 2, 4, 8, 16, 32, 64, &c., geometrical series (leading terms) of the twelfth term.

And 64×64-4096, the twelfth term.

But, when the first term of the series and the ratio are different, the indices must begin with a cipher, and the sum of the indices, made choice of, must be one less than the number of terms, given in the question; because 1 in the indices stands over the second term, and 2, in the indices, stands over the third term, &c.; and, in this case, the product of any two terms, divided by the first, is equal to that term beyond the first, signified by the sum of their indices.

Thus, 0, 1, 2, 3, 4, 5, 6, &c.; indices.

And, 6+5, index of the 12th term.

Then, 1, 3, 9, 27, 81, 243, 729, &c., geometrical series. And, 729+243, the twelfth term.

8. If the ratio of any geometrical series be double, the difference of the greatest and least terms is equal to the sum of all the terms, except the greatest; if the ratio be tripled, the difference is double the sum of all the terms, except the greatest, &c., &c.

9. In any geometrical series decreasing, and continued ad infinitum, half the greatest term is equal to the sum of all the remaining terms, ad infinitum. Let a first or least term, l= last or greatest term, s=sum of all the terms, n=number of terms, r-ratio, L-logarithm, and the different cases be represented as in the following Table. By this means, a summary of the whole doctrine of Continual Proportionals may be presented at a single view.

By an explanation of the first problem or case in the following Table, it is presumed the different problems will be easily understood. In the first case of the Table, read-The first term, ratio, and number of terms given, to find the sum of all the terms.

RULE. The ratio, less 1, raised to the power denoted by the number of terms, and divided by the ratio less 1, the result multiplied by the first term.