INCREMENTS

, Method of, a branch of Analytics, in which a calculus is founded on the properties of the successive values of variable quantities, and their differences, or Increments.

The inventor of the Method of Increments was the learned Dr. Taylor, who, in the year 1715, published a treatise upon it; and afterwards gave some farther account and explication of it in the Philos. Trans. as applied to the finding the sums of series. And another ingenious and easy treatise on the same, was published by Mr. Emerson, in the year 1763. The method is nearly allied to Newton's Doctrine of Fluxions, and arises out of it. Also the Differential method of Mr. Stirling, which he applies to the summation and interpolation of series, is of the same nature as the Method of Increments, but not so general and extensive.

From the Method of Increments, Mr. Emerson observes, “The principal foundation of the Method of Fluxions may be easily derived. For as in the Method of Increments, the Increment may be of any magnitude, so in the Method of Fluxions, it must be supposed infinitely small; whence all preceding and successive values of the variable quantity will be equal, from which equality the rules for performing the principal operations of fluxions are immediately deduced. That I may give the reader, continues he, a more perfect idea of the nature of this method: suppose the abscissa of a curve be divided into any number of equal parts, each part of which is called the Increment of the abscissa; and imagine so many parallelograms to be erected thereon; either circumscribing the curvilineal figure, or inscribed in it; then the finding the sum of all these parallelograms is the business of the Method of Increments. But if the parts of the abscissa be taken infinitely small, then these parallelograms degenerate into the curve; and then it is the business of the Method of Fluxions, to find the sum of all, or the area of the curve. So that the Method of Increments finds the sum of any number of finite quantities; and the Method of Fluxions the sum of any infinite number of infinitely small ones: and this is the essential difference between these two methods.” Again, “There is such a near relation between the Method of Fluxions, and that of Increments, that many of the rules for the one, with little variation, serve also for the other. And here, as in the Method of Fluxions, some questions may be solved, and the integrals found, in finite terms; whilst in others we are forced to have recourse to infinite series for a solution. And the like difficulties will occur in the Method of Increments, as usually happen in Fluxio<*>s. For whilst some fluxionary quantities have no fluents, but what are expressed by series; so some Increments have no integrals, but what infinite series afford; which will often, as in fluxions, diverge and become useless.”

By means of the Method of Increments, many cutious and useful problems are easily resolved, which scarcely admit of a solution in any other way. As, suppose several series of quantities be given, whose terms are all formed according to some certain law, which is given; the Method of Increments will find out a general series, which comprehends all particular cases, and from which all of that kind may be found.

The Method of Increments is also of great use in sinding any term of a series proposed: for the law being given by which the terms are formed; by means of this general law, the Method of Increments will help us to this term, either expressed in finite quantities, or by an insinite series.

Another use of the Method of Increments, is to find the sums of series; which it will often do in finite terms. And when the sum of a series cannot be had in finite terms, we must have recourse to infinite series; for the integral being expressed by such a series, the sum of a competent number of its terms will give the sum of the series required. This is equivalent to transforming one series into another, converging quicker: and sometimes a very few terms of this series will give the sum of the series sought.

1. When a quantity is considered as increasing, or decreasing, by certain steps or degrees, it is called an Integral.

2. The increase of any quantity from its present value, to the next succeeding value, is called an Increment: or, if it decreases, a Decrement.

3. The increase of any Increment, is the Second Increment; and the increase of the 2d Increment, is the 3d Increment; and so on.

4. Succeeding Values, are the several values of the integral, succeeding one another in regular order, from the present value; and Preceding Values, are such as arise before the present value. All these are called by the general term Factors.

5. A Perfect quantity is such as contains any number of successive values without intermission; and a Defective quantity, is that which wants some of the successive values. Thus x₁ x₃ x₄ x₅ is a Perfect quantity; and x₂ x₄ x₅, an Imperfect or defective one.

Notation. This, according to Mr. Emerson's method, is as follows:

1. Simple Integral quantities are denoted by any letters whatever, as z, y, x, u, &c.

2. The several values of a simple integral, are denoted by the same letter with small figures under them: so if z be an integral, then z, z₁, z₂, z₃, &c are the present value, and the 1st, 2d, 3d, &c, successive values of it; and the preceding values are denoted by figures with negative signs, thus z_-1, z_-2, z_-3, z_-4, are the 1st, 2d,| 3d, 4th preceding values; and the sigure denoting any value, is the characteristic.

3. The Increments are denoted with the same letters, and points under them: thus, x˙ is the Increment of x, and z is the increment of z. Also x_1<*> is the Increment of x₁; and x_n. of x_n, &c.

4. The 2d, 3d, and other Increments, are denoted with two, three, or more points: so z_.. is the 2d Increment of z, and z_... is the 3d Increment of z, and so on. And these are denominated Increments of such an order, according to the number of points.

5. If x be any Increment, then [x] is the integral of it; also ²[x] denotes the integral of [x], or the 2d integral of x; and ³[x] is the 3d integral of x, or an integral of the 3d order, &c.

6. Quantities written thus, x₁ . . . x₅ mean the same as x₁ x₂ x₃ x₄ x₅, or signify that the quantities are continued from the first to the last, without break or interruption.

Rule 1. If the proposed quantity be not fractional, and be a perfect integral, cons<*>sting of the successive values of the variable quantity which increases uniformly: Multiply the proposed integral by the number of factors, and change the lowest factor for an Increment. So the Increment of is - 3x˙ + 6z˙; for the Increment of the constant quantity a is 0 or nothing. So likewise,

The Increment of c x x₁ x₂ x₃, is 4c x˙ x₁ x₂ x₃.

The Increment of ax_-3 x_-2 x_-1, is 3ax˙ x_-2 x_-1.

The Increment of x_-m . . . x_n is .

Rule 2. In fractional quantities, where the denominator is perfect, and the variable quantity increases uniformly: Multiply the proposed integral by the number of factors, and by the constant Increment with a negative sign, and take the next succeeding value into the denominator. Thus,

The Increment of , is .

The Increment of , is .

Rule 3. The Increment of any power, as xⁿ is ; that is, the difference between the present value xⁿ and the next succeeding value ―(x + x˙))ⁿ. And generally, the Increment of any quantity whatever, is found by subtracting the present value, or the given quantity, from its next succeeding value. Also by expanding the compound quantity in a series, and subtracting xⁿ from it, the Increment will be either

So the Increment of x⁴, is .

The Increment of 1/x³ or x^{- 3} is

The Increment of a^x, a being constant, is .

The Increment of 1/a^x is .

The Increment of .

And so on for any form of Integral whatever, subtracting the given quantity from its next succeeding value. So,

The Increment of the log. of x is , which, by the nature os logarithms, is &c.

Schol. From hence may be deduced the principles and rules of fluxions; for the method of fluxions is only a particular case of the method of Increments, fluxions being infinitely small Increments; therefore if in any form of Increments the Increment be taken infinitely small, the form or expression will be changed into a fluxional one.

Thus, in , which is the Increment of the rectangle xz, if x˙ and z˙ be changed for x^. and z^., the expression will become for the fluxion of xz, or only zx^. + xz^., because x^.z^. is insinitely less than the rest.

So likewise, if x˙ be changed for x^. in this &c, which is the Increment of xⁿ, it becomes &c, or only nx^{n - 1}x, for the fluxion of the power xⁿ, as all the terms after the first will be nothing, because x^.2 and x^.3 &c are infinitely less than x^..

And thus may all the other forms of fluxions be derived from the corresponding Increments. And in like manner, the finding of the integrals, is only a more general way of finding fluents, as appears in wha<*> follows.

Rule 1. When the variable quantity in<*>reases uniformly, and the proposed integral consists of the successive values of it multiplied together, or is a perfect Increment not fractional: Multiply the given Incre-| ment by the next preceding value of the variable quantity, then divide by the new number of factors, and by the constant Increment.

Ex. Thus, the integral of 4cx˙x₁x₂x₃ is cxx₁x₂x_<*>.

The integral of 3ax x_-2 x_-1 is ax_-3 x_-2 x_-1.

Rule 2. In a fractional expression, where the variable quantity increases uniformly, and the denominator is perfect, containing the successive values of the variable quantity: Throw out the greatest value of the variable letter, then divide by the new number of factors, and by the constant Increment with a negative sign. So,

The integral of is .

The integral of is .

Rule 3. Various other particular rules are given, but these and the two foregoing are all best included in the following general table of the most useful forms of Increments and integrals, to be used in the same way as the similar table of fluxions and fluents, to which these correspond.

Integrals, when found from given Increments, are corrected in the very same way as fluents when found from given fluxions, viz, instead of every several variable quantity in the integral, substituting such a determinate value of them as they are known to have in some particular case; and then subtracting each side of the resulting equation from the corresponding side of the integral, the remaining equation will be the correct form of the integrals.

For an example of the use of the Method of Increments, fuppose it were required to find the sum of any number of terms of the series 1.2 + 2.3 + 3.4 + 4.5 &c. Let x be the number of the terms, and z the sum of them. Then, by the progression of the series, the last or the x term is x x₁, and the next term after that will be x₁ x₂, that is z˙ = x₁ x₂, where x˙ = 1. Hence the integral is , which is the sum of x terms of the given series. So if the number of terms x be 10, this becomes 1/3 . 10 . 11 . 12 = 440, which is the sum of 10 terms of the given series 1.2 + 2.3 + 3.4 &c. Or, when x = 100, the sum of 100 terms of the same series is .

Again, to find the sum z of n terms of the series &c.

Here the nth term is Put ; then is , and the nth term is ; and the n + 1th term or z is 1/x x₁ x₂; the general integral of which is . But this wants a correction; for when n = 0 or no terms, then x = - 1, and the sum z = 0, and the integral becomes z or ; that is - 1/12 is the correction, and being subtracted, the correct state of the integrals becomes , which is the sum of n terms of the proposed series. And when n is insinite, the lat<*>er fraction is nothing, and the sum of the infinite series, or the infinite number of the terms, is accurately 1/12.

When n = 100, the sum of 100 terms of the series becomes

For more ample information and application on this science, see Emerson's Increments, Taylor's M<*>thodus Incrementorum, and Stirling's Summatio & Interpolatio Serierum.

INCURVATION of the rays of Light. See Light, and Refraction.