326 Method of Progressions. 



constituted, one or more terms from which the adjoining ones' 

 on each side will either increase, or they will decrease. 



If there be a term from vvliich the terms decrease on both sides, 

 it is called a maximum value of the algebraical expression which 

 forms that term. 



If there be a term from wliich tlie terms increase on both sides, 

 it is called a minirmim value of the expression. 



As an algebraic expression can be variable* only by changing 

 the numerical value of one or more of its elements, it seems most 

 natural and most scientific to consider the different quantities 

 that arise from increasing or dimiuishing the value of the va- 

 riable quantity as a progression of terms. They are sometimes 

 compared to the ordinates of a plane curve ; the only difference 

 is, that in the one case we have a progression of geometrical mag- 

 nitudes ; and in the other, of algel)raical quantities. When 1 ex- 

 plain my Method of Tangents (which will most probably be in 

 my next Letter), it will appear that to determine the points in a 

 curve where the tangents are parallel to the axis, is nearly the 

 same thing as to find the maxima and minima of quantities. 



If we consider the m^^ term, in a progression of quantities, to 

 be that at which a maximum or a mininuun takes place ; and m 

 to be such a number that the difTerence between the adjoining 

 terms shall be extremely small ; then the term which precedes 

 the ?«"' term will be nearly equal to that which follows it. Re- 

 gard them as equal ; that is, make m— 1''^ term = 7?j+ i"^ term. 

 An equation from whence the value of the variable quantity may 

 be found, which gives a maximum or minimum value to an alge- 

 braical expression. 



These terms are absolutely equal in some particular cases only, 

 and in those cases our method is undoubtedly true. It is also 

 true when they are not equal in consequence of a compensation 

 of errors, in the manner I have shown to be the case in the qua- 

 drature of curves, &c. See Phil. Mag. vol. Ivii. p. 201. 



According to this method, it is very easy to find whether a 

 quantity be a maximum of a minimum at the to"' term, by com- 

 paring it with either the term which precedes, or that which 



follows it. If the m— 1''^ term be less than the ?«"' term, then 

 the to"" term is a m.aximuui, if it be greater then the m^^ terra 

 is a minimum. 



As an example of the application of the method of progressions 



* It is sometimes said that, while a yariable magnitude passes from one 

 state to another, it passes throuif!i all the intermediate states of magnitude ; 

 but this is true only of geometrical magnitudes, and not of nimibers, or the 

 symbols of numbers ; for numbers proceed by units, that is, by steps or 

 finite gradations. 



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