INC 



1ND 



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 pa- 

 rallelograms is the business of the method 

 of increments. But if the parts of the ab- 

 scissa 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 num- 

 ber of finite quantities ; and the method 

 of fluxions the sum of any infinite num- 

 ber 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 maybe solved, 

 and the integrals found, in finite terms ; 

 whilst in others we are forced to have re- 

 course to infinite series for a solution. 

 And the like difficulties will occur in the 

 method of increments, as usually happen 

 in fluxions. 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 curi- 

 ous and useful problems are easily re- 

 solved, which scarcely admit of a solu- 

 tion in any other way. As, suppose seve- 

 ral 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 finding 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 incre- 

 ments will help us to this term, either ex- 

 pressed in finite quantities, or by an in- 

 finite series. Another use of the method 

 of increments is to find the sum 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 some- 



times a very Few terms of this series will 

 give the sum of the series sought. See 

 Emerson's Increments. 



INCUBUS, or nightmare, in medicine, 

 the name of a disease, which consists in a 

 spasmodic contraction of the muscles of 

 the breast, usually happening in the night, 

 and attended with a very painful difficul- 

 ty of respiration and great anxiety. 



INCUMBENT, a clerk diligently resi- 

 dent on his benefice with cure ; and call- 

 ed incumbent ot that church, because he 

 does or ought to apply himself sedulously 

 to discharge the duty of his cure. 



INCURVATION of t/ie rays of light, 

 their bending out of a rectilinear or 

 straight course, occasioned by refraction. 



INDEFINITE, or INDETEHMINATE, that 

 which has no certain bounds ; or to which 

 the human mind cannot affix an}'. Des 

 Cartes makes use of this word in his phi- 

 losophy instead of infinite, both in num- 

 bers and quantities, to signify an incon- 

 ceivable number, or a number so great 

 that an unit cannot be added to it ; and a 

 quantity so great as not to be capable of 

 any addition. Thus, he says, the stars vi- 

 sible and invisible are in number indefi- 

 nite ; and not as the ancients held infi- 

 nite ; and that quantity may be divided 

 into an indefinite number of parts, not an 

 infinite number. 



INDEFINITE is also used* in the schools, 

 to signify a thing that has but one ex- 

 treme ; for instance, a line drawn from 

 any point and extended infinitely. 



INDEFINITE, in grammar, is understood 

 of nouns, pronouns, verbs, participles, ar- 

 ticles, &c. which are left in an uncertain 

 indeterminate sense, and not fixed to any 

 particular time, thing, or other circum- 

 stance. 



INDENTED, in heraldry, is when the 

 out-line of an ordinary is notched like the 

 teeth of a saw. 



INDENTED line, in fortification, the same 

 with what the French engineers call re- 

 dent ; being a trench and parapet run- 

 ning out and in, like the teeth of a saw ; 

 and is much used in irregular fortifica- 

 tion. 



INDENTURE, is a writing, containing 

 a conveyance between two or more, in- 

 dented or cut unevenly, or in and out, on 

 the top or side, answerable to another 

 writing that likewise comprehends the 

 same words. Formerly, when deeds were 

 more concise than at present, it was usual 

 to write both parts on the same piece of 

 parchment, with some words or letters 

 written between them, through which the 

 parchment was cut, either in a straight OP 



