INCREMENTS. 



entreat the friendly matheiiaticians, who are lovers of 

 fcicnce, to lend their kind affiftance for the advancement of 

 this uncultivated branch of knowledge, yet in its infancy, 

 or rather, as yet, in the hands of Lucina, either according to 

 the model I have here laid before them, or fome better if it 

 cm be found, fo that, by degrees, it may at length be 

 brought to perfcftion." Hence it appears that this author 

 WAS, in 1763, fenfible of the nesrledt that had been paid 

 to the theory of increments ; and even now, near half a cen- 

 tury after this date, the fubjeft has not been advanced, nay 

 foarcelv touched upon by any Englirti mathematician, while 

 foreign authors are fil ing quarto volumes with the theory of 

 differences finUs, and its almoft univerfal application to the 

 moil curious and important of mathematical inquiries. Eu- 



«- + 



0("'-^),.«-, 



1.2.3 



+ &c. 



by means of this general expreflion, for the increment of 

 K"^, we readily deiuce the increment of any function of Xy 

 as *•„ + a X" + i *, + &c. Thus, for example, if it 

 were required to find the increment of x» + 3 ■-• ^ + 4.V, we 

 have increment of x ' = 3 .v * n + 3 .v n ^ + " ' 



of 4A- = 411 



Hence the increment of .v^ + 71-^ -f 4X = 3 .»•-- fi + 

 *(3«^ + 6«) + „' + 3«- +4n. 



It is therefore not uecefTary, when we proceed thus, that 

 the propofed funftiou fliould be made up of fa<5tors in arith- 

 ler, whofe univerfal genius led him to the inveltigation of metical progreffion, as in the former rule. 



Again, the increment - is the difference between — and 



every fubjeCl that was ufeful and interefting, did not lea 

 the theory ot increments untouched, but hns ti-eated of it, 

 in his ufual mafterly ftyle, in a work entitled " Inflitutiones 

 Calculi Differentiales," &c. in which he has given a new 

 form, and much extended the bounds of this important 

 br-\nch of analyfis; and fubfequent authors have adopted his 

 ideas, and rendered permanent the form he gave it. Vari- 

 ous other works have fince appeared to illuftrate and render 

 famihar the principles of this doftrine ; the molt complete 

 of which is the " Traitedes Differences," &c. par Lacroix. 

 BolTut has likewife a chapter on this fubjeft in his " Traites 

 des Calcul Difierentiel,'' where the theory is treated in a 

 very elementary and comprehcnfive manner ; and the fame is 

 aifo done by Coufin, in chapter 3. of the introdudtion to his 

 «' Traitcs de Calcul Diffcrentiel," &c. 



Having thus given a brief flcctch of the hiftory and pro- 

 grefs of the method of increments, we Ihall now endeavour 

 to explain the principles and application of it : in order to 

 which, that we may prepare the reader for the more general 

 theory, it will be ufeful to confider the fame in a more limited 

 form in the firft inllance, in doing which we cannot have a 

 better model than that of M. Nichole above-mentioned. 



The method of this author is extremely fimple, but cer- 

 tainly lefs general than that of Dr. Taylor ; it is, however, 

 well calculated for conveying the firft clear and connefted 

 ideas of this theory, and prepares the reader for more general 

 refearches, by leadir.g him on, from ftep to ftep, with order 

 and precifion. 



If we confider x as any variable quantity, which is con- 

 tinually increafed by a conilant quantity n, fo that it becomes 

 fuccelhvcly x, x + n, x + zn, &c. : and if y be any func- 

 tion of .r, made up of failors, as ^ = ;i; (.v + n) (.r + 2 «) 

 (* + 3 n), then the difFcreRce between this value oi y, and 

 tiiat which it becomes when .v is again increafed by n, is the 

 increment of J', or of .r (.r + ?/1 (x 4- in) [x -\- 3n); v.'hich 

 increment is- readily obtamed, by obferv;n<r, that 



(x-mo -- « r. = « 



{X + /7j (.V + 2 «) - X (.V + „) = 2« (.V + «) 



(;r + n) {X -I- 2«) {x + 3») - .V {x + n) (.v + zn) = 3,, 

 .(jr H- n) (r -r 2 w) ; and hence generally, the incre- 

 ment of A- (.r + n) (.r + 2 h) ... (.r -;- rn) = (r + l) n 

 (x -f n) (x -f 2n) . . (j' -f rn) : And hence again con- 

 verfely, the integral of (r -4- l) b (a- + n) . . . (j: f 2n) 

 (a- -I- rii) ^ B (:r + n) (.v + 3 li^ ...(* + rn) ; or more 

 generally, the integral ai x i^x -\- ri) (x -i- 2b) ... (jc + rn) 

 (.-„)._(^r^(^J_2^)_^(^+r^^ Again. 



the increment of 

 (r-fd)-. Now(.r + n) 



- - 2) B 



s the dilTerence between x'^ and 

 m\jn~\) 



" + — r~ — 



X {X 4- «) (.V + 2 B) (^- + r - I »). 



This, as was before obferved, is not the moll general 

 mode of confidering the fubjecl, but we are miich miftaken 

 if it be not the moft obvious and natural ; and therefore the 

 bell adapted for illuflration, and for conveying to a beginner 

 the fird ideas of the theory. We will now (hew the applica- 

 tion of the above principles to an example or two, by way 

 of elucidation, and then proceed to,-i more general and ex- 

 tended inveiligation of the method of increments, and its 

 application to mathematical problems. 



^.v. I. — Let it be propofed to find the fum of 100 terms 

 of the feries 



1.2 + 2.34-3.4 + 4.5 100 . lOI. 



Each of thefe terms is of the form x {x + l), and it is 

 obvious that the next term to 100 .101, that is loi . 102, is 

 the inclement of tlie feries ; or, making 100 = .v, the lall 

 term is x (x + I ), and the fucceeding one is (x -h 1 ) (a- 4- 2 ) ; 

 which is evidently the increment of the feries, or the dif- 

 ference between the propofed feries in the firil cafe, and 

 what it becomes when x is increafed by the common dif- 

 fercnce i ; and therefore converfely, the integral of this in- 

 crement, that is, of (.V I- i) i.v -f- 21, will be the fum of 

 1 the 



