IXTEOEB. 



INTEGRAL CALCULI'S. 



804 



supposed to mean a turning machine or tool It U certain they were 

 acquainted with the UM of diamond-powder, though NaxUn dual 

 appears to hare been chiefly used. The atone waa firat brought 

 to the requiaite form and aurfaoe by a |>olisher (politor) ; when the 

 engraving waa executed by the scalptor who employed various kind* 

 of ateel instrument*, with emery, or Naxian duat, and oil, aa a medium, 

 or a diamond point act in stetl ; and in the finishing, to which the 

 Oreeka paid the utmost attention, probably uaing the diamond-duat 

 instead of emery. (See Muller, ' Arch, dor Kunat.,' 313 &c., and the 

 authoritiea there cited.) 



The modern practice of cutting stonei in intaglio is by an apparatus 

 similar in principle to the turning lathe, which give* the cutting tool, 

 placed horizontally, a quick rotatory motion, and the atone on which 

 the design ia to be engraved being brought in contact with it, the 

 aurface ia ground away or indented, till the effect required U produced. 

 Instrument* of various sizes are uaed, which can easily be removed 

 ad replaced, and it is usual, during the process of engraving, to supply 

 the points of the tools with diamond-dust mixed with a little sweet 

 oil. Aa the work proceeds proofs are occasionally taken in wax. 



IXTKGER, a whole number, as distinguished from a fraction. The 

 more common name for a multiple of unity ia " whole number," mean- 

 ing a number of units without any broken unit or fraction of a unit. 

 But if the student find any difficulty in separating the word " whole " 

 for this purpose from its common meaning, he may accustom himself 

 to the word integer. We are led to this remark by finding in a work 

 of celebrity an attempt to connect the word " whole," as used in 

 " whole number," with its general meaning, as when we say the whole 

 ia greater than its part, aa follows : " Integers may be considered as 

 numbers which refer to unity, aa a whole to a part." 



INTEGRAL CALCULUS. The integral calculus is the inverse of 

 the DIFFERENTIAL CALCCH:S ; that is to say, if A being given, it be a 

 question of the differential calculus to find B ; then B being given, it is 

 a question of the integral calculus to find A. 



The question of finding a differential coefficient requires the attain 

 ment of the limit of the ratio of two simultaneously diminishing 

 increments of y and x, y being a function of x : and therefore the 

 fundamental question of the integral calculus is as follows : knowing 

 the limit of ratio of the increment of y to the increment of x, required 

 the function of x which y is. Or, having given a function of .-, 

 required that function of which the given function is the differential 

 coefficient. 



But though this view of the fundamental question is sufficient in 

 pure mathematics, it is not calculated to connect the process of in- 

 tegration with those conceptions which the mind employs in applica- 

 tion to geometry or mechanics. We are here accustomed to a rough 

 speciea of integral calculus, with which the preceding seems at first to 

 have no connection. Thus a number of small straight lines joined 

 together appear to compose a curve with sufficient exactness [ARC] : 

 we arrive at the calculation of a body's variable motion by supposing 

 it uniform during small intervals, and accelerated at the end of each 

 interval [ACCELERATION] ; and we obtain the area of a curve with any 

 degree of exactness by subdividing it into a large number of small 

 curvilinear areas, for each of which we substitute a rectangle [AREA]. 

 It should seem then that when, making the proper use of the terms 

 [INFINITE], we say that every magnitude is made up of an infinite 

 number of infinitely small parts, we might add that every one of the 

 parts ia of a more simple kind than the whole. Thus we appear to 

 nave a right to aay that a curve ia made of infinitely small itraig/it Una; 

 that gradual variable motion ia made up of infinitely small separate 

 impulta; that the area of a curve ia made tip of infinitely small net- 

 anklet. A correct understanding of this connection is the key to that 

 of the integral calculus, and most completely so to that of its ready 

 application. 



The student who baa read the articles above cited may now 

 endeavour to connect the results, and others of the same kind, by the 

 following generalisation. Let a whole be divided into parts, and let 

 each part be capable of subdivision into two parts, one of which can 

 be simply explained and found, and the other of which would be aa 

 difficult to find aa the whole itaelf. Lot A + a be the first part, of 

 which A is of the former species, and a of the latter. Let B + 4, c + <, 

 ftc., be the other parts, of which B, c, Ac., are of the former, and 6, 

 f, Ac., of the Utter species. Then the whole question is 



A+B+C+ + o + 6-t-c + ....; 



by which nothing is gained u yet, for a, 4, c, Ac., are of as much diffi- 

 culty aa the whole which ia to be found. But suppose that when the 

 number of parU ia considerable, a ia very small compared with A, even 

 though A should be small ; and the same of It compared with B, and c 

 compared with c, Ac. Then the whole in question is nearly found by 

 adding A, B, c, Ac. : for say that a were less than the thousandth part 

 of A, 6 less than the thousandth part of B, and so on ; then a + 4 + c + 



is also leaa than the thousandth part of A + B + c , or the 



latter may be taken for the whole with an error of lea* than one in a 

 thousand. Further, suppose that by taking a number of parts suffi- 

 ciently great, we can make a, b. e, Ac., as small aa we please in com- 

 parison of A, B, o, to., theaa + 6 + c + ... may be as small a part u we 

 pleas* of A + B + C + &O. Consequently, by continuing this process 

 without limit, the limit of the summation of A + B + C+ ... ia the 



whole required, without the necessity of paying attention to the re- 

 maining portions. 



Now let ^r be a function of x, of which the differential coefficient 

 <t>'.c does not become infinite when x has any value between a and a + l. 

 Then [TAILOR'S TUTORUM] it may be shown that, whenever x and x + 

 A lie between those values, 



f> (x + h) +x=j'x. h+ rt, 



where r is not such a function of x and k as would hinder rh and A 

 diminishing without limit together. Let a become a + 4 by the steps 

 a T 9, a -r 20, ..... a + , being = 6. We have then 



6 



A , B, o, . . . z, being functions of the same species as P. Sum these, 

 remembering that t.=4, and we find that 



>(o (&) fa 

 is made up of the following series : 



{A+ Bfl + c + + z9 | 9 



If then we diminish $ without limit, or increase without limit the 

 number of steps by which we pass from a to a + 4, we have before us 

 such a case as has been already described. Let p be the least value of 

 <f'x corresponding to values of x between a and a + 4 : then the ratio 

 of any term in the first series to the corresponding term in the second 

 cannot fall short of that of j,e to A0-, or a<f, Ac , or the ratio of p to 

 A0 or B9, &c. But by diminishing 9 without limit, all the preceding 

 ratios are increased without limit ; that is, the ratio of the first series 

 to the second series is increased without limit. We have then the 

 following equation : 



if \.n + 4) $>a = limit of 2 (Q'x . Ax) beginning at x=a and ending 

 when x=a + 4 : or, if the interval from a to a + 4 be divided into 

 parts, each of the value Ax (called t in the preceding), and if .T be 

 made successively equal to a, o 1- Ax ..... a + 4 Ax, then the sum 

 of all the values of <f'x, each multiplied by Ax, approaches without 

 limit to <t> (a + 4) ^a, when is increased, or Ax diminished with- 

 out limit. Now the same sort of convention by which [DIFFERENTIAL 



CALCULUS] the limit of is expressed by ^rj. is here extended, and 

 the limit of 3 (fx . Ax) is written_/Vx rfx. The beginning and final 

 values of x are placed above and below the integral sign_/ : thus the 

 preceding equation is written 



It is common to represent the terminal value of .r by K itmlf , as 

 follows : 



/ 



and when the initial value of x is left indefinite, then a simple con- 

 stant is written for <pa, and the symbols of the limits are omitted, as 

 follows : 



= /Vrrfx + c, 



Let us now suppose a given function /x, upon which we wish to 

 perform the preceding summation, from x = a to ,r=a*4; namely, 

 making n Ax =4, we desire to find the limit of 



-Ax) JAx 



on the supposition that n is increased, or Ax diminished, without 

 imit. This process can be performed immediately, if we can find the 

 function which has /x for ita differential coefficient. Let/,* have the 

 diff. co. fx ; then, by the preceding theorem, the required limit of the 

 summation is 



/,( + &)-/,. 



For instance, so soon as we know that -L is the differential coeffi- 



x 



cientof log x, we know that log (a + It) log a is the limit of the 

 'allowing series, 



Az Ax Ax Ax 



a + Ax 



a + 2Ax 



a+4-Ax 



.he number of terms being n, Ax being the nth part of 4, and being 

 noreaacd without limit. 



