tion. If the condition causes decided symptoms surgical removal of the offending body is indicated. See LITHOTOMY; LITHOTRITY.

KARL M. VOGEL, M. D., New York City.

Calculus, The Infinitesimal.-The Infinitesimal, or Differential and Integral, Calculus is not so much a branch of mathematics as a method or instrument of mathematical investigation, of indefinite applicability. The masters now seldom try to treat it in less than a thousand large pages; here we may hope no more than to expose its basic principles, to illustrate its characteristic processes, and to exhibit some of its nearer-lying applications, with their results. Even so little will require utmost condensation and self-explaining abbreviations. We might define the Calculus as the Theory and Application of Limits, so central and dominant is this latter concept. We must, then, clear the ground for its full presentment. Successive addition of the unit 1, continued without end, gives rise to the Assemblage of positive integers, in which all additions, multiplications, and involutions are possible. This assemblage is ordered: i.e. of two different elements, a and b, either a <b or a>b; and if a <b and b<c, then a<c. To make all subtractions (inverses of addition) possible, we annex the symmetric assemblage of negative integers, any negative integer, as a' (or -a), being defined by the equation a+a'=o, this o itself being defined by a-a=o. To make all divisions (inverses of multiplication) possible, we annex the assemblage of Fractions, quotients of integers by integers. This total assemblage of integers and their quotients, both + and we may call the domain or assemblage of Of rational real numbers, wherein all direct operations (of addition and multiplication) and also the inverses (subtraction and division) are possible.

The operation of involution is direct, a special case of multiplication, but is not commutative like addition and multiplication: thus a+b=b+a, ab=ba, but in general abba. Hence the direct operation ab, yielding c, has two inverses: Given b and c, to find a, and given a and c, to find b. The former gives rise to roots or surds, the latter to logarithms. But neither of these can in general be found in the universe of rationals; to make such inversions always possible, we must still further enlarge the domain of number by annexing Irrationals. These demand exact definition. Divide the assemblage of rationals into two classes, A and B, any member a of the first being any member b of the second. Three possibilities present themselves:

1. A may contain and be closed by a number a> any other a but <any b.

2. B may contain and be closed by a number <any other b but > any a.

3. Neither A may contain a largest a, nor Ba smallest b. Thus we may form (1) A of 2 and all rationals <2; or (2) В of 2 and all rationals > 2, -in either case 2 is a border (frontière) number; or (3) A of all negatives and all positive rationals whose squares are <2 and B of all positive rationals whose squares are > 2. Here there is no border number among rationals. But a border does exist, defined as > any a but any b. We name it second root of 2 and denote it by

2 or 2. All such common borders are called Irrationals. The assemblage of irrationals is determined by all possible such partitions of rationals (A, B). The assemblage of all rationals and all such irrationals is the assemblage of Reals. It remains and is possible to extend the operations of arithmetic to all reals. In particular, the assemblage of rationals is dense; i.e. between every two there is an infinity of others; in the same sense the assemblage of reals also has density. Again, always on dividing all reals into A and B, each member of A each member of B, there will be a border 7, the greatest in A or the least in B, all less numbers being in A, all greater in B. Hence, and in this sense, the assemblage of reals is named continuous. In this continuum, admitting no further introductions, suppose a magnitude to assume successively an infinity of values: V, V,... Un, ... Un+k,...; it is then called a variable, V, and its values in order form a sequence, S. It often happens that V will approach some constant L, so that by enlarging n we may make and keep the modulus or absolute worth (ie., regardless of sign) of the difference V-L<any preassigned positive magnitude, e, for all following values of '; in symbols, |Vn+k-L] < ε for every positive k. Then L is called the Limit of 1: L=Lim. V. Plainly, V cannot have two limits as thus defined. It is easily seen that will have a limit when and only when Vn+k-Vn <ên. If V changes always in the same sense, by increase or decrease, it has a limit when and only when [V]< always than some fixed number n. When increases (positively or negatively) beyond any assignable n, it is often said to have as limit.

A perfect geometric illustration is found in the sequences I and C of inscribed and circumscribed regular polygons of the circle. Here every is > every I; also Cn-In<ɛ; also Cn+k−Cn < £, In+k − In < ¤, C n − A < ɛ, A − In< ɛ (4 being the circle-area); hence A is the common limit both of Сn and of In, for n increasing without limit (n = ∞).

as

I

I

I

1+1+1+1+6+1+

I

Algebraically, if c1, C3, Cg... C2nti · be the sequence (0) of odd convergents, and c2, c,... Can... the sequence (E) of even convergents in an interminate continued fraction then every Conta >Can, also C2n+1-C2(n+k)+1 < €, Cgn−C2(n+k) < £; The odd convergents from above and the even and C2+1-(VI3−3)<ɛ, (V13 − 3) − C 2n < ɛ. convergents from below close down endlessly upon their common limit, VT3-3-as quad. rants of an hyperbola and its conjugate close down upon their common asymptote.

will and is called Infinitesimal (o); its limit The difference VL is a variable small at

is o.

The quotient of two a's will generally be a variable; if it has a finite limit L, the o's are named of the same order; if the limit of the quotient is o (or), then the numerator (or denominator) is of higher order. If any o be chosen as standard, it is called principal infinitesimal; any other whose pth root is of the same order as the principal is itself said to be of pth order.

Easy theorems are now proved as to the limits of the sum, difference, product, quotient of variables. In general: If R(u, v, w, ...) be a rational function of simultaneous variables,

2,, w,

n

n.

, and if u, v, w have limits l, m, —then R(u, v, w...) has a limit R(l, m, .)—always provided that this latter does not involve a division by o, which has no sense. If two V's differ at most by a o, and one has a dimit, the other has the same limit. Herewith there becomes possible a Calculus of the Limits of Variables instead of the Variables themselves. These limits are often far the more important, as we shall soon see.

A variable V (or sequence v1, 2...) is bounded above when we may assign a value M that it cannot exceed; then there is a certain smallest number, its upper limit, which it cannot exceed. Similarly, it is bounded below when we may assign an m below which it cannot sink; then there is a certain greatest number, its lower limit, under which it cannot descend. If V may assume either of these limits as one of its values, then that limit is attainable and is a Maximum or a Minimum; otherwise it is unattainable. If V be a proper fraction, its limits, o and 1, are not attainable. When V may assume every value between its attainable limits, a and b, it is said to vary continuously in the interval [a, b]. But if a, or b, or both, be unattainable, we shall write [1+o, b] or [a, b−o], or [a+o, b-o].

When to values of one magnitude correspond values of another, the magnitudes are called Functions of each other (Leibnitz). The one to which arbitrary values may be supposed given is called the argument or independent variable; the other, whose corresponding values may be reckoned or observed (or which at least exist), is called the function. Such are a number and its logarithm or sine; the radius of a sphere and its surface or volume; the elasticity of a medium and the velocity of an undulation through it; etc.... The general functional connection of x and y is expressed by F(x, y) = =0. If this F be an entire polynomial in x and also in y, then F is algebraic, otherwise transcendental. If F be solved as to y, thus y=f(x), then y is an explicit function of x; otherwise, an implicit function. If f(x) be the quotient of two entire polynomials in x, then f(x) is a rational function of x; otherwise, irrational. If to any one value of x there corresponds only one value of y, then y is a one-valued or unique function of x; if x be also a unique function of y, then there exists between x and y a one-to-one correspondence.

If y=f(x) and x=(y) express the same correspondence between x and y, then f and denote inverse functions. A function may reduce to a constant; as xn=1, for every finite a when n=0.

As x ranges in [a, b], f(x) will also range. Similarly f(x) may have an upper limit M and a lower limit m; then f(x) is bounded in [a, b].[m, M]is its interval and M-m its oscillation. If either m or M be absent (or ∞), this oscillation is ∞. If we cut [a, b] into n sub-intervals (ak, bk) (k = 1, . . . n), then plainly the upper limit of f(x) will be M in at least one [ak, bk] and > M in none; the oscillation will not be > M-m in any [ak, bk].

If as x approaches c, no matter how, f(x) approaches f(c) as its limit, then f(x) is continuous at c (i.e. for x=c). Or, if f(x) be bounded in [c-o, c+o] and if the limit of its Oscillation be o for a vanishing, then f(x) is Continuous at c. That is, we must be able to

make and keep the oscillation of f(x) small at will by making and keeping the fluctuation in x small at will.

It may be that limit f(c+o) = f(c) only for +o, then f(x) is named continuous right of c; or that Lim. f(c+o)=f(c) only for -σ, then f(x) is named continuous left of c. Only when f(x) is continuous both right and left of c [f(c) being the same], is f(x) continuous at c.

If f(x) be continuous at all points (values of x) right of a and left of b, it is named continuous in [a, b].

The infinitesimal [c-o, c+] is called the (immediate) vicinity (or neighborhood) of c. A change in the value of a v is conveniently denoted by 4v, read difference-v or Delta-v; hence 4x and 4y will denote corresponding (simultaneous) differences or changes in x and y.

If now y=f(x) be continuous in [a, b], we may cut this latter up into finite sub-intervals, 4x, each so small that the oscillation of y in each shall be <ɛ. Hence Heine calls y uniformly (equably, gleichmässig) continuous in [a, b]. This corresponds to uniform convergence, as of a power-series, y Σ Cnxn, as

=

I x

1

x

Again, y=sin is not defined for x=0, but whatever value be assigned it there, it remains discontinuous, since sin vibrates infinitely fast between +1 and -1 for xo.-Again, f(c±) may approach a limit for a vanishing, yet not approach f(c). Thus, let f(x) = x2 x2 + + a decreasing geometrical series, 1+x2 ratio (1+x2)-1, hence Lim. f(x)=1+x2. Then as xo, f(x)=1, Lim. f(o±ø)=1; but for x=o, f(x) = f(o) =o.-There are many immediate consequences of continuity, which we have no space to discuss here, such as: function continuous in [a, b] attains its upper and lower limits (its maximum and minimum); it also assumes at least once every value between ƒ(a) and ƒ(b),— —a property, however, not peculiar to continuous functions (Darboux).

A

The notion of function is at once extended to several variables, u = f(x, y, . . .), one- or manyvalued, algebraic or transcendental, etc., as before. Here each variable, as x, has its range or interval [a, a']; so y its [b, b'], etc. All possible sets of values (x, y . . .) form an assemblage or the Domain (D) of variation. Any set (or point) for which any variable has an extreme or border value, as a or a', b or b', is a border

point; the assemblage of all such is the border or contour of D. A simple geometric depiction of D in rectangular coördinates for only two variables, x and y, would be a rectangle with sides xa, x=a', 3b, y=b'; of three variables, x, y, z, it would be a cuboid bounded by the planes xa, x=a', y=b, y=b', z=c, z=c'; etc. The point (x, y) or (x, y, z) may be anywhere in or on the rectangle or cuboid. Such a D may be thought cut up into elements, infinitesimal rectangles or cuboids. Suppose any point (a,,b,,...) within an element. If now f(x, y,...) approaches f(a,, b1,...) as limit, as point (x, y, ...) approaches point (a,, b1, . . .), no matter how, then f(x, y, .) is called continuous at (a, b, . . .). This amounts to saying that the oscillation of f shrinks towards zero as the element contracts, no matter how, about the point; that is, infinitesimal functionchanges correspond to any and all infinitesimal argument-changes in the immediate vicinity of the point.

Any f(x, y, ...) is called continuous within D when continuous at every point in D, border included; but on this border, as x, y, ... approach a, b,,... the point must not get without D. An f is continuous in the (immediate) vicinity of a point, when continuous within an infinitesimal D including that point. In general, theorems holding for functions of one variable may be extended, with proper modifications, to functions of several variables.

4x'

called

Derivatives:-In the study of functional dependence, the main subject of scientific inquiry, it is of first importance to know how corresponding changes in the magnitudes are related. To discover this, we form the quoΑν tient of corresponding differences, Difference-Quotient (DQ). In general, it is very complex, but breaks up into two parts, one independent of 4x, the other vanishing with 4x. The first is the important part and is named Derivative (D) or Differential Coefficient (DC). More formally, if y=f(x) be a unique continuous function of x in [a, b], and x ▲y _f(x ± 4x) − f(x) be any point therein, and if approaches a limit as 4x approaches o no matter how, then that limit is called Derivative (D) of f(x), as to x, at the point x. If f(x) has a D at every point of [a, b], the assemblage of them forms a new function, the Derivative of f(x) for [a, b], which we may write f'(x) with Lagrange, or Df(x) with Cauchy. Hence f(x)

Ax

=f'(x)+o.

4x

=

± 4x

Geometric Interpretation.-The Differential Calculus originated in the Problem of Tangents." Let P be any point of a curve referred to rectangular axes X, Y, and let P be between O and C. Draw secants PQ, PQ', sloped and to X, and to each other; draw ordinates through P, Q, Q'; through P and 2 draw parallels to X, meeting ordinates hrough Q and P at D and D'. Then PD=4x, D'Q' = 4'x, DQ=4y, PD' = 4'y. 4 tan 0, =tan ', tan (9' —9) : 4x tan . If now by approaching Q and Q' to P we can make and keep o, and therefore tan, small at will, then the secants settle

Also

Αν 4x

=

f′(x) =

Αν 4x

down into a common position called tangent to the curve at P (sloped to X) and the Ay A'y common limit of Tx and is tant; or 4'x =tan T. But if P were an angular point, then PQ and PQ' would not tend together, would tend to one limit, the progressive s'y differential coefficient, and to another 4'x limit, the regressive differential coefficient; only when these two coalesce is there a Derivative proper. Thus in y=x y = x (c2 – 1) / (62+1).

the progressive DC=1, the regressive DC = −1, the two limiting positions of the secants are perpendicular.

n = I

How thickly may such salients be strewn along a curve? To have a D, i.e. to be differentiable, plainly the function must be continuous; it was long thought that this necessary condition was sufficient, that the continuous function possessed in general a D, save at certain special points. It was Riemann who first suggested (at least as early as 1861) the astonishing possibility that such an 22=00 sin(2x) f(x) as Σ though everywhere continuous, was nowhere differentiable; but as he left no proof, it was generally thought he meant that it was possible to find such salients in every infinitesimal [x, x+4x], which was easy to show; but Weierstrass thought he meant strictly that the D did not exist for any value of x. In any case Weierstrass himself produced (July 18, 1872) an example of such a function, y = Σ bn сos(aпx),-where a is an odd integer, b a positive <1, and ab>1+37, which, though everywhere continuous, has nowhere a D, since the progressive and regressive Difference-Quotients are everywhere opposite in sign and increase oppositely toward

as they pass over into Differential Coefficients (Math. Werke von K. Weierstrass, II,'p. 71–4). Geometrically, in the graph of the differentiable function, the polygon formed by n consecutive chords tends towards the curve for n∞, PQ and PQ' tend to coalesce as Q and Qʻ both approach P, the triangle POQ' becomes flatter and fictter (we may suppose the arc QPQ' steadily enlarged under a microscope to its original length as Q and Q' close down on P), the curve we may say is elementally straight at P. But with Weierstrass's function the polygon remains always reentrant, zigzag, and consecutive chords, PQ and PQ', tend to separate at a straight angle. Such discontinuities may yet present themselves to the future student of nature. If y=x2 4y=2x4x+4x2,

whence Day= of a square whose side is x, and 2x is the border of the square perpendicular to which the square expands, the D is the front of variation.

( x + = 1);

; hence

yx=D sin x cos x=sin(x +·

=

= p′(u)· f'(x).

hence yx-Yu • Ux = 4x Au 4x' But y may exist even when the supposition fails, and this rule with it.

In particular, if y = f(x) and inversely D#0, x=(y), and if either variable has a

= COS

Hence D cos x = -
-- sin x

(x + 1). Hence derivation of sine and

cosine as to the angle merely adds:

2

2

to the angle.

AV 4x or f'(y)

Also, D tan x=1+tan x2=secx. If y=sin-1x, sin y=x,

I

yx

hence

VI-X

0,

I

to

4x I hence Lim. Ay f'(x) i.e., in general, the D's of x as f'(x)' y and of y as to x are reciprocals of each other:

Now, f(x) = (b − a) { f(x) − f(a)} − (x − a) 14(b)-4(a) is such an f(x), made to order, being differentiable within [a, b]. Hence f'(x) = (b− a) p′(x) — { p(b) — p(a)} must = o for somex in[a, b]. Hence p(b) (a) = (b− a) p′(x). Commonly we write a for x and x+h for b; then x=x+0h, where 0 is in [0, 1], so that p(x+h) − P(x)=hp'(x+0h), the extremely imHence portant formula for finite increments. we see at once that if the D is everywhere o within an interval, the function is constant in that interval; and hence that two functions whose D's are equal in an interval can themselves differ only by a constant in that interval a Theorem at the base of the Integral

a

a2+x2•

Va2-x2 cially, D sinSimilarly we treat the hyperbolic sine and cosine and tangent (hsx, hex, htx), and their inverses hs-1x, hc-'x, ht-1x, with the important results: Dhsx=hcx, Dhcx=hsx, Dhtx =1-hix2,

The Infinite Series cannot always be differ entiated by differentiating term by term, but only under certain conditions of equable convergence. If each term fn(x) of f(x) = L £fn(x) be unique and continuous and if converge, for every x in [a, a+d], and if Lf(x) = f(a),—

xa+o

for such a series the Theorem holds: If each term has a finite progressive DC, fn'(a), and if the series of PC's,” v° fn(a+4x) — fn(a) verges equably for every 4x>o and <d, then fn'(a) also converges and

--

4x

fn(a+1x) − ƒn (a), for 4x=+0.

Ax