Page:The New International Encyclopædia 1st ed. v. 04.djvu/33

* CALCULUS. 10 CALCULUS. work, Y, between tlie limits r, and v, is, there- fore, W = A-logr, + c — J;logi-; — c. The limits being defined, this 'integral' is called a Definite iDleuial, and the operation is usually denoted as follows: 'i /kdv I'l — Ij- = fclogi;. — A;logi;.= fclog— The constant /.-, coming from the law pv = k, depends on the amount of gas employed and on the temperature at which the compression is carried out. It may, of course, be found by actually measuring the pressure and volume of the given amount of gas at the given tempera- ture, and multiplying the pressure by the vol- ume. By substituting in the above expression this value of A", as well as the numerical values of the initial volume r, and the final volume r.„ involved in an actual compression, we will finally obtain the «ork which the problem required to calculate, and this is actually the way in which engineers determine that important quantity. Anotlwr ^yay of Stating and Solving the Third Problem. — The relation pr = k between the pres- .siires and volumes of a gas whose temperature is kept constant (i.e. the law of Boyle and Mari- otte) may be represented geometrically' by a curve called an equilateral hyperbola, every point of the curve corresponding to a definite pressure and volume. (See Asymptote.) Fur- ther, it is shown in text-books of natural philoso- phy that the work performed in compressing the gas from an initial volume represented by the line OA (see figure) to a final volume, OB, is represented by the area AA'B'B. The problem of determining the work may therefore be viewed as requiring to determine the area inclosed bj- the hyperbola and the axis OV between the limits OA = c, and OB = i;... To solve this problem the area AA'B'B may be imagined as made up of an infinite number of infinitely narrow strips. One such strip is roughly sliown in the figure between the lines marked p and p'. The diiTercnce be- tween p and p' woiId evidently be the greater, the greater the distance between them. But since the distance is supposed to be infinitely .small, the two lines may be taken as equal and the strip nuiy be consi<lcred as a rectangle. Calling its infinitely small base tlr, the area of the rectangle is seen to be pdr. The total area AA'B'B may now be obtained by summing up the infinite num- ber of "dilferentials of area' like pdc inclosed between the limits OA = c, and OB = y,. The summation may be performed by the integral calculus and is denoted by a definite integral, as pdi: "2 We have seen before that the result of this integration is tlog-^. The required area there- fore^ equals the natural logarithm of the ratio -- — 1 -, multiplied by the number (k) representing OB the product of any pair of coordinates, such as OB X BB', or OA X AA', etc. The calculus method is analogously employed whenever it is required to find the "area inclosed by a given curve, elementary geometry being in most cases powerless to furnish the "desired answer; and thus the calculus finds extensive application in the solution of many important problems of geometry. The above sketch outlines the methods of rea- soning by which the calculus attacks problems involving variable quantities. As to its limita- tions, it must be obsened that while the differ- ential calculus teaches how to differentiate read- ily any function whatever, the converse problem, viz. that of integrating a given differential, is often very ditticult, requiring all manner of alge- ' braic artifices, and is sometimes altogether im- possible. In other words, in their studies of nature, seientists are often led to construct dif- ferentials (just as in our third problem we con- structed the differential pdv) which they cannot - integrate, because they can conceive no "function which, on differentiation, would yield the given differential. Finally, it may be" observed that the Hour is no better than th'e grain, and if data that are made to pass through the mill of the calculus lead to doubtful resvilts. it is the fault not of the calculus, but of the data : the calculus itself is as exact as any other branch of mathe- matics, in spite of the fact that the things it deals with seem so often to dwindle away into nothing. History. The invention of the calculus meth- od is generally referred to the latter half of the Seventeenth Century, but the course of its early development really leads nuich further back. Thus, the 'method of exhaustion,' which, as first applied, consisted in comparing the area bounded by a given curve with the area of an inscribed or circumscrilied polygon whose luimber of sides is continually increased, is related to the present calculus through the doctrine of limits. Simi- larly, the surfaces of the sphere, cylinder, and cone were compared with prismatic and pyra- midal surfaces. By this method -Archimedes cal- culated the value of tt, obtained the areas of the parabola, ellipse, and one of the spirals, and found t!ie ratio of a sjjherical surface to the sur- face of the circumscribed cylinder. Kepler (ICl.T) was the first to improve this method by introducing into geometry the idea of infinity. He considered the circle as composed of an in- finite number of triangles (with their vertices at the centre and with their bases on the circum- ference), and the cone as composed of an infinite number of pyramids. The next advance is due to Cavalicri (q.v.), who effected quadrature by sununing the infinitesimal elements into which he