Page:The New International Encyclopædia 1st ed. v. 04.djvu/30

* CALCULUS. to any method of treating problems by means of a system of algebraic notation. Thus, the Cal- culus of Forms (see Forms) is a symbolic treat- ment of the properties of invariants ; Imaginary Calculus is the method of calculating by the use of the imaginary unit (see Complex Numbeb), and the Calculus of Quaternions (see Quater- nions) is the method of treating cei-tain prob- lems with tlie aid of the quaternion symbolism. Usually, however, the term is employed to desig- nate the Difterential and Integral Calculus, a branch of mathematical science afl'ording. by one general method, a solution for many of the most diflicult problems of pure and applied mathe- matics. The Differential and Integral Calculus. This is one of the most useful branches of mathe- matics. While elementary algebra and geometry deal with quantities whose value is fixed, the cal- culus investigates quantities whose value is con- tinually changing. Considering that all nature in all its aspects varies continually, the impor- tance of a mathematical method of dealing with variables is evident: and it is easy to see why science had made so little progress before the invention of the calculus, and why progress has been so rapid since. Three simple examples may serve to show the kind of problems usually attacked by the calcu- lus, and the manner in which it solves them. The first two of these examples can also, on account of their simplicity, be solved by means of ele- mentary algebra, without resorting to the calcu- lus. Nevertheless, they are typical calculus prob- lems, and furnish as good examples of the calcu- lus method as would be furnished by similar but much more complicated problems lying really beyond the power of elementary mathematics. Problem I. Suppose the sum of two adjoining sides of a rectangle known. What must be the length of each side so that the rectangle may have the greatest possible area? Problem II. A person in a boat 3 miles from the nearest point on a straight shore wishes to reach a place 5 miles away from that point. He can row 4 miles an hour and walk 5 miles an hour. Where should he land in order to reach his point in mininmm time'; Problem III. To determine the work per- formed when a gas is compressed at constant temperature is one of the fundamental problems of theoretical engineering. Work is generally defined as the force required to move a body, multiplied by the distance traversed. In the case of a gas compressed in a cylindrical vessel, the bodv moved is the piston. If at the beginning of the experiment the pressure exercised on the pis- l l 1 rr ton is, say, p pounds per square inch of surface, and the area of the piston is a ; then p X a is evi- dently the force acting on the piston. This force, however, multiidicd by the distance traversed by the piston during compression will not by any means give the work performed. For during compression the force will, of course, have t*) be continually increased; in other words, it will not retainits original value ap fi.ed, but will be a variable. In this case algebra and geometry fail to give a method of direct computation and the calculus has to be resorted to. 16 CALCULUS. In order to understand how the calculus deals with problems of this nature, it is necessary to gras]) clearly some fundamental ideas, which usually appear somewhat dillicuit to the beginner in calculus, just as the idea of any fixed num- ber being represented by the letters a. b, c, ap- pears difficult to the child first taking up the study of elementary algebra. Fundamental Ideas: Function, Differential, Differential Coefficient, Limit. — Variables are re])resented in calculus by the Latin letters x, ij, etc., or by the Greek letters J, f. etc., just as unknown quantities are represented in algebra. If the value of one variable continually depends on that of another variable, the first variable is said to be a function of the second, and the fact is denoted by writing: y = f{x). Thus, the variable area y oi a square is said to be a func- tion of the variable length x of its side, and in this case the expression y =: f{x) stands for the equation y = x'. In investigating the functions and their variables, the calculus catches them at a given moment for the purpose of determining the relative rate of their variation at that mo- ment. Consider the motion of a ball thrown up in the air. Its velocity changes from instant to instant. We might get a rough idea of its mo- tion by measuring the distance traversed during the first second, during the second second, during the third second, etc. But our results would be far from precise: for, however small an interval of time a second is, the velocity of our ball, changing continually, must be dill'crent at the end of that interval from what it is at its be- ginning. Our results would be even rougher if instead of the second we employed as a unit of time the minute. To render the results mathe- matically precise, we would have to take for our unit not a finite, but an infinitely, small interval of time, an instant. The distance traversed dur- ing such an interval would be called the differ- ential of distance and would be denoted in calcu- lus by the synil>ol (//, if I stand for distance. Similarly, our infinitely small interval of time would be called the differential of time and would be denoted by the sjnnbol dt, if t stand for time. But as this idea of what a ditVereutial is is somewhat vague, owing to the diHiculty of actually conceiving something that is •infinitely snuill.' the following considerations may l)e re- sorted to. Studying the motion of a ball thrown up in the air, we consider infinitely small inter- vals of time (// merely in order to be al>le to think of the motion as uniform: for withiii any finite interval the motion is variable. But if at a given instant the motion should actually be- come unifcum. and continue so, we might think of our ditl'erential dt as representing any finite length of time, be it 5 minutes, or 10 minutes, or .50(1 minutes. For when a bodv moves with perfectly iniiform speed, that speed may be read- ily determined by ascertaining the distance trav- ersed during any interval of time wh.atever; the result is the same whether we divide the dis- tance traversed in 5 minutes by 5, or that traversed in 10 minutes by 10. We may, accord- ingly, define the differential of distance dl as the distance that 'trould lie traversed by the ball in an arbitrary, finite interval of time, dl, beginning at a given instant, if at that instant the motion. became uniform. In Ibis manner we may avoid thinking of infinitely small quantities. The velocity would then be dl-^dt, no matter how