Silvanus Phillips Thompson

“ You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus. ”

“ Moreover, we are usually dealing with more than one variable at once, and thinking of the way in which one variable depends on the other: for instance, we think of the way in which the height reached by a projectile depends on the time of attaining that height. Or we are asked to consider a rectangle of given area, and to enquire how any increase in the length of it will compel a corresponding decrease in the breadth of it. Or we think of the way in which any variation in the slope of a ladder will cause the height that it reaches, to vary. ”

“ Now in what sense is it true that a speed of yards per second is the same as yards per minute? Ten yards is not the same as yards, nor is one second the same thing as one minute. What we mean by saying that the rate is the same, is this: that the proportion borne between distance passed over and time taken to pass over it, is the same in both cases. ”

“ If we want to go so far that not even a Whitworth’s measuring machine would detect it, we should merely have to go to about terms. A microscope would not show even the term! So the infinite number of operations is no such dreadful thing after all. The integral is simply the whole lot. But, as we shall see, there are cases in which the integral calculus enables us to get at the exact total that there would be as the result of an infinite number of operations. ”

“ Why does anybody want to differentiate twice over? We know that when the variable quantities are space and time, by differentiating twice over we get the acceleration of a moving body, and that in the geometrical interpretation,
as applied to curves, means the slope of the curve. But what can mean in this case? Clearly it means the rate (per unit of length ) at which the slope is changing—in brief, it is a measure of the curvature of the slope. ”

“ Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third (or higher) —orders, if only we take the small quantity of the first order small enough in itself.
But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred. ”

“ When an integral is to be thus defined between two limits, we call the lower of the two values the inferior limit, and the upper value the superior limit. Any integral so limited we designate as a definite integral, by way of distinguishing it from a general integral to which no limits are assigned. ”

“ By showing you that what one fool can do, other fools can do also, it lets you see that these mathematical swells, who pride themselves on having mastered such an awfully difficult subject as the calculus, have no such great reason to be puffed up. ”

“ To accelerate a mass requires the continuous application of force. The force necessary to accelerate a mass is proportional to the mass, and it is also proportional to the acceleration which is being imparted. ”

“ On one side is imaginary; that is, there is no value of that can be represented by a graph; the latter is therefore entirely on the right side of the axis of (see Fig. 30) .
On plotting the graph it will be found that the curve goes to the origin, as if there were a minimum there; but instead of continuing beyond, as it should do for a minimum, it retraces its steps (forming what is called a “cusp”) . There is no minimum, therefore, although the condition for a minimum is satisfied, namely . It is necessary therefore always to check by taking one value on either side. ”

“ And when such a differential equation has been found, one can get no further until it has been integrated. Generally it is much easier to state the appropriate differential equation than to solve it:–the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy. ”

“ If we think of the curve which will represent as a function of , it will be such that its slope at any point will be proportional to the ordinate at that point, and will be a negative slope if is positive. So obviously the curve will be a die-away curve (p. 156) , and the solution will contain as a factor. But, without presuming on this bit of sagacity, let us go to work. As both and occur in the equation and on opposite sides, we can do nothing until we get both and to one side, and to the other. To do this, we must split our usually inseparable companions and from one another. ”

“ You will surely realize from this chapter and the preceding, that in many respects the calculus is an art rather than a science: an art only to be acquired, as all other arts are, by practice. Hence you should work many examples, and set yourself other examples, to see if you can work them out, until the various artifices become familiar by use. ”

“ Find the area of the first quadrant of the curve (known as “Pascal’s Snail”) , the polar equation of which is .
Volumes by Integration.
What we have done with the area of a little strip of a surface, we can, of course, just as easily do with the volume of a little strip of a solid. We can add up all the little strips that make up the total solid, and find its volume, just as we have added up all the small little bits that made up an area to find the final area of the figure operated upon. ”

“ We see that an explicit function in , , , etc., is simply something the value of which changes when , , , etc., are changing, either one at the time or several together. Because of this, the value of the explicit function is called the dependent variable, as it depends on the value of the other variable quantities in the function; these other variables are called the independent variables because their value is not determined from the value assumed by the function. For example, if , and are the independent variables, and is the dependent variable. ”

“ If you are spending money, and the amount you spend in a short time be called , the rate of spending it will be , or rather, should be written with a minus sign, as , because is a decrement, not an increment. But money is not a good example for the calculus, because it generally comes and goes by jumps, not by a continuous flow–you may earn £ a year, but it does not keep running in all day long in a thin stream ”

“ Now any fool can see that if is considered as made up of a lot of little bits, each of which is called , if you add them all up together you get the sum of all the ’s, (which is the same thing as the whole of ) . The word “integral” simply means “the whole.” If you think of the duration of time for one hour, you may (if you like) think of it as cut up into little bits called seconds. The whole of the little bits added up together make one hour. ”

“ The rate at which a velocity is increasing is called the acceleration. If a moving body is, at any particular instant, gaining an additional velocity in an element of time , then the acceleration at that instant may be written
;
but is itself . Hence we may put ; and this is usually written ;
or the acceleration is the second differential coefficient of the distance, with respect to time. Acceleration is expressed as a change of velocity in unit time, for instance, as being so many feet per second per second ”

“ As before, let the owner begin with a capital of £, earning interest at the rate of per cent. per annum; but, instead of hoarding the interest, let it be added to the capital each year, so that the capital grows year by year. Then, at the end of one year, the capital will have grown to £; and in the second year (still at 10%) this will earn £ interest. He will start the third year with £, and the interest on that will be £. s.; so that he starts the fourth year with £. s., and so on. It is easy to work it out, and find that at the end of the ten years the total capital will have grown to £. ”

“ Think of any curve that has a minimum point in it (like Fig. 15) , or like Fig. 34, where the point of minimum is marked , and the curve is concave upwards. To the left of the slope is downward, that is, negative, and is
getting less negative. To the right of the slope has become upward, and is getting more and more upward. Clearly the change of slope as the curve passes through is such that is positive, for its operation, as increases toward the right, is to convert a downward slope into an upward one. ”

“ Now we want to find the particular value that has when the curve is neither sloping up nor sloping down, that is, at the particular place where . So, writing does not mean that it always is ; but you write it down as a condition in order to see how much will come out if is to be zero.
The second remark is one which (if you have any wits of your own) you will probably have already made: namely, that this much-belauded process of equating to zero entirely fails to tell you whether the that you thereby find is going to give you a maximum value of or a minimum value of . ”

“ The beginner, who now knows how easy most of those processes are in themselves, will here begin to realize that integration is an art. As in all arts, so in this, facility can be acquired only by diligent and regular practice. He who would attain that facility must work out examples, and more examples, and yet more examples, such as are found abundantly in all the regular treatises on the Calculus. Our purpose here must be to afford the briefest introduction to serious work.
Example 1. Find the solution of the differential equation
. ”

“ Now this process can certainly be reversed: that is to say, that if this last expression were given, it is certain that it can somehow be split back again into its original components or partial fractions. Only we do not know in every case that may be presented to us how we can so split it. In order to find this out we shall consider a simple case at first. But it is important to bear in mind that all which follows applies only to what are called “proper” algebraic fractions, meaning fractions like the above, which have the numerator of a lesser degree than the denominator ”

“ WHEN TIME VARIES. Some of the most important problems of the calculus are those where time is the independent variable, and we have to think about the values of some other quantity that varies when the time varies. Some things grow larger as time goes on; some other things grow smaller. The distance that a train has got from its starting place goes on ever increasing as time goes on. Trees grow taller as the years go by. Which is growing at the greater rate ”

“ Suppose we were to let grow at simple interest till it became ; then, if at the same nominal rate of interest, and for the same time, we were to let grow at true compound interest, instead of simple, it would grow to the value epsilon. ”

“ By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable. Often in the consideration of physical relations one is able to build up an expression for the law governing the interaction of the parts or of the forces that govern them, such expression being naturally in the form of a differential equation, that is an equation containing differential coefficients with or without other algebraic quantities. ”

“ Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can. ”

“ But when we come to the right-hand side of the equation we must remember that what we have got to sum up together is not all the ’s, but all such terms as ; and this will not be the same as , because is not a constant. For some of the ’s will be multiplied by big values of , and some will be multiplied by small values of , according to what happens to be. So we must bethink ourselves as to what we know about this process of integration being the reverse of differentiation. ”

“ What we have to investigate is the value of ; or, in other words, if the angle varies, we have to find the relation between the increment of the sine and the increment of the angle, both increments being indefinitely small in themselves. Examine Fig. 43, wherein, if the radius of the circle is unity, the height of is the sine, and is the angle. Now, if is supposed to increase by the addition to it of the small angle –an element of angle–the height of , the sine, will be increased by a small element . ”

“ To differentiate the product of two functions, multiply each function by the differential coefficient of the other, and add together the two products so obtained.
You should note that this process amounts to the following: Treat as constant while you differentiate ; then treat as constant while you differentiate u; and the whole differential coefficient will be the sum of these two treatments. ”

“ MAXIMA AND MINIMA. One of the principal uses of the process of differentiating is to find out under what conditions the value of the thing differentiated becomes a maximum, or a minimum. This is often exceedingly important in engineering questions, where it is most desirable to know what conditions will make the cost of working a minimum, or will make the efficiency a maximum. ”

“ If, while is, as before, the distance of the foot of the ladder from the wall, is, instead of the height reached, the horizontal length of the wall, or the number of bricks in it, or the number of years since it was built, any change in would naturally cause no change whatever in ; in this case has no meaning whatever, and it is not possible to find an expression for it. Whenever we use differentials , , , etc., the existence of some kind of relation between , , , etc., is implied, and this relation is called a “function” in , , , etc. ”