Calculus Made Easy quotes

CONSIDERING how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.

Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics—and they are mostly clever fools-seldom take the trouble to show you how easy the easy calculations are.

Some calculus-tricks are quite easy. Some are enormously difficult.

The fools who write the textbooks of advanced mathematics—and they are mostly clever fools-seldom take the trouble to show you how easy the easy calculations are.

On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.

On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.

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.

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.

… x is considered as made up of a lot of little bits, each of which is called d x, if you add them all up together you get the sum of all the d x's, (which is the same thing as the whole of x).

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 3600 little bits called seconds. The whole of the 3600 little bits added up together make one hour.

We shall have also to learn under what circumstances we may consider small quantities to be so minute that we may omit them from consideration.

Everything depends upon relative minuteness.

Before we fix any rules let us think of some familiar cases. There are 60 minutes in the hour, 24 hours in the day, 7 days in the week. There are therefore 1440 minutes in the day and 10080 minutes in the week.

it may certainly be regarded as a small quantity. But compare a farthing with pound sign 1000: relatively to this greater sum, the farthing is of no more importance than StartFraction 1 Over 1000 EndFraction of a farthing would be to a sovereign.

Even a golden sovereign is relatively a negligible quantity in the wealth of a millionaire.

Now if we fix upon any numerical fraction as constituting the proportion which for any purpose we call relatively small, we can easily state other fractions of a higher degree of smallness.

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.

Now in the calculus we write d x for a little bit of x. These things such as d x, and d u, and d y, are called "differentials," the differential of x, or of u, or of y, as the case may be.

The witty Dean Swift[2] once wrote:

An ox might worry about a flea of ordinary size—a small creature of the first order of smallness. But he would probably not trouble himself about a flea's flea; being of the second order of smallness, it would be negligible.

Even a gross of fleas' fleas would not be of much account to the ox.

We call the ratio StartFraction d y Over d x EndFraction "the differential coefficient of y with respect to x."

It is a solemn scientific name for this very simple thing.

But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frightened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names;

We call the ratio StartFraction d y Over d x EndFraction "the differential coefficient of y with respect to x." It is a solemn scientific name for this very simple thing.

But we are not going to be frightened by solemn names, when the things themselves are so easy.

Instead of being frightened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ratio StartFraction d y Over d x EndFraction.

(10) y equals RootIndex n StartRoot StartFraction 1 Over x Superscript m Baseline EndFraction EndRoot

You have now learned how to differentiate powers of x. How easy it is!

[3] N.B.—This ratio StartFraction d y Over d x EndFraction is the result of differentiating y with respect to x. Differentiating means finding the differential coefficient. Suppose we had some other function of x, as, for example, u equals 7 x squared plus 3.

… Over d x EndFraction 2nd Column equals StartFraction 3 b x Superscript 6 Baseline plus 5 a b x Superscript 4 Baseline minus 2 c x Over left parenthesis x squared plus a right parenthesis squared EndFraction period left parenthesis Answer period right parenthesis EndLayout

The working out of quotients is often tedious, but there is nothing difficult about it.

Some further examples fully worked out are given hereafter.

Take another example.

A man may have only a few pounds in his possession, and yet be able to spend money at the rate of millions a year-provided he goes on spending money at that rate for a few minutes only.

Suppose you hand a shilling over the counter to pay for some goods; and suppose the operation lasts exactly one second.

Try another simple problem in maxima and minima. Suppose you were asked to divide any number into two parts, such that the product was a maximum? How would you set about it if you did not know the trick of equating to zero?

I suppose you could worry it out by the rule of try, try, try again.

Let 60 be the number. You can try cutting it into two parts, and multiplying them together. Thus, 50 times 10 is 500; 52 times 8 is 416; 40 times 20 is 800; 45 times 15 is 675; 30 times 30 is 900.

Epsilon. To this mysterious number 2.7182818 etc., the mathematicians have assigned as a symbol the Greek letter epsilon (pronounced epsilon). All schoolboys know that the Greek letter pi (called pi) stands for 3.141592 etc.; but how many of them know that epsilon means 2.71828?

Yet it is an even more important number than pi!

Suppose we were to let 1 grow at simple interest till it became 2;

Thus integral d y equals y, and integral d x equals x.

Any one can understand how the whole of anything can be conceived of as made up of a lot of little bits; and the smaller the bits the more of them there will be.

Thus, a line one inch long may be conceived as made up of 10 pieces, each one tenth of an inch long; or of 100 parts, each part being StartFraction 1 Over 100 EndFraction of an inch long;

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 20 terms. A microscope would not show even the 18 Superscript th 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.

A microscope would not show even the 18 Superscript th 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.

"Integral dee-wy equals integral eks-squared dee-eks."]

We haven't yet integrated: we have only written down instructions to integrate—if we can.

Let us try. Plenty of other fools can do it-why not we also? The left-hand side is simplicity itself. The sum of all the bits of y is the same thing as y itself. So we may at once put:

We haven't yet integrated: we have only written down instructions to integrate—if we can. Let us try.

Plenty of other fools can do it-why not we also?

The left-hand side is simplicity itself. The sum of all the bits of y is the same thing as y itself. So we may at once put: y equals integral x squared d x period

Dodges.

A great part of the labour of integrating things consists in licking them into some shape that can be integrated.

The books-and by this is meant the serious books-on the Integral Calculus are full of plans and methods and dodges and artifices for this kind of work. The following are a few of them.

The equation which results from integrating a differential equation is called[7] its "solution"; and it is quite astonishing how in many cases the solution looks as if it had no relation to the differential equation of which it is the integrated form.

The solution often seems as different from the original expression as a butterfly does from the caterpillar that it was.

Who would have supposed that such an innocent thing as StartFraction d y Over d x EndFraction equals StartFraction 1 Over a squared minus x squared EndFraction could blossom out into y equals StartFraction 1 Over 2 a EndFraction log Subscript epsilon Baseline StartFraction a …

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.

… y Over d t squared EndFraction equals k StartFraction d squared y Over d x squared EndFraction comma the solution would have been the same, but the velocity of propagation would have had the value a equals StartRoot StartFraction k Over m EndFraction EndRoot period

You have now been personally conducted over the frontiers into the enchanted land.

And in order that you may have a handy reference to the principal results, the author, in bidding you farewell, begs to present you with a passport in the shape of a convenient collection of standard forms (see pp. 249-251).

Of that there can be, from their point of view, no possible manner of doubt whatever. It commits several most grievous and deplorable errors.

First, it shows how ridiculously easy most of the operations of the calculus really are.

Secondly, it gives away so many trade secrets.

Secondly, it gives away so many trade secrets.

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.

They like you to think how terribly difficult it is, and don't want that superstition to be rudely dissipated.

that there is an utter failure on the part of the author to demonstrate with rigid and satisfactory completeness the validity of sundry methods which he has presented in simple fashion, and has even dared to use in solving problems! But why should he not?

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.

But why should he not? 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.

But why should he not? 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.

One other thing will the professed mathematicians say about this thoroughly bad and vicious book: that the reason why it is so easy is because the author has left out all the things that are really difficult.

And the ghastly fact about this accusation is that—it is true!

That is, indeed, why the book has been written-written for the legion of innocents who have hitherto been deterred from acquiring the elements of the calculus by the stupid way in which its teaching is almost always presented.

That is, indeed, why the book has been written-written for the legion of innocents who have hitherto been deterred from acquiring the elements of the calculus by the stupid way in which its teaching is almost always presented.

Any subject can be made repulsive by presenting it bristling with difficulties.

The aim of this book is to enable beginners to learn its language, to acquire familiarity with its endearing simplicities, and to grasp its powerful methods of solving problems, without being compelled to toil through the intricate out-of-the-way (and mostly irrelevant) mathematical gymnastics so dear …

There are amongst young engineers a number on whose ears the adage that what one fool can do, another can, may fall with a familiar sound.

They are earnestly requested not to give the author away, nor to tell the mathematicians what a fool he really is.