*Calculus Made Easy*

* Calculus Made Easy: being a Very Simplest Introduction to those Beautiful Methods of Reckoning which are Generally Called by the Terrifying Names of the Differential Calculus and the Integral Calculus*, by Silvanus Phillips Thompson, was first published in 1910, and is considered a classic and elegant introduction to the subject. A 1998 update by Martin Gardner provides notes for modern readers and provides current versions of many obsolescent mathematical notations or terms. The following quotes are taken from his second edition, published in 1914.

## Contents

## Quotes[edit]

### Prologue[edit]

- 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.
- p.xi

- 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.
- p.xi

- Being myself a remarkably stupid fellow, I have had to teach 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.
- p.xi Noted as an ancient Simian proverb: "What one fool can do, another can."

### Ch.1 To Deliver You from the Preliminary Terrors.[edit]

- The preliminary terror, which chokes off most [students]... from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning—in common sense terms—of the two principal symbols that are used in calculating.
- p.1

- (1)
*d*... merely means "a little bit of." Thus*dx*means a little bit of x or*du*means a little bit of u. ...you will find that these little bits or elements may be considered to be indefinitely small.- p.1

- (2) which is merely a long S... may be called, if you like, "the sum of." Thus means the sum of all the little bits of
*x*or means the sum of all the little bits of*t*. Ordinary mathematicians call this symbol "the integral of."- p.1

- Now any fool can see that if
*x*is considered as made up of a lot of little bits, each of which is called*dx*, if you add them all up together you get the sum of all the*dx'*s (which is the same thing as the whole of*x*). The word "integral" simply means "the whole."- p.2

- When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow. That's all.
- p.2

### Ch.2 On Different Degrees of Smallness.[edit]

- We shall have... 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.
- p.3

- 1 minute is a very small quantity of time compared with a whole week, Indeed, our forefathers considered it small as compared with an hour, and called it "one minùte," meaning a minute fraction—namely one sixtieth—of an hour. When they came to require still smaller subdivisions of time, they divided each minute into 60 still smaller parts, which, in Queen Elizabeth's days, they called "second minùtes" (
*i.e.*small quantities of the second order of minuteness). Nowadays we call these... "seconds." But few people know why they are so called.- p.3-4

- If, for the purpose of time, 1/60 be called a
*small*fraction, then 1/60 of 1/60 (being a*small*fraction of a*small*fraction) may be regarded as a*small quantity of the second order*of smallness.- Footnote: The mathematicians talk about the second order of "magnitude" (
*i.e.*greatness) when they really mean second order of*smallness*. This is very confusing to beginners. - p.4

- Footnote: The mathematicians talk about the second order of "magnitude" (

- The smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become. 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.- p.5

- 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.
- p.5

- Now in the calculus we write
*dx*for a little bit of*x*. These things such as*dx*, and*du*, and*dy*, are called "differentials," the differential of*x*, or of*u*, or of*y*, as the case may be. [You read them as dee-eks or dee-you or dee-wy.] If*dx*be a small bit of*x*, and relatively small of itself, it does not follow that such quantities as or or are negligible. But would be negligible, being a small quantity of the second order.- p.6

- Let us think of
*x*as a quantity that can grow by a small amount so as to become , where*dx*is the small increment added by growth. The square of this is . The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of*x*.^{2}- Note that the third term can be interpreted as either "a bit of a bit of x," or "(a bit of x)
^{2}." - p.6

- Note that the third term can be interpreted as either "a bit of a bit of x," or "(a bit of x)

- The witty Dean Swift once wrote:

"So Nat'ralists observe, a Flea

"Hath smaller Fleas that on him prey.

"And these have smaller Fleas to bite 'em,

"And so proceed*ad infimitum.*

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.- Reference Jonathan Swift,
*On Poetry: A Rapsody* - p.8

- Reference Jonathan Swift,

### Ch.3 On Relative Growings.[edit]

- Let
*x*and*y*be respectively the base and the height of a right-angled triangle (Fig. 4), of which the slope... is fixed at 30º. If we suppose this triangle to expand and yet keep its angles the same as at first, then, when the base grows so as to become*x*+*dx*the height becomes*y*+*dy*. Here, increasing*x*results in an increase of*y*. The little triangle, the height of which is*dy*and the base of which is*dx*is similar to the original triangle; and it is obvious that the value of the ratio*dy*/*dx*is the same as that of the ratio*y*/*x*.- p.11

- We call the ratio
*dy*/*dx*"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*dy*/*dx*.- p.15

- The process of finding the value of
*dy*/*dx*is called "differentiating." But, remember, what is wanted is the value of this ratio when both*dy*and*dx*are themselves indefinitely small. The true value of the differential coefficient is that to which it approximates in the limiting case when each of them is considered as infinitesimally minute.- p.16

### Ch.4 Simplest Cases.[edit]

- Let us begin with the simple expression
*y*=*x*^{2}. ..Now remember that the fundamental notion about the calculus is the idea of*growing*. Mathematicians call it*varying*. ...the enlarged*y*will be equal to the square of the enlarged*x*. Writing this down we have*y*+*dy*= (*x*+*dx*)^{2}. Doing the squaring [see Ch.2, Fig.2 above] we get . ...*dx*^{2}will mean a little bit of a little bit of*x*; that is... a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms. Leaving it out, we then have: Now*y*=*x*^{2}; so let us subtract this from the equation and we have left Dividing across by*dx*, we find- p.18

- Try differentiating
*y*=*x*^{3}. We let*y*grow to*y*+*dy*while*x*grows to*x*+*dx*. Then we have y + dy = (x + dx)^{3}. ...[By a similar argument as above] ...Try differentiating*y*=*x*^{4}. Starting as before by letting both y and x grow a bit, we have:*y*+*dy*= (*x*+*dx*)^{4}. ...[By a similar argument as above]- p.20-21

- Let us collect the results to see if we can infer any general rule...

y |
dy/dx |
---|---|

x^{2} |
2x |

x^{3} |
3x^{2} |

x^{4} |
4x^{3} |

- Just look at these results: the operation of differentiating appears to have had the effect of diminishing the power of
*x*by 1 (for example in the last case reducing*x*^{4}to*x*^{3}), and at the same time multiplying by a number (the same number in fact which originally appeared as the power). Now, when you have once seen this, you might easily conjecture how the others will run. You would expect that differentiating*x*^{5}would give 5*x*^{4}or differentiating*x*^{6}would give 6*x*^{5}.- p.22

- Following out logically our observation, we should conclude that if we want to deal with any higher power,—call it
*n*—we could tackle it in the same way. Let*y*=*x*^{n}, then we should expect to find that*dy*/*dx*=*nx*^{(n-1)}. For example let*n*= 8 then*y*=*x*^{8}; and differentiating it would give*dy*/*dx*= 8*x*^{7}.- p.23

## Epilogue and Apologue[edit]

- Thirdly, among the dreadful things they will say about "So Easy" is this: 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 uent in the
*use*of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus.- p.247 and 248

## External links[edit]

- Silvanus Phillips Thompson,
*Calculus Made Easy: being a Very Simplest Introduction to those Beautiful Methods of Reckoning which are Generally Called by the Terrifying Names of the Differential Calculus and the Integral Calculus*(1914, originally 1910)