Different Degrees of Smallness | Calculus Made Easy

Chapter Summary (Express)

We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness.

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.

Obviously

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 minute,” 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 the 16th century, they called “second minutes” (i.e. small quantities of the second order of minuteness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called.

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.

Obviously $1$ 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 minute,” 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 the 16th century, they called “second minutes” (i.e. small quantities of the second order of minuteness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called.

Now if one minute is so small as compared with a whole day, how much smaller by comparison is one second!

An example about a small quantity of the second order of smallness

Again, think of a penny as compared with a ten dollar bill: it is only worth $\frac{1}{1000}$ part. A penny more or less is of precious little importance compared with a ten dollar bill: it may certainly be regarded as a small quantity. But compare a penny with $< / m i >< m n d a t a - l a t e x = " 10 " > 10 < / m n >< m o d a t a - l a t e x = ", " >, < / m o >< m n d a t a - l a t e x = " 000 " > 000 < / m n >< / m a t h >: r e l a t i v e l y t o t h i s g r e a t e r s u m, t h e p e n n y i s o f n o m o r e i m p o r t a n c e t h a n < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d a t a - l a t e x = " \frac{1}{1000} " >< m f r a c d a t a - l a t e x = " \frac{1}{1000} " >< m n d a t a - l a t e x = " 1 " > 1 < / m n >< m n d a t a - l a t e x = " 1000 " > 1000 < / m n >< / m f r a c >< / m a t h > o f a p e n n y w o u l d b e t o a t e n d o l l a r b i l l . E v e n10,000 is relatively a negligible quantity in the wealth of a billionaire.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. Thus if, for the purpose of time, \frac{1}{60} be called a small fraction, then \frac{1}{60} of \frac{1}{60} (being a small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness. 1Or, if for any purpose we were to take 1 percent (i.e. \frac{1}{100}) as a small fraction, then 1 percent of 1 percent (i.e. \frac{1}{10, 000}) would be a small fraction of the second order of smallness; and \frac{1}{1, 000, 000} would be a small fraction of the third order of smallness, being 1 percent of 1 percent of 1 percent.If we consider \frac{1}{10^{6}} (one millionth) as “small” for a specific purpose. In this context, \frac{1}{1, 000, 000} of \frac{1}{1, 000, 000}, that is \frac{1}{10^{12}} (one trillionth) will be a small quantity of the second order of smallness, and its impact can be completely disregarded in comparison. Lastly, suppose that for some very precise purpose we should regard \frac{1}{1, 000, 000} as “small.” Thus, if a first-rate chronometer is not to lose or gain more than half a minute in a year, it must keep time with an accuracy of 1 part in 1, 051, 200 . Now if, for such a purpose, we regard \frac{1}{1, 000, 000} (or one millionth) as a small quantity, then \frac{1}{1, 000, 000} of \frac{1}{1, 000, 000}, that is \frac{1}{1, 000, 000, 000, 000} (or one trillionth) will be a small quantity of the second order of smallness, and may be utterly disregarded, by comparison.Then we see that 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.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 penny 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. [You read them as dee-eks, or dee-you, or dee-wy .] If d x be a small bit of x, and relatively small of itself, it does not follow that such quantities as x \cdot d x, or x^{2} d x, or a^{x} d x are negligible. But d x \times d x would be negligible, being a small quantity of the second order.A very simple example will serve as illustration.Let us think of x as a quantity that can grow by a small amount so as to become x + d x, where d x is the small increment added by growth. The square of this is x^{2} + 2 x \cdot d x + (d x)^{2} . 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} . Thus if we took d x to mean numerically, say, \frac{1}{60} of x, then the second term would be \frac{2}{60} of x^{2}, whereas the third term would be \frac{1}{3600} of x^{2} . This last term is clearly less important than the second. But if we go further and take d x to mean only \frac{1}{1000} of x, then the second term will be \frac{2}{1000} of x^{2}, while the third term will be only \frac{1}{1, 000, 000} of x^{2} .Geometrically this may be depicted as follows: Draw a square (the following figure) the side of which we will take to represent x .Fig. 2.1Now suppose the square to grow by having a bit d x added to its size each way. The enlarged square is made up of the original square x^{2}, the two rectangles at the top and on the right, each of which is of area x \cdot d x (or together 2 x \cdot d x), and the little square at the top right-hand corner which is (d x)^{2} . In the next figure, we have taken d x as quite a big fraction of x —about \frac{1}{5} .Fig. 2.2But suppose we had taken it only \frac{1}{100} —about the thickness of an inked line drawn with a fine pen (the next figure). Then the little corner square will have an area of only \frac{1}{10, 000} of x^{2}, and be practically invisible. Clearly (d x)^{2} is negligible if only we consider the increment d x to be itself small enough.Fig. 2.3A simile Let us consider a simile. Suppose a millionaire were to say to his secretary: next week I will give you a small fraction of any money that comes in to me. Suppose that the secretary were to say to his boy: I will give you a small fraction of what I get. Suppose the fraction in each case to be \frac{1}{100} part. Now if Mr. Millionaire received during the next week < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d a t a - l a t e x = " 1000 " >< m n d a t a - l a t e x = " 1000 " > 1000 < / m n >< / m a t h >, t h e s e c r e t a r y w o u l d r e c e i v e 10 and the boy 10 cents. Ten dollars would be a small quantity compared with < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d a t a - l a t e x = " 1000 " >< m n d a t a - l a t e x = " 1000 " > 1000 < / m n >< / m a t h >; b u t 10 c e n t s i s a s m a l l s m a l l q u a n t i t y i n d e e d, o f a v e r y s e c o n d a r y o r d e r . B u t w h a t w o u l d b e t h e d i s p r o p o r t i o n i f t h e f r a c t i o n, i n s t e a d o f b e i n g < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d a t a - l a t e x = " \frac{1}{100} " >< m f r a c d a t a - l a t e x = " \frac{1}{100} " >< m n d a t a - l a t e x = " 1 " > 1 < / m n >< m n d a t a - l a t e x = " 100 " > 100 < / m n >< / m f r a c >< / m a t h >, h a d b e e n s e t t l e d a t < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d a t a - l a t e x = " \frac{1}{1000} " >< m f r a c d a t a - l a t e x = " \frac{1}{1000} " >< m n d a t a - l a t e x = " 1 " > 1 < / m n >< m n d a t a - l a t e x = " 1000 " > 1000 < / m n >< / m f r a c >< / m a t h > p a r t ? T h e n, w h i l e M r . M i l l i o n a i r e g o t h i s 1000, Mr. Secretary would get only $ 1, and the boy 0.1 cents! The witty Dean Swift 2 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 infinitum 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.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. ↩︎ On Poetry: a Rhapsody (page 20), printed 1733—usually misquoted. ↩︎Full Chapter We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness. 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. Obviously 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 minute,” 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 the 16th century, they called “second minutes” (i.e. small quantities of the second order of minuteness). Nowadays we call these small quantities of the second order of smallness “seconds.” But few people know why they are so called. Now if one minute is so small as compared with a whole day, how much smaller by comparison is one second! Again, think of a penny as compared with a ten dollar bill: it is only worth \frac{1}{1000} part. A penny more or less is of precious little importance compared with a ten dollar bill: it may certainly be regarded as a small quantity. But compare a penny with $ 10, 000 : relatively to this greater sum, the penny is of no more importance than \frac{1}{1000} of a penny would be to a ten dollar bill. Even 10,000 is relatively a negligible quantity in the wealth of a billionaire. 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. Thus if, for the purpose of time, ___MATH_BLOCK_10___ be called a small fraction, then ___MATH_BLOCK_11___ of ___MATH_BLOCK_12___ (being a small fraction of a small fraction) may be regarded as a small quantity of the second order of smallness.<a class="footnote-ref" href="#fn1" id="fnref1" role="doc-noteref">1</a> Or, if for any purpose we were to take ___MATH_BLOCK_13___ percent (i.e. ___MATH_BLOCK_14___) as a small fraction, then ___MATH_BLOCK_15___ percent of ___MATH_BLOCK_16___ percent (i.e. ___MATH_BLOCK_17___) would be a small fraction of the second order of smallness; and ___MATH_BLOCK_18___ would be a small fraction of the third order of smallness, being ___MATH_BLOCK_19___ percent of ___MATH_BLOCK_20___ percent of ___MATH_BLOCK_21___ percent. Lastly, suppose that for some very precise purpose we should regard ___MATH_BLOCK_22___ as “small.” Thus, if a first-rate chronometer is not to lose or gain more than half a minute in a year, it must keep time with an accuracy of ___MATH_BLOCK_23___ part in ___MATH_BLOCK_24___. Now if, for such a purpose, we regard ___MATH_BLOCK_25___ (or one millionth) as a small quantity, then ___MATH_BLOCK_26___ of ___MATH_BLOCK_27___, that is ___MATH_BLOCK_28___ (or one trillionth) will be a small quantity of the second order of smallness, and may be utterly disregarded, by comparison. Then we see that 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. 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 penny becomes important if only it is multiplied by a few hundred. Now in the calculus we write ___MATH_BLOCK_29___ for a little bit of ___MATH_BLOCK_30___. These things such as ___MATH_BLOCK_31___, and ___MATH_BLOCK_32___, and ___MATH_BLOCK_33___, are called “differentials,” the differential of ___MATH_BLOCK_34___, or of ___MATH_BLOCK_35___, or of ___MATH_BLOCK_36___, as the case may be. [You read them as dee-eks, or dee-you, or dee-wy.] If ___MATH_BLOCK_37___ be a small bit of ___MATH_BLOCK_38___, and relatively small of itself, it does not follow that such quantities as ___MATH_BLOCK_39___, or ___MATH_BLOCK_40___, or ___MATH_BLOCK_41___ are negligible. But ___MATH_BLOCK_42___ would be negligible, being a small quantity of the second order. A very simple example will serve as illustration. Let us think of ___MATH_BLOCK_43___ as a quantity that can grow by a small amount so as to become ___MATH_BLOCK_44___, where ___MATH_BLOCK_45___ is the small increment added by growth. The square of this is ___MATH_BLOCK_46___. 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 ___MATH_BLOCK_47___. Thus if we took ___MATH_BLOCK_48___ to mean numerically, say, ___MATH_BLOCK_49___ of ___MATH_BLOCK_50___, then the second term would be ___MATH_BLOCK_51___ of ___MATH_BLOCK_52___, whereas the third term would be ___MATH_BLOCK_53___ of ___MATH_BLOCK_54___. This last term is clearly less important than the second. But if we go further and take ___MATH_BLOCK_55___ to mean only ___MATH_BLOCK_56___ of ___MATH_BLOCK_57___, then the second term will be ___MATH_BLOCK_58___ of ___MATH_BLOCK_59___, while the third term will be only ___MATH_BLOCK_60___ of ___MATH_BLOCK_61___. Geometrically this may be depicted as follows: Draw a square (Fig. <a href="#fig:fig1" data-reference-type="ref" data-reference="fig:fig1">2.1</a>) the side of which we will take to represent ___MATH_BLOCK_62___. <figure id="block-blk-1oot511ik" class="figure" data-block-id="blk-1oot511ik" style="display: block; max-width: 350px; margin: 0 auto; text-align: center;"> <img src="/book-images/CME/Fig1.svg" alt="Fig. 2.1" style="max-width: 100%; height: auto; display: block; margin: 0 auto;"> <figcaption style="text-align: center; margin-top: 8px;">Figure 1 Fig. 2.1</figcaption> </figure> Now suppose the square to grow by having a bit ___MATH_BLOCK_63___ added to its size each way. The enlarged square is made up of the original square ___MATH_BLOCK_64___, the two rectangles at the top and on the right, each of which is of area ___MATH_BLOCK_65___ (or together ___MATH_BLOCK_66___), and the little square at the top right-hand corner which is ___MATH_BLOCK_67___. In Fig. <a href="#fig:fig2" data-reference-type="ref" data-reference="fig:fig2">2.2</a> we have taken ___MATH_BLOCK_68___ as quite a big fraction of ___MATH_BLOCK_69___—about ___MATH_BLOCK_70___. <figure id="block-blk-k3mkirhar" class="figure" data-block-id="blk-k3mkirhar" style="display: block; max-width: 350px; margin: 0 auto; text-align: center;"> <img src="/book-images/CME/Fig2.svg" alt="Fig. 2.2" style="max-width: 100%; height: auto; display: block; margin: 0 auto;"> <figcaption style="text-align: center; margin-top: 8px;">Figure 2 Fig. 2.2</figcaption> </figure> But suppose we had taken it only ___MATH_BLOCK_71___—about the thickness of an inked line drawn with a fine pen (Fig. <a href="#fig:fig3" data-reference-type="ref" data-reference="fig:fig3">2.3</a>). Then the little corner square will have an area of only ___MATH_BLOCK_72___ of ___MATH_BLOCK_73___, and be practically invisible. Clearly ___MATH_BLOCK_74___ is negligible if only we consider the increment ___MATH_BLOCK_75___ to be itself small enough. <figure id="block-blk-igh9rqfjl" class="figure" data-block-id="blk-igh9rqfjl" style="display: block; max-width: 350px; margin: 0 auto; text-align: center;"> <img src="/book-images/CME/Fig3.svg" alt="Fig. 2.3" style="max-width: 100%; height: auto; display: block; margin: 0 auto;"> <figcaption style="text-align: center; margin-top: 8px;">Figure 3 Fig. 2.3</figcaption> </figure> Let us consider a simile. Suppose a millionaire were to say to his secretary: next week I will give you a small fraction of any money that comes in to me. Suppose that the secretary were to say to his boy: I will give you a small fraction of what I get. Suppose the fraction in each case to be ___MATH_BLOCK_76___ part. Now if Mr. Millionaire received during the next week 1000, the secretary would receive $ 10 and the boy 10 cents. Ten dollars would be a small quantity compared with $ 1000; but 10 cents is a small small quantity indeed, of a very secondary order. But what would be the disproportion if the fraction, instead of being \frac{1}{100}, had been settled at \frac{1}{1000} part? Then, while Mr. Millionaire got his $ 1000, Mr. Secretary would get only $ 1, and the boy 0.1 cents! The witty Dean Swift 2 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 infinitum 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. 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. ↩︎ On Poetry: a Rhapsody (page 20), printed 1733—usually misquoted. ↩︎The Meaning of the Two Principal Symbols in Calculus The DerivativeLearn Most Efficiently Through Adaptive Books Quick Links Become an Author Teach a Book Privacy Policy Contact Us Blog Terms and Conditions Social Media Copyright © 2026 Avidemia. 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On Different Degrees of Smallness

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