On Different Degrees of Smallness

The Meaning of the Two Principal Symbols in CalculusThe Derivative

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  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 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  minutes in the hour,  hours in the day,  days in the week. There are therefore  minutes in the day and  minutes in the week.

Obviously  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 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 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 : relatively to this greater sum, the penny is of no more importance than of a penny would be to a ten dollar bill. Even
Fig. 2.1

Now suppose the square to grow by having a bit added to its size each way. The enlarged square is made up of the original square , the two rectangles at the top and on the right, each of which is of area (or together ), and the little square at the top right-hand corner which is . In the next figure, we have taken as quite a big fraction of —about .

Illustration for On Different Degrees of Smallness
Fig. 2.2

But suppose we had taken it only —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 of , and be practically invisible. Clearly is negligible if only we consider the increment to be itself small enough.

Illustration for On Different Degrees of Smallness
Fig. 2.3

A 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 part. Now if Mr. Millionaire received during the next week">You can't use 'macro parameter character #' in math mode10,000 is relatively a negligible quantity in the wealth of a billionaire.</p><p> </p><p>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, <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> be called a <i>small</i> fraction, then <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (being a <i>small</i> fraction of a <i>small</i> fraction) may be regarded as a <i>small quantity of the second order</i> of smallness.<a class="footnote-ref" href="#fn1" id="fnref1" role="doc-noteref"><sup>1</sup></a></p><p>Or, if for any purpose we were to take <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent (i.e. <mjx-container aria-label="\frac{1}{100}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="3.395ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1500.7 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{100}"><g data-mml-node="mfrac" data-latex="\frac{1}{100}"><g data-mml-node="mn" transform="translate(573.6,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="100"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><rect width="1260.7" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>) as a <i>small</i> fraction, then <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent of <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent (i.e. <mjx-container aria-label="\frac{1}{10,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="5.44ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 2404.3 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{10,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{10,000}"><g data-mml-node="mn" transform="translate(1025.4,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="10,000"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(1000,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(1278,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="2164.3" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>) would be a small fraction of the second order of smallness; and <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> would be a small fraction of the third order of smallness, being <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent of <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent of <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent.</p></details><p> </p><details><p> </p><summary>If we consider <mjx-container aria-label="\frac{1}{10^6}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.055ex;" xmlns="http://www.w3.org/2000/svg" width="3.294ex" height="3.012ex" role="img" focusable="false" viewBox="0 -864.9 1455.8 1331.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{10^6}"><g data-mml-node="mfrac" data-latex="\frac{1}{10^6}"><g data-mml-node="mn" transform="translate(551.1,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="msup" transform="translate(220,-450.9) scale(0.707)" data-latex="10^6"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mn" transform="translate(1033,393.1) scale(0.707)" data-latex="6"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use></g></g><rect width="1215.8" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (one millionth) as “small” for a specific purpose. In this context, <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>, that is <mjx-container aria-label="\frac{1}{10^{12}}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.055ex;" xmlns="http://www.w3.org/2000/svg" width="3.859ex" height="3.012ex" role="img" focusable="false" viewBox="0 -864.9 1705.8 1331.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{10^{12}}"><g data-mml-node="mfrac" data-latex="\frac{1}{10^{12}}"><g data-mml-node="mn" transform="translate(676.1,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="msup" transform="translate(220,-450.9) scale(0.707)" data-latex="10^{12}"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><g data-mml-node="TeXAtom" transform="translate(1033,393.1) scale(0.707)" data-latex="{12}" data-mjx-texclass="ORD"><g data-mml-node="mn" data-latex="12"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="32" xlink:href="#MJX-NCM-N-32" transform="translate(500,0)"></use></g></g></g><rect width="1465.8" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (one trillionth) will be a small quantity of the second order of smallness, and its impact can be completely disregarded in comparison.</summary><p> </p><p>Lastly, suppose that for some very precise purpose we should regard <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> 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 <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> part in <mjx-container aria-label="1,051,200" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.437ex;" xmlns="http://www.w3.org/2000/svg" width="9.931ex" height="1.943ex" role="img" focusable="false" viewBox="0 -666 4389.3 859" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1,051,200"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="051" transform="translate(944.7,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="35" xlink:href="#MJX-NCM-N-35" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-NCM-N-31" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2444.7,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="200" transform="translate(2889.3,0)"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g></g></svg></mjx-container>. Now if, for such a purpose, we regard <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (or one millionth) as a small quantity, then <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>, that is <mjx-container aria-label="\frac{1}{1,000,000,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="13.173ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 5822.5 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000,000,000}"><g data-mml-node="mn" transform="translate(2734.5,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(4056,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(4334,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(5834,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(6112,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="5582.5" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (or one trillionth) will be a small quantity of the second order of smallness, and may be utterly disregarded, by comparison.</p></details><p> </p><p>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 <i>in all cases we are justified in neglecting the small quantities of the second—or third (or higher)—orders</i>, if only we take the small quantity of the first order small enough in itself.</p><p>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.</p><p>Now in the calculus we write <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> for a little bit of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>. These things such as <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, and <mjx-container aria-label="du" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="du"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="u" transform="translate(520,0)"><use data-c="1D462" xlink:href="#MJX-NCM-I-1D462"></use></g></g></g></svg></mjx-container>, and <mjx-container aria-label="dy" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="2.285ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 1010 899" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dy"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="y" transform="translate(520,0)"><use data-c="1D466" xlink:href="#MJX-NCM-I-1D466"></use></g></g></g></svg></mjx-container>, are called “differentials,” the differential of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, or of <mjx-container aria-label="u" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="u"><g data-mml-node="mi" data-latex="u"><use data-c="1D462" xlink:href="#MJX-NCM-I-1D462"></use></g></g></g></svg></mjx-container>, or of <mjx-container aria-label="y" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="y"><g data-mml-node="mi" data-latex="y"><use data-c="1D466" xlink:href="#MJX-NCM-I-1D466"></use></g></g></g></svg></mjx-container>, as the case may be. [You <i>read</i> them as <i>dee-eks</i>, or <i>dee-you</i>, or <i>dee-wy</i>.] If <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> be a small bit of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, and relatively small of itself, it does not follow that such quantities as <mjx-container aria-label="x \cdot dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 572 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, or <mjx-container aria-label="x^2\, dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2\, dx"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g><g data-mml-node="mspace" data-latex="\," transform="translate(1008.6,0)"></g></g></g></svg><mjx-break size="0"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.848ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1259 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2\, dx"><g data-mml-node="mspace" data-latex="\,"></g><g data-mml-node="mi" data-latex="d" transform="translate(167,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(687,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, or <mjx-container aria-label="a^x\, dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.3ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1016.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="a^x\, dx"><g data-mml-node="msup" data-latex="a^x"><g data-mml-node="mi" data-latex="a"><use data-c="1D44E" xlink:href="#MJX-NCM-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(562,363) scale(0.707)" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g><g data-mml-node="mspace" data-latex="\," transform="translate(1016.5,0)"></g></g></g></svg><mjx-break size="0"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.848ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1259 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="a^x\, dx"><g data-mml-node="mspace" data-latex="\,"></g><g data-mml-node="mi" data-latex="d" transform="translate(167,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(687,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> are negligible. But <mjx-container aria-label="dx \times dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1092 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx \times dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.734ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2092.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx \times dx"><g data-mml-node="mo" data-latex="\times"><use data-c="D7" xlink:href="#MJX-NCM-N-D7"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(1000.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1520.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> would be negligible, being a small quantity of the second order.</p><p>A very simple example will serve as illustration.</p><p>Let us think of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> as a quantity that can grow by a small amount so as to become <mjx-container aria-label="x + dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 572 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x + dx"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.734ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2092.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x + dx"><g data-mml-node="mo" data-latex="+"><use data-c="2B" xlink:href="#MJX-NCM-N-2B"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(1000.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1520.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, where <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> is the small increment added by growth. The square of this is <mjx-container aria-label="x^2 + 2x \cdot dx + (dx)^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.688ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2072.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="mo" data-latex="+"><use data-c="2B" xlink:href="#MJX-NCM-N-2B"></use></g><g data-mml-node="mn" data-latex="2" transform="translate(1000.2,0)"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1500.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.481ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 3306.8 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="mo" data-latex="+"><use data-c="2B" xlink:href="#MJX-NCM-N-2B"></use></g><g data-mml-node="mo" data-latex="(" transform="translate(1000.2,0)"><use data-c="28" xlink:href="#MJX-NCM-N-28"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(1389.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1909.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="msup" data-latex=")^2" transform="translate(2481.2,0)"><g data-mml-node="mo" data-latex=")"><use data-c="29" xlink:href="#MJX-NCM-N-29"></use></g><g data-mml-node="mn" transform="translate(422,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. 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 <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. Thus if we took <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> to mean numerically, say, <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, then the second term would be <mjx-container aria-label="\frac{2}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{2}{60}"><g data-mml-node="mfrac" data-latex="\frac{2}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, whereas the third term would be <mjx-container aria-label="\frac{1}{3600}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="4.195ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1854.2 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{3600}"><g data-mml-node="mfrac" data-latex="\frac{1}{3600}"><g data-mml-node="mn" transform="translate(750.3,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="3600"><use data-c="33" xlink:href="#MJX-NCM-N-33"></use><use data-c="36" xlink:href="#MJX-NCM-N-36" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g><rect width="1614.2" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. This last term is clearly less important than the second. But if we go further and take <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> to mean only <mjx-container aria-label="\frac{1}{1000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="4.195ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1854.2 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1000}"><g data-mml-node="mn" transform="translate(750.3,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="1000"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g><rect width="1614.2" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, then the second term will be <mjx-container aria-label="\frac{2}{1000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="4.195ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1854.2 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{2}{1000}"><g data-mml-node="mfrac" data-latex="\frac{2}{1000}"><g data-mml-node="mn" transform="translate(750.3,394) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="1000"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g><rect width="1614.2" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, while the third term will be only <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>.</p><p>Geometrically this may be depicted as follows: Draw a square (the following figure) the side of which we will take to represent <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>.</p><figure id="fig:fig1"><figure class="image"><img style="aspect-ratio:350/346;" src="/book-images/CME/Fig1.svg" width="350" height="346"></figure><figcaption>Fig. 2.1</figcaption></figure><p>Now suppose the square to grow by having a bit <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> added to its size each way. The enlarged square is made up of the original square <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, the two rectangles at the top and on the right, each of which is of area <mjx-container aria-label="x \cdot dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 572 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> (or together <mjx-container aria-label="2x \cdot dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.425ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1072 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="2x \cdot dx"><g data-mml-node="mn" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(500,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="2x \cdot dx"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>), and the little square at the top right-hand corner which is <mjx-container aria-label="(dx)^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.561ex;" xmlns="http://www.w3.org/2000/svg" width="5.218ex" height="2.448ex" role="img" focusable="false" viewBox="0 -833.9 2306.6 1081.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="(dx)^2"><g data-mml-node="mo" data-latex="("><use data-c="28" xlink:href="#MJX-NCM-N-28"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(389,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(909,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="msup" data-latex=")^2" transform="translate(1481,0)"><g data-mml-node="mo" data-latex=")"><use data-c="29" xlink:href="#MJX-NCM-N-29"></use></g><g data-mml-node="mn" transform="translate(422,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. In the next figure, we have taken <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> as quite a big fraction of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>—about <mjx-container aria-label="\frac{1}{5}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="1.795ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 793.6 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{5}"><g data-mml-node="mfrac" data-latex="\frac{1}{5}"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="5"><use data-c="35" xlink:href="#MJX-NCM-N-35"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>.</p><figure id="fig:fig2"><figure class="image"><img style="aspect-ratio:350/340;" src="/book-images/CME/Fig2.svg" width="350" height="340"></figure><figcaption>Fig. 2.2</figcaption></figure><p>But suppose we had taken it only <mjx-container aria-label="\frac{1}{100}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="3.395ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1500.7 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{100}"><g data-mml-node="mfrac" data-latex="\frac{1}{100}"><g data-mml-node="mn" transform="translate(573.6,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="100"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><rect width="1260.7" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>—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 <mjx-container aria-label="\frac{1}{10,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="5.44ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 2404.3 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{10,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{10,000}"><g data-mml-node="mn" transform="translate(1025.4,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="10,000"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(1000,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(1278,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="2164.3" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, and be practically invisible. Clearly <mjx-container aria-label="(dx)^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.561ex;" xmlns="http://www.w3.org/2000/svg" width="5.218ex" height="2.448ex" role="img" focusable="false" viewBox="0 -833.9 2306.6 1081.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="(dx)^2"><g data-mml-node="mo" data-latex="("><use data-c="28" xlink:href="#MJX-NCM-N-28"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(389,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(909,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="msup" data-latex=")^2" transform="translate(1481,0)"><g data-mml-node="mo" data-latex=")"><use data-c="29" xlink:href="#MJX-NCM-N-29"></use></g><g data-mml-node="mn" transform="translate(422,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container> is negligible if only we consider the increment <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> to be itself small enough.</p><figure id="fig:fig3"><figure class="image"><img style="aspect-ratio:350/306;" src="/book-images/CME/Fig3.svg" width="350" height="306"></figure><figcaption>Fig. 2.3</figcaption></figure><details><p> </p><summary>A simile</summary><p> </p><p>Let us consider a simile.</p><p>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 <mjx-container aria-label="\frac{1}{100}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="3.395ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1500.7 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{100}"><g data-mml-node="mfrac" data-latex="\frac{1}{100}"><g data-mml-node="mn" transform="translate(573.6,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="100"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><rect width="1260.7" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> part. Now if Mr. Millionaire received during the next week, the secretary would receive You can't use 'macro parameter character #' in math mode<mjx-container aria-label="10" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.262ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 1000 688" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="10"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g></g></g></svg></mjx-container> and the boy <mjx-container aria-label="10" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.262ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 1000 688" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="10"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g></g></g></svg></mjx-container> cents. Ten dollars would be a small quantity compared with; 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 , had been settled at part? Then, while Mr. Millionaire got his You can't use 'macro parameter character #' in math mode<mjx-container aria-label="1000" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="4.525ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 2000 688" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1000"><g data-mml-node="mn" data-latex="1000"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g></g></g></svg></mjx-container>, Mr. Secretary would get only , and the boy 0.1 cents!

The witty Dean Swift2 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.

 

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  minutes in the hour,  hours in the day,  days in the week. There are therefore  minutes in the day and  minutes in the week.

Obviously  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 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 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 : relatively to this greater sum, the penny is of no more importance than of a penny would be to a ten dollar bill. Even
Fig. 2.1

Now suppose the square to grow by having a bit added to its size each way. The enlarged square is made up of the original square , the two rectangles at the top and on the right, each of which is of area (or together ), and the little square at the top right-hand corner which is . In Fig. 2.2 we have taken as quite a big fraction of —about .

Illustration for On Different Degrees of Smallness
Fig. 2.2

But suppose we had taken it only —about the thickness of an inked line drawn with a fine pen (Fig. 2.3). Then the little corner square will have an area of only of , and be practically invisible. Clearly is negligible if only we consider the increment to be itself small enough.

Illustration for On Different Degrees of Smallness
Fig. 2.3

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 part. Now if Mr. Millionaire received during the next week">You can't use 'macro parameter character #' in math mode10,000 is relatively a negligible quantity in the wealth of a billionaire.</p><p>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, <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> be called a <i>small</i> fraction, then <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (being a <i>small</i> fraction of a <i>small</i> fraction) may be regarded as a <i>small quantity of the second order</i> of smallness.<a class="footnote-ref" href="#fn1" id="fnref1" role="doc-noteref"><sup>1</sup></a></p><p>Or, if for any purpose we were to take <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent (i.e. <mjx-container aria-label="\frac{1}{100}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="3.395ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1500.7 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{100}"><g data-mml-node="mfrac" data-latex="\frac{1}{100}"><g data-mml-node="mn" transform="translate(573.6,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="100"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><rect width="1260.7" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>) as a <i>small</i> fraction, then <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent of <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent (i.e. <mjx-container aria-label="\frac{1}{10,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="5.44ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 2404.3 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{10,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{10,000}"><g data-mml-node="mn" transform="translate(1025.4,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="10,000"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(1000,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(1278,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="2164.3" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>) would be a small fraction of the second order of smallness; and <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> would be a small fraction of the third order of smallness, being <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent of <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent of <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> percent.</p><p>Lastly, suppose that for some very precise purpose we should regard <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> 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 <mjx-container aria-label="1" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g></g></g></svg></mjx-container> part in <mjx-container aria-label="1,051,200" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.437ex;" xmlns="http://www.w3.org/2000/svg" width="9.931ex" height="1.943ex" role="img" focusable="false" viewBox="0 -666 4389.3 859" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1,051,200"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="051" transform="translate(944.7,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="35" xlink:href="#MJX-NCM-N-35" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-NCM-N-31" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2444.7,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="200" transform="translate(2889.3,0)"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g></g></svg></mjx-container>. Now if, for such a purpose, we regard <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (or one millionth) as a small quantity, then <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>, that is <mjx-container aria-label="\frac{1}{1,000,000,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="13.173ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 5822.5 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000,000,000}"><g data-mml-node="mn" transform="translate(2734.5,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(4056,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(4334,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(5834,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(6112,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="5582.5" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> (or one trillionth) will be a small quantity of the second order of smallness, and may be utterly disregarded, by comparison.</p><p>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 <i>in all cases we are justified in neglecting the small quantities of the second—or third (or higher)—orders</i>, if only we take the small quantity of the first order small enough in itself.</p><p>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.</p><p>Now in the calculus we write <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> for a little bit of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>. These things such as <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, and <mjx-container aria-label="du" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="du"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="u" transform="translate(520,0)"><use data-c="1D462" xlink:href="#MJX-NCM-I-1D462"></use></g></g></g></svg></mjx-container>, and <mjx-container aria-label="dy" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="2.285ex" height="2.034ex" role="img" focusable="false" viewBox="0 -694 1010 899" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dy"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="y" transform="translate(520,0)"><use data-c="1D466" xlink:href="#MJX-NCM-I-1D466"></use></g></g></g></svg></mjx-container>, are called “differentials,” the differential of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, or of <mjx-container aria-label="u" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="u"><g data-mml-node="mi" data-latex="u"><use data-c="1D462" xlink:href="#MJX-NCM-I-1D462"></use></g></g></g></svg></mjx-container>, or of <mjx-container aria-label="y" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="y"><g data-mml-node="mi" data-latex="y"><use data-c="1D466" xlink:href="#MJX-NCM-I-1D466"></use></g></g></g></svg></mjx-container>, as the case may be. [You <i>read</i> them as <i>dee-eks</i>, or <i>dee-you</i>, or <i>dee-wy</i>.] If <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> be a small bit of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, and relatively small of itself, it does not follow that such quantities as <mjx-container aria-label="x \cdot dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 572 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, or <mjx-container aria-label="x^2\, dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2\, dx"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g><g data-mml-node="mspace" data-latex="\," transform="translate(1008.6,0)"></g></g></g></svg><mjx-break size="0"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.848ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1259 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2\, dx"><g data-mml-node="mspace" data-latex="\,"></g><g data-mml-node="mi" data-latex="d" transform="translate(167,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(687,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, or <mjx-container aria-label="a^x\, dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.3ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1016.5 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="a^x\, dx"><g data-mml-node="msup" data-latex="a^x"><g data-mml-node="mi" data-latex="a"><use data-c="1D44E" xlink:href="#MJX-NCM-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(562,363) scale(0.707)" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g><g data-mml-node="mspace" data-latex="\," transform="translate(1016.5,0)"></g></g></g></svg><mjx-break size="0"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.848ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1259 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="a^x\, dx"><g data-mml-node="mspace" data-latex="\,"></g><g data-mml-node="mi" data-latex="d" transform="translate(167,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(687,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> are negligible. But <mjx-container aria-label="dx \times dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1092 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx \times dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.734ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2092.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx \times dx"><g data-mml-node="mo" data-latex="\times"><use data-c="D7" xlink:href="#MJX-NCM-N-D7"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(1000.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1520.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> would be negligible, being a small quantity of the second order.</p><p>A very simple example will serve as illustration.</p><p>Let us think of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> as a quantity that can grow by a small amount so as to become <mjx-container aria-label="x + dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 572 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x + dx"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.734ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2092.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x + dx"><g data-mml-node="mo" data-latex="+"><use data-c="2B" xlink:href="#MJX-NCM-N-2B"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(1000.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1520.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, where <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> is the small increment added by growth. The square of this is <mjx-container aria-label="x^2 + 2x \cdot dx + (dx)^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="4.688ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2072.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="mo" data-latex="+"><use data-c="2B" xlink:href="#MJX-NCM-N-2B"></use></g><g data-mml-node="mn" data-latex="2" transform="translate(1000.2,0)"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1500.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="7.481ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 3306.8 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2 + 2x \cdot dx + (dx)^2"><g data-mml-node="mo" data-latex="+"><use data-c="2B" xlink:href="#MJX-NCM-N-2B"></use></g><g data-mml-node="mo" data-latex="(" transform="translate(1000.2,0)"><use data-c="28" xlink:href="#MJX-NCM-N-28"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(1389.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1909.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="msup" data-latex=")^2" transform="translate(2481.2,0)"><g data-mml-node="mo" data-latex=")"><use data-c="29" xlink:href="#MJX-NCM-N-29"></use></g><g data-mml-node="mn" transform="translate(422,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. 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 <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. Thus if we took <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> to mean numerically, say, <mjx-container aria-label="\frac{1}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{60}"><g data-mml-node="mfrac" data-latex="\frac{1}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, then the second term would be <mjx-container aria-label="\frac{2}{60}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="2.595ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1147.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{2}{60}"><g data-mml-node="mfrac" data-latex="\frac{2}{60}"><g data-mml-node="mn" transform="translate(396.8,394) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="60"><use data-c="36" xlink:href="#MJX-NCM-N-36"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><rect width="907.1" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, whereas the third term would be <mjx-container aria-label="\frac{1}{3600}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="4.195ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1854.2 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{3600}"><g data-mml-node="mfrac" data-latex="\frac{1}{3600}"><g data-mml-node="mn" transform="translate(750.3,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="3600"><use data-c="33" xlink:href="#MJX-NCM-N-33"></use><use data-c="36" xlink:href="#MJX-NCM-N-36" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g><rect width="1614.2" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. This last term is clearly less important than the second. But if we go further and take <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> to mean only <mjx-container aria-label="\frac{1}{1000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="4.195ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1854.2 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1000}"><g data-mml-node="mn" transform="translate(750.3,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="1000"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g><rect width="1614.2" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>, then the second term will be <mjx-container aria-label="\frac{2}{1000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="4.195ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1854.2 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{2}{1000}"><g data-mml-node="mfrac" data-latex="\frac{2}{1000}"><g data-mml-node="mn" transform="translate(750.3,394) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="1000"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g><rect width="1614.2" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, while the third term will be only <mjx-container aria-label="\frac{1}{1,000,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="7.484ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 3308 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{1,000,000}"><g data-mml-node="mn" transform="translate(1477.2,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="1,000,000"><g data-mml-node="mn" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mo" data-latex="," transform="translate(500,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(778,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(2278,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(2556,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="3068" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>.</p><p>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 <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>.</p><figure id="fig:fig1"><figure class="image"><img style="aspect-ratio:350/346;" src="/book-images/CME/Fig1.svg" width="350" height="346"></figure><figcaption>Fig. 2.1</figcaption></figure><p>Now suppose the square to grow by having a bit <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> added to its size each way. The enlarged square is made up of the original square <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, the two rectangles at the top and on the right, each of which is of area <mjx-container aria-label="x \cdot dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 572 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x \cdot dx"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> (or together <mjx-container aria-label="2x \cdot dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="2.425ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1072 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="2x \cdot dx"><g data-mml-node="mn" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(500,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg><mjx-break size="3"> </mjx-break><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="3.602ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1592.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="2x \cdot dx"><g data-mml-node="mo" data-latex="\cdot"><use data-c="22C5" xlink:href="#MJX-NCM-N-22C5"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(500.2,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(1020.2,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>), and the little square at the top right-hand corner which is <mjx-container aria-label="(dx)^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.561ex;" xmlns="http://www.w3.org/2000/svg" width="5.218ex" height="2.448ex" role="img" focusable="false" viewBox="0 -833.9 2306.6 1081.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="(dx)^2"><g data-mml-node="mo" data-latex="("><use data-c="28" xlink:href="#MJX-NCM-N-28"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(389,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(909,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="msup" data-latex=")^2" transform="translate(1481,0)"><g data-mml-node="mo" data-latex=")"><use data-c="29" xlink:href="#MJX-NCM-N-29"></use></g><g data-mml-node="mn" transform="translate(422,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>. In Fig. <a href="#fig:fig2" data-reference-type="ref" data-reference="fig:fig2">2.2</a> we have taken <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> as quite a big fraction of <mjx-container aria-label="x" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container>—about <mjx-container aria-label="\frac{1}{5}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="1.795ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 793.6 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{5}"><g data-mml-node="mfrac" data-latex="\frac{1}{5}"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="5"><use data-c="35" xlink:href="#MJX-NCM-N-35"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>.</p><figure id="fig:fig2"><figure class="image"><img style="aspect-ratio:350/340;" src="/book-images/CME/Fig2.svg" width="350" height="340"></figure><figcaption>Fig. 2.2</figcaption></figure><p>But suppose we had taken it only <mjx-container aria-label="\frac{1}{100}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="3.395ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1500.7 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{100}"><g data-mml-node="mfrac" data-latex="\frac{1}{100}"><g data-mml-node="mn" transform="translate(573.6,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="100"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><rect width="1260.7" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container>—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 <mjx-container aria-label="\frac{1}{10,000}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -1.089ex;" xmlns="http://www.w3.org/2000/svg" width="5.44ex" height="3.046ex" role="img" focusable="false" viewBox="0 -864.9 2404.3 1346.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{10,000}"><g data-mml-node="mfrac" data-latex="\frac{1}{10,000}"><g data-mml-node="mn" transform="translate(1025.4,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-345) scale(0.707)" data-latex="10,000"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" data-latex="," transform="translate(1000,0)"><use data-c="2C" xlink:href="#MJX-NCM-N-2C"></use></g><g data-mml-node="mn" data-latex="000" transform="translate(1278,0)"><use data-c="30" xlink:href="#MJX-NCM-N-30"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g></g><rect width="2164.3" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> of <mjx-container aria-label="x^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.282ex" height="1.912ex" role="img" focusable="false" viewBox="0 -833.9 1008.6 844.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x^2"><g data-mml-node="msup" data-latex="x^2"><g data-mml-node="mi" data-latex="x"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container>, and be practically invisible. Clearly <mjx-container aria-label="(dx)^2" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.561ex;" xmlns="http://www.w3.org/2000/svg" width="5.218ex" height="2.448ex" role="img" focusable="false" viewBox="0 -833.9 2306.6 1081.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="(dx)^2"><g data-mml-node="mo" data-latex="("><use data-c="28" xlink:href="#MJX-NCM-N-28"></use></g><g data-mml-node="mi" data-latex="d" transform="translate(389,0)"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(909,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g><g data-mml-node="msup" data-latex=")^2" transform="translate(1481,0)"><g data-mml-node="mo" data-latex=")"><use data-c="29" xlink:href="#MJX-NCM-N-29"></use></g><g data-mml-node="mn" transform="translate(422,363) scale(0.707)" data-latex="2"><use data-c="32" xlink:href="#MJX-NCM-N-32"></use></g></g></g></g></svg></mjx-container> is negligible if only we consider the increment <mjx-container aria-label="dx" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="2.471ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 1092 705" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="dx"><g data-mml-node="mi" data-latex="d"><use data-c="1D451" xlink:href="#MJX-NCM-I-1D451"></use></g><g data-mml-node="mi" data-latex="x" transform="translate(520,0)"><use data-c="1D465" xlink:href="#MJX-NCM-I-1D465"></use></g></g></g></svg></mjx-container> to be itself small enough.</p><figure id="fig:fig3"><figure class="image"><img style="aspect-ratio:350/306;" src="/book-images/CME/Fig3.svg" width="350" height="306"></figure><figcaption>Fig. 2.3</figcaption></figure><p>Let us consider a simile.</p><p>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 <mjx-container aria-label="\frac{1}{100}" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.816ex;" xmlns="http://www.w3.org/2000/svg" width="3.395ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 1500.7 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\frac{1}{100}"><g data-mml-node="mfrac" data-latex="\frac{1}{100}"><g data-mml-node="mn" transform="translate(573.6,394) scale(0.707)" data-latex="1"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)" data-latex="100"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use></g><rect width="1260.7" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container> part. Now if Mr. Millionaire received during the next week, the secretary would receive You can't use 'macro parameter character #' in math mode<mjx-container aria-label="10" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.262ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 1000 688" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="10"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g></g></g></svg></mjx-container> and the boy <mjx-container aria-label="10" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="2.262ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 1000 688" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="10"><g data-mml-node="mn" data-latex="10"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use></g></g></g></svg></mjx-container> cents. Ten dollars would be a small quantity compared with; 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 , had been settled at part? Then, while Mr. Millionaire got his You can't use 'macro parameter character #' in math mode<mjx-container aria-label="1000" class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="4.525ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 2000 688" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="1000"><g data-mml-node="mn" data-latex="1000"><use data-c="31" xlink:href="#MJX-NCM-N-31"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1000,0)"></use><use data-c="30" xlink:href="#MJX-NCM-N-30" transform="translate(1500,0)"></use></g></g></g></svg></mjx-container>, Mr. Secretary would get only , and the boy 0.1 cents!

The witty Dean Swift2 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 Meaning of the Two Principal Symbols in CalculusThe Derivative
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