# 11.2: Footnotes - Mathematics

1

However, neither of the two volumes contains more than the principal fragment of BM 15285. A new edition based on the three fragments that are known today can be found in Eleanor Robson, Mesopotamian Mathematics 2100–1600 BC. Technical Constants in Bureaucracy and Education. Oxford: Clarendon Press, 1999.

2

In other words, the edition is almost useless for non-specialists, even for historians of mathematics who do not understand the Old Babylonian tradition too well; several general histories of mathematics or algebra contain horrendous mistakes going back to Evert Bruins’s commentary.

## 11.2: Footnotes - Mathematics

2 When John, (A) who was in prison, (B) heard about the deeds of the Messiah, he sent his disciples 3 to ask him, “Are you the one who is to come, (C) or should we expect someone else?”

4 Jesus replied, “Go back and report to John what you hear and see: 5 The blind receive sight, the lame walk, those who have leprosy [a] are cleansed, the deaf hear, the dead are raised, and the good news is proclaimed to the poor. (D) 6 Blessed is anyone who does not stumble on account of me.” (E)

7 As John’s (F) disciples were leaving, Jesus began to speak to the crowd about John: “What did you go out into the wilderness (G) to see? A reed swayed by the wind? 8 If not, what did you go out to see? A man dressed in fine clothes? No, those who wear fine clothes are in kings’ palaces. 9 Then what did you go out to see? A prophet? (H) Yes, I tell you, and more than a prophet. 10 This is the one about whom it is written:

“‘I will send my messenger ahead of you, (I)
who will prepare your way before you.’ [b] (J)

11 Truly I tell you, among those born of women there has not risen anyone greater than John the Baptist yet whoever is least in the kingdom of heaven is greater than he.

The number /> is a square number if and only if one can arrange /> points in a square: Template:How

 m = 1 2 = 1 m = 2 2 = 4 m = 3 2 = 9 m = 4 2 = 16 m = 5 2 = 25

The expression for the th square number is . This is also equal to the sum of the first odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows:

So for example, 5 2 = 25 = 1 + 3 + 5 + 7 + 9.

There are several recursive methods for computing square numbers. For example, the th square number can be computed from the previous square by . Alternatively, the th square number can be calculated from the previous two by doubling the (n − 1) -th square, subtracting the -th square number, and adding 2, because . For example,

2 × 5 2 − 4 2 + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 6 2 .

A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.

Another property of a square number is that it has an odd number of divisors, while other numbers have an even number of divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs.

Lagrange's four-square theorem states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form /> . A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form /> . This is generalized by Waring's problem.

A square number can end only with digits 0, 1, 4, 6, 9, or 25 in base 10, as follows:

1. If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the digits preceding the ending 0s must also form a square.
2. If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four.
3. If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
4. If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four.
5. If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
6. If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.

In base 16, a square number can end only with 0, 1, 4 or 9 and

In general, if a prime divides a square number then the square of must also divide   if fails to divide , then is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number is a square number if and only if, in its canonical representation, all exponents are even.

Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given and some number , if is the square of an integer then divides . (This is an application of the factorization of a difference of two squares.) For example, 100 2 − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from to where covers some range of natural numbers .

A square number cannot be a perfect number.

The sum of the series of power numbers

can also be represented by the formula

The first terms of this series (the square pyramidal numbers) are:

0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201. (sequence A000330 in OEIS).

All fourth powers, sixth powers, eighth powers and so on are perfect squares.

It seems that you and your friend lack the mathematical knowledge to handle this delicate point. What is a proof? What is an axiom? What are $1,+,2,=$?

Well, let me try and be concise about things.

A proof is a short sequence of deductions from axioms and assumptions, where at every step we deduce information from our axioms, our assumptions and previously deduced sentences.

An axiom is simply an assumption.

$1,+,2,=$ are just letters and symbols. We usually associate $=$ with equality that is two things are equal if and only if they are the same thing. As for $1,2,+$ we have a natural understanding of what they are but it is important to remember those are just letters which can be used elsewhere (and they are used elsewhere, often).

You want to prove to your friend that $1+1=2$, where those symbols are interpreted as they are naturally perceived. $1$ is the amount of hands attached to a healthy arm of a human being $2$ is the number of arms attached to a healthy human being and $is the natural sense of addition. From the above, what you want to show, mathematically, is that if you are a healthy human being then you have exactly two hands. But in mathematics we don't talk about hands and arms. We talk about mathematical objects. We need a suitable framework, and we need axioms to define the properties of these objects. For the sake of the natural numbers which include$1,2,+$and so on, we can use the Peano Axioms (PA). These axioms are commonly accepted as the definition of the natural numbers in mathematics, so it makes sense to choose them. I don't want to give a full exposition of PA, so I will only use the part I need from the axioms, the one discussing addition. We have three primary symbols in the language: , S, +$. And our axioms are:

1. For every $x$ and for every $y$, $S(x)=S(y)$ if and only if $x=y$.
2. For every $x$ either $x=0$ or there is some $y$ such that $x=S(y)$.
3. There is no $x$ such that $S(x)=0$.
4. For every $x$ and for every $y$, $x+y=y+x$.
5. For every $x$, $x+0=x$.
6. For every $x$ and for every $y$, $x+S(y)=S(x+y)$.

This axioms tell us that $S(x)$ is to be thought as $x+1$ (the successor of $x$), and it tells us that addition is commutative and what relations it bears with the successor function.

Now we need to define what are $1$ and $2$. Well, $1$ is a shorthand for $S(0)$ and $2$ is a shorthand for $S(1)$, or $S(S(0))$.

Finally! We can write a proof that $1+1=2$:

1. $S(0)+S(0)=S(S(0)+0)$ (by axiom 6).
2. $S(0)+0 = S(0)$ (by axiom 5).
3. $S(S(0)+0) = S(S(0))$ (by the second deduction and axiom 1).
4. $S(0)+S(0) = S(S(0))$ (from the first and third deductions).

And that is what we wanted to prove.

Note that the context is quite important. We are free to define the symbols to mean whatever it is we want them to mean. We can easily define a new context, and a new framework in which $1+1 eq 2$. Much like we can invent a whole new language in which Bye is a word for greeting people when you meet them, and Hi is a word for greeting people as they leave.

To see that $1+1 eq2$ in some context, simply define the following axioms:

Now we can write a proof that $1+1 eq 2$:

1. $1+1=1$ (axiom 2 applied for $x=1$).
2. $1 eq 2$ (axiom 1).
3. $1+1 eq 2$ (from the first and second deductions).

If you read this far, you might also be interested to read these:

Those interested in pushing this question back further than Asaf Karagila did (well past logic and into the morass of philosophy) may be interested in the following comments that were written in 1860 (full reference below). Also, although Asaf's treatment here avoids this, there are certain issues when defining addition of natural numbers in terms of the successor operation that are often overlooked. See my 22 November 2011 and 28 November 2011 posts in the Math Forum group math-teach.

$[ldots]$ consider this case. There is a world in which, whenever two pairs of things are either placed in proximity or are contemplated together, a fifth thing is immediately created and brought within the contemplation of the mind engaged in putting two and two together. This is surely neither inconceivable, for we can readily conceive the result by thinking of common puzzle tricks, nor can it be said to be beyond the power of Omnipotence, yet in such a world surely two and two would make five. That is, the result to the mind of contemplating two two’s would be to count five. This shows that it is not inconceivable that two and two might make five but, on the other hand, it is perfectly easy to see why in this world we are absolutely certain that two and two make four. There is probably not an instant of our lives in which we are not experiencing the fact. We see it whenever we count four books, four tables or chairs, four men in the street, or the four corners of a paving stone, and we feel more sure of it than of the rising of the sun to-morrow, because our experience upon the subject is so much wider and applies to such an infinitely greater number of cases.

The above passage comes from:

Stephen’s review of Mansel's book is reprinted on pp. 320-335 of Stephen's 1862 book Essays, where the quote above can be found on page 333.

(ADDED 2 YEARS LATER) Because my answer continues to receive sporadic interest and because I came across something this weekend related to it, I thought I would extend my answer by adding a couple of items.

The first new item, [A], is an excerpt from a 1945 paper by Charles Edward Whitmore. I came across Whitmore's paper several years ago when I was looking through all the volumes of the journal Journal of the History of Ideas at a nearby university library. Incidentally, Whitmore's paper is where I learned about speculations of James Fitzjames Stephen that are given above. The second new item, [B], is an excerpt from an essay by Augustus De Morgan that I read this last weekend. De Morgan's essay is item [15] in my answer to the History of Science and Math StackExchange question Did Galileo's writings on infinity influence Cantor?, and his essay is also mentioned in item [8]. I've come across references to De Morgan's essay from time to time over the years, but I've never read it because I never bothered trying to look it up in a university library. However, when I found to my surprise (but I really shouldn't have been surprised) that a digital copy of the essay was freely available on the internet when I searched for it about a week ago, I made a print copy, which I then read through when I had some time (this last weekend).

[A] Charles Edward Whitmore (1887-1970), Mill and mathematics: An historical note, Journal of the History of Ideas 6 #1 (January 1945), 109-112. MR 6,141n Zbl 60.01622

(first paragraph of the paper, on p. 109) In various philosophical works one encounters the statement that J. S. Mill somewhere asserted that two and two might conceivably make five. Thus, Professor Lewis says $^1$ that Mill "asked us to suppose a demon sufficiently powerful and maleficent so that every time two things were brought together with two other things, this demon should always introduce a fifth" but he gives no specific reference. <<>$^1$ C. I. Lewis, Mind and the World Order (1929), 250.>> C. S. Peirce $^2$ puts it in the form, "when two things were put together a third should spring up," calling it a doctrine usually attributed to Mill. <<>$^2$ Collected Papers, IV, 91 (dated 1893). The editors supply a reference to Logic, II, vi, 3.>> Albert Thibaudet $^3$ ascribes to "a Scottish philosopher cited by Mill" the doctrine that the addition of two quantities might lead to the production of a third. <<>$^3$ Introduction to Les Idées de Charles Maurras (1920), 7.>> Again, Professor Laird remarks $^4$ that "Mill suggested, we remember, that two and two might not make four in some remote part of the stellar universe," referring to Logic III, xxi, 4 and II, vi, 2. <<>$^4$ John Laird, Knowledge, Belief, and Opinion (1930), 238.>> These instances, somewhat casually collected, suggest that there is some confusion in the situation.

(from pp. 109-111) Moreover, the notion that two and two should ["could" intended?] make five is entirely opposed to the general doctrine of the Logic. $[cdots]$ Nevertheless, though these views stand in the final edition of the Logic, it is true that Mill did, in the interval, contrive to disallow them. After reading through the works of Sir William Hamilton three times, he delivered himself of a massive Examination of that philosopher, in the course of which he reverses his position--but at the suggestion of another thinker. In chapter VI he falls back on the inseparable associations generated by uniform experience as compelling us to conceive two and two as four, so that "we should probably have no difficulty in putting together the two ideas supposed to be incompatible, if our experience had not first inseparably associated one of them with the contradictory of the other." To this he adds, "That the reverse of the most familiar principles of arithmetic and geometry might have been made conceivable even to our present mental faculties, if those faculties had coexisted with a totally different constitution of external nature, is ingeniously shown in the concluding paper of a recent volume, anonymous, but of known authorship, Essays, by a Barrister." The author of the work in question was James Fitzjames Stephen, who in 1862 had brought together various papers which had appeared in Saturday Review during some three previous years. Some of them dealt with philosophy, and it is from a review of Mansel's Metaphysics that Mill proceeds to quote in support of his new doctrine $[cdots]$

Note: On p. 111 Whitmore argues against Mill's and Stephen's empirical viewpoint of "two plus two equals four". Whitmore's arguments are not very convincing to me.

(from p. 112) Mill, then, did not originate the idea, but adopted it from Stephen, in the form that two and two might make five to our present faculties, if external nature were differently constituted. He did not assign it to some remote part of the universe, nor did he call in the activity of some maleficent demon neither did he say that one and one might make three. He did not explore its implications, or inquire how it might be reconciled with what he had said in other places but at least he is entitled to a definite statement of what he did say. I confess that I am somewhat puzzled at the different forms in which it has been quoted, and at the irrelevant details which have been added.

[B] Augustus De Morgan (1806-1871), On infinity and on the sign of equality, Transactions of the Cambridge Philosophical Society 11 Part I (1871), 145-189.

Published separately as a booklet by Cambridge University Press in 1865 (same title i + 45 pages). The following excerpt is from the version published in 1865.

(footnote 1 on p. 14) We are apt to pronounce that the admirable pre-established harmony which exists between the subjective and objective is a necessary property of mind. It may, or may not, be so. Can we not grant to omnipotence the power to fashion a mind of which the primary counting is by twos, ,2,4,6,$&c. a mind which always finds its first indicative notion in this and that, and only with effort separates this from that. I cannot invent the fundamental forms of language for this mind, and so am obliged to make it contradict its own nature by using our terms. The attempt to think of such things helps towards the habit of distinguishing the subjective and objective. Note: Those interested in such speculations will also want to look at De Morgan's lengthy footnote on p. 20. (ADDED 6 YEARS LATER) I recently read Ian Stewart's 2006 book Letters to a Young Mathematician and in this book there is a passage (see below) that I think is worth including here. (from pp. 30-31) I think human math is more closely linked to our particular physiology, experiences, and psychological preferences than we imagine. It is parochial, not universal. Geometry's points and lines may seem the natural basis for a theory of shape, but they are also the features into which our visual system happens to dissect the world. An alien visual system might find light and shade primary, or motion and stasis, or frequency of vibration. An alien brain might find smell, or embarrassment, but not shape, to be fundamental to its perception of the world. And while discrete numbers like$1,2,3,$seem universal to us, they trace back to our tendency to assemble similar things, such as sheep, and consider them property: has one of my sheep been stolen? Arithmetic seems to have originated through two things: the timing of the seasons and commerce. But what of the blimp creatures of distant Poseidon, a hypothetical gas giant like Jupiter, whose world is a constant flux of turbulent winds, and who have no sense of individual ownership? Before they could count up to three, whatever they were counting would have blown away on the ammonia breeze. They would, however, have a far better understanding than we do of the math of turbulent fluid flow. ## 2 Running Scheme This chapter describes how to run MIT/GNU Scheme. It also describes how you can customize the behavior of MIT/GNU Scheme using command-line options and environment variables. ### 2.1 Basics of Starting Scheme Under unix, MIT/GNU Scheme is invoked by typing at your operating system&rsquos command interpreter. In either case, Scheme will load itself and print something like this: This information, which can be printed again by evaluating tells you the following version information. &lsquo Release &rsquo is the release number for the entire Scheme system. This number is changed each time a new version of Scheme is released. Following this there may be additional names for specific subsystems. &lsquo SF &rsquo refers to the scode optimization program sf &lsquo LIAR/ ARCH &rsquo is the native-code compiler, where ARCH is the native-code architecture it compiles to &lsquo Edwin &rsquo is the Emacs-like text editor. There are other subsystems you can load that will add themselves to this list. ### 2.2 Customizing Scheme You can customize your setup by using a variety of tools: • Command-line options. Many parameters, like memory usage and the location of libraries, may be varied by command-line options. See Command-Line Options. • Shell scripts. You might like to write scripts that invoke Scheme with your favorite command-line options. For example, you might not have enough memory to run Edwin or the compiler with its default memory parameters (it will print something like &ldquoNot enough memory for this configuration&rdquo and halt when started), so you can write a shell script that will invoke Scheme with the appropriate --heap and other parameters. • Scheme supports init files: an init file is a file containing Scheme code that is loaded when Scheme is started, immediately after the identification banner, and before the input prompt is printed. This file is stored in your home directory, which is normally specified by the HOME environment variable. Under unix, the file is called .scheme.init . In addition, when Edwin starts up, it loads a separate init file from your home directory into the Edwin environment. This file is called .edwin under unix (see Starting Edwin). You can use both of these files to define new procedures or commands, or to change defaults in the system. The --no-init-file command-line option causes Scheme to ignore the .scheme.init file (see Command-Line Options). ### 2.3 Memory Usage Some of the parameters that can be customized determine how much memory Scheme uses and how that memory is used. This section describes how Scheme&rsquos memory is organized and used subsequent sections describe command-line options and environment variables that you can use to customize this usage for your needs. Scheme uses four kinds of memory: • A stack that is used for recursive procedure calls. • A heap that is used for dynamically allocated objects, like cons cells and strings. Storage used for objects in the heap that become unreferenced is eventually reclaimed by garbage collection. • A constant space that is used for allocated objects, like the heap. Unlike the heap, storage used for objects in constant space is not reclaimed by garbage collection any unreachable objects in constant space remain there until the Scheme process is terminated. Constant space is used for objects that are essentially permanent, like procedures in the runtime system. Doing this reduces the expense of garbage collection because these objects are no longer copied. • Some extra storage that is used by the microcode (the part of the system that is implemented in C). All kinds of memory except the last may be controlled either by command-line options or by environment variables. MIT/GNU Scheme uses a two-space copying garbage collector for reclaiming storage in the heap. The second space, used only during garbage collection, is dynamically allocated as needed. Once the storage is allocated for the constant space and the heap, Scheme will dynamically adjust the proportion of the total that is used for constant space the stack and extra microcode storage is not included in this adjustment. Previous versions of MIT/GNU Scheme needed to be told the amount of constant space that was required when loading bands with the --band option. Dynamic adjustment of the heap and constant space avoids this problem. If the size of the constant space is not specified, it is automatically set to the correct size for the band being loaded it is rarely necessary to explicitly set the size of the constant space. Additionally, each band requires a small amount of heap space this amount is added to any specified heap size, so that the specified heap size is the amount of free space available. The Scheme expression &lsquo (print-gc-statistics) &rsquo shows how much heap and constant space is available (see Garbage Collection). ### 2.4 Command-Line Options Scheme accepts the command-line options detailed in the following sections. The options may appear in any order, with the restriction that the microcode options must appear before the runtime options, and the runtime options must appear before any other arguments on the command line. Any arguments other than these options will generate a warning message when Scheme starts. If you want to define your own command-line options, see Custom Command-line Options. Note that MIT/GNU Scheme supports only long options, that is, options specified by verbose names, as opposed to short options, which are specified by single characters. All options start with two hyphens, for compatibility with GNU coding standards (and most modern programs). These are the microcode options: Specifies the initial world image file (band) to be loaded. Searches for filename in the working directory and the library directories, using the full pathname of the first readable file of that name. If filename is an absolute pathname (on unix, this means it starts with / ), then no search occurs&mdash filename is tested for readability and then used directly. If this option isn&rsquot given, the filename is the value of the environment variable MITSCHEME_BAND , or if that isn&rsquot defined, all.com in these cases the library directories are searched, but not the working directory. Specifies the size of the heap in 1024-word blocks. Overrides any default. The size specified by this option is incremented by the amount of heap space needed by the band being loaded. Consequently, --heap specifies how much free space will be available in the heap when Scheme starts, independent of the amount of heap already consumed by the band. Specifies the size of constant space in 1024-word blocks. Overrides any default. Constant space holds the compiled code for the runtime system and other subsystems. Specifies the size of the stack in 1024-word blocks. Overrides any default. This is Scheme&rsquos stack, not the unix stack used by C programs. Causes Scheme to write an option summary to standard error. This shows the values of all of the settable microcode option variables. Specifies that Scheme is running as a subprocess of GNU Emacs. This option is automatically supplied by GNU Emacs, and should not be given under other circumstances. This detaching behavior is useful for running Scheme as a background job. For example, using Bourne shell, the following will run Scheme as a background job, redirecting its input and output to files, and preventing it from being killed by keyboard interrupts or by logging out: This option is ignored under non-unix operating systems. Specifies that Scheme should not generate a core dump under any circumstances. If this option is not given, and Scheme terminates abnormally, you will be prompted to decide whether a core dump should be generated. This option is ignored under non-unix operating systems. Sets the library search path to path . This is a list of directories that is searched to find various library files, such as bands. If this option is not given, the value of the environment variable MITSCHEME_LIBRARY_PATH is used if that isn&rsquot defined, the default is used. On unix, the elements of the list are separated by colons, and the default value is /usr/local/lib/mit-scheme- ARCH . Specifies that a cold load should be performed, using filename as the initial file to be loaded. If this option isn&rsquot given, a normal load is performed instead. This option may not be used together with the --band option. This option is useful only for maintenance and development of the MIT/GNU Scheme runtime system. The following options are runtime options. They are processed after the microcode options and after the image file is loaded. This option causes Scheme to ignore the$/.scheme.init file, normally loaded automatically when Scheme starts (if it exists).

Under some circumstances Scheme can write out a file called scheme_suspend in the user&rsquos home directory. 1 This file is a world image containing the complete state of the Scheme process restoring this file continues the computation that Scheme was performing at the time the file was written.

Normally this file is never written, but the --suspend-file option enables writing of this file.

This option causes Scheme to evaluate the expression s following it on the command line, up to but not including the next argument that starts with a hyphen. The expressions are evaluated in the user-initial-environment . Unless explicitly handled, errors during evaluation are silently ignored.

This option causes Scheme to load the file s (or lists of files) following it on the command line, up to (but not including) the next argument that starts with a hyphen. The files are loaded in the user-initial-environment . Unless explicitly handled, errors during loading are silently ignored.

This option causes Edwin to be loaded and started immediately when Scheme is started.

The following options allow arguments to be passed to scripts via the command-line-arguments procedure.

procedure: command-line-arguments

Returns a list of arguments (strings) gathered from the command-line by options like --args or -- .

This option causes Scheme to append the argument s, up to (but not including) the next argument that starts with a hyphen, to the list returned by the command-line-arguments procedure.

This option causes Scheme to append the rest of the command-line arguments (even those starting with a hyphen) to the list returned by the command-line-arguments procedure.

### 2.5 Custom Command-line Options

MIT/GNU Scheme provides a mechanism for you to define your own command-line options. This is done by registering handlers to identify particular named options and to process them when Scheme starts. Unfortunately, because of the way this mechanism is implemented, you must define the options and then save a world image containing your definitions (see World Images). Later, when you start Scheme using that world image, your options will be recognized.

The following procedures define command-line parsers. In each, the argument keyword defines the option that will be recognized on the command line. The keyword must be a string containing at least one character do not include the leading hyphens.

procedure: simple-command-line-parser keyword thunk [help]

Defines keyword to be a simple command-line option. When this keyword is seen on the command line, it causes thunk to be executed. Help , when provided, should be a string describing the option in the --help output.

procedure: argument-command-line-parser keyword multiple? procedure [help]

Defines keyword to be a command-line option that is followed by one or more command-line arguments. Procedure is a procedure that accepts one argument when keyword is seen, it is called once for each argument. Help , when provided, should be a string describing the option. It is included in the --help output. When not provided, --help will say something lame about your command line option.

Multiple? , if true, says that keyword may be followed by more than one argument on the command line. In this case, procedure is called once for each argument that follows keyword and does not start with a hyphen. If multiple? is #f , procedure is called once, with the command-line argument following keyword . In this case, it does not matter if the following argument starts with a hyphen.

procedure: set-command-line-parser! keyword procedure

This low-level procedure defines keyword to be a command-line option that is defined by procedure . When keyword is seen, procedure is called with all of the command-line arguments, starting with keyword , as a single list argument. Procedure must return two values (using the values procedure): the unused command-line arguments (as a list), and either #f or a thunk to invoke after the whole command line has been parsed (and the init file loaded). Thus procedure has the option of executing the appropriate action at parsing time, or delaying it until after the parsing is complete. The execution of the procedures (or their associated delayed actions) is strictly left-to-right, with the init file loaded between the end of parsing and the delayed actions.

### 2.6 Environment Variables

Scheme refers to many environment variables. This section lists these variables and describes how each is used. The environment variables are organized according to the parts of MIT/GNU Scheme that they affect.

Environment variables that affect the microcode must be defined before you start Scheme others can be defined or overwritten within Scheme by using the set-environment-variable! procedure, e.g.

#### 2.6.1 Environment Variables for the Microcode

These environment variables are referred to by the microcode: the executable C program called mit-scheme- ARCH - VERSION . The values they specify are overridden by the corresponding command-line options, if given.

The initial band to be loaded. The default value is all.com .

A list of directories. These directories are searched, left to right, to find bands and various other files. On unix systems the list is colon-separated, with the default /usr/local/lib/mit-scheme- ARCH - VERSION .

The size of constant space, in 1024-word blocks overridden by --constant . The default value is computed to be the correct size for the band being loaded.

The size of the heap, in 1024-word blocks overridden by --heap . The default value depends on the architecture: for 32-bit machines the default is &lsquo 3072 &rsquo, and for 64-bit machines the default is &lsquo 16384 &rsquo.

The size of the stack, in 1024-word blocks overridden by --stack . The default value is &lsquo 1024 &rsquo.

#### 2.6.2 Environment Variables for the Runtime System

These environment variables are referred to by the runtime system.

Directory in which to look for init files, for example /home/joe . Under unix HOME is set by the login shell.

Directory for various temporary files. The variables are tried in the given order. If none of them is suitable, built-in defaults are used: /var/tmp , /usr/tmp , /tmp .

Directory containing the debugging information files for the Scheme system. Should contain subdirectories corresponding to the subdirectories in the source tree. By default, the information is searched for on the library path.

Specifies the location of the options database file used by the load-option procedure. The default is optiondb.scm on the library path.

#### 2.6.3 Environment Variables for Edwin

These environment variables are referred to by Edwin.

Directory where Edwin expects to find files providing autoloaded facilities. The default is edwin on the library path.

Directory where Edwin expects to find files for the &lsquoinfo&rsquo documentation subsystem. The default is edwin/info on the library path.

Directory where Edwin expects to find utility programs and documentation strings. The default is edwin on the library path.

Filename of the shell program to use in shell buffers. If not defined, the SHELL environment variable is used instead.

Filename of the shell program to use in shell buffers and when executing shell commands. Used to initialize the shell-path-name editor variable. The default is /bin/sh on unix systems.

Used to initialize the exec-path editor variable, which is subsequently used for finding programs to be run as subprocesses.

Used when Edwin runs under unix and uses X11. Specifies the display on which Edwin will create windows.

Used when Edwin runs under unix on a terminal. Terminal type.

Used when Edwin runs under unix on a terminal. Number of text lines on the screen, for systems that don&rsquot support &lsquo TIOCGWINSZ &rsquo.

Used when Edwin runs under unix on a terminal. Number of text columns on the screen, for systems that don&rsquot support &lsquo TIOCGWINSZ &rsquo.

### 2.7 Leaving Scheme

There are several ways that you can leave Scheme: there are two Scheme procedures that you can call there are several Edwin commands that you can execute and there are graphical-interface buttons (and their associated keyboard accelerators) that you can activate.

Two Scheme procedures that you can call. The first is to evaluate

which will halt the Scheme system, after first requesting confirmation. Any information that was in the environment is lost, so this should not be done lightly.

The second procedure suspends Scheme when this is done you may later restart where you left off. Unfortunately this is not possible in all operating systems currently it works under unix versions that support job control (i.e. all of the unix versions for which we distribute Scheme). To suspend Scheme, evaluate

If your system supports suspension, this will cause Scheme to stop, and you will be returned to the shell. Scheme remains stopped, and can be continued using the job-control commands of your shell. If your system doesn&rsquot support suspension, this procedure does nothing. (Calling the quit procedure is analogous to typing C-z , but it allows Scheme to respond by typing a prompt when it is unsuspended.)

## Methods

### Study Design and Participants

This study was an investigator-initiated, phase II, open-label, single-arm type II expansion-platform trial (16, 21, 22) performed at The University of Chicago along with two of its community-based satellite sites. The study protocol and all amendments were approved by The University of Chicago institutional review board. The protocol was conducted in accordance with the Declaration of Helsinki and was overseen by an internal data and safety monitoring committee. All patients provided written informed consent before enrollment.

Eligible patients were ages 18 years or older with histologically proven metastatic GEA from a biopsy of a stage IV site (cytology was acceptable from effusions/ascites). Patients were required to have newly diagnosed advanced disease, or recurrence after previous curative-intent therapy if completed more than six months prior. Key inclusion criteria included ECOG performance status of 0 to 2, and no grade 2 or higher peripheral edema, peripheral neuropathy, or diarrhea. Patients had measurable or evaluable nonmeasurable disease as per RECISTv1.1. Key exclusion criteria included history of known or suspected autoimmune disease, active second malignancy, intercurrent illness/infection, cardiac ejection fraction less than 50%, or history of cerebral vascular accident or myocardial infarction within six months. Full eligibility details are in the protocol (see Appendix in the supplementary files).

### Biomarker Assessment and Prioritized Treatment Assignment

Biomarker profiling assays were performed in parallel on all samples, including baseline primary and metastatic biopsies, as well as first (PD1) and second (PD2) progressive disease biopsies. Analyses included NGS using FoundationOne, including MSI and TMB testing (85), along with CPS of PD-L1 by IHC using the 22C3 pharmDx assay (86), all from Foundation Medicine. PD-L1 was considered positive at CPS ≥10, and TMB was high if ≥15 mutations per megabase (34). Genes were considered amplified by NGS if eight copies or higher were observed. HER2 status was assessed and considered positive if IHC3 + or IHC2 + together with FISH amplification (ratio of HER2:CEP17 probes greater than or equal to 2 ref. 87). Circulating tumor DNA (ctDNA) was obtained and analyzed using Guardant360 at baseline and serially at each disease progression time point, as previously described (29, 88). If EGFR or MET amplification was identified in one of a patient's tissue- or ctDNA-NGS results, all of that patient's samples were analyzed for these two genes by FISH at Neogenomics (40). If a sample demonstrated PD-L1 CPS ≥ 10 and was not MSI-H, then Epstein–Barr virus (EBV) status was determined by ISH using probes against Epstein–Barr encoded RNA1. EGFR expression by selected-reaction-monitoring mass spectrometry (SRM-MS) was quantified and considered positive if above the limit of detection (attomols/microgram), as previously described (40, 89). To address the possibility of insufficient tissue to perform all intended analyses, testing was prioritized on each sample by a set of rules in accordance with the treatment assignment algorithm described below.

Based on the results of this extensive molecular profiling, a tumor sample was assigned to one of eight biological categories based on a predefined algorithm (Table 2): Group 1 IO, including MSI-H, EBV+, TMB high [≥15 mutations/megabase (mt/Mb)], and/or PD-L1 IHC CPS ≥10 groups 2–5 RTK amplification of HER2, EGFR, FGFR2, and MET, respectively group 6 genomic activation of the MAPK/PIK3CA/GNAS pathways group 7 EGFR expressing by SRM-MS group 8 all negative. The group 1 IO was prioritized second to group 2 HER2-positive tumors in the first-line setting only, but was then first priority in second line and later. For groups 2–5, if two or more RTKs were concurrently amplified, then the gene with the highest copy number would take priority, given evidence that higher gene copies correlated with higher expression, which correlated with higher efficacy of matched targeted therapy (3, 29, 40, 90–93). If the final biomarker assignment was discordant between the primary and metastatic tumors at baseline prior to first-line therapy, then the metastatic tumor would take precedence. If the quantity of metastatic tissue was not sufficient (QNS) to complete all assays and biomarker assignment, then ctDNA could be used for biomarker assignment. If there were no alterations actionable by ctDNA per the algorithm, then the primary tumor profile was used. If QNS despite these steps, then the patient would be assigned to group 8. Temporal (PD1, PD2, and PD3) biopsies were obtained from progressing lesions.

### Therapeutic Procedures

Cytotoxic doublets were administered as biweekly treatment cycles in each of up to three therapy lines (Supplementary Fig. S3). First-line cytotoxic therapy of modified FOLFOX6 entailed day 1 oxaliplatin 85 mg/m 2 i.v. with leucovorin 200 mg/m 2 i.v. over 2 hours, then 5-fluorouracil (5FU) bolus 400 mg/m 2 i.v., then 2,400 mg/m 2 i.v. continuous infusion over 46 hours. An 8/2016 amendment permitted, at the discretion of the treating investigator, omission of the 5FU bolus and leucovorin from the onset of treatment (modified FOLFOX7). Second-line cytotoxic therapy of modified FOLFIRI (no 5FU bolus) entailed irinotecan 180 mg/m 2 i.v. with leucovorin 200 mg/m 2 i.v. over 2 hours, then 5FU 2,400 mg/m 2 over 46 hours. Third-line cytotoxic therapy of FOLFTAX entailed docetaxel 50 mg/m 2 i.v. with leucovorin 200 mg/m 2 i.v. over 2 hours, then 5FU 2,400 mg/m 2 over 46 hours (94–96). Palliative radiotherapy to the primary tumor of 30 Gy over two to three weeks was allowed if patients experienced worsening dysphagia and/or bleeding consistent with localized disease progression while all other systemic disease was controlled systemic therapy was held during this time and then resumed one to two weeks after completion of radiotherapy.

All adverse events were graded according to the NCI Common Toxicity Criteria for Adverse Events version 4.0. Intrapatient dose reductions of oxaliplatin, irinotecan, docetaxel, and 5FU were allowed depending on the type and severity of toxicity omitting the 5FU bolus was the first modification per protocol for most toxicities, upon which resuming the bolus in any subsequent cycle or line was not permitted. Additionally, any dose modifications to the 5FU or leucovorin were carried over to next-line therapies.

Patients began first-line FOLFOX therapy immediately while biomarker testing was initiated. Upon obtaining biomarker group assignment according to the algorithm (Table 2), the appropriate monoclonal antibody was added to the next scheduled dose of cytotoxic therapy, continuing every two weeks. Upon each disease progression (PD1 and PD2), patients changed to the next-line cytotoxic doublet while remaining on the prior assigned monoclonal antibody until PD1/PD2 molecular profiling results were obtained, upon which the appropriate antibody would be incorporated at the next scheduled dose of cytotoxic therapy.

To limit cumulative toxicity, oxaliplatin, irinotecan, and docetaxel were permitted to be stopped and resumed intermittently (“OPTIMOX,” ref. 99 “OPTIMIRI,” ref. 100 and “OPTITAX”), while continuing maintenance 5FU plus monoclonal antibody. Each line of therapy was considered to have failed only upon disease progression on the full cytotoxic doublet or progression on maintenance therapy but inability to resume the cytotoxic doublet for any reason. Patients were assessed for disease progression by imaging of the chest, abdomen, and pelvis every two months (four cycles). Patients had study treatment discontinued if they developed progressive disease as defined by RECIST1.1 after three lines of therapy, or earlier if unable to continue to the next treatment line for any reason. Other criteria for removal included withdrawal of consent or treatment-related adverse events not resolving after nine weeks of treatment interruption.

### Outcomes

The primary efficacy endpoint of the study was one-year OS, defined as the proportion of patients treated with ITT alive at 12 months. All patients were followed for survival to the final data lock on August 20, 2020. Other primary endpoints were safety and feasibility the molecular approach would be deemed safe if less than a 5% serious adverse event rate was observed from baseline and serial biopsies. The molecular approach would be deemed feasible if at least 85% of patients were assigned to therapy within two months of enrollment and if at least 85% of patients obtained a successful biopsy at PD1. Secondary endpoints included overall safety and tolerability progression-free survival for each line of therapy (PFS1,2,3) calculated as the time from starting each cytotoxic doublet until documentation of clinical or radiologic disease progression or death, whichever occurred first objective response rate (ORR1,2,3) by RECIST1.1 and disease control rate (DCR1,2,3) for each line of therapy and time to PANGEA treatment failure (TTF) among the patients treated with ITT. Outcomes were compared with historical outcomes and also those non-ITT patients having lack of availability of monoclonal antibodies (group 4 FGFR2 and group 5 MET). A preplanned mITT analysis included patients within group 5 able to get off-label tyrosine kinase inhibitors during their treatment course. Prespecified secondary analyses included analysis of OS, PFS, ORR, and DCR by individual biomarker group by treatment line, as well as contrasting the pooled outcomes of higher-priority groups 1–4 (or groups 1–5 for mITT) compared with lower-priority groups 6–8 of the algorithm, and also evaluating outcomes after excluding the effect of group 2 (HER2). Characterization of biomarker heterogeneity at baseline spatially and over time after targeted therapy were also secondary endpoints. Given the association with prognosis, an ad hoc characterization of baseline absolute NLR was performed, as previously described (75).

### Statistical Analyses

Using a z-test based on the Greenwood standard error to accommodate censoring, 68 patients treated per ITT provided 80% power to detect an improvement in one-year OS rate from 50% historically to 63% with a one-sided alpha of 10%. Assuming exponential survival, this corresponds to an HR of 0.67. The historical 50% one-year rate implies a median of 12 months and was obtained as a weighted average of a sample comprised of 20% of patients having HER2-positive disease with an anticipated H0-HER2+ mOS of 16 months and 80% of patients having HER2-negative disease with an anticipated H0-HER2- mOS of 11 months. [Of note, 16 of 80 (20%) of all patients enrolled, or 68 (23.5%) of the ITT, or 16 of 70 (22.9%) of the mITT patients in our study were HER2-positive.] Patients receiving at least one dose of first-line FOLFOX therapy and having availability of monoclonal antibody (though not necessarily receiving it) were considered evaluable for the primary outcome by ITT. Analysis of OS, PFS (during first, second, and third-line treatments), and TTF was estimated using Kaplan–Meier methods. All secondary endpoints including safety were assessed in all ITT patients who received at least one dose of first-line cytotoxic therapy. The log-rank test was used to compare OS, PFS, and TTF between various subgroups. All statistical analyses were done using Stata version 16.0 (StataCorp). This trial is registered with ClinicalTrials.gov (NCT02213289).

## A Categorization Model for Educational Values of the History of Mathematics

There is not a clear consensus on the categorization framework of the educational values of the history of mathematics. By analyzing 20 Chinese teaching cases on integrating the history of mathematics into mathematics teaching based on the relevant literature, this study examined a new categorization framework of the educational values of the history of mathematics by combining the objectives of high school mathematics curriculum in China. This framework includes six dimensions: the harmony of knowledge, the beauty of ideas or methods, the pleasure of inquiries, the improvement of capabilities, the charm of cultures, and the availability of moral education. The results show that this framework better explained the all-educational values of the history of mathematics that all teaching cases showed. Therefore, the framework can guide teachers to better integrate the history of mathematics into teaching.

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## Contents

The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction.) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola. The name "parabola" is due to Apollonius, who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved. [1] The focus–directrix property of the parabola and other conic sections is due to Pappus.

Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope. [2] Designs were proposed in the early to mid-17th century by many mathematicians, including René Descartes, Marin Mersenne, [3] and James Gregory. [4] When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. [5]

A parabola can be defined geometrically as a set of points (locus of points) in the Euclidean plane:

### Axis of symmetry parallel to the y axis Edit

This parabola is U-shaped (opening to the top).

The horizontal chord through the focus (see picture in opening section) is called the latus rectum one half of it is the semi-latus rectum. The latus rectum is parallel to the directrix. The semi-latus rectum is designated by the letter p . From the picture one obtains

The latus rectum is defined similarly for the other two conics – the ellipse and the hyperbola. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, p is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, p , is the distance of the focus from the directrix. Using the parameter p , the equation of the parabola can be rewritten as

1. In the case of f < 0 the parabola has a downward opening.
2. The presumption that the axis is parallel to the y axis allows one to consider a parabola as the graph of a polynomial of degree 2, and conversely: the graph of an arbitrary polynomial of degree 2 is a parabola (see next section).
3. If one exchanges x and y , one obtains equations of the form y 2 = 2 p x =2px> . These parabolas open to the left (if p < 0 ) or to the right (if p > 0 ).

### General case Edit

(the left side of the equation uses the Hesse normal form of a line to calculate the distance | P l | ).

The implicit equation of a parabola is defined by an irreducible polynomial of degree two:

The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of a function

The general function of degree 2 is

which is the equation of a parabola with

Two objects in the Euclidean plane are similar if one can be transformed to the other by a similarity, that is, an arbitrary composition of rigid motions (translations and rotations) and uniform scalings.

A synthetic approach, using similar triangles, can also be used to establish this result. [7]

The general result is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. [6] Therefore, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas do not.

The pencil of conic sections with the x axis as axis of symmetry, one vertex at the origin (0, 0) and the same semi-latus rectum p can be represented by the equation

If p > 0 , the parabola with equation y 2 = 2 p x =2px> (opening to the right) has the polar representation

If one shifts the origin into the focus, that is, F = ( 0 , 0 ) , one obtains the equation

Remark 1: Inverting this polar form shows that a parabola is the inverse of a cardioid.

Remark 2: The second polar form is a special case of a pencil of conics with focus F = ( 0 , 0 ) (see picture):

### Diagram, description, and definitions Edit

The diagram represents a cone with its axis AV . The point A is its apex. An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle θ , as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola.

A cross-section perpendicular to the axis of the cone passes through the vertex P of the parabola. This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. Its centre is V, and PK is a diameter. We will call its radius r .

Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described. It has a chord DE , which joins the points where the parabola intersects the circle. Another chord BC is the perpendicular bisector of DE and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry PM all intersect at the point M.

All the labelled points, except D and E, are coplanar. They are in the plane of symmetry of the whole figure. This includes the point F, which is not mentioned above. It is defined and discussed below, in § Position of the focus.

Let us call the length of DM and of EM x , and the length of PM y .

### Derivation of quadratic equation Edit

The lengths of BM and CM are:

Using the intersecting chords theorem on the chords BC and DE , we get

For any given cone and parabola, r and θ are constants, but x and y are variables that depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as Cartesian coordinates of the points D and E, in a system in the pink plane with P as its origin. Since x is squared in the equation, the fact that D and E are on opposite sides of the y axis is unimportant. If the horizontal cross-section moves up or down, toward or away from the apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function in the equation.

### Focal length Edit

It is proved in a preceding section that if a parabola has its vertex at the origin, and if it opens in the positive y direction, then its equation is y = x 2 / 4f , where f is its focal length. [b] Comparing this with the last equation above shows that the focal length of the parabola in the cone is r sin θ .

### Position of the focus Edit

In the diagram above, the point V is the foot of the perpendicular from the vertex of the parabola to the axis of the cone. The point F is the foot of the perpendicular from the point V to the plane of the parabola. [c] By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is complementary to θ , and angle PVF is complementary to angle VPF, therefore angle PVF is θ . Since the length of PV is r , the distance of F from the vertex of the parabola is r sin θ . It is shown above that this distance equals the focal length of the parabola, which is the distance from the vertex to the focus. The focus and the point F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, the point F, defined above, is the focus of the parabola.

This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape.

### Alternative proof with Dandelin spheres Edit

An alternative proof can be done using Dandelin spheres. It works without calculation and uses elementary geometric considerations only (see the derivation below).

The intersection of an upright cone by a plane π , whose inclination from vertical is the same as a generatrix (a.k.a. generator line, a line containing the apex and a point on the cone surface) m 0 > of the cone, is a parabola (red curve in the diagram).

The reflective property states that if a parabola can reflect light, then light that enters it travelling parallel to the axis of symmetry is reflected toward the focus. This is derived from geometrical optics, based on the assumption that light travels in rays.

Consider the parabola y = x 2 . Since all parabolas are similar, this simple case represents all others.

### Construction and definitions Edit

The point E is an arbitrary point on the parabola. The focus is F, the vertex is A (the origin), and the line FA is the axis of symmetry. The line EC is parallel to the axis of symmetry and intersects the x axis at D. The point B is the midpoint of the line segment FC .

### Deductions Edit

The distances EF and EC are equal because E is on the parabola, F is the focus and C is on the directrix. Therefore, since B is the midpoint of FC , triangles △FEB and △CEB are congruent (three sides), which implies that the angles marked α are congruent. (The angle above E is vertically opposite angle ∠BEC.) This means that a ray of light that enters the parabola and arrives at E travelling parallel to the axis of symmetry will be reflected by the line BE so it travels along the line EF , as shown in red in the diagram (assuming that the lines can somehow reflect light). Since BE is the tangent to the parabola at E, the same reflection will be done by an infinitesimal arc of the parabola at E. Therefore, light that enters the parabola and arrives at E travelling parallel to the axis of symmetry of the parabola is reflected by the parabola toward its focus.

This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram. This is the reflective property.

### Other consequences Edit

There are other theorems that can be deduced simply from the above argument.

#### Tangent bisection property Edit

The above proof and the accompanying diagram show that the tangent BE bisects the angle ∠FEC. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus and perpendicularly to the directrix.

#### Intersection of a tangent and perpendicular from focus Edit

Since triangles △FBE and △CBE are congruent, FB is perpendicular to the tangent BE . Since B is on the x axis, which is the tangent to the parabola at its vertex, it follows that the point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex. See animated diagram [8] and pedal curve.

#### Reflection of light striking the convex side Edit

If light travels along the line CE , it moves parallel to the axis of symmetry and strikes the convex side of the parabola at E. It is clear from the above diagram that this light will be reflected directly away from the focus, along an extension of the segment FE .

### Alternative proofs Edit

The above proofs of the reflective and tangent bisection properties use a line of calculus. Here a geometric proof is presented.

In this diagram, F is the focus of the parabola, and T and U lie on its directrix. P is an arbitrary point on the parabola. PT is perpendicular to the directrix, and the line MP bisects angle ∠FPT. Q is another point on the parabola, with QU perpendicular to the directrix. We know that FP = PT and FQ = QU . Clearly, QT > QU , so QT > FQ . All points on the bisector MP are equidistant from F and T, but Q is closer to F than to T. This means that Q is to the left of MP , that is, on the same side of it as the focus. The same would be true if Q were located anywhere else on the parabola (except at the point P), so the entire parabola, except the point P, is on the focus side of MP . Therefore, MP is the tangent to the parabola at P. Since it bisects the angle ∠FPT, this proves the tangent bisection property.

The logic of the last paragraph can be applied to modify the above proof of the reflective property. It effectively proves the line BE to be the tangent to the parabola at E if the angles α are equal. The reflective property follows as shown previously.

The definition of a parabola by its focus and directrix can be used for drawing it with help of pins and strings: [9]

A parabola can be considered as the affine part of a non-degenerated projective conic with a point Y ∞ > on the line of infinity g ∞ > , which is the tangent at Y ∞ > . The 5-, 4- and 3- point degenerations of Pascal's theorem are properties of a conic dealing with at least one tangent. If one considers this tangent as the line at infinity and its point of contact as the point at infinity of the y axis, one obtains three statements for a parabola.

The following properties of a parabola deal only with terms connect, intersect, parallel, which are invariants of similarities. So, it is sufficient to prove any property for the unit parabola with equation y = x 2 > .

### 4-points property Edit

Any parabola can be described in a suitable coordinate system by an equation y = a x 2 > .

Proof: straightforward calculation for the unit parabola y = x 2 > .

Application: The 4-points property of a parabola can be used for the construction of point P 4 > , while P 1 , P 2 , P 3 ,P_<2>,P_<3>> and Q 2 > are given.

Remark: the 4-points property of a parabola is an affine version of the 5-point degeneration of Pascal's theorem.

### 3-points–1-tangent property Edit

Application: The 3-points-1-tangent-property of a parabola can be used for the construction of the tangent at point P 0 > , while P 1 , P 2 , P 0 ,P_<2>,P_<0>> are given.

Remark: The 3-points-1-tangent-property of a parabola is an affine version of the 4-point-degeneration of Pascal's theorem.

### 2-points–2-tangents property Edit

Proof: straight forward calculation for the unit parabola y = x 2 > .

Application: The 2-points–2-tangents property can be used for the construction of the tangent of a parabola at point P 2 > , if P 1 , P 2 ,P_<2>> and the tangent at P 1 > are given.

Remark 1: The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem.

Remark 2: The 2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is not related to Pascal's theorem.

### Axis direction Edit

Proof: can be done (like the properties above) for the unit parabola y = x 2 > .

Application: This property can be used to determine the direction of the axis of a parabola, if two points and their tangents are given. An alternative way is to determine the midpoints of two parallel chords, see section on parallel chords.

Remark: This property is an affine version of the theorem of two perspective triangles of a non-degenerate conic. [10]

### Parabola Edit

Steiner established the following procedure for the construction of a non-degenerate conic (see Steiner conic):

This procedure can be used for a simple construction of points on the parabola y = a x 2 > :

Proof: straightforward calculation.

Remark: Steiner's generation is also available for ellipses and hyperbolas.

### Dual parabola Edit

A dual parabola consists of the set of tangents of an ordinary parabola.

The Steiner generation of a conic can be applied to the generation of a dual conic by changing the meanings of points and lines:

In order to generate elements of a dual parabola, one starts with

The proof is a consequence of the de Casteljau algorithm for a Bezier curve of degree 2.

In the following, the angle of two lines will be measured by the difference of the slopes of the line with respect to the directrix of the parabola. That is, for a parabola of equation y = a x 2 + b x + c , +bx+c,> the angle between two lines of equations y = m 1 x + d 1 , y = m 2 x + d 2 x+d_<1>, y=m_<2>x+d_<2>> is measured by m 1 − m 2 . -m_<2>.>

Analogous to the inscribed angle theorem for circles, one has the inscribed angle theorem for parabolas: [11] [12]

(Proof: straightforward calculation: If the points are on a parabola, one may translate the coordinates for having the equation y = a x 2 > , then one has y i − y j x i − x j = x i + x j -y_><>-x_>>=x_+x_> if the points are on the parabola.)

y = 2 a x 0 ( x − x 0 ) + y 0 = 2 a x 0 x − a x 0 2 = 2 a x 0 x − y 0 . (x-x_<0>)+y_<0>=2ax_<0>x-ax_<0>^<2>=2ax_<0>x-y_<0>.>

on the set of points of the parabola onto the set of tangents.

This relation is called the pole–polar relation of the parabola, where the point is the pole, and the corresponding line its polar.

By calculation, one checks the following properties of the pole–polar relation of the parabola:

• For a point (pole) on the parabola, the polar is the tangent at this point (see picture: P 1 , p 1 , p_<1>> ).
• For a pole P outside the parabola the intersection points of its polar with the parabola are the touching points of the two tangents passing P (see picture: P 2 , p 2 , p_<2>> ).
• For a point within the parabola the polar has no point with the parabola in common (see picture: P 3 , p 3 , p_<3>> and P 4 , p 4 , p_<4>> ).
• The intersection point of two polar lines (for example, p 3 , p 4 ,p_<4>> ) is the pole of the connecting line of their poles (in example: P 3 , P 4 ,P_<4>> ).
• Focus and directrix of the parabola are a pole–polar pair.

Remark: Pole–polar relations also exist for ellipses and hyperbolas.

### Two tangent properties related to the latus rectum Edit

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f . Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then (1) the distance from F to T is 2f , and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle. [13] : p.26

### Orthoptic property Edit

If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents that intersect on the directrix are perpendicular.

### Lambert's theorem Edit

Let three tangents to a parabola form a triangle. Then Lambert's theorem states that the focus of the parabola lies on the circumcircle of the triangle. [14] [8] : Corollary 20

Tsukerman's converse to Lambert's theorem states that, given three lines that bound a triangle, if two of the lines are tangent to a parabola whose focus lies on the circumcircle of the triangle, then the third line is also tangent to the parabola. [15]

### Focal length calculated from parameters of a chord Edit

Suppose a chord crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola be c and the distance from the vertex of the parabola to the chord, measured along the axis of symmetry, be d . The focal length, f , of the parabola is given by

### Area enclosed between a parabola and a chord Edit

The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram that surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola. [16] [17] The slope of the other parallel sides is irrelevant to the area. Often, as here, they are drawn parallel with the parabola's axis of symmetry, but this is arbitrary.

A theorem equivalent to this one, but different in details, was derived by Archimedes in the 3rd century BCE. He used the areas of triangles, rather than that of the parallelogram. [d] See The Quadrature of the Parabola.

If the chord has length b and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord is h , the parallelogram is a rectangle, with sides of b and h . The area A of the parabolic segment enclosed by the parabola and the chord is therefore

In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord. This can be done with calculus, or by using a line that is parallel to the axis of symmetry of the parabola and passes through the midpoint of the chord. The required point is where this line intersects the parabola. [e] Then, using the formula given in Distance from a point to a line, calculate the perpendicular distance from this point to the chord. Multiply this by the length of the chord to get the area of the parallelogram, then by 2/3 to get the required enclosed area.

### Corollary concerning midpoints and endpoints of chords Edit

A corollary of the above discussion is that if a parabola has several parallel chords, their midpoints all lie on a line parallel to the axis of symmetry. If tangents to the parabola are drawn through the endpoints of any of these chords, the two tangents intersect on this same line parallel to the axis of symmetry (see Axis-direction of a parabola). [f]

### Arc length Edit

If a point X is located on a parabola with focal length f , and if p is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola that terminate at X can be calculated from f and p as follows, assuming they are all expressed in the same units. [g]

This quantity s is the length of the arc between X and the vertex of the parabola.

The length of the arc between X and the symmetrically opposite point on the other side of the parabola is 2s .

The perpendicular distance p can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of p reverses the signs of h and s without changing their absolute values. If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of s . The calculation can be simplified by using the properties of logarithms:

s 1 − s 2 = h 1 q 1 − h 2 q 2 f + f ln ⁡ h 1 + q 1 h 2 + q 2 . -s_<2>=q_<1>-h_<2>q_<2>>>+fln +q_<1>>+q_<2>>>.>

This can be useful, for example, in calculating the size of the material needed to make a parabolic reflector or parabolic trough.

This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the y axis.

S is the focus, and V is the principal vertex of the parabola VG. Draw VX perpendicular to SV.

Take any point B on VG and drop a perpendicular BQ from B to VX. Draw perpendicular ST intersecting BQ, extended if necessary, at T. At B draw the perpendicular BJ, intersecting VX at J.

For the parabola, the segment VBV, the area enclosed by the chord VB and the arc VB, is equal to ∆VBQ / 3, also B Q = V Q 2 4 S V ><4SV>>> .

The area of the parabolic sector SVB = ∆SVB + ∆VBQ / 3 = S V ⋅ V Q 2 + V Q ⋅ B Q 6 =<2>>+<6>>> .

Since triangles TSB and QBJ are similar,

A circle through S, V and B also passes through J.

Conversely, if a point, B on the parabola VG is to be found so that the area of the sector SVB is equal to a specified value, determine the point J on VX and construct a circle through S, V and J. Since SJ is the diameter, the center of the circle is at its midpoint, and it lies on the perpendicular bisector of SV, a distance of one half VJ from SV. The required point B is where this circle intersects the parabola.

If a body traces the path of the parabola due to an inverse square force directed towards S, the area SVB increases at a constant rate as point B moves forward. It follows that J moves at constant speed along VX as B moves along the parabola.

If the speed of the body at the vertex where it is moving perpendicularly to SV is v, then the speed of J is equal to 3v/4.

The construction can be extended simply to include the case where neither radius coincides with the axis SV as follows. Let A be a fixed point on VG between V and B, and point H be the intersection on VX with the perpendicular to SA at A. From the above, the area of the parabolic sector S A B = 2 S V ⋅ ( V J − V H ) 3 = 2 S V ⋅ H J 3 <3>>=<3>>> .

Conversely, if it is required to find the point B for a particular area SAB, find point J from HJ and point B as before. By Book 1, Proposition 16, Corollary 6 of Newton's Principia, the speed of a body moving along a parabola with a force directed towards the focus is inversely proportional to the square root of the radius. If the speed at A is v, then at the vertex V it is S A S V v >>v> , and point J moves at a constant speed of 3 v 4 S A S V <4>>>>> .

The above construction was devised by Isaac Newton and can be found in Book 1 of Philosophiæ Naturalis Principia Mathematica as Proposition 30.

The focal length of a parabola is half of its radius of curvature at its vertex.

Image is inverted. AB is x axis. C is origin. O is center. A is (x, y) . OA = OC = R . PA = x . CP = y . OP = (Ry) . Other points and lines are irrelevant for this purpose.

The radius of curvature at the vertex is twice the focal length. The measurements shown on the above diagram are in units of the latus rectum, which is four times the focal length.

Consider a point (x, y) on a circle of radius R and with center at the point (0, R) . The circle passes through the origin. If the point is near the origin, the Pythagorean theorem shows that

But if (x, y) is extremely close to the origin, since the x axis is a tangent to the circle, y is very small compared with x , so y 2 is negligible compared with the other terms. Therefore, extremely close to the origin

Compare this with the parabola

which has its vertex at the origin, opens upward, and has focal length f (see preceding sections of this article).

Equations (1) and (2) are equivalent if R = 2f . Therefore, this is the condition for the circle and parabola to coincide at and extremely close to the origin. The radius of curvature at the origin, which is the vertex of the parabola, is twice the focal length.

A concave mirror that is a small segment of a sphere behaves approximately like a parabolic mirror, focusing parallel light to a point midway between the centre and the surface of the sphere.

Another definition of a parabola uses affine transformations:

The focal length can be determined by a suitable parameter transformation (which does not change the geometric shape of the parabola). The focal length is

Hence the focus of the parabola is

The definition of a parabola in this section gives a parametric representation of an arbitrary parabola, even in space, if one allows f → 0 , f → 1 , f → 2 >!_<0>,>!_<1>,>!_<2>> to be vectors in space.

This curve is an arc of a parabola (see § As the affine image of the unit parabola).

In one method of numerical integration one replaces the graph of a function by arcs of parabolas and integrates the parabola arcs. A parabola is determined by three points. The formula for one arc is

The following quadrics contain parabolas as plane sections:

• elliptical cone,
• parabolic cylinder,
• elliptical paraboloid,
• hyperbolic paraboloid, of one sheet,
• hyperboloid of two sheets.

Hyperboloid of two sheets

A parabola can be used as a trisectrix, that is it allows the exact trisection of an arbitrary angle with straightedge and compass. This is not in contradiction to the impossibility of an angle trisection with compass-and-straightedge constructions alone, as the use of parabolas is not allowed in the classic rules for compass-and-straightedge constructions.

This trisection goes back to René Descartes, who described it in his book La Géométrie (1637). [18]

If one replaces the real numbers by an arbitrary field, many geometric properties of the parabola y = x 2 > are still valid:

Essentially new phenomena arise, if the field has characteristic 2 (that is, 1 + 1 = 0 ): the tangents are all parallel.

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates (x, x 2 , x 3 , …, x n ) the standard parabola is the case n = 2 , and the case n = 3 is known as the twisted cubic. A further generalization is given by the Veronese variety, when there is more than one input variable.

In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2 . Generalizations to more variables yield further such objects.

The curves y = x p for other values of p are traditionally referred to as the higher parabolas and were originally treated implicitly, in the form x p = ky q for p and q both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula y = x p/q for a positive fractional power of x . Negative fractional powers correspond to the implicit equation x p y q = k and are traditionally referred to as higher hyperbolas. Analytically, x can also be raised to an irrational power (for positive values of x ) the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic and cannot be analyzed by algebraic geometry.

In nature, approximations of parabolas and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a ball flying through the air, neglecting air friction).

The parabolic trajectory of projectiles was discovered experimentally in the early 17th century by Galileo, who performed experiments with balls rolling on inclined planes. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences. [19] [h] For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless moves along a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and doesn't resemble a parabola.

Another hypothetical situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th centuries by Sir Isaac Newton, is in two-body orbits, for example, the path of a small planetoid or other object under the influence of the gravitation of the Sun. Parabolic orbits do not occur in nature simple orbits most commonly resemble hyperbolas or ellipses. The parabolic orbit is the degenerate intermediate case between those two types of ideal orbit. An object following a parabolic orbit would travel at the exact escape velocity of the object it orbits objects in elliptical or hyperbolic orbits travel at less or greater than escape velocity, respectively. Long-period comets travel close to the Sun's escape velocity while they are moving through the inner Solar system, so their paths are nearly parabolic.

Approximations of parabolas are also found in the shape of the main cables on a simple suspension bridge. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary, but in practice the curve is generally nearer to a parabola due to the weight of the load (i.e. the road) being much larger than the cables themselves, and in calculations the second-degree polynomial formula of a parabola is used. [20] [21] Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola (see Catenary#Suspension bridge curve). Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola. Suspension-bridge cables are, ideally, purely in tension, without having to carry other forces, for example, bending. Similarly, the structures of parabolic arches are purely in compression.

Paraboloids arise in several physical situations as well. The best-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point, or conversely, collimates light from a point source at the focus into a parallel beam. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a dubious legend, [22] constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite-dish receiving and transmitting antennas.

In parabolic microphones, a parabolic reflector is used to focus sound onto a microphone, giving it highly directional performance.

Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid-mirror telescope.

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet", follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

### Gallery Edit

A bouncing ball captured with a stroboscopic flash at 25 images per second. The ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.

Parabolic trajectories of water in a fountain.

The path (in red) of Comet Kohoutek as it passed through the inner Solar system, showing its nearly parabolic shape. The blue orbit is the Earth's.

The supporting cables of suspension bridges follow a curve that is intermediate between a parabola and a catenary.

The Rainbow Bridge across the Niagara River, connecting Canada (left) to the United States (right). The parabolic arch is in compression and carries the weight of the road.

Parabolic arches used in architecture

Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of transparent plastic. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre. (See Rotating furnace)

Parabolic microphone with optically transparent plastic reflector used at an American college football game.

Edison's searchlight, mounted on a cart. The light had a parabolic reflector.

Physicist Stephen Hawking in an aircraft flying a parabolic trajectory to simulate zero gravity

In previous writing developing this work, we have used the term “Mathematical Discourse in Instruction—Primary,” (MDI-P). The history of this work is a co-development of MDI frameworks between Hamsa Venkat and Jill Adler, which shared roots in sociocultural theory but differed in specific formulations across work in secondary and primary mathematics. In order to avoid confusions between the secondary and primary level models, we have changed our titling of the framework to MPM. Writing with Adler and her team is underway, detailing the histories and trajectories of development of both MPM and MDI.

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## 11.2: Footnotes - Mathematics

New York Institute of Technology
Department of Mathematics

Course Outline

Course Number: 260
Course Name: Calculus III
Text: James Stewart, Multivariable Calculus, 5 th Ed., Brooks-Cole, 2003 (ISBN 0-534-39357-8)

Curves Defined by Parametric Equations

Calculus with Parametric Curves

Areas in Polar Coordinates

The Integral Test p-series

Absolute Convergence and the Ratio Test

Representations of Functions as Power Series

Taylor and Maclaurin Series

p. 806: 1-7*, 11-15*, 23-27*, 33, 35, 39, 43

Three Dimensional Coordinate Systems

Equations of Lines and Planes

Functions of Several Variables

Tangent Planes and Linear Approximations

Double Integrals over Rectangles

Double Iterated Integrals

Double Integrals over General Regions

p. 1038: 1, 7, 13, 15, 20, 21, 27, 37-49*

Double Integrals in Polar Coordinates

Footnotes
:
1) The # of hours column is designed to approximate the appropriate pace of the course. An hour translates into 50 minutes of classroom instructional time.
2) Homework exercises indicate the type of problems that a student should be able to do. The instructor will determine the particular exercises that are assigned. The asterisk, *, indicates that the group of indicated exercises include only the odd numbers. For example, 7-13* indicates the problems 7, 9, 11, and 13.

Calculators :
This course requires that each student own a graphing calculator. The Department of Mathematics recommends the Texas Instruments models, TI-86, TI-89, or TI-92. The calculator will be needed for homework, quizzes, and exams, including the final exam.

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