## Tuesday, 12 May 2020

### Volume F- H

F

Factorial/ Factorial
In his book on The Art of Computer Progamming, Donald Knuth points to an example of the factorial (in particular 8!) in the Hebrew book of creation.

The first use of a multiplication of long strings of successive digits for a specific problem may have been by Euler in solving the questions of derangements. "The Game of Recontre (coincidence), also called the game of treize (thirteen), involves shuffling 13 numbered cards, then dealing them one at a time, counting aloud to 13. If the nth card is dealt when the player says the number 'n,' the dealer wins (this is known in combinatorics as a derangement of 13 objects.). Euler calculated the probability that the dealer will win.

It should be noted that this problem was solved earlier, by P.R. de Montmort, in 1713, though his work was unknown to Euler."

In an article entitled, "Calcul de la probabilité dans le jeu de rencontre" published in 1753, Euler wrote.

which is translated by Richard J. Pulskamp as "The number of cases $1^. 2^. 3^. 4 \dotsb m$ being put for brevity =M." Cajori points out that this was probably not intended to be a general notation, but a temporary expedient.

In 1772 A T Vandermonde used [P]n to represent the product of the n factors p(p-1)(p-2)... (p-n+1). With such a notation [P]p would represent what we would now write as p!, but I can imagine this becoming, over time, just [p] (De Morgan would do just such a thing in his 1838 essays on probability). Vandermonde seems to have been the first to consider [p]0 (or 0!) and determined it was (as we now do) equal to one. Vandermonde's notation included a method for skipping numbers, so that [p/3]n would indicate p(p-3)(p-6)... (p-3(n-1)). It even allowed for negative exponents.

Vandermonde's symbol for [P]n would today represent what is generally called the "falling factorial." The common symbols seem to be [n]k or Donald Knuth's suggestion of $n^{\underline{k}}$. Similar symbols exist for a "rising factorial", (n) (n+1) (n+2)...(n+k-1). Knuth's pleasing mnemonic version $n^{\overline{k}}$ and (n)k which is common in working with hypergeometric series and is called the Pochammer symbol, although he never seemed to have used it for that, and used it for the combination of n things taken k at a time $\binom{n}{k}$. I think either approach could be easily extended to using  (n/s) as the base with the "s" representing the "skip rate". So (5/3)4 could represent the rising step factorial 5 * 8 * 11 * 14.

The word factorial is reported to be the creation of Louis François Antoine Arbogast (1759-1803). The symbol now commonly used for factorial seems to have been created by Christian Kramp in 1808 according to a note I found in Lectures on fundamental concepts of algebra and geometry (1911), by John Wesley Young with a note on "The growth of algebraic symbolism" by Ulysses Grant Mitchell. It was in the Note by Mitchell (pg 239) that I found the credit for the symbol to Kramp. Kramp had previously used the word "facultes" for the process, but deferred in favor of Arbogast's term instead. Here is a translation from Jeff Miller's page, "I've named them facultes. Arbogast has proposed the denomination factorial, clearer and more French. I've recognised the advantage of this new term, and adopting its philosophy I congratulate myself of paying homage to the memory of my friend". Both Kramp and Arbogast were working with sequences of products. (Kramp's more general notation allowed for "the product of the factors of an arithmetic progression, that is,$a(a+r)(a+2r)\dotsb(a+nr−r)$, I use the notation $a^{n|r}$ is well described in the post mentioned above by Hartzer)

In his Dictionary of Curious and Interesting Numbers, David Wells tells the following story: "Augustus de Morgan ... was most upset when the " ! " made its way to England. He wrote:'Among the worst of barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! ... which gives their pages the appearance of expressing admiration that 2, 3, 4, etc should be found in mathematical results.'"

Another early symbol (shown below) was also used. Here is the discription of its origin from the web page of Jeff Miller,

An early factorial symbol, was suggested by Rev. Thomas Jarrett (1805-1882) in 1827. It occurs in a paper "On Algebraic Notation" that was printed in 1830 in the Transactions of the Cambridge Philosophical Society and it appears in 1831 in An Essay on Algebraic Development containing the Principal Expansions in Common Algebra, in the Differential and Integral Calculus and in the Calculus of Finite Differences (Cajori vol. 2, pages 69, 75).

I later found a copy of the 1830 paper on Google Books, and here is the way Jarrett presented the notation:

The symbol persisted and both symbols were in use for some time. Cajori suggests that the Jarrett |n symbol was little used until picked up by I. Toddhunter in his texts around 1860, and it was the use of his texts in America that may have influenced its use in the USA where it was more popular than the current symbol until around WWI.

The image below is from the 1889 textbook, A College Algebra by J.M. Taylor of Colgate.

A second image shows that the symbol was still in use even after the textbooks had adopted the "n!" symbol. This image is a note on the top a page on combinations in the 1922 text College Algebra by Walter Burton Ford of the University of Michigan. The book uses the exclamation point notation, but the hand written reminder is in the notation of Jarrett (and perhaps the teacher of Ms. Mabel M Walker whose signature is in the front of the book).

I recently found even a later date of the use of the Jarrett symbol. In the Mathematics Teacher for February of 1946 the symbol is used in an article by C. V. Newsome and John F. Randolph in illustrating Newton's power series for Sin(x). The fact that it is done with no comment indicates it must have still been commonly used.

I also came across an Arabic use of a very similar symbol, that is apparently still current. A note from an AP calculus teacher in February of 2009 indicated that a transfer student from Egypt uses something like this symbol currently.

A variation of Vandermonde's [p/3]n which allow the symbol to be extended to the idea of multiplying every other number, or every third, etc. What is today called the double factorial, triple factorial etc. The earliest use I can find of either the "!!" notation or the term double factorial is by B. E. Meserve, in 1948 (Double Factorials, American Mathematical Monthly, 55 (1948)) His usage indicates he is using a well understood term, and symbol so I suspect there is earlier usage. For example the use of a double factorial, as in 7!! means multiply 7*5*3*1; and 7!!! would be 7*4*1 (every third multiple). This seems to be little or no improvement to my mind from the notation Vandermonde used for the same purpose. It is important not to confuse these symbols with (7!)! which is the factorial of 7! or 5040!.

I received a comment to this post from Maurizio Codogno who had an even later use of Jarrett's symbol for factorial. He writes, "I found the L notation for the factorial in the book The Math Entertainer,(by Philip Heafford) which is dated 1959 (I have the 1983 reprint) He even shared a digital copy from the book.

This may seem a big number of arrangements. It is the product of 6 x 5 x 4 x 3 x 2 x 1. Another way of writing this product is $\lfloor6$, or, as it is often printed, 6!. It is called factorial 6.

I am now wondering if the notation is still in use in some part of the globe.

A good approximation to n! for large values of n is given by Stirling's Formula, which probably ought to be named for De Moivre. $n! \approx \sqrt{2\pi n} (\frac{n}{e})^n$ "The Factorial can also be generalized to the real and complex numbers using the Gamma Function

There is also a subfactorial and symbol in math. I am still searching for links to early uses, variations, etc. What little I knew a few years ago (and today) is here. Would love to have your input.

Unfortunately the same symbol, !n, often used for the subfactorial, was applied in 1971 by D. Kurepa for the sum of factorials,$!n=\sum _{k=0}^{n-1} k!$ so !5 would be 4! + 3! + 2! + 1! + 0! = 24 + 6 + 2 + 1 + 1=34 .

Amazingly these two seemingly unconnected sequences are related. For clarity if we call the subfactorial seqeunce S(n) and the factorial sum sequence F(n) then it can be shown that $F(n) \equiv (-1)^{k-1} S(n-1)$ Mod n.

There are other variations on the factorial. The primorial is the product of all the primes less than or equal to n, and is usually expressed as n#, so 5# = 5*3*2. They are useful to prime hunters, and the term was created by the very successful prime finder, Harvey Dubner. I would love to have a source for it's use, or the creation of the symbol.

There is an alternating factorial which is the sum of the terms of a factorial sequence alternately added and subtracted. For example af(5) = 5!- 4! + 3! - 2! + 1!. The only symbol I have seen is af(n), but I think something like $\pm n!$ would be somewhat elegant. Go forth and use it. Donald Knuth, are you reading this?

There is also a superfactorial, the product of the factorials from 1 to n, $\prod\limits_{k=1}^n k!$. I have seen the symbol of a heart suggested, so 3 ♥ would be 1!*2!*31.

There is even a hyperfactorial, although I have never seen it in use. H(n) = $\prod\limits_{k=1}^n k^k$ These get big in a hurry. (If you have information on the origin and uses of any of these, please advise.)

The term factorial is drawn from the more common math (and English) term factor. The roots of both these words are in the word fact and its Latin root facere, to do. To know the facts, is to know what has been done. The person who does something is then called the factor. In business a factor was once a common term for one who buys or sells for another. Today the word agent is more common. Colonial businesses often employed a person to do various menial tasks, as a factotum, literally one who does everything (today we might call them a "gopher").

Things that were necessary in order to "do something" became factors in the event, and today you may hear a coach say, "Defense was the most important factor in our victory."

Factors then became the parts of the whole, and a factory was where they were put together to make a final product. These words run over into the mathematical meanings. The factors are the numbers that are put together (by multiplication) to make the product. Because the product is made up by putting together parts, it is called a composite number.

The word "measure" has often been used in much the same way we now use the word factor. In his Universal Arithmetick Newton distinguishes three kinds of numbers, "integer, fracted, and surd", and defines an integer as "what is measured by Unity."

Frederick Emerson's North American Arithmetic(1850)says "One number is said to MEASURE another, when it divides it without leaving any remainder." (pg 18) Later it states," A number which divides two or more numbers without a remainder is called their COMMON MEASURE." This is after the definition of factor on page 12, and immediately precedes "A square number is the product of two equal factors" on page 19.

Other English words from the "to do" meaning of fact include facility (the ability to do), faction (a group working to do the same thing), facilitate (make easy to do) and faculty.

Farey Sequence John Farey was a geologist, not a mathematician, but he is better remembered today for a single short (four paragraphs) paper he wrote in 1816 than for all his good works in geology. In that year he sent a paper to the Philosophical Magazine (or at least it was published in that year) called On a curious property of vulgar fractions .and described a pattern that appears in sequences of what we would today call common (vulgar is the Latin term for common) fractions, like 3/8 etc, which are in simplest terms. They observation has almost NO practical use, not even to prove other things mathematically, and yet, it seems to have all kinds of interesting properties that tend to keep us fascinated with it. If you have never been introduced to it, here is a brief description, and some novel relationships that I think are interesting, with some links to places where they are made clearer than I could do in this brief space.

As so often happens in math, it is named for Farey, but someone else did it first. Charles Haros, had published similar results in 1802 which were not known to Farey.

So First... What is it we are talking about? If you take ALL the fractions that could be written in simplest terms with a denominator less than some number n, say n=5 (since that is the one Farey used in his paper), and put them in order from lowest to highest... you get The "curious" thing that Farey noticed is that if you ignore the way your fifth grade teacher taught you to add fractions, and do it the way YOU would have added them, "add the tops, add the bottoms", then each number in the sequence is the sum of the terms on each side of it... for example 1/4 and 2/5 are on each side of 1/3, and if you add by this approach you get (1+2)/(4+5) = 3/9 and that simplifies to 1/3 . The number obtained by adding two fractions in this fashion is often called the mediant
OK, so that is how you make them. Our first question might be, how many of them are there? F(5) obviously has eleven terms (I counted). If we picked a value of N, what would be the number of fractions in the set F(N). A little investigation would show that F(1) = 2 (0 and 1); and F(2) = 3 (0, 1/2, and 1). So how many would be added to the next set... and the next... it turns out that each new set will have all the values of the previous set (of course) and will add one for every value of one through n that is co-prime (has no common divisors) to n. These numbers that are co-prime with n are called the totients of n, and yes, there is a cool symbol, $\psi (n)$ for how many there are for each n.     So the set for N=6 will have the eleven terms of Ferry(5) plus 1/6 (one has no common factor with six), and 5/6. (notice that 2/6, 3/6, and 4/6 are already in the sequence in F(5) in simplified form)... thus F(6) has thirteen terms.. and in general we get a recursive formula that say the Order of F(N) = Order of F(n-1) + φ(n).
A second nice curiosity related to Farey sequences are the Ford Circles. "Ford circles are named after American mathematician Lester R. Ford, Sr., who described them in an article in American Mathematical Monthly in 1938, volume 45, number 9, pages 586-601" (from Wikipedia). In fact, the wikipedia article is a nice place to see how they work, and I need say no more as it shows the circles for F(5). What is amazing is that each circle is tangent to every other circle for a fraction it will be adjacent to in ANY sequence.... ahhh, go on..say "cool".

Another interesting thing about the Farey sequence is that If you treated each fraction a/b as a point (b,a) then none of the lines cross. If you make a triangle with the origin and any two adjacent Farey fractions, since each of the triangles have a determinant of one (meaning the area is 1/2) and therefore, by Pick's theorem, they cannot contain any other lattice points in their interior. A nice explanation of this, including the photo below, is at the Cut-The-Knot web site Fermat Point  The Fermat point, often called the Torricelli point, was first proposed, according to many sources, by Fermat in a letter to Torricelli in 1640.  Torricelli eventually solved the problem but it was only published years later in 1659 by his student, Viviani.  It is also said that Fermat had written about (solved?) the problem in the early 1600's. (Information about a letter, or earlier proof by Fermat would be warmly welcomed.)

The problem was to find the point that would minimize the distances to the three vertices of the triangle, thus the geometric median of the triangle.  If all the triangles angles are less than 120 degrees, then the point can be found by constructing equilateral triangles exterior to the triangle on all three sides, then connecting the remote vertices of these triangles to the opposite vertices of the primary triangle. In this case the Fermat/Torricelli point will lie interior to the triangle.  If the triangle has an angle greater than 120 degrees, then the equilateral triangles must be drawn internally to the triangle and the point sought lies in the exterior of the triangle.  I also do not know if Torricelli found both these solutions, or only the ones that lie inside the triangle.
Fibonacci/Brahmagupta/Diophantus Identity All the way back to before 500 BC when the Pythagoreans decided to take credit for the triangle relation called the Pythagorean Theorem, there has been interest in numbers that were the sum of two squares, and thus the square of the hypotenuse of a right triangle.  Each of the names above had some part in making public a method of creating a sum of two squares in two different ways that had a common sum. Some part of this idea was known to Euclid, Proclus, and Diaphontus of Alexandria included it in his 200 AD Arithmetica.  Brahamagupta extended it a little past this basic by adding a multiplier, but Fibonacci stuck with the basics in introducing the identity to the west in his 1225 Book of Squares, perhaps his second most popular publication.

The basic idea is that if you take any four numbers, not in proportion, so that the first is smaller than the second, and the third is smaller than the fourth, then you can use them to create two different sums of two squares that have the same sum, (and would thus have the same hypotenuse if they were the legs of a right triangle).  If the four numbers were the legs of already known Pythagorean triangles, then the identity would give two Pythagorean triangles with a common hypotenuse that was the product of the hypotenuii of the two original triangles.

The identity for any four numbers a, b, c, d under the restrictions previously mentioned give the following two relations.

(a+b)^2 (c+d)^2 = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-b)^2.  Student's familiar with the old mnemonic for multiplying two binomials, (FOIL)  will recognize these two  products as the first and last in one case, and the inside and outside, in the other.

The results also follow the pattern of multiplying complex numbers (a+bi)(c+di) in the first instance, and using the conjugate in the second, (a+bi)(c-di); although we remind students that none of the individuals credited with these ideas were alive when complex numbers were formulated.

Viete went even farther in his description of this identity.  He described the two techniques with different names, the second (First and Last negative in my example)  he called synaereseos, from a Greek verb meaning 'taken together'. This is still a term in language for when two vowels are combined into a dipthong.

The first (First and Last positive),  he referred to as diaeresos, meaning 'taken to pieces.' He stated clearly that if you took the difference between the two smaller angles in the first two triangles,  the result in his diaeresos method, would be the difference of  the two original angles, the exact result of conjugate divisions.  And if you maintain the angle orientation so that you focus on the angle between the x-axis and the hypotenuse, you would get the sum of the angles in the synaereseos method.

Fibonacci's Sequence This sequence is named for Leonardo of Pisa (1175-1230)  He is also called Leonardo Fibonacci, which seems to have been a contraction of fillius Bonacci, son of Bonacci.  He is one of the three men most responsible for introducing the Arabic numerals into the Western World.  In one of his famous books, The Liber Abacci (book of calculating) he poses and solves the famous rabbit problem which produces the now famous sequence {0, 1, 1, 2, 3, 5, 8, 13, 21, ...} in which each value after the first two, are the sum of the two which precede it.

According to Paul J Nahin, author of An Imaginary Tale, the name Fibonacci was not common until centuries after Leonardo's death, and during his lifetime he was called Bigollo, a slang term for a loafer drawn from the word bighellione.

Although it is almost certain that he knew, Fibonacci never wrote that each value was the sum of the other two.  The first written statement of this observation is credited to Kepler almost 400 years later. The formal definition as we now used it, was first written to Albert Girard (1535-1632) who edited the words of Simon Stevin, was the first to write $f_n + f_{n+1} = f_{n+2}$ It would be almost another century before Robert Simson, for whom the Simson line is mis-named, recognized that each term was the convergent of the continued fraction which is known as the Golden Mean or Golden Ratio:

Since all the entries are one, this is the slowest of all convergents to converge.

Yet another 150 years would pass before the explicit formula for $f_n$, the nth Fibonacci number would be found by J. Binet.

Using $\phi$ to represent the quantity $\frac{\sqrt {5} + 1}{2}$ Binet found that the value could be found by $f_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$

As late as the mid-1800's French mathematician Gabrielle Lame proved that the number of n-digit Fibonacci numbers is at least four, and at most 5, but Lame did not use the name "Fibonacci numbers". In fact, they were often called Lame numbers because of his proof.

The Fibonacci sequence was given its name in May of 1876 by the outstanding French
mathematician Francois Edouard Anatole Lucas, who had originally called it
“the series of Lame,” after the French mathematician Lame, mentioned earlier. (*Thomas Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, New York, 2001.)

The relationship between the Fibonacci Sequence and the Pythagorean Theorem was first observed by Charles W. Raine, and published in a paper in Scripta Mathematica in 1948. Fifteen Puzzle
Before there was Sudoku, before there was Rubik's Cube, there was the fifteen puzzle. In 1880 it was THE hot game to play, and seemingly everyone did, "About the year 1880, everyone in Europe and America was engaged in the solution... ". The image above shows an old copy that was for sale somewhere (sorry, I should have made a note).
The object of the game was to slide the fifteen squares into the open space and by doing so put them in correct order. There are 15! or 1,307,674,368,000 ways to put the fifteen squares into the box, and exactly half of them are impossible to solve.
Late at night as I was thumbing through my 1917 copy of H. E. Licks, "Recreations in Mathematics", and while glancing through the section on the fifteen puzzle came across the following: "It has been stated that this interesting puzzle was invented in 1878 by a deaf and dumb man as a solitaire game. "
Ok, maybe, but it just seemed too improbable.... Occam's Razor and all that... besides, I thought I had heard that it was a Sam Loyd puzzle. A litte quick research and I found a reveiw of The 15 Puzzle: How It Drove the World Crazy by Jerry Slocum and Dic Sonneveld at MAA online.
Sam Loyd was almost certainly the premier puzzle master of the late Nineteenth Century, and he was not shy about claiming credit for almost anything, and did claim the invention of the Fifteen Puzzle in his books. The more likely truth, is that , "The puzzle was "invented" by Noyes Palmer Chapman, a postmaster in Canastota, New York, who is said to have shown friends, as early as 1874, a precursor puzzle consisting of 16 numbered blocks that were to be put together in rows of four, each summing to 34. Copies of the improved Fifteen Puzzle made their way to Syracuse, New York by way of Noyes' son, Frank, and from there, via sundry connections, to Watch Hill, RI, and finally to Hartford (Connecticut), where students in the American School for the Deaf started manufacturing the puzzle ((So there was the connection to deaf and dumb inventor story) and, by December 1879, selling them both locally and in Boston (Massachusetts). Shown one of these, Matthias Rice, who ran a fancy woodworking business in Boston, started manufacturing the puzzle sometime in December 1879 and convinced a "Yankee Notions" fancy goods dealer to sell them under the name of "Gem Puzzle". In late-January 1880, Dr. Charles Pevey, a dentist in Worcester, Massachusetts, garnered some attention by offering a cash reward for a solution to the Fifteen Puzzle. The game became a craze in the U.S. in February 1880, Canada in March, Europe in April, but that craze had pretty much dissipated by July. Apparently the puzzle was not introduced to Japan until 1889. Noyes Chapman had applied for a patent on his "Block Solitaire Puzzle" on February 21, 1880. However, that patent was rejected, likely because it was not sufficiently different from the August 20, 1878 "Puzzle-Blocks" patent (US 207124) granted to Ernest U. Kinsey."(from Wikipedia)... ( Kinsey's patent has an application date in November of 1877, and includes interlocking pieces so they stayed in the box like modern fifteen puzzles do An image from the patent is here.
What Sam Loyd did do, was realize that in 1/2 of the 15! possible ways to put the 15 squares in the puzzle, it could not be solved. He immediatly offered a \$1000 prize to anyone who could solve it in his newspaper aritcle, magazines, and books, and draw a ton of free advertising.
Eventually it became known that the puzzle could not always be solved, and some of the interest waned, but the puzzle remained popular enough that it was a subject for methaphor for writers. "But Big Jack Fish Lake was two days' travel away, and meanwhile my ankle made life intolerable, and the map proved more maddening than the fifteen puzzle.", from On Snow-Shoes to the Barren Ground, by Caspar Whitney ; 1896 - page 72)

One of the things that was discovered about the Loyd Puzzle, the one that could NOT be done, was that if you left the blank in the top left hand corner (putting 1-2-3 in the top row, then 4-5-6-7 next, etc), it would be possible to put all the numbers in the correct order. This would be the same on the solvable puzzle as getting all the numbers in the right order except reversing the 14 and 15... try it.

Here is an additional "15 Puzzle" not related to the physical puzzle above.  I found it on Greg Ross' Futility Closet a while back:
A problem from the 1999 Russian mathematical olympiad:
Show that the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.

Fifteen Peg Puzzle

The puzzle shown above, often called a triangular peg solitaire game is so common you may have last seen it on your table at Cracker Barrel Restaurants.
Below I will give a solution to a common form of the puzzle, so don't look too far below if you want to test out your skills without clues or a solution. If you want to play the online version of the game, try your luck here
If you get really interested, you can try games removing any one of the pegs instead of a corner (which is NOT the easiest possible solution).    If you get frustrated, here are some good hints about the game from an excellent page by George Bell.

Note the symmetry of the triangular board: there are three corner holes, three interior holes , and three holes at the midpoint of each edge , plus six "other" holes .
The following rules of thumb are based on a mathematical analysis of the game and should help you solve the puzzle
1. Avoid jumping into a corner. Of course, in some situations (such as beginning without a corner peg) this is the only jump possible.
2. Avoid any jump which starts from one of the  interior holes. Such a move is almost always a dead end (none of the solutions include this jump).
3. The easiest place to begin the game is with the missing peg (hole) at one of the  midpoint locations. The hardest place to begin is with the missing peg at one of the  interior holes.

Complete solution below:::

The truth is, there are thousands of possible solutions to the game. The solution I just came across is for the case in which you start with the top corner (the one hole) empty, and finish with a single peg in that hole (which I just learned is called a single vacancy complement solution). If you number the holes starting at an apex of the triangle with one, then two and three on the next row, and continue with 11 through 15 on the bottom row, then moves can be described with (x,y) coordinates where the term (x,y) means move the peg in hole x to hole y.
A winning solution to the 15-hole triangular peg solitaire game using this method is: (4,1), (6,4), (15,6), (3,10), (13,6), (11,13), (14,12), (12,5), (10,3), (7,2), (1,4), (4,6), (6,1).

Not only does this solution leave the final peg in the original empty hole, but the sum of all the x,y hole numbers in the solution is prime (179) and if I time this right it should be first posted on the 179th day of the year 2012. By the way, if you only sum the values of the landing holes, that's prime also (73).

I first came across this curious little fact at the Prime Curios page.

Haven't pursued it yet, but since there are thousands of solutions to this puzzle, I assume that there would be other sequences which produced a prime also.

Good luck, and share the ones that you find with me.

First is the first ordinal number (how's that for a circular definition) and means the leading person, or item by some characteristic, chronological, value, or order.  The child born before all others is the "first" child.
It is drawn from the Old English fyrst, which was a variant of fore(front).

Focus comes from the Latin word for hearth, or fireplace.  The first use of focus as a mathematical word seems to have been with Kepler, perhaps inspired by the fact that the Sun was at one focus of the planets elliptical orbits.  It may also have been that in many European homes in the Middle Ages, the hearth was the center of activity.  Carl Boyer writes in A History of Mathematics, "As the curves(conic sections) are now introduced in textbooks, the foci play an important role, but Apollonius (author of one of the first great treatises on the conic sections) had no names for these points, and her referred to them only indirectly."  In 1656, Thomas Hobbes wrote "the focus of an hyperbole, is in the axis." Just as with ellipse and ellipsis, the hyperbola and hyperbole seem to have been interchangeable in early English.

Formula comes from the Latin forma, from which we get the word form for shape or style.  The Romans were very formal in many of their religious and social practices, and the correct forma, or procedure was taught to the young. Brief, or small, rituals used the diminutive of forma, formula.  Over time short list of directions and instructions began to be called formulas also.  Today we apply the word to any collection of words or symbols used in a ceremony or procedure, from logging into a website, to a chemical description of water, to a function for generating the Fibonacci numbers. The first use cited by OED is by R. Kirwan in 1794.

Fractal appears to be a very modern term created by Benoit B Mandelbrot in an article published in Scientific American magazine around 1975. Mandelbrot discussed his choice of names in The Fractal Geometry of Nature.  In it he wrote "I coined fractal from the Latin adjective fractus."  Fractus is a derivative of frangere, for broken, which is also the root of fraction. The definition can appear confusing but the basic idea is about self-similarity at different orders of magnitude.  Merriam-Webster dictionary give the definition as :  any of various extremely irregular curves or shapes for which any suitable chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.  A classic example (not a perfect one) is a single broccoli floret looks greatly like a whole head of broccoli.

Fraction comes from the Latin root, frangere, to break.  A fraction then, representing the broken portion of some whole.  The word was first used in English by Geoffrey Chaucer in the work he was most famous for in his life, but now is little known to the general public, a Treatise on the Astrolabe in 1440. Add caption
Frustum (sometimes called frustrum with an extra r) is from the Latin and means "a piece broken off".  The English adoption was as early as 1658.  The Philosophical Transactions of 1669 includes," a figure of different bases which he calls a frustum of a pyramid."

Mathematically it usually refers to a part of a solid cut off between two parallel planes, as opposed to truncated, cut off by an oblique plane.The Indo-European root of the word is bhreus and is related to cutting, crushing, or pounding.  Related words from the same root are fraction, fragment, bruise, and possibly brush (from a bundle of cut twigs).  A frustum of a square based pyramid is shown, but you can have a frustum of any right cone, or any pyramid.

Function The origin of function is in the Latin word fungi, which means perform. It is not related to the modern word fungi, for mushrooms, which is from the Greek word for sponge.  "Perform" is an appropriate root for the idea of a mathematical function as an operation, or operations, to be performed on a set of values, the domain of the function.  Leibniz first applied the word to mathematics around 1675, "de functionibus" ,and according to Morris Kline, he is the first to use the phrase "function of x." Later Euler developed the notation "f(x)" around 1735. The word emerged in English in 1758 paper by J. Landen with a slightly different meaning.  The first English use of the word with it's current meaning in print, according to the OED, was in 1779 in Chamber's Cylopedia

The performance part of a function helps explain the meaning of related language terms like defunct, for "no long performing", and dysfunctional, not performing properly.

G

Galley Division ----- See Division

Gamma Function The factorial function only applies to non-negative integers.  The gamma function is an extension of the factorial function to cover the rest of the reals and the complex plane.   and Euler described the function, but not the name in his letters to Goldbach around 1730.

The name, and the symbol, $\Gamma (n)$, were created by Legendre in 1808.  The Gamma functions is given by the formula $\Gamma (z) = \int_0^\inf x^{z-1}e^{-x}dx$ for non-zero z.  For converting Gamma to Factorials the rule is $n! = |Gamma(n-1)$ .

Gas  See Chaos

Gaussian function         See Normal Distribution

Geocenter See Centroid  also see Just an Average Point

Geometry is derived from the Greek word for the earth, geos, from the Goddess Gaia, and the term for "to measure", metros.  Literally then, geometry means "to measure the earth" and that was it's original purpose.  Although the Greeks and Latins pronounced the name of the Earth Goddess as gay' yuh, it came into English pronounced more like Jee' uh, from which we get many geos related words like geography, and geology.
Just as the study of the Earth, geology, recalls the ancient Greek God of the sky, the term Uranology, from the Greek God Ouranos, is the study of the sky, but the more common term today is astronomy.  But do not despair for the lost memory of the god of the sky, for he is preserved in the name of the planet discovered by Wiliam Herschel in 1781.  Although Herschel wanted to name the planet in honor of King George III, remembered as the bad guy of the American revolution by US History, Johann Bode (see Bode's Law) came up with the name that stuck, Uranus.   The sky god is also remembered through a discovery a few years later (1798) by chemist Martin Klaproth.  It was a tradition to name metals after planets, so Klaproth named his new metal after the new planet, calling it Uranium.  Later he found another metal and decided to name it after the Earth, but instead of using the Greek Goddess of the Earth, he chose to use the Roman equivalent and called the new metal tellurium. The Roman Goddess of the Earth was known by two names.  The first was Terra, from which we draw words like terrestrial, and its better known opposite, extra-terrestrial. The other name was Tellus which is almost non-existent today, except in the name of Klaproth's metallic discovery.

Geometric Mean          See Mean
Gergonne Point/Triangle

Googol is a number invented by the nine year old nephew of Dr Edward Kasner, Milton Sirotta, when asked to think of a name for a 1 followed by 100 zeros.  10^100 is an incredibly large number.  The largest estimates for the number of particles in the universe is only about 10^85.  A googol is a million times a billion times this quantity.  A googol is thus 101 digits long in base ten.

Graham's Number  may have been the first number to appear in the Guiness Book of World Records as an entry.  The number was created by the namesake, Ronald Graham in a conversation with Martin Gardner, in trying to simplify (Gardner was not actually a trained mathematician) his explanation of an upper bound on a problem in Ramsey Theory he was working on.  It was an incredibly large number, and at the time, the largest specific positive integer ever published in a mathematical proof.  In 1977 Gardner shared the number with the world, and three years later, it shows up in the Book of World Records.  The universe is too small to contain an explicit digital recitation of the number.

Graph has come to have multiple meanings in mathematics, but for most students it relates to the graph of functions on the coordinate axes.  The origin is from the Greek graphon, to write, perhaps with earlier references to carving or scratching. Jeff Miller's web site suggests that the use of graph as a verb may have first been introduced as late as 1898.

In a post to a history newsgroup, Karen Dee Michalowicz commented on the history of graphing:
It is interesting to note that the coordinate geometry that Descartes introduced in the 1600's did not appear in textbooks in the context of graphing equations until much later.  In fact, I find it appearing in the mid 1800's in my old college texts in analytic geometry.  It isn't until the first decade of the 20th century that graphing appears in standard high school algebra texts. [This matches rise of  graph paper in the same periods].  Graphing is most often found in books by Wentworth.  Even so, the texts written in the 20th century, perhaps until the 1960's, did not all have graphing.  Taking Algebra I in the middle 1950's, I did not learn to graph until I took Algebra II

Math historian Bea Lumpkin has written about the early use of graphs by the Egyptians in what was an early use of what painters call the grid method:

In my article ... I suggest, "It is possible that the concept of coordinates grew out of the Egyptian use of square grids to copy or enlarge artwork, square by square.  It needs just one short, important step from the use of square grids to the location of points by coordinates.
In the same posting she comments on the finding of graphs in Egyptian finds dating back to 2700 BC:
"An architect's diagram of great importance has lately been found by the Department of Antiquities at Saqqara.  It is a limestone flake, apparently complete, measuring about 5 x 7 x 2 inches, inscribed on one face in red ink, and probably belongs to the IIIrd dynasty"  Here is the reason that Clark and Engelbach attached great importance to the diagram.  It shows a curve with vertical line segments labeled with coordinates that give the height of points on the curve that are equally spaced horizontally.  The vertical coordinates are given in cubits, palms and fingers.  The horizontal spacing, the authors write "... most probably that is to be understood as one cubit, an implied unit elsewhere."  To clinch their analysis, Clarke and Engelbach observe:  "This ostrakon was found near the remains of a solid saddle-backed construction, the top of which, as far as could be ascertained from its half-destroyed condition, closely approximated tot he curve obtained from the data on the ostrakon.

This certainly lays claim to the oldest line graph I have ever heard.

Gross The common measurement meaning of gross related to its origins in the Latin grossus thick means thick, or large. The present use of gross for 144, or a dozen dozens is also drawn from this meaning.  The OED cites 1411 as the earliest use of the written term; and mentions a great gross  for a dozen gross.

The Germans seemed first to have used the idea of a gross or great dozen to mean 144.  The use of gross for large also gives us the word grocer, orignally a grocer sold wholesale, in large quantities, and thus the name.  Something so big that it was visible to anyone gives us such phrases as gross injustice, and the medical term gross lesion, for one that is visible to the naked eye.  There was a famous mathematician named Robert Grosseteste whose last name literally means Big Head.

Group – A group is a set with some binary operation that obeys 4 requirements: closure, associativity, invertibility, and identity. The most common example is Z (the integers) with the + operator. The earliest study of groups was with Lagrange in the 18th century when he studied groups of permutations to understand roots of polynomial equations higher than 4. In 1799, Paolo Ruffini attempted to prove the impossibility of solving a quintic by investigating what are now primitive and imprimitive groups (it would later be completed by Abel in 1824). Later, groups arose with the study of polynomial equations from Évariste Galois in the 1830s who extended the work of Ruffini and Lagrange and studied the symmetry group between the roots of a polynomial, which is now called a Galois group. He gave solvability criterion based on the symmetry groups of these roots. Unfortunately Galois’ ideas were rejected (at 18 years of age in 1829) and published posthumously in 1846. He was honored with connecting group theory with field theory and his name defines this link, now called “Galois theory”. More general symmetry groups were studied by Cauchy and his 1854 paper gives the first formal definition of a finite group.

In geometry, Felix Klein used group theory to organize hyperbolic and projective geometry in a more coherent way. Additionally, Sophus Lie began the study of what are now Lie groups in 1884. Further, discrete group theory was continued by Klein and others with connections to modular forms and monodromy.

Thirdly, in number theory, Gauss had been implicitly using abelian groups in 1798 textbook and later, Leopold Kronecker did the same. In 1847, Ernst Kummer attempted to prove Fermat’s Last Theorem using groups describing factorization into prime numbers.

These contributions from number theory and geometry formalized the notion of a group by the 1870s. Camille Jordan began this unified theory with his Traité des substitutions et des équations algébriques in 1870. This unification also began with Walther von Dyck in 1882 who first defined a group in the modern sense. Lastly, textbooks by Weber and Burnside helped group theory develop as a discipline. *Derek Orr

H

Happy Numbers A happy number is a number deﬁned by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay),or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).For example, 19 is happy, as the associated sequence is:
1^2 + 9^ 2 = 82
8^ 2 + 2^ 2 = 68
6 ^2 + 8^2 = 100 1^ 2 + 0^2 + 0^2 = 1.  Many recreational mathematicians extend the idea to Happy Numbers in any base by converting the number to that base and then squaring and continuing in similar fashion, but always converting each sum of squares to the appropriate base.  Numbers that do not go to 1, may enter into a cycle and repeat the same numbers continuously.

The origin is unclear, although Wikipedia credits knowledge of it to Reg Allenby, a british author and lecturer at Leeds University when it was brought to his attention by his daughter, who learned of them at school.  It also says that Richard Guy suggested they may have started in Russia.

Although no date is given in the Wikipedia citation, I once suspected from an observation of David Wells' Curious and Interesting Numbers that the term entered English between 1987, and 1997 . I draw this conclusion from the fact that the term is not included in the '87 edition, but is in the '97 edition.

I have more recently found an earlier usage of the term.  The earliest I have ever found this term was in an article in The Arithmetic Teacher, Feb 1974.  "Happiness is some Intriguing numbers"  by Billie Earl Sparks of George Peabody College in Nashville, Tn.

Google Books shows an earlier use in 1970 in the Bulletin of the California Mathematics Council but does not show the work.  There is however in  the Spring 1973 issue of Pi Mu Epsilon Journal, an article byDaniel P. Wensing, of John Carroll University with the simple title, Happy Numbers that concerns the same idea and mentions it in many bases.

Harshad "Joy-giver" Numbers were created by Indian Mathematician D. R. Kaprekar, to apply to any number that is divisible by the sum of its digits.  12 is a Harshad number since 12 is divisible by the sum of 1+2.  One well known famous number, the Hardy-Ramanujan Taxi-cab number, 1729 is also a Harshad number, since 1729 is divisible by 1+7+2+9 = 19, and almost too nicely, the other factor is 91, the reverse of 19. See, can you feel the Joy it gave you?

Hectare is a unit of land equal to 100 ares or about 2.47 acres, or 10,000 square meters.  The prefix hecto is from the Greek word for one hundred, hekaton.  The prefix is common in units of measure such as hectogram, and hectometer.  The OED in its earliest citation in the Naval Chronicles of 1810, "Hectar, square hectometer".

Height comes into English through the German and is related to words like hop and heap.   Even early in the 20th century it was written and pronounced like heighth, or highth (compare to length and width in present use) but the "th" ending seems to have mostly, but not completely disappeared from common usage.The height of an object represents the distance from one base to the opposite vertex, or parallel face.

The Helen of Geometers  See Cycloid *Wikipedia
Helix is preserved from the Greek and has maintained its meaning since antiquity.  The word seems to have been used to apply to ideas about wrapping or twisting but only its mathematical meaning has survived. The Helix is a curve in three space wrapping (think of a cork screw) itself around a central fixed axis at a constant distance. It was, and sometimes may still be, used to describe a coil in a single plane.  The tangent to the helix always maintains the same angle to the fixed axis.  Do not confuse with the two space spiral which wraps around a point in the plane at an increasing distance.

Heptagon/Septagon A polygon with seven sides and angles.  The hepta is from the Greek for seven, and the gon is from the Greek word for bend, which gives us knee, gnaw, and angle.

Septagon is sometimes used, substituting the Latin root for seven, but this mix of Latin and Greek roots is frowned on.  Many simply resort to 7-gon.

The figure shows the regular heptagon, with all sides equal, and all angles equal.  The British 20 and 50 pence coins are reuleaux heptagons with the angles curved and the sides bowed out gently to make it roll smoother for vending, etc.

The term was first written by Henry billingsley in 1570 in his translation of Euclid's Elements.

Heron's (Hero's) formula Heron of Alexandria, sometimes called Hero, lived around the year 100AD and is most remembered for a formula for finding the area of a triangle given the three sides.  If we call the sides a, b, and c, and the semi-perimeter s = (a+b+c)/2; then the area is given by $\sqrt{s(s-a)(s-b)(s-c)}$

Documents from the Arabic writers indicate Archimedes may well have known this formula 300 years before Heron.
Heron's formula was extended in the 7th century by Brahamagupta who adjusted the formula to the sides of inscribable quadrilaterals. Since around 1880, the triangular method of Heron has been known as Heron's formula, or Hero's Formula. It emerged in French, formula d'Heron (1883?) and German, Heronisch formel (1875?) and in George Chrystal's Algebra in 1886 in England.

In 1986 a copy of Heron's Metrica was recovered in Constantinople (now Istanbul) that had been copied around 1100 AD.  It contains the oldest known demonstration of the formula.  Heron is also remembered for his invention of a primitive steam engine, one of the earliest forerunners of the thermometer, and a vending machine to dispense holy water at temples.

Heronian Triangles are defined as triangles that have integer sides, and integer area.  Some use the a less restricive definition of all these being rational numbers, since some multiple will make rationals into integers.

The discovery of the 3,4,5 right triangle which would meet the definition, seems lost in antiquity back before 500 BC.

The second integer sided triangle, the 13, 14, 15; was known to Heron of Alexandra as early as 70 AD, almost 2000 years ago. That connection is reason enough for many mathematicians to use his name for them.  The earliest usage of Heronian triangle I can find is from a 1916 puzzle in the American Mathematical Monthly, to divide the triangle whose sides are 52, 56, and 60 into three Heronian Triangles by lines drawn from the vertices to a point within. The problem was posed by Norman Anning, Chillwack, B.C. It then includes a description that suggests it is introducing a new term, "The word Heronian is used in the sense of the German Heronische (with a German citation) to describe a triangle whose sides and area are integral. The only other mention of a Heronian triangle in a google search before the midpoint of the 20th century revealed a 1930 article from the Texas Mathematics Teacher's Bulletin. It credits a 1929 talk, it seems, by Dr. Wm. Fitch Cheney Jr. who, "discusses triangles with rational area K and integral sides a, b, c, the g.c.f of the sides 1, under the name Heronian triangles."

L E Dickson's History of Number Theory states that Heron stated the 13, 14, 15 triangle and gave its area as 84. Brahmagupta is cited in the same work for giving an oblique triangle composed of two right triangles with a common leg a, stating that the three sides are $\frac{1}{2}(\frac{a^2}{b}+ b)$ , $\frac{1}{2}(\frac{a^2}{c}+ c)$, and $\frac{1}{2}(\frac{a^2}{b}- b) + ( \frac{1}{2}(\frac{a^2}{c}- c)$

Histogram/Histograph The roots of histogram is probably from the Greek, histo, for tissue, and gram, for write or draw.  The suffixes gram and graph are almost interchangeable, and both have to do with the act of writing or drawing.  E. S. Pearson, the first known user, (1891) apparently thought of each vertical bar as a cell.  Some have suggested that the root is from the word "history" since a histogram provides a record, and certainly Pearson knew of this meaning also.  The Greek root, histor, is for "a learned man".  The implication is that a learned man is aware of history, but it is more direct than just good advice.  The Indo-European root is the same root that gives us wise.

Hexagon The geometric polygon with six sides and six angles is named from the Greek roots for *Wikipedia
six, hex, and bend or angle, gon. The gon is the same root that gives us knee (think genuflect) and gnaw.

The image shows the regular hexagon with all  equal sides an angles.  The division shows that a regular hexagon can be formed by six equilateral triangles.

Hexagons have a property reminiscent of  Napoleon's theorem for triangles.   On any hexagon, if you construct equilateral triangles on each edge, and find the centroid of each, then the midpoints of the line segment connecting centroids of opposite sides of the hexagon, will form an equilateral hexagon.

The word seems to have entered English first in the writings of Leonard Digges in 1571, "the lesser heagonum."  In 1570 Henry Billingsley shortened this to hexagon in his translation of Euclid's Elements, and it became the standard use.

Hollerith cards In 1889 Dr. Herman Hollerith of New York City received patent #395,782 for the ﬁrst tabulating machine. It used punched cards and electrical counters operated by electromagnets. Its ﬁrst extensive use was in the compilation of the population statistics for the eleventh U.S. census in 1890.  His system was designed to record separate statistical items by means of combinations of holes in a punched card to carry information about an individual. The information contained on numerous cards could then be tallied by passing the cards through electrical counters operated by electromagnets. The patent described its application in compilation of the statistics of the population for the U.S. Census. The first extensive application of this system was for the 1890 census counting data items such as age, sex, occupation, etc., of which tallies could be made in combinations such as how many males of certain ages.*TIS (These punched cards were once a principle element of writing computer programs, which I, and many others, do not remember fondly.)

Also See Stanine

Horner’s Method – Method that approximates the roots of a polynomial. It was introduced by William Horner in 1819 and was the most popular way to approximate polynomials until computers in the 1970s. It is still a quick way to evaluate polnomials. The method is essentially the same as Newton's method and synthetic division. Although it is dubbed “Horner’s Method”, this way of finding roots was known to Ruffini in 1809 and Newton in the 1600s as well as even further back to Chinese and Persian mathematicians in the early teen centuries.
If you are not familiar with the root approximation method is is nicely explained at Wolfram Mathword. A simple approach to the evaluation is to use the compacted method of expressing the polynomial. An easy evaluation of 2x^3 + 3x^2 -5x + 3 is to write the expression as (2x+3(x - 5(x + 3) then to evaluate at, say x=2, simply do each operation using two each time you get to an x, so 2*2+3*2 -5*2+3. Proceed from left to right doing each operation to get 2+2=4, +3 = 7, *2 = 14, -5 = 9, *2 = 18, +3 =21. WIth a little practice, you don't even have to rewrite the polynomial. *Derek Orr
Hour/Year are both derived from the Greek root horo, which was applied to idea about time and the seasons.In the old Germanic language, horo became yoro, and year was derived from the same root that gave us hour.  Today horoscopes are for telling fortunes, but the practice is rooted in the original meaning, measuring the aspect of the stars and planets to measure the seasons. Horology is still the name for the science of making timepieces.

Hundred is from the German root hundt. The quantity that it represents has not been consistent over the years, or from place to place.  It has ranged in value from its common present value of 100, to 112, 120, 124 and 132 at different times and places.  The remnants of these old measures still persist in the hundredweight in some countries representing 112 or 120 pounds, depending on the country. Shortly after the US adopted 100 pounds as the  hundredweight as a standard, the UK formalized 112 pounds, and outlawed the US measure. The weight is still used, I believe, in the Imperial system. In Germany the hundredweight was 120 pounds, and has since been modified to Long Hundred.
A hundred has also been used to represent an area of land equal to 100 hides (the general size of a hide came from the German for the amount of land necessary to support a family, the land measure of Hundreds usually measured from 60 to 120 acres {there's that Long hundred again.} ). The measure of area was frequently used in colonial US and parts of England in place of the "shire" or "ward".   A curious custom related to one hundred as a unit of land occurs in England when a member of the house of commons wishes to resign his seat, which is illegal.  An MP accepts stewardship of the "Chiltern Hundreds", an area of chalk hills near Oxford and Buckingham, and effects his release.

Hyperbola/Hyperbole  The hyperbola, is drawn *Wikipedia
from the Greek roots hyper, beyond or in excess, and bola, to throw.  Hyperactive is a well known example of the one root, and ball is drawn from the other, as well as parabola.  The meaning seems to be something like "thrown too far".  The most common explanation is that means the plane was thrown so far away from the angle that the sides of the cone makes with the axis, that it cut both nappes of the cone.  The para in parallel means beside, and it only cuts one nappe because it is cut by a plane that is beside (parallel) the the angle of the sides of the cone with the axis.  The ellipse, of course, falls short of that angle, and thus makes a closed curve on one nappe.  As with ellipse and ellipsis, early English use of hyperbola was interchangeable with hyperbole.

Hypercube in geometry the n-dimensional analog of a square, or a cube, is called a hypercube.  It is also called a tessaract.  Tessarect was a creation of Charles H Hinton in 1888, the son in law of George Boole, and one of the earliest writers on the idea of a fourth dimension.  For more about Hinton, his wife, and her sisters and their progeny, see "Those Amazing Boole Girls".

Hyperfactorial   See Factorial

Hypocycloid of Four Cusps -  See Astroid

Hypocycloid of Three Cusps - See Deltoid

Hypotenuse The term we use for the longest side in a right triangle comes from the common Greek root hypo, for under, (as in hypodermic-under the skin) and the less common tein, for stretch.  This last is the source of the word tension.  Although today we draw triangles in any orientation that suits us, it seems the Greeks generally drew them with the largest side on the bottom, hence the literal root of hypotenuse, to stretch below.  So why is it only the right angle that is called a hypotenuse? Well I'm not sure (If you know, please write and explain) but I imagine it may relate to the fact that of all the triangles inscribed in a given circle, the 90o angle has a longer "longest" side than any other such triangle.  When the angle is a right angle, the hypotenuse is the diameter of the circle.