Math Words, pg 9

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**Affine Geometry** is the study of measures that are preserved under a transformation that carries each point (x,y) to a new point (ax + cy + e, bx + dy + f). This is sometimes described as a parallel projection from one plane to another. Euler was one of the first to study affine geometry. In an affine geometry Euclid's third and fourth postulates no longer apply.

The main root of affine comes from the Latin root *finis* for end or border. The prefix is a mutation of *ad* with the meaning near and the combination was used to mean sharing a common boundary. This was generalized to sharing a common interest of most any kind. Today affinity refers to an attraction of almost any kind.

**Axiom** In the language of mathematics and logic, an axiom is a statement that is considered to be true without need for proof. The word is often used interchangeably with **postulate**. The origin comes from the Greek root *axios* for worthy. An axiom is something that is accepted as worthy of its own accord, without proof or risk of refutation. The root shares a common origin with words relating to leadership or influence such as agitate, agent, and examine. Some other words from this relationship include agony, which originally meant a great conflict or contest, and ambassador (one who is sent around).

Although we often speak of Euclidean axioms or Euclidean postulates, math historian D.E. Smith has written that Euclid seems to have used a more general phrase meaning "common notion". Certainly he would have known of the word axiom as it was used by many of the ancient Greeks, including Aristotle.

The origin of **postulate** is from the Latin *postulare* which means to request or entreat. It is related to the word pray also.

**Binary Numbers** The word
binary refers to something that has two main parts. The
root is from the Late Latin word *binarius* which
literally meant "two by two." Binary numbers are
numbers built up with the use of only two numerals, ones
or zeros. The counting sequence 0,1,2,3,4,5 in binary
would be 0, 1, 10, 11, 100, 101. According to the
Oxford English Dictionary, the word first appeared in
English in 1796.

Most people are so familiar with the fact that computers use binary numbers that they assume it started out that way. There were several far-sighted individuals in the 1930's who saw the advantage of binary earlier, but most of the earliest electronic computers used decimal arithmetic. John von Neumann pressed for the use of binary arithmetic in 1946, and the rest is history.

A binary logic system based on true and false conditionals was developed in mid-19th century by George Boole.

**Bisect** means to cut into
two equal pieces. The word combines the common Latin
*bi* for two, and *secare*, to cut, which also
is the root for secant. Although the roots are based in
the ancient Latin, the word is actually an English
creation first appearing around 1645.

**Catalan Numbers ** first appeared in disguise in a problem Euler first proposed to Christian Goldbach in 1751. The problem is now called "Euler's Polygon Division Problem", and asks, in how many ways may
a plane convex polygon of n+2 sides be divided into triangles by diagonals. Euler gave a solution that looked like The numbers in the sequence are now called Catalan Numbers after Eugene Catalan. Catalan, it seems, wrote a paper in 1838 with an improved proof and the sequence became associated with his name. He may have been the first to write about the relationship between the sequence and the number of expressions containing n pairs of parentheses which are correctly matched. I have read that he discovered the relationship while playing with the "Towers of Hanoi" problem, but have not found when this was done. I have recently found a paper that suggests that a Chinese mathematician/scholar, Antu Ming, had found the sequence in connection with the expansion of sin(mx) as a power series of sin(x) at least by 1730.

The numbers in the Catalan sequence also answer questions such as the following. In how many paths can you move from the origin of a coordinate axis to the point (n,n) if each move consists of either an upward move or a move to the right one unit between two lattice points and you cannot cross the line y=x . Another application answers the question in how many ways can 2N beans (for an application, think votes) be divided into two containers if one container can never have less than the second?

The sequence of numbers is given by 1,2,5,14,42,132 ... or in general by the function . It may seem hard to figure out where the 1/(n+1) comes from in the formula. I found a really nice explanation with graphics at this Wikipedia page(go down to the third proof)

The numbers also can be found from Pascal's triangle by taking (2n Choose n) - (2n Choose n+1). These are two adjacent numbers in each even numbered row

The function is named for Eugene Catalan of Belgium (1814-1894).

I recently (March 2009) came across a note on the "God Plays Dice" blog pointed out that the Catalan sequence has an asymptotic function, (a value it approaches closer and closer as n gets bigger and bigger) of

The page also had an interesting joke of the kind that only a mathematician could appreciate; "asks you to identify the sequence "una, dues, cinc, catorze, quaranta-dues, cent trenta-dues, quatre-cent vint-i-nou,...", which are the Catalan numbers 1, 2, 5, 14, 42, 132, 429... in the Catalan language. (I'm reporting the spellings as I found them in my sources; apparently the spelling is not totally standardized.) ". The source of the joke is the American Mathematical Monthly, vol. 103 (1996), pages 538 and 577

**Coefficient** A coefficient is a number, or variable, that is multiplies a variable term. In a common linear equation like ** 2x-3y=5 ** the 2 and 3 are the coefficients of the variables x and y. In the typical equation of a general quadratic polynomial we write ** Ax ^{2} + Bx + C =0 ** but we call the letters A, B, and C the coefficients of the terms. Even though they are variables, the represent some constant, but unknown value unlike the variable x which is variable of the expression. The origin of the word reaches back to the early Latin word

**Collinear** . Two or more points are said to be collinear if they lie on the same line. The roots are easy to identity as the Latin *con* or *com* for with, and the word line. The double letter is a common in words where the last letter of the prefix is changed to match the first letter of the root word. Here is a comment on this practice from John Conway that was posted to a discussion group:

The "co" here is from the Latin "con", meaning "with". Such Latin prefixes have a habit of merging with the next consonant in certain circumstances. For instance "in", meaning "not", has done this in words like "irreparable" and "immiscible" which should logically be "inreparable" and "inmiscible", which would be much harder to pronounce, as would "conlinear".

There are lots of other examples, most of which are so familiar that one no longer analyses them - for instance "offer" should logically be "obfer", meaning (roughly) "to bring out", and "differ", which should be "disfer" ("to bring apart"), "accept" = "adcept" ("to capture towards"), and so on.

John Conway

**Component** When students first study vectors they think of the component parts primarily in terms of the horizontal and vertical components of the vector, but any two vectors that can be added to make a third are the components of the vector. This word also begins with the Latin root *com* for with and joins it to *ponere*, to place. Literally then, the components of the vector are the vectors that are "placed together" to form the resultant. Other math words coming from the same base include compound and exponent.

**Conjecture** A mathematical conjecture is a tentative conclusion in the absence of formal proof. We think it is so, but have not, or can not prove it. After looking at the three altitudes in dozens of triangles, and observing that in each case they intersect at a single point, a student may make the conjecture that this happens in all triangles. His repeated examination gives him reason to believe the statement may be true, but not sufficient evidence to declare the statement as proven. The word seems to have not been common until the beginning of the 20th century, but the roots are very old. The prefix is the often use *com* for with or together, and the main root is from *jacere*, to throw. When we make a conjecture we throw together two ideas, the hypothesis and conclusion, before the relationship is fully tested.

**Constant ** The
mathematician uses the word constant to represent an
unchanging or fixed quantity. Often the quantity is
unknown, and is thus represented by a symbolic variable
such as the use of "b" in **y = mx+b** to represent
the value of the y-intercept, which may change from line
to line, but is "constant" in any one line. This use of
a variable/constant seems to be a contradiction to
students at times, and being at ease with this usage is
one of the ingredients of mathematical maturity.

The word may have first been used mathematically by
Leibniz in the Late 17th century. The roots come from
the Latin *com* for with and *stare* for
stand. The union literally means to stand together and
came to represent something dependable, faithful, and
unchanging.

**Continuous** The mathematical creation of continuous is credited to Euler in his __Introduction to Infinite Analysis__. For the beginning student, it is easiest to think of a continuous curve as one that can be drawn completely without picking up the pencil; that is, one with no jumps or breaks in the curve. The prefix is the common Latin *com* for with. The main root *tingere*, touch or pull, is from the same origin as tension and tangent. The word contingent, for something that is related to the situation is from the same root and literally means "touching on all sides."

**Corollary ** The
mathematical use of corollary is to describe a
proposition that can be shown to be true with little or
no effort from some theorem already proven. It is
sometimes also applied to the results of deductive and
inferential reasoning and more generally as the
consequence or result of something else.

The root
of the word is found in the Greek word for curved
*koronos*. Variations of the word produced
*corona* in Latin for crown, and the diminutive of
that, *corolla* for a small crown or garland. This
last use generalized to a small amount of money that was
passed as a gratuity; what we would call a tip today.

"So where does that get into math?" you ask. Well
it seems that when Euclid wrote he tended to ignore
these little insignificant extra results of a theorem.
Editors of his work would insert these results in the
margin marked with a little garland, as if to honor the
theorems which produced them. It was from this source
that we came to call these secondary results corollaries
of the original proof.

**Critical Points ** The
critical points of a function are usually determined to
be the points where the graph of the function takes on a
maximum value, a minimum value, or has a point of
inflection. Sometimes the word is generalized to
"important" points, and the zeros of the function may be
included. The first use of "critical point" with this
meaning was by Edwin Bidwell Wilson in his 1912
__Advanced Calculus__, in which he defined them as the
points where f'(x)=0.

The word comes from the Greek
*kritikos*, to discern or distinguish.

**Cube ** A cube is now the common name for a regular polyhedron with six faces. The cube is one of the five Platonic solids. The word was used by the ancient Greek geometers, but not always in the way we use it now. Euclid used cube in the same sense it is used today, but Heron (Hero) used the term for any right parallelepiped (box shape) and used hexahedron for the regular polyhedron of six faces.

The root of cube is in the Greek word *kubos* which was the name of dice used to play games of chance as well as by the oracles to divinate the future. The early root of the word comes from the Indo-European *keu* root which relates to bending, rolling or turning. This can be seen in other words from the same root such as incubate (to lie down on) and concubine (lie down with). The early Latin *cupa* which meant tub or vat gave rise to cupola, and later to cup as we use it now. The elbow, where the arm bends, was called *cubitum* and left its name in the unit of measure called a **cubit**

**Cycloid** A cycloid is the path of a point on the circumference of a circle as it rolls along a straight line (see figure). It was apparently named by Galileo in a letter from Galileo to Cavalieri written in 1640 where he says that he has studied the curve for more than 50 years. He may also have been the inspiration for Torricelli's interest in the cycloid, as they communicated closely in the last few years of Galileo's life. The word seems to be formed on the Greek word *kukloeides* for "circle like."

The parametric equations for a cycloid of a circle with radius r and with initial cusp on the origin is given by x(t)= r (t - sin(t)) and y(t)= r(1 - cos(t))

The cycloid gained the attention, and interest of Mersenne, perhaps through his communication with Galileo. Mersenne circulated questions about it among other mathematicians. One of the mathematicians whose attention he brought to the cycloid was the young Gilles de Roberval. Roberval used the term ** trochoid** from the Greek word for wheel. Roberval was able to prove several new theorems about the area and tangents of the curve, but did not publish his work.

His choice not to publish was probably because of the method of appointing positions, or chairs, at universities by competitive examination. Roberval's appointment to the "Chair of Ramus" was from just such a competitive appointment, which he had to recapture every three years. Fortunately, the incumbent was allowed to pose the questions for each new competition.

This lack of public sharing of information lead to duplication and repetition of efforts, and the bickering that subsequently followed over who discovered what. This was so common in the early 17th century, particularly over the question related to the cycloid, that the curve was nicknamed the ** Helen of Geometers**, and Montucla referred to it as "la pomme de discorde" in __Histoire des mathematiques, II__.

Another interesting story about the cycloid, and another name for it, are related to Blaise Pascal. On the evening of Nov 23, 1654 Pascal experienced what he called a "religious ecstasy", and he subsequently abandoned the study of Math and Science. Several years later, however, he was unable to sleep as the result of a toothache. Trying to distract himself, he began to study the properties of the cycloid, which he called by the French word * roulette*, from the diminutive of the Latin

The roulette wheel used in casinos is drawn from the same French root, and the addition of the word wheel is redundant.

The
area under a single arch of the cycloid is equal to three times the area
of the generating circle [Area=3P r^{2}]
and the length of the path is equal to the perimeter
of a square which would just contain this circle [perimeter=8r]. Galileo tried to find the area under the curve by cutting the shape out of metal and weighing it. His answer was a little over the true value. His student, Torricelli, was the first to get the correct area. His method is shown here along with a clever method Christopher Wren used to find the arc length.

If the point is moved to the interior of the circle, the path is called a **curtate** cyloid from the Latin *curtus*, for short. The root is also found in such words as curt (rudely brief) and curtail (to cut short). The path of a point on a radius extended beyond the circumference of the circle is called a prolate cycloid, almost unchanged from the Latin *prolatus*, stretched out.

The parametric equations of the curtate and prolate cycloid for a circle with radius r, with point at a distance h from the center, are given by x = r t - h sin(t), y = r - h cos(t)

The bridge over the Cam river at Trinity College in Cambridge, built by James Essex in 1764-1765, is sometimes suggested to have cycloidal arches. [for example, David Singmaster in his BSHM Gazetteer, "The bridge over the Cam in the grounds of Trinity has cycloidal arches"] A recent post by John Harper to the Historia Matematica newsgroup suggests this is not true. He writes, "Are they really cycloids, or half-ellipses? The Inventory of the Historical Monuments in the City of Cambridge (Royal Commission on the Historical Monuments of England, HMSO London 1959) says on p244 that they're semi-elliptical, and the photo in plate 38 suggests they really are. " Judge for yourself at this image link.

**Cylinder** A cylinder is a surface generated by the locus of lines parallel to a given line, the generatrix, and passing through a curve in a plane, the directrix. In the most common case the curve is a circle, although it could be any closed plane curve. The root of cylinder is from the Greek *kulindros* for roll or revolve. The term is ancient and appears in the *Conic Sections* of Apollonius.

**Decimal Fractions** Some math historians suggest that the decimal fraction can be traced back to the 14th century BC in China. However far back they began, the cultural tendency toward measurement system in the base ten led to an early use of decimal fractions in China. By the 1500's they had moved across the Arab nations into Renaissance Europe. During the middle ages most fractions were still expressed in the sexagesimal (base sixty) system, but there were exceptions. In his ^{10} in his *Opus palatinum*, published after his death. By the latter half of the 16th century, mathematicians like Viete were championing the use of decimal fractions in mathematics, but the common man did not use them at all. In 1579 in his *Canon Mathematicus* Viete wrote

Sexagesimals and sixties are to be used sparingly or never in mathematics, and thousandths and thousands, hundredths and hundreds, tenths and tens, and similar progressions, ascending and descending, are to be used frequently or exclusively.When he wrote decimal fractions Viete used a vincula below the fractional part, such as 356,704

The popularization of the decimal fractions was primarily due to Simon Stevin, a Dutch mathematician and engineer who also was the founder of the science of hydrostatics. In 1585 he published *De Thiende* (The Tenth) in Flemish. He also issued a French version entitled *La Disme*, and it became one of the most influential math books of all time popularizing computation among the masses. Stevin did not put denominators on his fractions, but marked each number with a circle above it, or behind it, that marked the power of ten in the divisor. For example, for the first few digits of pi, 3.141, he would write

except that he put circles around the exponent values in the top line. Here is an image from

It was the translation of Stevin's book into English that produced the first use of the word Decimal in English. Robert Norman translated the title as "Decimall Arithmetike".

**The decimal point** is most often created to Christoph Clavius for his use of them in a table of sines in 1593. Others attribute the creation to Magini. Both men were friends of Kepler. Clavius is more remembered for being the head of the commission appointed by Pope Greogory XIII to correct the Julian calendar. It was not until its use by Napier that it became commonly accepted. By 1616 Napier was using decimal fractions with the decimal point, and occasionally a comma. The point became the common usage in England, but much of Europe still maintains a comma so that the 3.14159 of English and American schoolchildren may look like 3,14159 on the Continent. [Actually the British are more likely to write the decimal point raised off the line so that the number would read 3^{.}14156.]

The International Standards Organization, which promotes the use of a single technical standard around the world, supports the use of a decimal comma and a space between periods, so that the American 3.14159 would look like 3,141 59 in an ISO tech manual, and where we in America would write 1,234.5678 the ISO recommends 1 234,567 8.

One additional remark about Stevin, he experimented with falling bodies by dropping two large lead spheres, one weighing ten times the other, onto a board from a height of thirty feet and determined that they fell at the same rate. This was published in 1586, well before Galileo supposedly did his similar experiment at Pisa. He is often credited with being the father of the study of Hyrdrostatics as well as dynamics. There is a statue and a square in his honor in Brugge, where he was born. One of the panels on the statue is from the frontpiece of his book, *Wiskonstighe Ghedachtenissen* (Mathematical Memoirs). The word Wiskond, which is now the Dutch word for mathematics was created by Stevin. A Wikipedia site adds, "Stevin thought the Dutch language to be excellent for scientific writing, and he translated a lot of the mathematical terms to Dutch. As a result, Dutch is the only Western European language that has a lot of mathematical terms that do not stem from Latin, including its Dutch name: wiskunde."
"

According to a post from Bruce Burdick to the Historia Matematica discussion list, the first book printed in the Americas using decimal fractions was Carlos de Siguenza y Gongora's __Libra Astronomica__ printed in Mexico in 1690.

You can see an English translation of *De Thiende* here and a photo of the cover sheet to the original French version here

**Radix Expansion**Decimal fractions are a special case of *radix expansions*. Just as integers can be expressed in bases other than base ten, fractions can be expressed in other bases as well. For example, 5/8 in base two is .101[2] where the [2] indicates the "radix" or base is two. For more about expressing fractions in other bases, see this blog.

**Domain and Range** The domain of a function is the set of all possible values of the independent variable(s). To the basic algebra student this is often reduced to "the set of all possible x-values." In general usage the word is applied to mean the territory or area of control. The origin of the word is close to this meaning, coming from the same root as dominate. The mathematical use seems to have begun around the end of the 19th century. Jeff Miller's excellent web site on the first use of some mathematical words states that, "DOMAIN was used in 1886 by Arthur Cayley in "On Linear Differential Equations" in the Quarterly Journal of Pure and Applied Mathematics: ".

The set of images, or function values, of the domain of a function is called the **range** of the function. In the function f(x,y)=2x+y^{2} the image of the domain point (3,2) would be the value 8. In most first year algebra classes (at least in the USA) students work with functions generally expressing y as a function of x; y=f(x); and the possible or allowed x values are usually considered the domain, the possible or allowed y-values are the range.

The term **range** is used in statistics to indicate the width, or difference between the maximum and minimum value in the set of all possible data values in a sample or population. Occasionally the term is misused to indicate the set of actual values, probably in confusion with the way it is used in functions. It appears that the statisticall term preceeded the function usage. Miller has, "RANGE (in statistics) is found in 1848 in H. Lloyd, 'On Certain Questions Connected with the Reduction of Magnetical and Meteorological Observations,' Proceedings of the Royal Irish Academy, 4, 180-183. " For a function he has, "RANGE (of a function). In 1865, The Differential Calculus by John Spare has: 'It is useful to become acquainted with the methods of fully examining the entire history of a function of one or more variables, in respect to the range of values which the function and its variable may sustain, and to their mutual dependence'"

**Equal and Equation** The mathematical use of equal means that two things are related in a transitive, symmetric and reflexive way in relation to some specified properties. The meaning is rooted in the Latin word *aequus* for level.

When we use the root in **equation**, we often do so in a subjective or conditional sense. Some equations may be true for any value, such as 2x = x + x and these are often referred to as an**identity**. Some are true only under certain conditions, such as y = 2/x which is true for certain values of x and y if neither x nor y is equal to zero; or 3x = 6 which is only true if x = 2.

According to Jeff Miller's web site on the first use of terms, for the first use in English he includes, "Equation appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: 'Many rules...of Algebra, with the equations therein vsed.' It also appears in the preface to the translation, by John Dee: 'That great Arithmeticall Arte of Aequation: commonly called...Algebra.' "

From a string of posts on the topic in the Historia Matematica discussion group I found, "Equatio appears in the ordinary high school sense of the word in Fibonacci's Liber Abbaci, Ch. 15, section 3 " [Barnabus Hughes]; "And Bhaskara's _Bijaganita_ in the twelfth century uses Skt. "samikarana", "making equal", to refer to an equation with quantities on both sides." [Kim Plofker]

When the root is united with the Latin root

The **=** symbol that is now universally accepted was first used by Robert Recorde in The Whetstone of Wit (1557). According to the St Andrews Math History site,"He justifies using two parallel line segments:
... bicause noe 2 thynges can be moare equalle ". They also point out that "
The symbol = was not immediately popular. The symbol || was used by some and ae ( or oe), from the word 'aequalis' meaning equal, was widely used into the 1700s."

Sometime in the last half of the 19th Century, a symbol with three horizontal lines, ≡, was introduced to seperate the idea of identity from simple equality. Cajori gives credit for this notation to Riemann. The earliest example I have found is from Simon Newcomb's Analytic Geometry text of 1895 shown below. Note there is no suggestion that this is new notation, so it probably did exist in somewhat common use prior to this publication.

The ≡ symbol is also used in geometry for identically equal, and also for other uses. See more at congruent.

Heron of Alexandria, sometimes called Hero, lived around the year 100 AD and is most often remembered for a formula for the area of a triangle. The formula gives a method of computing the area from the lengths of the three sides. If we call the sides a, b, and c; then the area is given by where the "s" stands for the semi-perimeter, . You can find a nice geometric proof of Heron's formula at this link to the Dr. Math site. The proof was done by Dr. Floor, who credits the method to Paul Yiu of Florida Atalantic University.

Documents from the Arabic writers indicate Archimedes may well have known this formula 300 years before Heron. In 1896 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 and one of the earliest forerunners of the thermometer. The image at right shows a picture of a reconstruction of Heron's steam engine. The image is from the
Smith College meuseum of Ancient Inventions where you can find more about Heron's, and many other's, interesting creations.

Heron's *Metrica* also contains one of the earliest examples of a method of finding square roots that is called the **divide and average** model. To find an approximate square root of a number, N, think of any number smaller than N, which we will call M. Then find a new approximation by letting E = (M + N/M)/2. Another approximation can be found by repeating the method with this new approximation. For example, beginning with N=20 and M= 2, we get E= (2 + 20/2) / 2 or E= (2+10)/2 = 6.

Repeating with M= 6 we get E= (6+ 20/6)/2 = ( 6 + 3 1/3 )/2 = 14/3 or 4 2/3. After only two iterations from a very bad starting guess the approximation is within .2 of the correct value.

Heron is also remembered for a problem he solved in *Catoprica*; Given two points, A and B, on the same side of a line, find a point X on the line so that the total distance AX+XB is a minimum. The solution may come quickly if you know that the translation of *Catoprica* is "About Mirrors". The solution given by Heron is to find the mirror reflection of point B in the line, B', and draw a straight line from A to B'. Where it intersects the line is the choice of point X.

**Occam's Razor** William of Ockham was an English Philosopher of the Early 14th Century. He is most remembered today for the quotation "*Entia non sunt multiplicanda praeter necessitatem *. The direct translation is close to "Entities ought not to be multiplied except from necessity." Occam's razor has become a scientific rule of thumb for deciding between two theories to explain a single phenomenon. Given two otherwise equal theories, the more simple one is the better.

The modern spelling is Ockham, and the remains of the estate is located off the M25 in London near Woking. All Saints Church, which dates to the 13th century, contains a modern stained-glass window of William of Occham. There is also a statue. Behind the church is a gate into the grounds of Ockham Park, but it is private land. It may be of interest to students of mathematics and computer science that Ada Lovelace husband,also named William, was the Baron of Ockham in the 19th century.

Ada's mother, Lady Byron, had intentionally schooled Ada in the Sciences and Mathematics to counteract the "poetic tendencies" she might have inherited fom her father. Ada knew Mary Somerville and Augustus de Morgan socially and received some math instruction from both. She is known to have assisted Charles Babbage in the design of an "analytical engine", an early mechanical computing device. She is often credited with writing the first computer program. She died of cancer in the womb in November of 1852, only 36 years of age, and was buried beside Lord Byron, the father she never knew, in the parish church of St. Mary Magdalene, Hucknall in the UK. In 1980, 165th years after Ada's birth, the US Defense Department announced a powerful new computer language. They named it Ada in honour of the Countess of Lovelace's important role in the history of computing.

** Octahedron** The octahedron is one of the five regular polyhedra called Platonic Solids. The Octahedron was one of the latter ones discovered, perhaps coming several hundreds of years after even the dodecahedron was known. The octahedron has eight equilateral triangles for faces, and can be constructed by placing the bases of two square based pyramids together. A link to a web site which allows you to print out nets of the octahedron, as well as the rest of the regular polyhedra, is at the Platonic Solids page.

**Reciprocal ** In mathematics the reciprocal of a number is the quotient of one divided by the number. In the more clear language of algebra, the reciprocal of n is 1/n. The origin of the word comes from the Latin *reciprocus* for returning, or alternating. Reciprocal occurs in the Encyclopedia Britannica as early as 1797.

**Solve & Solution ** The middle root of solve and solution finds its way to us form the Greek *luein* which meant to loosen or release. The Latin prefix *seu* in front came to represent actions like pulling apart or untie, and gave us our word for solve. A solution was something then, that allowed us to pull apart the problem. The same roots appear in absolute, dissolve, and soluble.

**Sphere ** The word sphere is from the Greek *sphaira* for a globe or ball shape. The word is little changed in use or application from its earliest usage.

Two types of sphere-like, or spheroid, shapes that are common in mathematics are the **oblate** and **prolate** spheroids. Oblate is applied to shapes like the Earth in which the distance between the poles is less than the diameter at the equator. Prolate is applied to shapes in which the distance between the poles would be greater. Both figures are examples of what are more formally called **ellipsoids**. The prolate spheroid is formed by rotating an ellipse around its major transverse axis. The oblate spheroid is formed by rotating about the minor axis.

The word oblate is also used as the name for a lay person dedicated to a religious life, which indicates its original relation to the word offer. The word is built on the Latin prefix *ob* for to or toward, joined with *latus*, to carry, and literally means," to carry forward or bring to". The origin of prolate is very similar, as the prefix *pro* means forward, giving it the literal meaning of "carry forward" also. Prolate was used to indicate something that was strectched from its original shape and this explains its use for the "stretched" sphere.

The formula for the volume of a sphere is v=4/3 pi r^3. For an ellipsoid it is 4/3 pi a r^2 where a is the ratio of the "polar" diameter to the "equitorial" diameter. Additional formulas can be found at the Dr. Math Formulas page.

**Subfactorials** The name subfactorial was
created by W A Whitworth around 1877. The symbol for the subractorial is !n,
a simple reversal of the use of the exclamation for n-factorial, although this symbol is relativly newer than the word. Whitworth himself used a symbol something like __|| n __ in imitation of the symbol for factorial introduced by Jarrett that was then common. Cajori's classic on the symbols of mathematics, (published in 1928) gave no mention of the use of the !n notation, but does credit G Crystal with the use of an inverted exclamation mark **after** the n. Perhaps the age of the typewriter ushered in the move to placing the exclamation mark at the front.

Here is Whitworth's comments on what may have been the first use of the symbol in print:

Whitworth was wrong about the symbol preceeding the name for the term. The word factorial was the creation of Louis François Antoine Arbogast (1759-1803). The "!n" symbol now commonly used for factorial was created by Christian Kramp in 1808, and the Jarrett symbol that Whitworth seems to think came first was atually not used until 1827. Chrystal was well aware of Whitworth's symbol choice, as he indicates in the remarks on the page he first used the inverted symbol in his "Algebra, an Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges" which is still being republished to this day:

I had thought that Chrystal's inverted exclamation was a one-off symbol not picked up by anyone else until I found this "Note on Derangements", by M. T. L. Bizley © 1967 The Mathematical Association."

The earliest use of the more common current notation, "!n" that I can find was sent to me by Dave Renfro. It appears in the Problems and Questions section of the MAA , by C. W. Trigg, #342 page 285 © 1958

Others eschew all of these for a function notation approach. I have seen P(n) used frequently and many uses of D(n) for Derangements, I assume. The D(n) notation is often used with partial derangements in which r of the n objects are in their proper places, and written then as D(n,r); thus D(5,2) would give the number of ways of haing five objects with three mis-placed and two others in their correct position.

The formula for subfactorial n is given by !n = . A more simple way to compute the value is to round the answer of n!/e. It can also be found by a recursive rule... using F(0) = 1 and for each new value F(n)= n F(n-1) + (-1)^{n}. So F(1) = 1(1) - 1, and F(2) = 2(0)+1 and F(3)= 3(1)-1.

The sequence was first produced by Nicolaus Bernoulli in trying to answer the following problem, which was posed by P R de Montmort (it may be that Montmort already had a solution). If N letters and N Envelopes to contain them are prepared, in how many ways may ALL the letters be placed in the wrong envelopes. The solution is !n, and the first few answers are 0, 1, 2, 9, 44, 265, 1854... .

Euler used the same method to develop the probability of winning in the game of *rencontre*, now called "coincidences" in his paper "Calcul de la Probabilite dans le jeu de Rencontre", published around 1751.
An English translation of the paper by Richard J. Pulskamp is available at this site and the original document can be seen here.

The method of placing an ordered set in such a way that no element falls in the correct order is called a **derangement**. Here is a link to a nice geometric approach to illustrate the first few derangements, at the Mathforum. You can find a way to simulate the problem using a Ti Graphing calculator here.