Math Words, pg 15

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The word apside, and its related terms apoapsis and periapsis are most commonly applied to conic sections, particularly the ellipse. The term apside can actually be applied to any motion in reference to a fixed reference point, and refers to a point on the orbit where, for at least an instant, the point is neither approaching nor receding from the reference point.

In an ellipse the distance from a focus to the nearest vertex is called apoapsis. In the case of the elliptic orbit of the Earth around the sun, the point where the Earth is closest to the sun is called aphelion (from helios) and in the orbit of the moon or other satellite about the Earth the closest point is called apogee (from Geo for Earth).

The distance from the focus of the ellipse to the most distant vertex is called the periapsis, and the special case for orbits about the sun is called perihelion, and for orbits around the Earth the term is perigee.

The root apsis is sometimes written apse in architecture where it is used to represent a (usually) domed projection of a building. The original Greek root is from the word haptein for "to fasten". The Latin translation became apsis which related to both arches, and the orbit or path of motion of an object. The root also appears in the word synapse. The prefixes apo is from the Greek for "away" and is similar to the Latin prefix ab and the Germanic ebbe. The Greek root is still found in words like apothecary, apology, and apocalypse. The peri prefix is also Greek for close or about, and remains in words like perimeter and periscope.

Bernoulli's Theorem, the Law of Large Numbers

Jakob Bernoulli's Law of Large numbers is so simple that, in his words, "even the stupidest man knows by some instinct of nature" that it is true; yet a rigorus proof took him over 20 years to complete. A cornerstone of modern probability and statistics, the theorem simply says that the proportion of successes after a large number of trials will approach the true probability of the event on a single trial. (It actually says something like that only with limit notation and epsilons, but we will let you pursue that burden on your own.). Bernoulli referred to it as his "golden theorem" because of its combined simplicity, usefullness, and the difficulty of its proof.

The name "Law of Large Numbers" was not created by Bernoulli. Although the problem was published in Ars Conjectandi(The Art of Conjecturing) in 1713, we find in Jeff Miller's web site on the first use of some math words :

LAW OF LARGE NUMBERS. La loi de grands nombres appears in 1835 in Siméon-Denis Poisson (1781-1840), "Recherches sur la Probabilité des Jugements, Principalement en Matiére Criminelle," Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 1, 473-494 (James, 1998).
. Teachers (or students) who are interested in a picture of the titlepage to Ars Conjectandi can find one here.

The paradox of the Imp in the Bottle

Would you pay $100.00 for a magic bottle that would grant your every wish except prolonged life? All the wealth you could dream of, fame, and beauty could all be yours. Would you pay if you knew that you must sell the bottle for less than you paid or you would be damned for all eternity? This is a paradox from the story The Bottle Imp by Robert Louis Stevenson.

What paradox? Well consider that no one would buy the bottle for one cent since they could not sell it for less. Certainly no one would pay two cents because they would have to sell it for one cent and we already see that no one would buy it for one cent, so they would not buy it for two cents. By the same argument the price is raised to any king's ransom, and yet, at some high price, wouldn't you buy it?

This type of paradox where the result of many little logical truths are contradictory to their chained consequence is often called a heap paradox or a sorities paradox from the Greek word for a heap or pile. The paradox is similar to the following idea: one grain of rice is not a pile, two grains of rice is not a pile, but eventually if we keep adding a single grian, we get a pile, yet it seems impossible to define WHEN it became a pile.

A paradox is something that seems to go against logic or thought. The word paradox is from the Greek paradoxon joining the roots para for beyond with doxa for belief or opinion. The latter root remain in words like orthodox, decent, and decorous

According to a Chronology of Recreational Mathematics by David Singmaster, the first examples of Paradoxes appeared around 330 BC in the writing of Eubulides. Others cite earlier examples include the liars paradox created by Epimenides in the 6th century BC.

Epimenides the Cretan. In verses 12-13 of Chapter 1 of his Epistle to Titus, St. Paul writes: `One of them, a prophet of their own, said “The Cretans are always liars, evil beasts, slothful bellies.” This testimony is true’. This allegation about Cretans is supposed to have been made by Epimenides, a native of Cnossus, which is the capital of Crete. So here we have a Cretan saying that all Cretans are liars. On this paradox and its connection with the standard Liar, see Alan Ross Anderson’s beautiful introduction to R.L. Martin (ed.), The Paradox of the Liar.

The Liar (and its strengthened variants). The standard Liar (attributed to Eubulides) is `This statement is false’. The strengthened version is `This statement is either false or neither true nor false’, or, more simply `This statement is not true’.

In spite of their long history, the use of the term paradox for such logical contradictions seems to have been created only in the 20th century by Bertrand Russel. [1903 B. RUSSELL Princ. Math. xliii. 358 This paradox, which, as I shall show, is strictly correlative to the Achilles, may be called for convenience the Tristram Shandy.]

I have also written a word document about several types of paradoxes.

Ceres, Goddess of Agriculture

According to legend, Ceres came into being after a terrible drought in Italy around 500 BC. The Roman priests demanded tribute and sacrifice to this new goddess. Sure enough, the drought ended, and so each year the first harvested grains and fruits were brought to the temple in her honor. These tributes were called cerealus, of ceres, and begat our modern day word for cereal. Because of the relationship between agriculture and growing, her name was soon related to other growth ideas. Creare was the Latin origin of our word "create", to bring forth or cause to grow.

Closly related to Creare and Ceres was the word crescere which meant to increase; and this root appears in several math related words. Through the French and into English it became the word increase. Decrease, of course, used the "de" to show a reversal of growth. Increment means to grow in steps, and is from the same root. The waxing and waning of the moon produced a visual curved surface which also took its name from the Latin word for grow; and today we call the shape, and the lighted portion of the moon, a crescent. The same shape is often called a lune, a word derived from the name of another Roman Goddess, Luna, the Goddess of the Moon.

Clue or Clew

One evening as my wife and I sat reading in the Library of our home she turned to me and said, "Hey, Wordman," (she never calls me wordman unless she has a word she thinks she can stump me with)"Do you know where the word 'clue' comes from?" I pondered for a moment, and had to admit I didn't have a clue.

The origin, as she gleefully told me, goes back to the ancient Greek myths about a Monster who lived on the Island of Crete during the reign of king Minos. Each year fourteen people were sacrificed to the monstor, a Minotaur. It seems that Theseus volunteered to be part of the group andwent into the Labyrinth hoping to kill the Minotaur. The Minotaur was a half man-half bull, son of Minos’s wife Pasiphae and a beautiful bull, sent to Minos by Poseidon, to whom Minos was to sacrifice it. He didn’t do the sacrifice, so Poseidon made Pasiphae fall in love with it. The Minotaur was the result. Minos confined it in a Labyrinth built by the architect Daedalus. As he makes his way into the labyrinth, Theseus leaves a trail to find his way back out by unwinding a ball of string. In Old English, a ball of thread or twine was called a clewe, and so a method of guiding us toward the solution to a mystery or puzzle became known as a clew, or clue. Both spellings seem acceptable according to my dictionary, but I cannot recall ever seeing any other than clue in American English.

Estimate, Estimator

The most common use of the word estimate, both as a verb and a noun, are very similar to their common use in everyday English, and their origin in the Latin word aestimare which meant to value, or place a value on something. When the roofer estimates the cost of repairs, he is placing a value on the cost of his labor and materials. In much the same way we use the word informally in mathematics to talk about giving an approximate value for the result of some mathematical operation. Within the last century, however, the words estimate has been used in statistics to describe values formally calculated as predictions of the outcome of a random event based on a sample from the population of outcomes. The statistical use seems to have be created by R A Fisher around 1922.

The word esteem is from the same root and also relates to something which is valued, usually in a positive sense.

Fields Medal
      The Fields Medal is named for John Charles Fields, a Canadian mathematician who left instructions in his will to create a medal to honor "outstanding achievements in mathematics".  The idea of such a mathematical prize was adopted at the 1924 meeting of the  International Congress of Mathematicians.  Fields was the secretary of the convention in 1924.   The first medals were awarded in 1936. The true name of the awards is "International medals for outstanding discoveries in mathematics." The prize includes a cash award of $15,000 Canadian.
    You can see a picture of the medal, and a list of the recipients of the medal at these links.   That last link also includes the words of the actual letter by Fields to fund the award..  More extensive biographies of the winners are also available.   The inscription on the medal, transire sum pectus mundoque potiri, means "rise above your spirit and grasp (master) the world".  A picture of Archimedes is on the face of the medal, and the winner's name is imprinted on the edge of the medal.

  The Fields Medal is often referred to as the "Nobel Prize for Math."  The Nobel Prize does not recongnize contributions in mathematics.  There have been lots of stories about why including one that Nobel was angry because of an affair his wife had with a mathematician.  This one must be false as Nobel was a bachelor all his life. It seems more likely he was thiniking of physical discoveries and inventions. The mathematician most often named as suspect of the non-affair with the non-wife of Nobel, Mittag-Leffler, campaigned for many years to get a Nobel Prize for Poincare.

According to Steven Kranz, Mittag-Leffler was so upset about the lack of a prize in mathematics that he created one, modestly called the Mitag-Leffler Prize. With his wealth, or that of his wife, he established a lavish prize that included a medal twice as large as the Nobel Prize and a fancy dinner cooked by a well known chef. Alas the prize only lasted a couple of years before the endowment was depeleted since it had been invested primarily in German World War I bonds and Italian rail stock. In 1980 a donation by Holger Crafoord, and his wife Anna-Greta created a new prize called the Crafoord Prize. Holger Crafoord is the inventor of the artificial kidney. The fund is administered by the Royal Swedish Academy of Sciences, the same academy that gives the Nobel prize with the aim of rewarding and promoting research in scientific areas that fall outside the awards by the Nobel Prize, such as mathematics, but including geoscience, bioscience, and astronomy. In 1982, the first two prizes were awarded to mathematicians Louis Nirenberg and V.I. Arnold. Krantz says that when Nirenberg was apporached by a New York Times reporter and asked how he felt about being the first recipient of the award replied,"Is this some kind of joke?" A mathematician was not rewarded again by the prize until 1988. A list of the recipients can be found at his Wikipedia page



The Romans had hardly hightailed it out of Britain in the fifth century, when the Saxons started moving in. They brought with them many of the units of measure that would be the foundation of the later English system of measure, including the rod, the acre and the furlong. The word furlong is a contraction of "furrow long" and related to the length of a furrow. In Early England the common Green was usually set out as a square of ten acres. A furrow plowed on this green would then have a length equal to one furlong. [so a square furlong, is ten acres] By the time of the early Tudor kings the furlong was set at 220 yards. Later Elizabeth would adjust the lenth of the Roman Mile from 5000 feet to 5280 feet so that a mile would be 8 furlongs. That remains its length today. It can also be expressed as 40 rods, or 10 chains. Today it seems only horse races still use furlongs as a measure.

This length happened to coincide, to some degree, with the ancient Greek unit of a Stadia or Stadion which was the standard length of the Greek athletic fields. The stadion at Olympus, where the original Olympic Games were held, measures 630.8 feet or 192.3 meters. Stadium was the Roman adaptation of the word, and they set the distance equal to 1/8 of a Roman mile or about 625 feet.


A macron is a straight line or bar used above a vowel in the phonetic guide to a word to indicate a long sound. My dictionary uses to show the pronounciation of the word maize. Macron seems perfectly named for this usage as it is the Greek word for long.

In mathematics the macron is often used to name a bar placed over a single character, such as , which is read as "X-bar", to indicate the mean of a set of data in statistics. Other uses include the complement of a set, and the conjugate of a complex number If Z = a + bi then . When a bar is used over a number, group of numbers, or more than one character, it is usually called a vinculum.

Natural Numbers

It seems at one time in mathematical history the Natural Numbers were almost universally understood to describe the set of Positive integers; but of late, the issue has become more clouded. The same set was also frequently referred to as the Counting Numbers and, usually in more advanced papers, some times referred to as the cardinal numbers.

In 1484, Nicolas Chuquet referred to the set as the "Natural Progression". Jeff Miller points out in web page on the first Use of some mathematical words that the Encycolpedia Britannica defined the Natual Numbers as not including zero in the 1771 edition. By the end of the Nineteenth Century there had developed an alternative view that included zero in the set of natural numbers. Others chose to refer to the Naturals plus zero as the "non-negative integers"; and even developed various symbols such as N_0 and N0 for this different set. In 1889 Peano (Arithmetices principia nova methodo exposita)had excluded zero from the set in his publications, but in 1899 he included it. Bertrand Russell included zero in his Principles of Mathematics in 1903. By 1960 the Swedish Schools had formally legislated that textbooks would define the Natural numbers to include zero, but in countries like the US, both schools existed side by side, and one writer recently confessed to a newsgroup of having unintentionally defined it both ways in seperate chapters of the same book. In the same discussion some other mathematical authors suggeted that they now avoided the term choosing to use "Positive integers" in one case and "Non-negative Integers" in the other. Many modern textbooks in America now use the term "Whole Numbers" to describe the set of non-negative integers and Natural numbers for the set without zero.

The word Natural comes from Latin word natura from the root natus for birth. The word Counting is obviously from count, which is drawn from the French compter from the Latin computare which is the root of our word compute. The noble rank of Count is not from the same root. Whole seems to be derived from the Old Germanic term haile which is the ancient root of other words like Holy, Health, and the name Heloise (or Eloise), all of which refer to something being complete.

  Normal Probability Distribution

The normal curve was developed mathematically in 1733 by Abraham De Moivre as an approximation to the binomial distribution, however his paper was not discovered until 1924 when it came to the attention of Karl Pearson. Laplace had used the normal curve in 1783 to describe the distribution of errors, and the curve is sometimes called the "normal curve of error". Gauss used the normal curve to analyze astronomical data in 1809 and wrote much about the curve so it also called the Gaussian distribution. The ten mark bill shows Gauss and an illustration of the normal curve The term bell-shaped curve is often used to mean the normal curve, but many symmetric distributions, such as the t-distributions, are also bell-shaped but are not normal.

The first person to apply the normal distribution to social data was Adolph Quetelet (1796-1874). He collected data on the chest measurements of Scottish soldiers, and the heights of French soldiers, and found that they were normally distributed. His conclusion was that the mean was nature's ideal, and data on either side of the mean were a deviation from nature's ideal.

Robert Adrain was born in Ireland and emmigrated after the Irish Rebellion of 1798. Adrain and N. Bowditch were probably the two premier mathematicians in America before 1876. An 1804 article by Adrain was the first attempt to introduce Diophantine analysis into America. In 1809, while analyzing errors in surveying and dead reckoning at sea, Adrain discovered the Gauss Distribution in Probability Theory, demonstrating that errors are distributed according to a bell-shaped curve f(x) = C exp(-hx^2). Adrain was unaware that the French mathematician Adrien Legendre had asserted this without proof in 1805, and Karl Friedrich Gauss was to give a more rigorous proof later on, but Adrain's was the first proof.

I recenty read that the term "bell curve" was first applied by Jouffret in 1872 who used the term "bell Surface". Jeff Miller's web site on the first use of math words has, "Bell-shaped curve is found in 1876 in Catalogue of the Special Loan Collection of Scientific Apparatus at the South Kensington Museum by Francis Galton ". Wikipedia gives the attribution to Jouffret, and also adds that, "The name "normal distribution" was coined independently by Charles S. Peirce, Francis Galton and Wilhelm Lexis around 1875." In a recent note to the Historia Matematica group, William R Everdell added that the curve was sometimes named after Galton. Here is a clip from his post

Francis Galton did a lot of original work on the statistical normal curve after 1869 and demonstrated a device he called a "Quincunx" to the Royal Society in 1874. It's a pachinko-type machine that distributes shot like pinballs through about 20 pins into a bell curve. Indeed, the "bell" curve was called a "Galtonkurve" by Hugo De Vries who later used it in rediscovering Mendel's segregation law. cf. De Vries, 'Ueber halbe Galtonkurven als Zeichen diskontinuierlicher Variation', Berichte der Deutschen botanischen Gesellschaft. 12(1894), p197-207. By 'Galton' curve, De Vries means 'Gauss' or normal curve. Professor Jules MacLeod at Ghent U., may have introduced De Vries to it.

The formula for the Normal curve depends only on the Mean, , and the standard deviation, , of the population. The height of the probability distribution is given by .

The general shape is shown in the image below of a Standard Normal curve with a mean of 0 and a standard deviation of one. The percentages show the relative frequency of the population in each of the sections of the standard normal curve.

Normal NumberThe term "normal" has a very different meaning to the number theorist than it has for the statistician. In number theory we talk about an irrational number being "normal" if its digits occur with equal frequency. For example, a number expressed in decimal notation in which the number of digits that are 0,1, 2, etc all occur with the equal frequency. In this case we would say the number was base ten normal. If the expansion in binomial form had equal numbers of zero and one digits, it would be normal in base two. It seems that the term is only used for irrational numbers, but numbers like 123456789/9999999999 would seem to meet the requirements for normal base ten. It seems that numbers like Pi, and the Pythagorean constant (square root of two) are normal, but there is no formal proof. To the statistician, the digits are "uniformly distributed." For a more complete and formal definition, see the mathworld web page. In a post to a history discussion on the MathForum, William c. Waterhouse of Penn State wrote, "Emile Borel (according to Hardy and Wright) introduced the term "normal number" in 1909, proving that the non-normal numbers were a set of Lebesgue measure zero. So we can speak of normal numbers, or the normality of a number; but this is not the same as a "normal distribution" of digits (whatever that might mean). ...There are also many other uses of "normal" in mathematics: normal vectors to surfaces, normal subgroups, normal complex matrices, and so on. Clearly confusion is bound to be normal."

Reflex Angle

A reflex angle is an angle that measures more than 180o and less than 360o. The word seems to have been created to replace the earlier term re-entrant angle which seems to have disappeared around the end of the 19th century just as reflex was becoming common.

Prior to the late 19th century the term for common angles (smaller than 180o) was salient angle. The words make sense in terms of angles on a closed polygon. Salient means "jutting out or projecting" and the vertex of a salient angle is where the polygon projects out the most. The word is from the Latin salire for leap or jump. Reflex is from the Latin past participle of reflect, and means to bend back.

Here is the history of the first use of the word from Jeff Miller's Web site on the first use of math words
REFLEX ANGLE. An earlier term was re-entering or re-entrant angle.
Re-entering angle appears in Phillips in 1696: "Re-entering Angle, is that which re-enters into the body of the place" (OED2).
Re-entrant angle appears in 1781 in Travels Through Spain by Sir John T. Dillon: "He could find nothing which seemed to confirm the opinion relating to the salient and reentrant angles" (OED2).
The 1857 Mathematical Dictionary and Cyclopedia of Mathematical Science has re-entering angle: "RE-ENTERING ANGLE of a polygon, is an interior angle greater than two right angles."
Reflex angle is defined in 1889 in the Century Dictionary [Mark Dunn]. It also appears in the 1913 edition of Plane and Solid Geometry by George A. Wentworth, and may occur in the earliest edition of 1888, which has not been consulted.

I recently (2010) received a note from Kathy (no last name given) that pointed out that Wentworth's book is now available on line, and does indeed have the term "re-entrant" on page 68. Here is a clip from the page.

Spherical Triangle

"Straight" segments on a Sphere are created by arcs of great circles, which cut the sphere into two equal hemispheres. Since the length of an arc of a great cirlce depends only on the radius of the sphere and the central angle cutting the arc, it is common to use a sphere of radius one, and measure the length of a spherical segment with the angle measure of its central angle. The figure at the right shows a spherical right triangle CDE.

One of the differences between spherical triangles and the plane triangles studied in introductory geometry class is that a spherical triangle always has more than 180 degrees. As the triangle gets smaller, the sum of the angles gets closer to the lower limit of 180o or p radians. The sum of the angles in a spherical triangle will be less than 540o or p radians.

The use of a unit circle and the relation between the size and the sum of the angles allows mathematicians to measure the surphace area of a spherical triangle by using the measure of the angles in radians. Girard's Spherical Excess Thm Girard's Thm states that the area of a triangle is equal to the sum of the angles reduced by p. For the triangle in the image we would write K(CDE)= C+D+E - p. To find the surface area of a spherical triangle on a non-unit sphere, just multiply the spherical excess by the square of the radius.

Girard's Theorem is named for Albert Girard, a French mathematician who is also remembered for publishing the first math book using sin, cos, and tan for the trignometric ratios.


Surface is a frequenly used math word with a wide range of technical meanings. Perhaps the most common meaning is the idea of a two dimensional subset of the Euclidean three space. This could be a complete plane, but more often is the "shell" of a sphere or the combined faces of a polygon. The roots are from the French prefix sur for above or over and the Latin facie for form, shape, or more recently face.

For those interested in exploring some other mathematical meanings of surface, or a more technical explanation of this one, I suggest The world of Mathematics

Syracuse Algorithm

The syracuse algorithm is a iteration problem that deals with the following algorithm..
If a number n is odd, then f(n)= 3n+1
if n is even, then f(n) = 1/2 (n)
Each answer then becomes the new value to input into the function. The problem, or should I say problems, resolve around what happens to the sequence of outcomes when we keep putting the answer back into the function. For example if we begin with 15 we get the following sequence, also called the orbit of the number:
15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1...
One of the unproven conjectures is that for any number n, the sequence will always end in the number 1. This has been shown to be true for all numbers up to just beyond 1016. A second interesting question is how long it takes for a number to return to the value of 1. For the example above, the number 15 took 17 steps to get back to the unit value. Questions such as which three (or other n) digit number has the longest orbit. There are many vairations of the problem, but if you are interested in a good introduction, check this link from Simon Fraser University"

The Syracuse Algorithm is often also called Collatz's Problem, Hasse's problem, Thwaite's problem, and Ulam's problem after people who have worked and written on the problem. It is unclear where the problem originated, as it seems to have had a long history of being passed by word of mouth before it was ever written down. It is often attributed to Lothar Collatz from the University of Hamburg who wrote about the problem as early as 1932. The name "Syracuse Problem" was applied by after H. Hasse, an associate of Collatz, visited and discussed the problem at Syracuse University in the 1950's. During the 1960's Stan Ulam circulated the problem at Los Alamos laboratory. One famous quote about the problem is from Paul Erdos who stated, "mathematics is not yet ready for such problems".


Trace has a wide range of mathematical meanings ranging from the technical to the commonplace. All of them seem to stem from the common idea of a path or a mark left by a passing object. The World of Mathematics website has multiple pages of definitions of the mathematical meanings of list and htis link will take you to the one related to the diagonal of a matrix, with links to many more.

The origin of trace is from the Latin root trahere, to pull or draw. The same word is the root of tractor, trailer, and trait. Another math word from the same root is Tractrix. Rural readers may know that the lines from the harness of a horse to the object to be pulled are still called traces.