Math Words, pg 4

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**Add / Addition ** The arithmetic word add is from the Latin root *addere*, to give or to do. The *dere* part of the root is the same root that gives us Data/Datum and the name for **dice**. Donation and condone also share the same root. The first recorded use of the word in English is from "The Crafte of Nombrynge" according to Jeff Miller's web site . The document was one of the first English language documents dealing with mathematics. The symbol "+" for addition was used for surplus and deficit before it was used as an operation symbol. It has been shown that the symbol was used on barrels to mark them as over or under weight prior to the earliest known use in manuscripts. Some even suggest it dates back to the Early Greeks, but others suggest it is a short form of the script word *et*, Latin for and. The symbol seems to have been first used for an operation in the late 1400's and only came into common usage after it was used by Robert Recorde in his 1557 book,** The Whetstone of Witte**.

**Altitude** is the geometric name for the perpendicular distance from one base of a geometric figure to the opposite vertex, or parallel face. The word altitude is also sometimes used for the line segment representing this distance in a figure. The word comes from the Latin *altus* for high with relations to ideas about growth and nourishment. English words like elder, exalt, and adult are derived from the same root. In astronomy the word altitude refers to the angle of a celestial object above the horizon.

**Calculate / Calculus** The origin of both these words is in the Greek word *kalyx*, for pebble or small stone. The manipulations of small stones on counting boards to do arithmetic operations led to the present mathematical meanings of calculate and calculus. [You can see some beautiful old counting tables at this page from Convergence ]
The pebble root is still present in the medical use of the word calculus, a name for an accretion of mineral salts in the body into a small "stone" such as kidney stones. The name for the element calcium comes from the same root. The prefix *calci* usually relates to calcium or limestone in some way. The furnace used for annealing in the glass making process is called a calcar, and the same word is used for a spur or projection on the heel. Interestingly, the material created in the calcar (glass furnace) is called frit, and is the root of the food product called a fritter; both words coming from the Latin word for fry.

Students who struggled through a first semester calculus course may appreciate the following tale from Steven Krantz's __Mathematical Apocrypha__

"An American mathematician of some note was returning from a trip abroad and had to go through Customs. The US Customs Officer asked him what he had been doing during his one-week sojourn. The reply was that he had been at a mathematics conference. The Customs Officer then took this man aside and detained him for some time with a great many tedious questions about exactly where he head been and what he had been doing during his travels. The mathematician kept glancing nervously at his watch, worried that he would miss his connecting flight. The Customs Official finally got to a point of asking our friend what he had had for dinner each day. Finally the mathematician threw up his hands and exclaimed, 'Why are you doing this to me?' The Customs Officer smiled and said, 'Ah, Now you know how I felt when I took calculus.'"

**Compute** joins the com (with) and the Latin root *putare*. This root is often cited as related to thinking or reckoning, but its meaning comes from an early word for cut or slice. The same root appears in amputate. This goes back to the earliest use of numbers in commerce and the idea of comparing values to a counting or tally stick. The sticks were notched to record values for future reference. Computing, then, was comparing the quantity of items to the marks on the tally stick. This helps to understand the word dispute. *Dis* is from the root *duis* for two is used to represent the idea of two things separated. If one is disgraced, they are separated from grace. Dispute literally means "a separate or different count". A difference in the records could easily lead to what we now think of as a dispute.

**Correlation** The long road to the long word correlation, begins with the ancient Sanskrit word for scales, *tula*. The action of a scale to lift or support the weight led to the Greek word *telamon* for one who supports or bears something. This flows into the Latin word *latus* for to carry. The *re* prefix for back gives us relate, literally to carry back. Finally the *co* is added to indicate something the same or shared. Two things are correlated if there is some common thing back to which we can carry them both.

The first use of the term in the true statistical sense is attributed to Francis Galton. In __The History of Statistics__ by Stephen Stigler, he writes, "In December 1888,... Galton read a short paper to the Royal Society, 'Co-relations and their measurement, chiefly from anthropometric data'." In it was the first use of what he called his "index of co-relation." Stigler also explains, "The initial spelling of the term *co-relation* seems to have been a conscious attempt on Galton's part to distinguish his term from the word *correlation*, which was already in common use. In particular, the physicist W. R. Grove had published *The Correlation of Physical Forces* in 1846, .... Grove wrote that two correlated ideas would be 'inseperable, even in mental conception," but he emphasized .... 'one .... cannot take place without involving the other'."

In another book describing the same idea Galton has written in the margin, "nice wd." In spite of his efforts to create a different word, the statistical idea seem began to be spelled *correlation*.

Here is more about the term Correlation Coefficient, and the symbol r.

**Difference** The common *dis* (away) prefix is masked here, as it sometimes is, by a double consonant. The second root, *ferre* is from the Latin for to carry. The difference between two numbers is the amount that one has been "carried away" from the other. The same root is present in fertile, but not in ferry.

**Distance** Distance is the union of *dis* ,to separate or move away, and the *sta*, stand, that shows up in statistics.

**Gross** The common meanings of gross relate to its origins in the Latin *grossus* which meant thick or large. The present mathematical use of gross for 144 or a dozen dozens is also drawn from this meaning. The Germans seem 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. Originally a grocer sold wholesale, in large quantities, and thus the name. Something so big that it was visible to anyone gives us such uses as gross injustice, and the medical term gross lesion for one that is visible to the naked eye.

**Height** The word height comes into English through the German and is related to words like hop *hupp* and heap. Even early in the 20th century it was frequently pronounced and written heighth or highth (compare to length and width in present use) but the "th" ending seems to have disappeared from common usage. The height of an object represents the perpendicular distance from one base to the opposite vertex or parallel face.

**Line** The roots of the word line are hidden in the genus of the flax plant, *lenum* and the fibers of the plant that were spun together to make a thread called linen. The idea of stretching the thread to mark a straight segment between two points leads easily to the name of the imaginary one dimensional object passing through the two points.

The Greek word for line was *gramma* which is related to our modern words for grammer and graph. Heath's translation of Euclid's definition is, "A line is a breadthless length". Proclus describes the line as the path of a point when moved. His definition of line obviously included more than just the straight line. Heron classifies non-straight lines into three categories; circular, spiral, and general curves (ie, not circles or spirals).

**Magnitude** The Roman goddess *Maia* (she who is great) from whom we get the name for the month of May, is also the root of magnitude, and many other words that relate to greatness or size. The greatest value in a set, the** Maximum **is drawn from the same root. The root actually runs back to origins in Sanskrit and has relations in words like omega, maharajah, and Almagest, which was the title of Ptolemy's great work (about 150 AD) which chronicled the knowledge of the time about geography and astronomy. Current English words related to the same root include magnificent, magnanimous, maestro, mayor, magistrate, matador, and master

I found an interesting note for all those calculus and analysis students who have laboured over problems of maxima and minima in Heinrich Dorrie's __100 Great Problems of Elementary Mathematics__ (which I highly recommend).

At what point of the Earth's surface does a vertically suspended rod appear longest? (I.e. at what point is the visual angle at a maximum?). This problem was posed in 1471 by the mathematician Johannes Muller, called Regiomontanus.... The problem, which in itself is not difficult, nevertheless deserves special attention as theThe problem appears still in textbooks related to the viewing angle of a picture, or the best position to attempt a field goal. I've even co-authored a version of it myself. Eli Maor points out that there is no evidence to show that Regiomontonous knew the solution.first extreme problemencountered in the history of mathematics since the days of antiquity.

** Mean / Median** If you sometimes get mean and median confused it is no wonder. Both words come from the same origin, and originally meant the same thing. Both words, as well as the more common word middle, come from the Indo-European root *medhyo*. When most students think of the mean they think of it as the same as the average, but this is only one of the many means of mathematics, called the arithmetic mean. Here are the definitions of some of the most common:**ARITHMETIC MEAN **--- The sum of the values divided by the number of values added, also called the arithmetic average. The average of 3, 8 and 10 is (3+8+10)/3 = 7. The arithmetic mean of coordinates of a point give the coordinates of the midpoint. In statistics the arithmetic mean of a population is usually represented with the Greek letter mu, _{}. Here is the earlist use as reported on Jeff Miller's web site, " _{}, as the symbol for the mean of the normal distribution, was surprisingly late in becoming established. Fisher adopted it in the 1936 (sixth) edition of the Statistical Methods for Research Workers. He had been using m since 1912. He used x-bar for the sample mean throughout.
**GEOMETRIC MEAN** --- The product of the numbers taken to the nth root where n is the number of values multiplied. The geometric mean of 3, 8, and 10 is (3*8*10)^(1/3) or the cube root of 240. If a rectanle is formed with side lengths a, and b, the geometric mean of a and b gives the side length of a square with the same area. In a similar way the geometric mean of three numbers gives the side length of a cube with the same volume as a rectangular parallelepiped (box) with sides of lengths a, b, and c.

**HARMONIC MEAN** --- The reciprocal of the mean of the reciprocals of the values. The harmonic mean of 3, 8, and 10 is 1/[(1/3+1/8+1/10) /3]. The harmonic mean is the answer to a problem that often perplexes students; what is the average speed if you travel from here to there at speed a and back at speed b. The "average" in this case is the harmonic mean of the speeds a and b. As long as the distance traveled is constant, the same method works to find the average speed for any number of trips **HERONIAN MEAN** --- The Heronian mean of two numbers a and b is given by (A+sqrt(AB)+B)/3. If a frustum of a pyramid or cone has one base with area A, and the other with area B, the volume is found by multiplying the height of the frustum by the Heronian mean of A and B.

The Heronian mean can also be found by combining the Geometric mean and the arithmetic mean in a weighted form. That is, the Harmonic mean is (Geometric Mean + twice the arithmetic mean)/3. I assume this can be extended for more than two numbers. **ROOT MEAN SQUARE** --- The RMS-mean or Root-mean-square is named for its formula, take the square root of the (arithmetic) mean of the squares of the values. For two numbers a and b, the RMS-mean is given by sqrt((a^2 + b^2)/2). The RMS-mean is common in statistics where the RMS-mean of the distances of the data values from the mean of the population is called the standard deviation. In statisitics the standard deviation of a population is usually sybolized with a lower case Greeek letter sigma, _{}.*{more detail on the history of the use of sigma can be found at the link to sigma}* I find it interesting that in a trapezoid with bases of length a and b, the segment parallel to the bases that divides the trapezoid into equal areas has a length equal to the RMS-Mean.

I came across a note from Sam Kutler that gives some notes and references on the early language on means

The term "geometric" was first used by the pythagoreans according to Iamblichus ON NICHOMACHUS'S *INTRODUCTION TO ARITHMETIC*"

In ancient days in the time of Pythagoras & the mathematicians of his school there were only three means (mesotetes) the arithmetic, the geometric (gometrike) , and a third in order which was then called subcontrary...renamed harmonic.

Archytas cited by Porphyry:

Now there are 3 means in music: 1st the arithmetic (the first exceeds the second by the same amount that the second exceeds the third; 2nd the geometric (gametrika) (the first term is to the second as the second is to the third); and 3rdly the subcontrary, the so-called harmonic (whaterver part of itself the first term exceeds the second, the middle term exceeds the third by the same part of the third.

These quotations are from SELECTIONS ILLUSTRATING THE HISTORY OF GREEK MATHEMATICS by Ivor Thomas, 2 volumes in the Loeb Classical Library & thus containg the Greek on facing pages with the English.

Samuel S. Kutler St. John's College Annapolis, MD 21404

The

It is said that the ancient Greeks studied ten different means. Other words drawn from the same root include mediate (in the middle of the opposing parties), medium, mitten (put your hands in the middle), meridian, and Mediterranean (the middle of the world, at least to the ancients).

**Ordinate** The ordinate of a point is its distance from the x-axis. The origin is the Latin root *ordinare*, to put in order. All the words you would expect, order, ordinary, and orderly; are from the same root. The ordinal numbers are the numbers used for indicating position (first, second, third, etc.). The first use of the word ordinate for the y-coordinate of a point seems to have been by Leibniz around 1695, although it had earlier been used in a more general sense for distances by others.

**Point** The word point comes from the Latin word *pungere* which means to pierce or prick. The word became generalized to things related both to the tip of the sharp object, or the mark left by its use. Our modern word puncture comes from the same root, as do pungent (a sharp odor) and punctual.

The Greek word for point was *semeion*, for mark or symbol. Euclid's first definition in the __Elements__ was "Sęmeion estin, hou meros outhen"; which Heath translates as "A point is that which has no part." Our modern word, **semantics** comes from the same root.

**Pound** The unit of weight now common in the US came from a long string of language ideas that at first seem poorly related. The original Indo-European *spen* related to spinning or twisting, and gives us our current word for spin. Hanging things sometimes spin, and so the Romans attached the word to things that were hung up and begat modern words like pendant, penthouse, and suspend. The act of hanging things on a balance rod to measure weights led to the use as an object of weight, and gave us peso, pound, and pendulum. The Romans first used divisions of twelve ounces in their weight system and had a large unit called the As. Eventually the As became a coin and the word for the unit of weight became the *Libre*. This is the origin of the ** LB** abbreviation for pounds.

**Radius** The word radius was a Latin word for the spoke of a wheel, and no resource I can find seems to take the origin back any farther. Some include comments about an "uncertain origin". Please note that I have found no documentation for the following conjecture, but undaunted by that, I give full license to my imagination. The similarity of radius to the Indo-European root *werad* from which we get radical and rhizome and the image of the roots of a plant spreading out in all directions like the spokes of a wheel seem too much to resist. Whatever its origin, the word radius was shortened to produce ray, and is also the source of the word radio, named because the electromagnetic rays radiate out in every direction.

**Square root symbol or Radix** The math historian Florian Cajori divides the modern history of root symbols into four types, the radix, , *l* for latus, the more modern symbol , and the fractional exponent, x^{1/2}.

The word radix is used by Leonardo of Pisa in his translations of Arabic manuscripts, and he uses the symbol in his __Practical Geometry__ (1220). It became very common in Italy and Spain and was used in various forms by Tartaglia, Cardan, and Bombelli. It was never widely used In England, and by the end of the 17th century it was passing from use.

The Latin *l* for latus was introduced by the Roman Surveyor Junius Nipsus in the 2nd century BC and was used in the arithmetic of Ramus (1592) and by Viette and Briggs (1624).

The modern radical originated in Germany. An image of the use in Stifel's *Arithmetica Integra* can be here. It had been thought to be a deformation of the R in radix, but Cajori points out that more recent investigations of a series of German manuscripts indicate it evolved from a dot and a single pen stroke right and up. The symbol spread throughout the continent and was embellished and expanded. Wallace placed the index for higer roots inside the radix. Gerard suggested placing the index in the modern position in the opening of the radical and by the 18th century this was the common practice. It was not until the __Geometrie__ (1637) of Descarte that the vinculum or horizontal bar was added with the radical. By the time that Descarte employed the union of radical and vinculum that has become so popular, Orsme and Stevin had already suggested the use of fractional exponents to indicate roots and powers. Cajori points out that generations of students might have been spared the duplicity of notations if Descarte had adopted the notation of x^{A/B} to represent the Bth root of the Ath power of x

The ancient root of ratio comes from the same early Indo-European root that gave us arithmetic. It is sometimes given as *ar* and sometimes *ree*. In its earliest incarnations the word may have related to "fitting together", but quickly took on a meaning related to counting (putting all the items together into one group, perhaps). By the Latin *reri* it had taken on the ideas of "reason", from which comes **rational**, and **ratio** for a comparison of two magnitudes. Rate is a synonym for ratio and comes from the same source. The word **rational** is used in common language to mean a method of thinking based on logic and reason, and in mathematics to describe a comparison of two magnitudes. A rational number is a number that may be expressed as a ratio of two integers. The letter **Q** is generally used to represent the set of rational numbers. At Jeff Miller's web site on the earliest use of some math symbols, I found the statement, "Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. (N. Bourbaki was a group of mostly French mathematicians which began meeting in the 1930s, aiming to write a thorough unified account of all mathematics.) The letters stand for the German Quotient and Zahlen. These notations occur in Bourbaki's Algébre, Chapter 1. "

The real numbers may be divided into two sets by separating numbers into the **rational numbers**, and the **irrational numbers**. Rational numbers are numbers that may be expressed as the ratio of two integers. All common fractions would be in this category, 2/3 or 5/4, as well as the integers themselves since 3 can be expressed as the ratio of 3/1. Any decimal expression that terminates after some time can be expressed as a rational number also. As an example, .35 can be written as 35/100 or 7/20. Decimal numbers that repeat the same string of digits forever are also rational numbers. Expressions like .444….. can also be represented as 4/9, and in general it is easy to express any decimal fraction that repeats right from the decimal point by writing the repeating string in the numerator and as many nines as there are digits in the repeat string in the denominator. For example the three digit repeat sequence .453453453…. can be written as 453/999. A little algebra allows us to show that if the number repeats after some initial non-repeating sequence, it can still be rewritten as a rational. For example .23453453…. can be written as a rational with the numerator equal to 23453-23= 23430; and the denominator equal to 99900 (note that three digits repeated, hence three nines, and two did not, hence two zeros).

Irrational numbers are real numbers that can NOT be expressed as a ratio of two rational numbers. The story of the irrationals probably starts with the Pythagorean discovery that the diagonal of a square could not be expressed as a ratio of the sides in any way. If the sides of the square are 1 unit in length, the diagonal will have a length that is the square root of two, so is irrational. The is approximately equal to 1.41423156… but the decimal expansion never reaches a point where some cycle repeats itself forever. In fact all square roots of integers that are not perfect squares (numbers like 1, 4, 9, 16, etc) are irrational. Other famous numbers that are irrational include Pi, which is appx 3.14159265… and e, which is appx 2.7182818284590… and the golden ratio which is appx 1.6180339… .

A recent discussion on the Historia Matematica list explains the origin and development of irrational. I have clipped parts of several documents.

In the eminent website Earliest Known Uses of Some of the Words of Mathematics I read about the history of the word irrational:

Cajori (1919, page 68) writes, "It is worthy of note that Cassiodorius was the first writer to use the terms 'rational' and 'irrational' in the sense now current in arithmetic and algebra."

Irrational is used in English by Robert Recorde in 1551 in The Pathwaie to Knowledge: "Numbres and quantitees surde or irrationall."

Heath mentions (Vol 1 p. 92) that Magnus Aurelius Cassiodorius presented Greek geometry in his encyclopaedia "De artibus ac disciplinis liberalium literarum" (about 475 A.D.). I suppose it is there Cajori has found the terms 'rational' and 'irrational'. Perhaps in his writing about proposition 47, book I? But did Cassiodorius really mention irrational numbers? Does 'now current' mean that he in some sense had a numberline?

I know that Jacques Peletier in 1563 uses the word irrational number and he means that the sum of a rational and a irrational number is always irrational according to, what he calls a philosophical axiom. [Staffan Rodhe]

Let me intervene in this learned discussion with the following remarkable observation made by Johan Kepler in the first pages of his Harmoniae Mundi. The Greek words translated by the Latin "rational" and "irrational" (segments, i.e., numbers) are "logos" and "alogos", resp. When in Greek mathematics one mentions "logoi" or "alogoi" in connection with segments (such as the side and diagonal of a quadrilateral) it means "expressible" or "un-expressible", resp., Hence the Latin translation "rational" and "irrational" is a mis-translation, and it should better be translated as "expressible" or "inexpressible", resp., when appearing in the mathematical context. But now is too late for such a reformation of terminology. Yaakov S. Kupitz

There are actually three Greek words having similar meaning. In Plato one finds occasionally "arrhetos" (unspeakable, inexpressible, related to "rhetoric") and "alogos" (irrational, "illogical"). The word in Euclid is "a-sym-metra" (plural) referring to two in-com-mensurables (a piece-for-piece translation of "asymmetra", and somewhat distinct from the English cognate "asymmetric"). I find it interesting that the English word "unspeakable" carries a heavy emotional connotation of being "too horrible for words," but that connotation is not in the Greek "arrhetos". [Roger Cooke]

**Repunit** The term repunit, for a number made up of all unit digits, was created by Albert Beiler in the 1960's as a contraction of __rep__eated __unit__. It still does not appear in most dictionaries, and is therefore difficult to trace... more to come, I hope.

One of the most famous puzzle masters of the early 1900's was Henry Ernest Dudeney. In his puzzle book,__The Canterbury Puzzles__, of 1907 he poses a problem about Repunit factoring, without using the term repunit. The problem is posed thus:

"It used to be told at St Edmondsbury," said Father Peter on one occasion, "that many years ago they were so overrun with mice that hte good abbot gave orders that all the cats from the country round should be obtained to exterminate the vermin. A record was kept, and at the end of the year it was found that every cat had killed an equal number of mice, nad the total was 1,111,111 mice. How many cats do you suppose there were?"I will begin to discuss the answer shortly, so those who wish to solve the problem first might stop reading and take a few moments to work on a solution.

Later the problem provides that there is more than one cat, and each cat kills more than one mouse, in fact each cat killed more mice than there were cats.

Dudeney points out that for the solution to be unique, there must be only two factors of 1,111,111. In truth, the factors of 1,111,111 are 239 and 4649, so there must have been 239 cats who each caught 4649 mice.

In the solutions in the second edition in 1919, Dudeney discusses the general problem of finding solutions of repunits (again without using the term) and gives some interesting tables and remarks which I have tried to recopy accurately

In a footnote He points out that "Mr. Oscar Hoppe, of New York, informs me that, after reading my statement in the introduction [where he says n=19 is prime], he was led to investigate the case of n=19, and after long and tedious work he succeeded in probing the number to be a prime. He submitted his proof to the London Mathematical Society, and a specially appointed committee of that body accepted the proof as final and conclusive. He refers me to the Proceedings of the Society for 14th February, 1918.Lucas, in his

L'Arithmetique Amusante, gives a number of curious tables which he obtained from an arithmetical treatise, called theTalkhys, by Ibn Albanna, an arabian mathematician and astronomer of the first half of the thirteenth century. In the Paris National Library are several manuscriptes dealing with theTalkhys, and a commentary by Alkalacadi, who died in 1486. Among the tables given by Lucas is one giving all the factors of numbers of the above form (repunits) up to n=18 (eighteen ones in a row). It seems almost inconceivable that Arabians of that date could find the factors where n=17 as given in my introduction [On page 18 of the introduction he gives the factors of 11,111,111,111,111,111 as 2,071,723 and 5,363,222,357.]. But I read Lucas as stating that they are given inTalkhys,though an eminent mathematician reads him differently, and suggest to me that they were discovered by Lucas himself. This can, of course, be settled by an examination of the Talkhys, but this has not been possible during the war.(WWI)The difficulty lies wholly with those cases where n is a prime number. If n=2, we get the prime 11. The factors when n=3, 5, 11, and 13 are respectively (3x37), (41 x 271), (21,649x513,239), and (53 x 79 x 265,371,653)[Dudeney used a raised dot for multiplication which I have replaced with an x for my convenience]. I have given in these pages the factors where n=7 and n=17.

The factors where n=19, 23, and 37 are unknown, if there are any.[emphasis added]^{*}

Dudeney also points out three "curious series of factors" that he thought would "doubtless interest the reader." They are shown here:
n=2 --> 11

n=6 --> 11 x 111 x 91

n=10 --> 11 x 11,111 x 9091

n=14 --> 11 x 1,111,111 x 909,091

Can you guess n=18?

Or the same numbers can be written as

n=6 --> 111 x 1,001

n=10 --> 11,111 x 100,001

n=14--> 1,111,111 x 1,000,000,001

For the cases where n is a multiple of 4, we get

n=4 --> 11 x 101

n=8 --> 11 x 101 x 10,001

n=12--> 11 x 101 x 100,010,001

n=16--> 11 x 101 x 1,000,100,010,001

**Residual** Sit back, stay right there, and I will tell you the origin of residual. Wait! I just did. The common *re* prefix means back and *sid* is from the Latin *sedere* which means to sit, so the literal meaning of residual is one who sits back or, more appropriately, stays seated. In statistics we use it in the same sense as residue, that which remains (stays seated) when something else is taken away; what remains from the observed amount when the predicted amount is removed. Another closely related word is residence. Other words drawn from the *sedere* root include sedentary [one who sits around a lot], sediment [stuff that settles], and sedative [something that keeps you from moving around].

**Robust** A statistical test is robust if it holds true even when the underlying assumptions are not met. Although the word itself is very old, the statistical use was introduced in 1959 by George Box, the son-in-law of R. A. Fisher. The word comes to the English language from the Indo-European root *reudh* for things that are red or reddish, but its meaning comes from the Latin *robur*, the Roman name for the Red Oak. Whether it was because the Oak is such a hardy, vigorous tree, or the idea that red cheeks are a sign of health, the word began to take on meanings related to strong or healthy. The word corroborate, to support, is drawn from this relation to strength. Many other words from the original root and related to red are still common in our language. Rust and ruby are examples of objects named for their reddish color.

**Root and Radical** The Indo-European root *werad* was used for the branches or roots of plants. Later it was generalized to mean the origins, or beginnings of something whether it was physical or mental. In arithmetic the **root** of a number is the number that is used to build up another number by repeated multiplication. Since 8 = 2*2*2 we say that 2 is the third root of eight. The word root is also used in the mathematics of functions to indicate the value that will produce a zero (a ground level number) for the function. If f(x)= x^2-9 then x=3 is one of the roots or zeros of the function. The word was used by al-Khowarizmi in his writings and was translated as radix in the Latin translations of his algebra.

The same root word also gave us the word **radical ** which is used for the symbol indicating a root. Students often call this the square root sign. The symbol actually has two parts, the radical and the horizontal bar. The symbol is written so that the two run together and appear as a single symbol, but the actual radical looks more like a check mark. The horizontal bar is a vinculum , just like the bar over a repeating fraction. According to Jeff Miller's web site, "The radical symbol first appeared in 1525 in Die Coss by Christoff Rudolff (1499-1545). "

Other English words related to the Indo-European root *werad* are rutabaga, radish, race, radix, and eradicate. From the Greek equivalent we get rhizome and licorice (honest). Strangely, I can find no written record of a relation between this root and the word radius, which seems so very much alike in form and usage.

**Sesquicentennial** The prefix *sesqui* is from the Latin and means one and one-half of whatever follows. **Sesquicentennial** refers to a period of one and one-half centuries, 150 years. The prefix is actually a contraction of two parts, *semi*, which means one-half, and the enclitic *que* which means "and". About the only other surviving word with this root is the usually derisive **sesquipedalian**. The *ped* root is for the word foot, and refers to very long words; words that are literally one and one-half feet long.

**Sine** The name sine came to us from the Latin *sinus*, a term related to a curve, fold, or hollow. It is often interpreted as the fold of a garment, which was used as we would use a pocket today. The use in mathematics probably comes about through the incorrect translation of a Sanskrit word. Here is a brief description of how that came about from Passage to China by Amartya Sen. "In his Sanksrit mathematical treatise completed in 499 AD, Aryabhata used jya-ardha (Sankscrit for “chord half”), shortened later into jya, for what we now call “sine.” Arab mathematicians in the eighth century transliterated the Sanskrit word jya into the proximate sound of jiba and then later changed it into jaib (with the same consonants as jiba), which is a good Arabic word, meaning a bay or a cove, and it was this word that was later translated by Gherardo of Cremona (circa 1150) into its equivalent Latin word for a bay or a cove, viz., sinus, from which the modern term “sine” is derived."

When Leonardo de Fibonacci used the term in his writing, it became permanent. According to Carl Boyar's "A History of Mathematics", the idea of the sine of an angle came from an Indian book written around the year 400. The early use of sine referred to a length of the chord in a circle. It was not until the 1700's and Leonid Euler [pronounced Oiler] that it became common to use the sine as a ratio.

**Skew ** The word skew seems to be a contraction or a derivative of the word eschew, which means to avoid or turn away from. Skew lines in geometry seem to meet this description. They are not in the same plane and they don't intersect. In more general terms something is skewed, or askew, if it is out of place or misdirected. This seems to be the use in statistics where the word is used for distributions that are not symmetrical or centered.

In statistics, skew also is expressed as a numeric parameter of a probability density function with the formula . Skew seems to have been used in relation to the matrices earlier than for the statisitcal shape of a distribution. Jeff Miller's web site lists "Skew symmetric determinant appears in 1849 in Arthur Cayley, Jrnl. für die reine und angewandte Math. XXXVIII. 93: ". For the first known use of terms related to statistical distributions he gives,"SKEW DISTRIBUTION and SKEW CURVE appear in 1895 in Karl Pearson’s Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material, Philosophical Transactions of the Royal Society A, 186, 343-414. ".

A **skew quadrilateral** is a quadrilateral formed when all four points do not lie in a single plane. Two faces of a tetrahedron, for example, would form a skew quadrilateral.

**Standard Deviation** The creation of the statistical term **standard deviation** is credited to Karl Pearson around 1893. The term deviation is the union of the Latin roots *de* and *via*. *Via* is the Latin word for road, and **deviate** literally means "away from the road". This fits the mathematical idea of something at a distance away from a center.

The following is from a posing by Stephen Stigler to the Math History discussion group

Actually, the TERM standard deviation was introduced by Karl Pearson in 1893-94. Other terms preceded it, but for slightly different quantities. The QUANTITY goes back to Gauss and Laplace. There is some on this in my 1986 book 'The History of Statistics'.

Stephen Stigler

For students, the confusion is exacerbated by the fact that we use "standard deviation" frequently for both a parameter, sigma, the true, known only to God, measure for a population and also for the standard deviation of a sample, s, a statistic intended to estimate the value of the true population parameter. I even read in a posting by J Stuart Hunter of Princton, "There’s a story that Prof. Frank Anscomb at Yale wanted to call s the standard deviation and sigma the scedistic, thus two words for two different things. You can’t win them all." Opportunities missed.

**Variable/Variance** The word vary, and its many variants (oops, there is one now) come from the speckled fur of animals used in early apparel. At first applied to changes of color, the word was eventually used for things that involve change of any kind. The original meaning is preserved in the word miniver. Today miniver is the term for the ermine trim on the ceremonial robes of British Peers, but the term originally referred to any decorative fur trim on a robe, and was common in medieval times. Sir Ronald Fisher created the statistical term variance around 1918 as a measure of the variability of a data set.

The versine of an angle, A, is an almost extinct expression for the quantity 1-cos(A). Up to the 1600's this was probably the second most common trigonometric value used. The Latin word *versed* relates to turning, and the "versed sine" was, in essence, the sine turned 90 degrees.

In 1835, James Inman introduced the term **haversine** to describe a value of 1/2 of the versine, "half-versine". The haversine was an important formula in spherical geometry and navigation, since it gave a simple way to find the approximate distance between two points on the earth using the Longitude and Longitudes. If we consider two points on the unit sphere, with positions given as (lat1, long1) and (lat2, long2) in radians, then the distance between them is given by where dLat and dLong are the differences in the latitiudes and longitudes. Tables for Navigation contained both Hav(x) and its inverse invHav(x) and the logs of these values to assist in prosthaphaeresis . To find the distances on the earth, the answer would be multiplied by the radius of the earth.
According to Jeff Miller's web site, the word first appeared in the third edition of __Navigation and Nautical Astronomy for the use of British Seamen.
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The mathematical terms converse and inverse are both from the same root. Many other words come less directly from this root. A plow turns dirt up and over and creates a furrow, a straight line of dirt along the ground. Things laid out along a straight line were sometimes said to resemble the furrow and called verses, and thus words in a line of a poem became a verse. To reverse is to turn back, and the obverse side is the side you see when you turn something over, and your vertebra are the joints that allow you to turn.

**Volume** Volume is from the Latin root *volvere*, to turn or roll. The idea comes from the practice of writing on scrolls that were then rolled into a cylinder. A volume referred to a roll of written material. According to Steven Schwartzman's **The Words of Mathematics**, the use of the word for an amount of space began around the 16th or 17th century. The more ancient Indo-European root is *wei* and relates to turning or twisting in general. Modern derivatives from this root include twist, wire, thread, vise and vine.