Sunday, 10 May 2020

Volume D-E


Data, Datum Datum is from the Latin dare, to give, and literally means "that which is to be given."  It is an English borrowing from the same word in Latin. Data is technically a plural word, but it has become common, and thus acceptable, to use it in a singular sense; "This data supports my hypothesis."  Many modern words use the same root.  Endow, dose, and donation have obvious connections to the "give" meaning.  Less obvious, but still related, are the words betray ,giving counsel or information to the enemy) , and surrender, literally the giving over of the weapons.  The word die, the singular of dice, is from the same root. The OED earliest citation for data in English is 1630.  With application to computing, the first use is 1940, W. J. Eckert, Punched Cards Methods 9.  

Dean The term now used for the head of a department or faculty at a school is derived from the Latin word deaconun which meant "chief of ten".  The similar sounding deacon, for a church leader is not related and comes from the Greek root diakonos for a servant.  According to John H Conway, the literal meaning is "one who raises the dust."

Decagon joins the greek root deka, for ten, with the suffix gonus, for angle. A closed polygon with ten sides and angles. Like all polygons, if the sides and angles are all equal, then it is called regular. Jeff Miller's website on the first use of math terms describes a Latin translation of Euclid's Elements as contining latus decagon and decagon equlaterus in 1509. The OED gives a French use of decagone in 1652. The earliest English citation is for Thomas Digges in his Patometria in 1571, but he uses "decagonum." J. Harris' Lexiconum Technicum in 1704 is the eariest given citation with the current English spelling.

Decahedron joins the Greek deka with the hedra for seat or base, for the name of the ten faced solid. It first appears in English over 200 years after the decagon, in the 1828 Noah Webster American Dictionary of the English Language. Whenever I used terms involving "hedra" I would remind students often that it is the same as the ending in cathedral, because that's where the Bishop's Seat was. It seemed ot help.

Decakis is a rarely used Greek term taken directly into Enlgish for ten times. The only known use to my knowledge is the stellation of a dodecahedron called a deckisdodecahedron, where there are 120 faces, multiplying the 12 faces of the dodecagon by ten of each star point. The "kis" also appears in the stellated tetrahedron which is called a triakis tetrahedron, where each stellar point multiplys the number of faces by 3, to produce a total of 12 faces.

Decile is a statistical term for the divisions of a distribution at each 10% level. The deciles are the nine dividers between each ten percent of the population. The second decile for instance, marks the break between the 10-20% group, and the 20-30% group, Thus there are only nine of them. "Decile appears in 1882 in Francis Galton, Rep. Brit. Assoc. 1881 245: "The Upper Decile is that which is exceeded by one-tenth of an infinitely large group, and which the remaining nine-tenths fall short of. The Lower Decile is the converse of this" [OED]." Jeff Miller's Web Site

Decimal Fractions are fractions whose denominators are a power of ten.  They are often written with a decimal point, or in some countries, a decimal comma.  The point is usually written with the bottom of the numbers in western cultures, but sometimes is raised to the middle of the line.  Some infinite decimal fractions are sometimes written as repeating decimal periods, and actually have denominators other than 10; for example, the decimal fraction \(\bar{.09} \) is equal to the fraction \( \frac{1}{11} \)

The Decimal Point " In 1617 in his Rabdologia John Napier referred to the period or comma which could be used to separate the whole part from the fractional part, and he used both symbols....According to Cajori (vol. 1, page 329), "Probably as early as the time of Hutton the expression ‘decimal point’ had come to be the synonym for ‘separatrix’ and was used even when the symbol was not a point." *Jeff Miller's Web Site

Degree is the union of the Latin roots de, down, and gradus, step.  Gradus is actually derived from the Greek word for "to walk" or "go".  Related words with the same root are congress (come together), regress (go back), and of course grade (the step you are on in school, or earned on an evaluation).  Degree as the measure (or step) of an angle dates back at least to the writings of Chaucer who used the word both in his Canterbury Tails (Squires Tale)  in 1386, and his more famous (then) book on the Astrolabe in 1400.

Degrees, why 360 in a circle I have seen many responses to why we use 360o in a circle, but the one that most impressed me was by the one below by the late Alexander Bogomolny.  I have copied the entire thing from his response to a question on  geometry discussion site, so here's why there are 360o in a circle,

"Babylonians used base 60 notation, which is convenient to divide a whole into 2, 3, 4, ... 30 parts.  Early Greeks then probably divided the radius of a circle into 60 parts. Hence, the diameter must have 120 parts. As Pi was known to be close to 3, the circumference would have 360 parts.

This argument may be used to exonerate the Bible (I Kings. 7:23 and II Chronicles, 4:2) which is said to quote 3 as the value of Pi. Not being a geometry manual, the Bible just picked out a simple approximation to Pi to convey the order of magnitude of the measured quantity. ...

Some history of the sexagesimal (base 60) notaions appear in D. E. Smith, History of Mathematics, v2, Dover"

*By Sam Derbyshire at the English Wikipedia, 
Deltoid In geometry, a deltoid is a hypocyloid (figure formed by one circle rolling inside a larger)  that has three cusps, which means it's radius must be one-third of the larger circle. Other cycloids were studied much earlier by Mersenne and Pascal, but the deltoid was first considered by Euler in 1745 in connection with optical problems. It is named after the Greek letter delta, \( \delta \) The area of the deltoid is twice the area of the rolling circle, and it fills 2/9 of the larger circle.

deMoivre's Formula is a method of raising any complex number, \( z=a+bi=r(cos\theta + i sin\theta)\), to any power.  The part in parenthesis is sometimes written with the abbreviation cis, for cosine + i sine.   The forumla says \( (r *cis\theta)^n = r^n(cis(n\theta).\$)

 deMoivre's Formula is named for Abraham de Moivre, although it was never stated in his works.  Welsh mathematician, William Jones, often credited with the creation of Pi for its current meaning, seemed to be referring to this theorem in his 1707 Synopses palmariorum matheseos, writing about a "beautiful theorem... of Mr. DeMoivre". Jeff Miller's Web site on Earliest use of math words has, "The earliest use in English of Demoivre’s formula which unmistakably refers to (cis θ)n = cis nθ which James A. Landau could find in a Google print search is in 1809 in Treatise on Plane and Spherical Trigonometry by Robert Woodhouse."  Even though de Moivre lived in England much of his life, the first use of " deMoivre's Theorem" appears 42 years after his death in the Monthly Magazine of 1796, "by de Moivre's theorem we have... "  *OED

Denominator The bottom number in a rational fraction serves to name the type of fraction being counted by the numerator. The root nomen for name carries this meaning. Nominate, noun, nomenclature and of course, name are all derived from this same root.  The de root means "complete", so the denominator names the size of the pieces of the fraction completely.  They are usually an ordinal number, for instance "fifths", and describe the number of such pieces required to construct a unit whole.  Like so many arithmetic terms, this one is from Robert Recorde in his Grounde of Arts in 1552.

Density – Usually mass/volume. The common (probably incorrect) tale is of Archimedes needing to measure the density of an irregular shape of gold, due to the rumor that King Hiero’s goldsmith was stealing gold from the king. Archimedes knew that he could compress it into a cube and measure the density easily but the king did not want this. Annoyed with what to do, Archimedes took a bath and noticed the water rose as he sunk in the tub. This gave him the fantastic idea to submerge the gold in water to measure its volume. Once he discovered this, so the story goes, he leapt from the bath and ran naked in the streets shouting “Eureka! Eureka!” and as a result, the word “eureka” is used to indicate a moment of enlightenment. *Derek Orr

Derivative/ Derive means to obtain a result from a set of logical steps.  It is not uncommon to hear mathematicians say things like "I derived the result by the fundamental laws of geometry."  The word is also used to describe a particular operation in calculus, to describe a function found from another function that describes the rate at which it changes. We derive the f(x) and get its derivative, usually labeled f'(x) which tells us the slope of the tangent of f(x) at any point.  The curve y= x^2 describes the curve of a parabola.  It's derivative, y' = 2x tells use the slope of the tangent to the parabola at any point along the curve.  You can even derive the derivative and get a function for the rate of change of the slope of the tangent.  And if you do it again, yes, you can, you get the rate of change of the rate of change of the original curve.... and on you go as far as you desire,  The roots of derive are de, for down or away, combined with the French rivus, brook or stream, from the same roots as river, and riviera.  The essence of the meaning is "to come from or out of". Newton did not use the term but used "fluxions" instead.  His influence and the rivelry between Newton and Liebniz kept the term from being much in use in England until the late 18th century.

Derivation  is another term for the result of deriving something.  We speak of the outcome we get from deriving as the derivation of the original information. The automobile can be thought of as a derivation of coaches.

Descartes' Circle See The Kiss Precise, here

Determinant in mathematics, particularly in linear algebra, means a dimensionless number, or scaler, that can be assigned to a square matrix.  Square matrices are often used to solve systems of equations in which the number of unknown variables and the number of equations are identical. One use of the determinant is to tell you if the system has a unique solution.  If it does not, the determinant will be zero. In transformation matrices, the Determinant also gives the volume transformation scale of the transformation. The symbols for a determinant vary from author to author, some writing the abbreviation det(A) , others using the vertical bars used in absolute value |A|.  The term was coined by Gauss in 1801, but not in its current meaning.  It was Cauchy who first used it in the modern meaning, in 1812. The OED reports that the first use in English was by Cayley in a book bearing the term in the title, On the Theory of Determinants, 1843.

Devil's Curve
In geometry, a Devil's curve is a planer curve with the Cartesian equation

\( y^{4}-x^4 + ay^2 + by^2 = 0 \) Gabriel Cramer, of Cramer's law fame, first studied the curves around 1850, and Lacroix extended this work in 1810. The curve has a central lemniscate, or ribbon shape, when graphed. The shape is named for a juggling game diabolo, using two sticks, a string, and a spinning prop that resembles this center of the graph. It was misunderstood, most likely, because it sounded like the Italian word diavolo, for "devil".

Diagonal comes from the Greek roots dia (to pass through or join) and gonus (angle) and describes the line segment which passes from the vertex of one angle to another in in a polygon.  In the Historia Matematica discussion group, Julio Cabillion credits Heron of Alexandria for the creation of the term around 100 AD.    In common use it is often now applied to cutting across at a path not parallel to an edge, or walk or some other boundary.  The term is also used for a plane cutting across two non-adjacent faces of a polyhedron.The OED gives the first use in English as "J. Shute, First Grounds Archit.  'The diagonal marked b".

Diagram joins the roots of dia (to pass through or join) with gram (written or drawn and in earlier times, carved).  It literally means "that which is marked out" as by the crossing of two lines.  This leads me to wonder how old the expression, "X marks the spot." could be. The OED credits the first use in English in 1645 to N. Stone in book about fortifications, "A word used by the Mathematiks for anything that is demonstrated by lines."

Diameter is an apt name for the measure across a circle.  The word comes from the union of the Greek roots dia, across and metros (to measure).  Euclid used diameter in relation to the bisecting chord of a circle, and also the diagonal of a square.  While students think of this last use as unusual, in geometry and engineering the word is often used to describe the longest straight line of a plane figure.It also appears in Physics as the strength of magnification of a lens.The OED gives the first use in English as 1387, and spelled "dyameter".

Diamond is not a formal mathematical term, but young students often use it for both the kite, and the rhombus.  The word diamond is derived from the Greek word adamas, which is so close to the origin of atom from atomos that one wonders if they are not related, but I can't find a direct link.  Adamas seems to have meant the hardest substance known, and atomos for something that can not be divided. Diamond for the shape was first used in 1496 to describe a rhombohedron, and only applied to the plane figure in 1496 in Dickson's translation of Euclid.

Dice/Die A least as frar back as 2000BC, the rich and the mystical have had dice to play with.  Very early dice were often more like the shape of a tetrahedron than a cube.  The modern cube shape came later, but was common by the birth of Christ. The tombs of Ancient Egyptian kngs have produced square based cylinders (think of a square pencil) that were apparently used much like dice, for games and predicting the future.  Dice were so common that when Jjulius Ceaser ordered his troops across the Rubicon River to begin his invasion of Italy, he declared, "Iacto alea est", "the die is cast".

The alea of the phrase was the Latin word for a die.  The only word I know in modern English with the same root is aleatory, "depending on chance."  Our word die (singular of dice) seems to come from the Greek word kubos which is also the root of cube. The comparison of dice in cooking, to chop into cubes, seems strongly indicative of this hard to trace etymology.

The OED says that the use of dice for both the singular and the plural is a modern adaptation.  The use of die is first credited to 1393, in J. Gowers, Confessio Amantis.


The common Latin root, dis(away), is masked here, as it sometimes is, by a double consonant. The second root, ferre is from the Latin root 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, surprisingly, in ferry. Difference is mentioned in the common arithmetic sense in 1400 in Geoffrey Chaucer's Treatise on the Astrolabe, "diffrense be-tween 1 and 1."


is another word that reflects back to the earliest uses of matehmatics. The word digit still refers both to the fingers (and toes) as well as the arabic number symbols for 0 to 9. The Indo-European root, deik, is related to many other words that reflect back to the use of the hands and fingers to point out things. Index, indicate, dictate, indict, token, dice, and strangly judge and teach are all related to the same word. Teacher is drawn from the Greek deiknunai "to point out". I much prefer the Latin term for educator, educare, which literally means to draw out.  The earliest use to the whole numbers less than ten is in 1400, where the digits are written in Roman numerals, "Somme is callyd nombre of digitys ... wihine ten as , ix, viii, vii, vi, v.... "

Dimension is used in several different ways in math and science.  When we talk about the dimension of a space we mean the number of coordinates needed to identify a point or location in that space. We may speak of a two dimensional plane, or a three dimensional spacial object.  We also use the word to describe the units by which we measure objects.  It is the units of dimension that bring reality to a mathematical problem, distinguishing four miles from four feet.  The origin of the word is indicative of this measurement theme,  Dimension is a weathered and worn version of the union of  dis (intense/strong)  and meteri (measure), with a combination meaning "measure carefully".  The OED gives the earliest use applying to spatial dimensions as 1413 in Pilgrims Sowle, "There is no body parfit withouten thre dymensions.."

In the great markets of Italy, the competition could be harsh, and so the enterprising businessman would off a little incentive to the reluctant buyer. If a merchant buying spices for shipment surveyed the counting board and seemed hesitant, the seller would sweep a few stones to the side, away from the count. These would be included free if the buyer bought the ones counted. The Latin roots of discount reflect the early counting table origins, the dis indicating a moving away, and the count from computare. When the French picked up the practice they called it d'escompte. By the middle of the 17th century, the English had begun to applly the practice in the pepper trade with Holland using the current word discount.  The OED gives the first English use 1622 in regard to East India Business.

Discriminant To most algebra students the discriminant is a number related to a quadratic equation that determines the nature of the solutions, real or complex.  The value of the discriminant in a quadratic is given by the expression B2 - 4 AC.  A positive discriminant indicates two positive roots.  A zero indicates one root, generally referred to as a double root (if the root is 3 the the factors are (x-3)(x-3).  If the discriminant is negative, then the two roots are a pair of complex conjugates.

The discriminant can also be found in higher order polynomials. The value at all orders can be expressed as the product of the squares of all the difference of any two zeros of the polynomial.  A cubic polynomial Ax3 + Bx2 + Cx + D for instance, is given by

\( \frac{B^2C^2 - 4DB^3 - 4AC^3 + 18 ABCD - 27 A^2D^2}{A^4} \)

Discriminantes also exist for many other relations in math other than the roots of polynomials. The discriminant of a general conic equaton, for example, distinguishes if a formula is for an ellipse, parabola or hyperbola.

The term was created in the last half of the 19th century.  The origin is from the union of dis for away, and the very early Indo-European root , skeri which referred to seperating by sifting, scraping or cutting, through  the Greek krinein to separate.  Descendents of the skeri root show up in diverse words today including script, crime, decree, secret, and endocrine.  The term first appears in English in relation to the quadratic discriminant, according to OED, in 1898.  D. A. Murray's Introduction to Differential Equations has, "the discriminant is b2 - 4 a c...".


Distance is from the union of the Latin roots dis (away) and sta, stand, that is the root of statistics. The literal meaning is standing apart. Related terms from these roots are discord and debate.
Density – Usually mass/volume. The common (probably incorrect) tale is of Archimedes needing to measure the density of an irregular shape of gold, due to the rumor that King Hiero’s goldsmith was stealing gold from the king. Archimedes knew that he could compress it into a cube and measure the density easily but the king did not want this. Annoyed with what to do, Archimedes took a bath and noticed the water rose as he sunk in the tub. This gave him the fantastic idea to submerge the gold in water to measure its volume. Once he discovered this, so the story goes, he leapt from the bath and ran naked in the streets shouting “Eureka! Eureka!” and as a result, the word “eureka” is used to indicate a moment of enlightenment.

Divergence – see Convergence.

Distributions – Generalized functions in mathematical analysis. These make it possible to differentiate functions that may not have a “regular” derivative in the undergraduate calculus sense. They can be traced back to the 1830s for solving ordinary differential equations using Green’s functions but were formalized much later. Kolmogorov/Fomin in 1957 say distributions originated in 1936 with Sergei Sobolev investigated second order hyperbolic partial differential equations. Later in the 1940s, Laurent Schwartz extended some of these ideas and it was his brazen attitude that really sparked their use in mathematics. *Derek Orr

Divide shares its major root with the word widow. The Latin root vidua refers to a separation. In widow the meaning is obvious, one who is separated from the spouse.  A similar version of the word was often meant to describe the feeling of bereavement that a widow would feel.  The Prefix, di, of divide is a contraction of dis,  a two based word meaning apart or away, as in the process of division in which equal parts are separated away from the whole, and each other into groups.  Note that the vi part of vidua also is derived from a "two" word, and is the same root as in vigesimal (two tens), for things related to twenty.  An individual is one who can not be divided.

I have written Some Notes on Division, and its History, which includes notes on early division algorithms, such as the Galley method, the Obelus  ÷, and its (mis)use and "alien division" .  

Divide, Symbols for

There are several different symbol names used or associated with division. The most common looks like a close parenthesis with a horizontal bar extending to the right at the top. There is a reason it looks like that, because that is what it was.



The parenthesis was introduced in the early 1500's without the bar, and in some cases with two reversed parentheses and the number to be divided between them, and a problem might look like 4) 128 ( 32 when finished. Both Stifel and Oughtred used the parenthesis without the overbar, or vincula. The bar was a in common use to bind things together much like we now do with parentheses, so (2+3)*5, might be written as \( \underline {2 + 3} *5\). This use of a bar, under or over a grouping dates back to Nicolas Chuquet, and remains today in marking the repeating cycle of a decimal fraction.
Another early division symbol, \( \div \) is the obelus, which dates from around 1650. \( 8 \div 2 = 4\) Its invention is often credited to British Mathematician John Pell, and it became the preferred symbol in both England and the colonies, but never seemed to have much success in Europe. (The use is a perversion, in my mind, of Pell's original intent for it's use. See my post, The Agony and the Obelus, Much ado about Notation. ). The colon, : was used in much the same way 8:2 = 4. The horizontal fraction bar precedes all of these symbols, but it itself was preceded by Hindu mathematicians who used the dividend written above the divisor without a line. The line first appeared in Arabic mathematical writing around 1200. Shortly thereafter it was used by Fiboancci and gained popular, and permanent, status.
When printed type came into use, the horizontal bar was very difficult to print because it used up three lines of space. Printers took to using the slanted bar, or solidus. This is reflected in the keys on your calculator or computer .


A Regular Dodecahedron is one of the five Platonic Solids. It is formed by twelve faces that are each a regular pentagon meeting three at a vertex. There are 20 of these vertices, joined by 30 edges. The name is from the Greek duodeka, literally two and ten. It is possible to from dodecahedra that are NOT regular.

Dots and Boxes -  see La Pipopipette/ Pigs in a Pen/ Dots and boxes

Domain The Domain of a function is that set of all possible values of the independent variable(s). To the basic algebra student this often means "the set of all possible x values."  The domain might be restricted by definition, x= {2, 3, 5, 17}  or by implied restriction, for example, y = 2/x can never be zero.  In general English the word  is applied to mean the territory or area of control.  The origin of the word is from the same root, dominus (master) as dominate. The mathematical use seems to have begun around the end of the 19th century.

Double Dummy Although it sounds more like the ultimate classroom insult, "double dummy" is the name of a particular technique in the design of experiments. I had never heard of the name until a request was placed on the AP Stats Newsgroup, and answered by Debra Balm.  Here is her response

Double dummy refers to a procedure that is used when the two treatments are so obviously different that it would undermine the idea of a double blind experiment.  For instance, you want to test a medication that is in capsule form to a medication that is a liquid.  Obviously, the subject would know what that they were getting if just given one of the medications.  So each subject receives both treatments with groups structured as follows:

Group One:  active capsule, placebo liquid

Group Two: placebo capsule, active liquid

Group Three: placebo capsule, placebo liquid

Dozen The word dozen is a contraction of the Latin Duodecim (two + ten).This root also appears in dodecagon and duodenum, the first part of the intestine that seems to be about 12 inches long. Some math and language historians think that a dozen is one of the earliest primitive groupings, perhaps because there are approximately a dozen cycles of the moon in a cycle of the sun.  It appears to be the basis of many larger values that were developed by many cultures.  A Shock was 60, or five dozen(a dozen for each finger on a hand), and many cultures had a great hundred of 120, or ten dozen (a dozen for all the fingers on both hands.  The Romans used a fraction system based on 12, and the smallest part, an uncil, became our word for an ounce, and also for inch by an i mutation, or umlaut.  Charlemagne established a monetary system that had a base of twelve and twenty and the remnants persist in many places.  In English money today 100 pence equal a pound, but only a short time ago a pound was divided into 20 shillings of 12 pence each. The word appears as early as 1340 in English.


East The mythical Greek god Hyperiod had three children, each ruling over a source of light.  Helios was god of the sun, Selene, his sister, governed the mood, and Eos, a second sister, was the goddess of dawn. It is understandable that the ancients used Eos not only for the dawn, but the direction in which it occurred, and sound drifted from eos, to east as it passed down through several languages.  The early spellings of east in English include ost, oost and eest. This last, almost surely from the same word in Dutch.  Early spellings in English include "aest" and Chaucer's "est".

The prefix, eo, came to symbolize not only the dawn of a day, but hte dawn, or beginning, of anything.  For this reason we have scientific terms like eohippus , "dawn horse", for the ancient ancestors of our modern horse. The cenozoic (litterally "new animals") period in history after the demise of the dinosaurs was rich with the emergence of new forms of mammals and birds, and its early part is called the eocene (dawn of new life)

Ellipse An ellipse, in mathematics is a plane closed curve with each point on the curve having the same sum of distances to a pair of points, called the foci (plural) or independently each is called a focus.  Some definitions include a circle as an ellipse with both foci on the same point (the center of the circle) in the same way you could say squares are a subset of rectangles.  The eccentricity of the ellipse is a measure of how stretched out, or  non-circular, the ellipse is, and ranges from 1, a circle, to 0, when it reaches to infinity and actually is a parabola.  The formula for an ellipse with axes parallel to the x and y axis is given by \( \frac{(x-p)^2}{a^2} + {(y-q)^2}/b^2 = 1 \) The point (p,q) is the center of the ellipse, the  a and b give the distance from the center to the vertices on the major(longer)  and minor(shorter) axis. The distance from the center to either of the foci is given by the Pythagorean relationship, \( a^2 + b^2 = c^2 \) The foci will lay on the major axis.

An ellipse is formed from a plane cutting a cone (or a cylinder) to form a closed curve.  They have many similarities to the other "conic" sections, the parabola and the hyperbola, which are also cut from a cone, but form open curves.

An ellipse can also be formed using only one of the foci, and a line called the directrix external to the ellipse and perpendicular to the major axis.  Each of the parabola, and hyperbola can also be formed by the directrix, focus method.

The first known use of the term ellipse was in Greek, élleipsis by Appolonius of Perga who wrote a famous work about the conic sections.

The word ellipse is from the Greek work Appolonius used and means a deficit, or something falling short,  Interestingly, the three common conic sections all have a "throwing"  aspect to their names. The ellipse falls short, the parabola (falling beside) and the hyperbola (over thrown). The OED gives the earliest citation for use in English as 1753, and points out that the terms ellipse and ellipsis were almost interchangeable until after 1815.

Ellipsis The Greek term for the ellipse was the same root for the common language term ellipsis.  Even if you don't know the word, you recognize the use when three dots ... indicate that something is missing, a deficit of words or numbers.  It can also apply to spoken language when describing a comment that falls short of being complete or leaves out important information.

Enantiomorphic The Greek root enantios (against, anti) and morph (form) and describes things like the opposite hands which are not copies, but mirror images of each other.  The word seems to have come to English only a few years before 1900, when it appeared in a dictionary.

There are many chemical structures, including many in the human body which exist as spacial mirror images of each other, and some of them react very differently in the human body.  You can make a pair of enatiomorphic dice if you take two identical dice and replace any pair of adjoining faces (for instance, switch the 3 and 2) on one of them.  The two dice can never be set so that all the faces are in the same position.  It was once the case that many oriental dice had a different direction of rotation about the vertex joining the faces 1,2,3 from clockwise to counter-clockwise.  Enantiomorphic solids are often called Chiral, for handed.

Eocene/ Eohippus        See East

Equal 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 aeques for level. Chaucer's 1400 Treatise on the Astrolabe uses the term.  Cajori's History of Mathematical Notation gives many early symbols used in the West, both before and after Recorde wrote the now ubiquitous "=" for the symbol in 1557, but even for another century, many mathematicians used no symbol at all, or used a collection of other symbols.In fact, after 1557, Recorde's symbol  did not appear again in print until 1618.  Viete, writing 1n 1571 used the same symbol for difference (the absolute  value of a-b) and this symbol was widely repeated in this use across Europe. Another early symbol for equality was the script conjunction of ae.

  When we use the term in equation, we often do so in a subjective or conditional sense. Some equations may be true for any value, say 2x = x+x, and these are often referred to as identities. Others are only true for select values, 3x-6=0 is only true if x=2.

Equation in US Mathematical circles means any assertion that two expressions form and equality. In France it seems, they only allow the definition to apply to situations in which at least one of the expressions has a variable in it.  It doesn't seem the kind of difference that would impair understanding on either side.  Equations may contain relations that are always true, such as 2x=x + x, and are generally referred to as identities.  Others may be true for only some values and not others and are called conditional equations.

When asked to solve an equation, we are being asked to find those values for which the two expressions are true.  Equations for lines, curves, or surfaces will have two or more variables and the solutions for these variables represent points on a plane or space.  For instance the curve y= x^2 has a solution at x=2, y=4, and the point (2,4) is a point on the parabola defined by this equation. The word equatio appears in Fibonacci's Liber Abaci in 1202, almost certianly out of the Arabic. The word seems to have entered English from the pen of John Dee around 1570.

equivalent is drawn from the same root as equal, and represents a sense of equality in some way.  We may say two military officers are of equivalent rank, and a dollar is equivalent in value to four quarters (US money). In geometry, the idea of congruence is shown with the three line symbol for equivalence \( \equiv \) Modular arithmetic often uses the term and sign for relationships which are equivalent, or congruent. For example,  \( 13 \equiv 1 mod(4) \) simply says that 13 and 1 have the same remainder when dividing by four.


Long before the first use of the anesthetic we now call ether, it was the name for the heavens, and later for the medium through which electromagnetic waves could propagate (after all, the reasoning went, everything must have something in which to move.) We can trace the use of ether as a name for the heavens back to Aristotle's explanation of the nature of matter (about 350 BC) Earthly things, such as a stone, would fall to the Earth because it was their natural place, the philosopher proclaimed. Fiery things would rise to the sky, witness the smoke. But the stars in the sky did not move either up, or down. They seemed to move in circles around the sky, so they must be made of something very different from the objects of earth and sky. The Sun, Moon, stars and comets all seemed to be ablaze, and so Aristotle called the heavenly material, aether, that which is ablaze. Eventually when scientists first tried to explain how light got here from the stars, they used Aristotle's word for a mass-less medium through which the light waves could move. Then they spent years trying to measure it.

Euler Line is the name of a line that passes through several important "centers: in a non-equilateral triangle.  The line contains the orthocenter, the centroid, the circumcenter, and the center of the nine-point circle.


Excenter/Excircle  Every triangle has three distinct circles that are tangent to one of its sides, and also tangent to the extensions of the other two sides.  The center of this triangle , the excenter, lies on the angle bisector of the opposite angle of the triangle, which passes through the incenter of the triangle as well.


I first heard of the word explementary in July of 1999. It was "re-created" by Steve Wells of a company called Think3 while working on a new CAD program, thinkdesign. The word was needed to represent the angle required to complete a 360o angle. They wanted a word that would be a natural sounding extension of complement and supplement. The Latin explementum means "filling" or "stuffing" and it is in the OED as "that which fills up". This is actually very similar to the meanings of complement and supplement. After a couple of days, he found hte word was not as new to mathematics as we had thought. Several days later he wrote to tell me that hte word already appeared on the "Dictionary of Technical Terms for Aerospace Use (Web Edition by Daniel R. Glover, Jr at the NASA Lewis Research Center, Cleveland, Ohio. Here is there definition, as sent to me by Mr. Wells:
Explement -- An angle equal to 360o minus a given angle. Thus, 150o is the explement of 210o and the two angles are called explementary angle--Two angles whose sum is 360o. My thanks to Mr. Wells for his advice and corrections as much of this content came directly from his emails.

Exponent The union of the Latin roots exo(out of) + poners(place).  The literal interpretation is to make something visible, or obvious.  The English word expound, from the same source, means to make clear. An exponent is also used in English to describe a person who explains or interprets.  Exponent as a mathematical term was introduced by Michael Stifel (1487-1567) in his book, Arithmetica Integra, in 1544. The OED credits the first use in English to George Berkeley in 1734.  Strangely, the related word exponential would precede it by 30 years, being used by John Harris in a paper on curves.

Expression A mathematical expression is a statement involving at least two numbers, variables, or a collection of numbers and variables that represent some value.  An equation connects two expressions that are equal.  3+5 is an expression, as is 4n+1.

Extrapolate/ Interpolate

In mathematics, interpolation is the process of estimating a value of a point between two known values of a function. This is often done by approximating a line or smooth curve between the values, and thus is the literal meaning of the word, "to smooth between". The inter root is familiar to most students as the Latin prefix for "between", and is found in words like interpose (to put between) , and intersect (to cut between). The second roots is polire and means to adorn or polish. Polish is one modern word from the root, and polite is another. The early Indo-European root pol for finger and feel" seems to be the distant ancestor. Things are polished until they are smooth to the touch. One dictionary suggested that this is the root of psalm through the Greek root psallein, to pluck, because the psalms were accompanied by a harp.
Extropolate was created as an extension of interpolate and suggests the smoothing of a line outside the known points. This often is done in statistics when we study a pattern over time to predict future events or outcomes.
Students should be aware that outside of mathematics, interpolate may have the negative connotation of giving false information, as in "filling in" false details.

Extrapolation in a mathematical sense, seems to have been first written in 1874 by W. S. Jevons, who credits the term to Sir George Airy.

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