Math Words, pg 7


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Abacus

The early Greeks used to design small trays and cover the bottom with sand to help hold pebbles steady when they were calculating. The Greek word for these trays and in general for round trays and large platters was abax and this became our word for abacus. The word may have been influenced by the semitic word for dust, but most feel this not probable. The Romans used the word abacus for any flat table or board, including the top plate in the capital of a Doric column.

Here is a picture of a Chinese Abacus from the excellent web site of Luis Fernandes. Click on the abacus and it will link you to his site where you can learn more about the history and use of the abacus, and even use a java applet of several different types of abaci (The plural of abacus may either be abacuses or abaci. In the second version the "ci" sounds like "sigh").

The word abacus, sometimes with the spelling abbacus, is often used to mean arithmetic or computation in general. In a posting to the Historia-Matematica newsgoup Heinz Lueneburg wrote,

"Concerning the Italian use of the word "abbaco" my Italian-German dictionary of the 19th century says that abbaco means 'elementary reckoning'. ... One finds the word with this meaning in Carlo Collodi, Le avventure di Pinocchio. Lots of editions.
Geppetto enters the room, where Maestro Ciliegia is sitting on the ground, and asks him what he is doing there. "Insegno l'abbaco alle formicole = I teach ciphering to the ants" was the reply.
My 20th century dictionary gives the meaning 'primer to arithmetic' ... ".

The origin of this double meaning seems to trace back to the use by Leonardo Pisano in his Liber Abbaci

Teachers may be interested in this interactive java abbacus . You can adjust to different bases and set the decimal point. You can also choose between Japanese, Korean, Russian and other types.

For those who aspire to advanced skills or historical knowldege on the topic, this link has some nice instrucion.


Astronaut The first root of astronaut is aster from the Greek word for star. It still appears in many scientific words such as Asteroid (star like). The naut is from the Greek word for a sailor, and retains this connection to the sea today in the word nautical. An astronaut, then, is literally a "sailor on the stars". The word seems to have been created around 1929 but did not become popular until 1961 according to one source.



AtomThe early Greek word for something that was so small it could not be subdivided was atomos. They used the term in reference to objects of time as well as matter. In the 1800's, chemist adopted the term atom to refer to what they thought was the indivisible unit of every element. The word actually is made up of two parts, the negative prefix a meaning not, and the root tom for cut, hence the meaning "cannot be cut". The Greek root is preserved in the word anatomy, "cut into pieces". The Greeks were not the only early culture to hit upon the idea of an indivisible smallest unit. In China this idea was represented by the word fen, and the Babylonians used shar.



Axis seems to be one of the older math words. The early Greek root aks for a point of turning or rotation. Words like axle and axon are derived from the same root. The Latins used the diminutive forms axilla, axla, ala for the shoulder or a wing. From this extension we get aisle, which separates a wing of a building, and axial, for the point where a bud grows from a branch.
The word seems to have first been used in English by Thomas Digges around 1570 in reference to the rotational axis of a right circular cone



Census Today a census is a complete enumeration or counting of a population. The Latin root of the word is censere which probably was derived from the Greek cosmos for order. The original meaning was of one who judged, assessed, or estimated and was used in judging the degree of wrong and assessing the penalty (tax) to be paid. It is clear from these combined activities that our word for censor (one who judges the acceptability of materials) and censure (expression of blame or disapproval) are both related. The first modern attempt at a census might well have been the ecclesiastical census of England in 1547. In 1694 an act was passed calling for a complete census of the English people, but it was never completed. The MAA's Historical Modules for the Teaching and Learning of Mathematics contain the following note on the US Census:

When the Constitution of the United States of America was written in 1787 the framers included a requirement that a national census be taken at regular intervals. It reads, “The actual Enumeration shall be made within three years after the first meeting of the Congress of the United States, and within every subsequent Term of ten years, in such Manner as they Shall by law direct.” (Article 1, Section 2.) Today every major country in the world conducts some sort of census on a periodic basis so that it can learn about its citizenry.
As late as the 18th century some people objected to the census on religious grounds, protesting their fear that the "Sin of David" would bring God's wrath upon them (see "Sin of David"). The MAA historical modules mentioned above comments on some of the biblical enumerations which were the earliest forms of a census.
The first evidence that we have of a census took place in Egypt around 3050 BCE when the rulers recorded information relating to the construction of the pyramids. (Meitzen, p. 15) Evidence that a census was conducted around the year 1500 BCE is recorded in the book of Numbers.
"Then the Lord spoke to Moses in the wilderness of Sinai, saying: Enroll the Levites by ancestral houses and by clans. You shall enroll every male from a month old and upward. So Moses enrolled them according to the word of the Lord, as he was commanded." (Numbers 3:4-5.)

Later in The Bible there is evidence that another census was taken around the year 1000 BCE. So the king said to Joab and the commanders of the army, who were with him,
“Go through all the tribes of Israel, from Dan to Beersheba, and take a census of the people, so that I may know how many there are.” (2 Samuel 24:1-2)[inserted note Some translations of the bible list this as " "Go throughout the tribes of Israel from Dan to Beersheba and enroll the fighting men, so that I may know how many there are." ]

About 400 years later in Egypt the police registered all heads of families. (Herodotus II., 177). And Emperor Augustus of Rome conducted a census around the year 7 BCE.
"In those days a decree went out from Emperor Augustus that all the world should be registered. This was the first registration and was taken while Quirinius was governor of Syria. All went to their own towns to be registered." (Luke 2:1-3)




Cone seems to come to us almost unchanged, in form or usage, from its early Greek roots. konos was the Greek word for a pine cone. It is probably related to an earlier proto Indo-European root ko, ke that related to sharpening. Objects having been sharpened to a point look like the tip of a cone.

Students are often confused between a cone, a conic surface, and a conic section because, in truth, their use even in textbooks seems to be imprecise. The following definitions are fashioned on those from Websters New World Dictinoary of Mathematics. :

A cone is a closed surface defined by a closed plane curve, C (called the directrix), and a point P(the vertex or apex) which is NOT in the plane of C. The cone consists of the lateral (side) surface of all line segments (called elements) from V to C, and the base which is C and its interior. The most commonly used version of the shape is the circular cone, where the closed curve C is a circle, but the base could be an ellipse, or any closed curve you could imagine.

A conic surface is an infinitely extended surface that is formed when the line segments of a cone are replaced with an infinite line. That means there is no base. The cone extends forever in both directions from the vertex and each of the coinical lateral surfaces is called a nappe of the conical surface.

The conic section [frequently just called a conic] is a curve determined by the intesection of a plane with a conical surface. They may also be defined by a point, called the focus, and a line, called the directrix [not the same as directrix used above for a cone], or sometimes two points called the foci (plural of focus). Students know these as the circle, ellipse, parabola and hyperbola.

If the directrix of a cone or a conic surface has a center, the line passing from the vertex through this center is called the axis of the cone/conic surface. If the axis is perpendicular to the plane containing the directrix then the cone/conic surface is called a right cone/conic surface otherwise it is called oblique.

A good source for information about definitions and formulas related to cones and conic sections is at the Dr. Math Faq page on formulas.



Cusp The word cusp comes from the Latin cuspis and was used to mean sharp or pointed. The points of a crescent moon are called its cusps, and the sharp pre-molar teeth of children are called bicuspids for their two points. In mathematics, a cusp is a point on a curve where two branches meet to have a common tangent line. The figure at left shows a traditional "pointed" cusp. Some mathematicians refer to figures like the one at right as a cusp also, but I find that an unsatisfying term. The curve on the left is called a first order or keratoid cusp from the Greek keras for horn. If the two branches of the curve approach the tangent from the same side it is called a ramphoid or second order cusp.


________


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

"Babylonians used base 60 notations 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 had 120 parts. As Pi was known to be close to 3, the circumference was taken to comprise 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 for 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. The fault is rather in omitting to mention that the quantity thus obtained is an approximation. Nowadays, the same sin is often committed by users of calculators many of whom can't tell the difference between 1/3 and 0.3333333333."
"Some history of the sexagesimal (base 60) notations appears in D.E.Smith, History of Mathematics, v2, Dover, 1958. This is a classic book, cheap and immediately available."
Alexander Bogomolny
http://www.cut-the-knot.com

Kim Mackey, a teacher from Valdez, Alaska, wrote to comment that he wondered that there might not be another explanation for why a culture with a base 60 system would use 360 degrees for a circle. I quote with his permission from a personal correspondence

I was going through your list of terms from the direction of the www.math.com website and came across your idea that you presented about why there are 360 degrees in a circle. I have a different idea. An equilateral triangle was easily constructed, even by the Babylonians. It makes sense to me that this powerful symbol would result in the Babylonians designating one of its angles as 1 base 60 unit. Since no matter what your units there are exactly 6 equilateral triangles in a complete circle (the circle of course circumscribing the triangles), then you would have 6*60 or 360 degrees in a circle. It makes even more sense when you consider the number of factors in 60.



Function

The origin of Function is in the Latin word fungi which means perform. It is not related to the modern word fungi, which came from the Greek word for sponge.
Perform is an appropriate root for the idea of a mathematical function as an operation, or operations, to be performed on a appropriate set of values, the domain of the function.

Leibniz first applied the word to mathematics around 1675, and according to Morris Kline, he is the first to use the phrase "function of x". Cajori's A History of Mathematical Notation credits Euler with the first use of f for a function:

The use of parentheses for this purpose [to enclose the argument of a function] occurs in Euler in 1734, who says "Si f (x/a + c) denotet functionem quameunque ipsius x/a + c." This is the first appearance of f for "function."{emphasis added}
. However, in e: The Story of a Number, Eli Maor credits Joseph Louis Lagrange with the creation of the symbol f. He also was responsible for the prime notation, f', f" etc; to represent the cascade of derivatives of a function.

In Euler, The Master of Us All, William Dunham has written, "Referring to the Introductio [Euler's 1748 book, Introductio in analysin infinitorium], historian Carl Boyer wrote, 'It was this work which made the function concept basic in mathematics.' Before Euler, analysis was about properties of 'curves'; afterward, it was about properties of 'functions'. The change was profound, forever altering the mathematical landscape.

The performance root base of function helps explain the meaning of related language terms like defunct for "no longer in use", that is, it is no longer performing. Dysfunctional, not performing properly, is another simple example.



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

It is interesting to note that the coordinate geometry that Decartes introduced in the 1600's did not appear in textbooks in the context of graphing equations until much later. In fact, I find it appearing in the mid 1800's in my old college texts in Analytical Geometry. It isn't until the first decade of the 20th century that graphing appears in standard high school algebra texts. Graphing is most often found in books by Wentworth. Even so, the texts written in the 20th century, perhaps until the l960's, did not all have graphing. Taking Algebra 1 in the middle 1950's, I did not learn to graph until I took Algebra II.

Math Historian Bea Lumpkin has written about the early use of graphs by the Egyptians.

In my article in : Arthur B. Powell and Marilyn Frankenstein eds, ETHNOMATHEMATICS, CHALLENGING EUROCENTRISM IN MATHEMATICS EDUCATION, SUNY, 1997, 113, I suggest: "It is possible that the concept of coordinates grew out of the Egyptian use of square grids to copy or enlarge artwork, square by square. It needs just one short, important step from the use of square grids to the location of points by coordinates."
In the same posting she comments on the finding of graphs in Egyptian finds dating back to around 2700 BC,
From Somers Clarke and R. Engelbach, ANCIENT EGYPTIAN CONSTRUCTION AND ARCHITECTURE, 1990, Dover, originally ANCIENT EGYPTIAN MASONRY, 1930,pp 52-3, "An architect's diagram of great importance has lately been found by the Department of Antiquities at Saqqara (Fig. 53). It is a limestone flake, apparently complete, measuring about 5x7x2 inches, inscribed on one face in red ink, and probably belongs to the IIIrd dynasty." Here is the reason that Clarke and Engelbach attached great importance to the diagram. It shows a curve with vertical line segments labeled with coordinates that give the height of points on the curve that are equally spaced horizontally. The vertical coordinates are given in cubits, palms and fingers. The horizontal spacing, the authors write "..most probable that it is to be understood as one cubit, an implied unit elsewhere (Journal of Egyptian Archaeology, xii, 134)." To clinch their analysis, Clarke and Engelbach observe: "This ostrakon was found near the remains of a solid saddle-backed construction, the top of which, as far as could be ascertained from its half-destroyed condition, closely approximated to the curve obtained from the data on the ostrakon (Fig. 54)."
This certainly lays claim to the oldest line graph I have ever heard of. (Bea Lumpkin comments have also contributed to the citation on Statistics with an early Eqyptian statistical record.)

Although coordinate graphs are usually described as starting with Des Cartes, there were some prototypes of graphs before him. Nicole Oresme who was the Bishop of Lisieus, France (1323-1382) displayed what might well be called function graphs in his writing. An image of a page with his graphs is here

One of the first statistical time series graphs is probably the one in The Commercial and Political Atlas, 1786, by William Playfair, brother of the more famous Scotish mahtematician John Playfair. A copy of the Playfair graph is found here. William Playfair lived from 1759-1823 and was a wandering Scot who introduced into statistics the line graph, bar chart and pie chart. He lived for a while in revolutionary France, but after a falling out with the revolutionaries he left there also. His falling-out with his Native Edinborough society occurred as a result of his publishing some negative remarks about Adam Smith. He should not be confused with the two William Playfairs (father and son) who lived about the same time and were both architechts.(The son was actually raised by the famous Scottish mathematician, John Playfair, an uncle, for whom Playfair's axiom is "mis"-named... and about which I will someday take the time to add a notation and provide a link here..) I recently became aware that there was another Scottish Playfair, Lyon, an eventual Baron, who lived near and slightly after these others who is somewhat known in mathematics because the Playfair Cipher was named for him by its inventor, Charles Wheatstone, who is better remembered for the Wheatstone Bridge [a laboratory instrument for precision measureing of electrical resistance]. There seems to be no close family relationship between him and any of the previously mentioned.

Just as a sidenote.. Wheatstone was a remarkably diverse inventor, who not only invented the resistance bridge, but is credited with invention of the accordion, the stereoscope, and a paper tape drive for sending Morse code telegraphs. This last is the first know use of paper tape for storage and transfer of data (1857).

The first topographic map by Marcellin Du Carla-Boniface appeared about the same time (1782) in France.

Florence Nightingale was an innovator in the creation of statistical charts. One of her most innovative is now called the coxcomb plot or polar area chart. In it she displayed information on deaths by cause during the different months of the year, but with the months in a circle around a center point.

Some Notes on graph paper:
from an unknown source:

Graph paper, a math class staple, became popular between 1890 and 1910, when the number of high school students in the U.S. quadrupled, and math professors took an active interest in improving high school education. Research mathematicians also took an active interest in improving high school education. E. H. Moore, a distinguished mathematician at the University of Chicago, served on mathematics education panels and wrote at length on the advantages of teaching students to graph curves using paper with “squared lines.”
The actual date of the first commercially published "coordinate paper" is usually attributed to Dr. Buxton of England in 1795 (if you know more about this man, let me know). The earlist record I know of the use of coordinate paper in published research was in 1800. Luke Howard (who is remembered for creating the names of clouds.. cumulus, nimbus, and such) included a graph of barometric variations. [On a periodical variation of the barometer, apparently due to the influence of the sun and moon on the atmosphere. Philosophical Magazine , 7 :355-363. ]
The increased use of graphs and graph paper around the turn of the century is supported by a Preface to the "New Edition" of Algebra for Beginners by Hall and Knight. The book, which was reprinted yearly between the original edition and 1904 had no graphs appearing anywhere. When the "New Edition" appeared in 1906 it had an appendix on "Easy Graphs", and the cover had been changed to include the subhead, "Including Easy Graphs". The preface includes a strong statement that "the squared paper should be of good quality and accurately ruled to inches and tenths of an inch. Experience shews that anything on a smaller scale (such as 'millimeter' paper) is practically worthless in the hands of beginners." He finishes with the admonition that, "The growing fashion of introducing graphs into all kinds of elementary work, where they are not wanted, and where they serve no purpose - either in illustration of guiding principles or in curtailing calculation - cannot be too strongley deprecated. (H. S. Hall, 1906)" The appendix continued to be the only place where graphs appeared as late as the 1928 edition.

The term "graph paper seems not to have caught on quickly. I have a Hall (the same H S Hall as before) and Stevens A school Arithmetic, printed in 1919 that has a chapter on graphing on "squared paper". Even later is a 1937 D. C. Heath text, Analytic Geometry by W. A. Wilson and J. A. Tracey, that uses the phrase "coordinate paper" (page 223, topic 153). Even in 1919 Practical mathematics for Home Study by Claude Irwin Palmer introduced a section on "Area Found by the Use of Squared Paper" and then defined "paper accurately ruled into small squares" (pg 183). It may be that the term squared paper hung on much longer in England than in the US. I have a 1961 copy of Public School Arithmetic ("Thirty-sixth impression, First published in 1910) by Baker and Bourne published in London that still uses the term "squared paper" but uses graphs extensively.

One of the most impressive graphic displays of information came from Charles Joseph Minard (1781-1870) in 1869, and depicted the size of Napoleon's army during the winter campaign against Moscow. For a very complete web page on the history of graphic innovation, see the page by Michael Friendly and Daniel J. Denis called Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization . In addition to some of the links given also includes this image of what is believed to be the first use of isometric lines (lines marking points with equal values of some variable). In the graph shown, credited to Sir Edmund Halley, the lines show points with equal magnetic declination from true North.

In "Blood, Dirt, and Nomograms, A Particular History of Graphs"; which holds the notes from a lecture by Thomas L. Hankins to the History of Science Society, he presents a nice exposition of the development of the term "graph" in English,

We have to be careful about using the word "graph" because it has had several different meanings. The word was coined in the English language in 1878 by the mathematician J. J. Sylvester, who used it to describe diagrams that he believed showed striking analogies between the chemical bonds in molecules and graphical representations of mathematical invariants and covariants of binary quantics. (See Figure 3.) In 1906 Charles Sanders Peirce wrote: "by a graph (a word overworked of late years), I, for my part, following my friends Clifford and Sylvester, the introducers of the term, understand in general a diagram composed principally of spots and of lines connecting certain of the spots."4 Peirce called his own logical diagrams "existential graphs" because they were also composed of spots and lines. Obviously these "spots and lines" were not what we now usually mean by a graph. Peirce's statement in 1906 that the term "graph" was being overworked of late indicates that at that time the word was beginning to take on its present, more general, meaning of a curve showing the relationship between coordinates. That leaves us with the question of what is to count as a graph. If we include maps and geometrical and astronomical diagrams, graphs are very old indeed. What was new in the late eighteenth century was a diagram with rectangular coordinates that showed the relationship between two measured quantities. Lambert called them "Figuren," Watt called them "diagrams," and William Playfair called them "lineal arithmetic." William Whewell, who seemed to rename everything that he came into contact with, called them the "method of curves."5



Interpolate and Extrapolate 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 a smooth curve between the values, and this is the literal meaning of the word, "to smooth between." The word was first used mathematically by John Wallis in the 1865 English language version of Arithmetica infinitorum, his algebra book Robin Wilson, in a Gresham College lecture says, "Another was his Arithmetica infinitorum, which introduced the mathematical term ‘interpolation’ ".

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 root is polire and means to adorn or polish. Polish is one modern word from the same root, 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 suggests that this is the root of psalm through the Greek word psallein, "to pluck", because the psalms were accompanied by a harp.

Extrapolate was created as an extension of interpolate and suggest the smoothing of a line outside the known points. This often is done in statistics when we study a pattern over time in order to predict future events.

Students should be aware that outside of mathematics, interpolate may have the negative connotation of giving false information, as in "filling in" false details.



Number Our word for number comes from the Latin numerus and probably earlier from the Greek nemein, to apportion or allot and also to take. The variations of the "take" form of the word produced numb (one who has been seized) and nimble (one who is quick to take). "The apportion" part seems to be the root of our words like astronomy, and binomial. The word numerist was the title of a Roman officer who kept registers.

Numeracy Here is a description of the origins of numeracy from the book A CALCULATING PEOPLE by Patricia Cline Cohen, which I felt was too good to paraphrase. " The word numeracy is a relatively recent addition to the English language. It has the awkwardness of a concocted word not yet weathered to smoothness through frequent use. The ... Oxford English Dictionary has leanly defined it as 'ability with or knowledge of numbers' and locates its origins in a 1959 report on English education that contrasts illiterate scientists with innumerate humanists. It was intended to be the analogue of 'literacy' ... In this context 'illiterate' did not mean 'unable to read', nor did 'innumerate' mean 'unable to add'; rather the words referred to an unspecified degree of deficiency at high levels that hindered communication among scholars."



Obelus The symbol "÷" which is used to indicate the operation of division is called an obelus. The word comes from the Greek word obelos, for spit or spike, a pointed stick used for cooking. Perhaps because both are sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger. The root also gives rise to the word obelisk for a pointed pillar of stone. The symbol was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out". It had also found occasional use as a symbol for subtraction. It was first used as a division symbol by the Swiss mathematician Johann H Rahn in his Teutsche Algebra in 1659. By a misunderstanding of a credit to John Pell about other material in the book, many English writers started using the symbol and calling it "Pell's notations". It is one of the most territorial of all math symbols, appearing in regular use in both the US and Great Britain and yet nearly non-existant in the rest of the world.



Planet In the days before every city had public lights that dimmed the heavens to our view, and every family had television to bring us in out of the night, the ancients could look up into a sky that seemed filled with a blanket of stars rotating in lockstep around the heavens. But when they looked up from what they believed was an Earth at the center of the universe, some lights in the sky did not act like the stars. The Sun and the Moon seemed to almost race across the day and night respectively. They were certainly different, and over time the ancients noticed that five more bodies seemed to "wander" across the sky against the background of the stars. These were the planets that are, even now, visible to the naked eye, Mercury, Venus, Mars, Jupiter, and Saturn. The Greek word planasthai meant, to wander, and so they took the name planet for these wanderers in the night sky. The ancient astronomers who would predict your fate studied the position of the seven heavenly wanderers at the time of your birth, and other important events in your life. Many believe these seven visible celestial objects are the reason we have a seven day week, and also account for the idea of "Lucky Seven". You were indeed lucky if you were born with all seven wanderers in favorable positions.



Prism A prism, from its origin, is the shape you would get if you took a saw and cut off the edges of some other shape. Mathematically, a prism is a shape with congruent polygons for bases and parallelograms for sides.
The root word is the Greek prizma which relates to cutting or sawing. A prismatoid is a term for a polyhedron with all its vertices lying in two parallel planes. Cut pieces of glass and crystal sometimes refracted light into a rainbow of colors, and so any multicolored or iridescent object is called prismatic. A prismoid is similar to a prism, but the two bases need only be similar, and the sides would be trapezoids rather than parallelograms. An example of a prismoid would be a frustum of a pyramid.
You can see images of prisms, and the associated geometric formulas for their volume and area at Dr. Maths Geometric Formula page



Product past participle of produce, which is the union of the Latin pro for "forward" and ducere which means, "to lead". The original meaning was thus to lead forward. Product, then, referred to that which was lead forward. It has been suggested that the mathematical meaning of the result of a multiplication may arise from the fact that multiplication of whole numbers leads "forward" to a larger result. I think it is more probable that the usage comes from the more modern meaning of the word product, an item produced as a direct result of some other action, in this case the multiplicative unification of two factors.

The ducere root is also the ancestor of duct, a tube or pipe for leading air or water. Duke, Duchess, conduct, seduce (literally, to lead away) and induce. And one other special word from the same root is educate, which came from Exducate, to lead out.



Pyramid is from the Greek root peramus. The same word also referred to a type of wheat bread or cake made in the shape we now call a pyramid. Some think that the Greeks adopted this name for the great Egyptian objects because they looked like the wheat cake. Others suggest that the Greeks adopted the Egyptian name for the objects, and then named, or renamed, the wheat cakes because they looked like the architectural objects.
I recently read a posting that suggested that the original word came from the Egyptian word for fire, because a pyramid was shaped like a fire. This may have some merit as the Greek word for fire was pura. This same root gives us the name pyrite (firestone in Greek), and the root for lots of fire words like pyromaniac. The ancient Egyptian term for pyramid was mer.



The Sin of David In the bible stories in Samuel and Chronicles, God sent a plague on the people of Israel because of David's "numbering the people". It is not clear what the word meant in this context, or why God was displeased, but thousands of years later religious people fought to defeat the idea of public numeration (census) based on the fear that it would bring God's wrath.

[from 2nd Samual 24]
"David was conscience-stricken after he had counted the fighting men, and he said to the LORD, "I have sinned greatly in what I have done. Now, O LORD, I beg you, take away the guilt of your servant. I have done a very foolish thing."
11 Before David got up the next morning, the word of the LORD had come to Gad the prophet, David's seer: 12 "Go and tell David, 'This is what the LORD says: I am giving you three options. Choose one of them for me to carry out against you.' "
13 So Gad went to David and said to him, "Shall there come upon you three [b] years of famine in your land? Or three months of fleeing from your enemies while they pursue you? Or three days of plague in your land? Now then, think it over and decide how I should answer the one who sent me."
14 David said to Gad, "I am in deep distress. Let us fall into the hands of the LORD, for his mercy is great; but do not let me fall into the hands of men."
15 So the LORD sent a plague on Israel from that morning until the end of the time designated, and seventy thousand of the people from Dan to Beersheba died.


Solidus The slanted bar, "/", that is used for fractions, and division is often called a solidus. If you think that looks too much like solid to be a coincidence, you are right. The word comes from the same root. From the glory days of Rome to the Fall of the Byzantine Empire, the solidus was a gold coin ("solid" money). The origin of the modern word "soldier" is from the custom of paying them in solidus. According to Steven Schhwartzman's The Words of Mathematics, the coins reverse carried a picture of a spear bearer, with the spear going form lower left to upper right. He suggests that this is the relation to the slanted bar. Cajori seems to indicate (footnote 6, article 275, Vol 1) that the symbol is derived from the old version of the latin letter s. This / symbol is also frequently called a virgule. Prior to the conversion to decimal coinage in the United Kingdom, it was common to use the symbol as a division between shillings and pence; for exampel 6/3 would indicate six shillings, three pence. Because of this use the symbol is also sometimes referred to as the shilling mark.



Transcendental A transcendental number is a number that can not be described by algebraic equations with rational coefficients. They were named because they "transcend" (go beyond) the bounds of algebra. A well known transcendental number was found by Liouville in 1844. The number has all zeros except for the digits at position 1,2,6,24, 120.. n!, which are ones.

Algebraic numbers are those which are solutions to a polynomial of the form a1xn+a2n-1++...+an=0. In the preface of When Least Is Best by Paul J. Nahin, he includes a good illustration of the strange interplay between rational, irrational-algebraic, and transcendental numbers. The number 2 is a solution to the algebraic equation x-2=0 and is a rational number. The square root of two,, is irrational, but is still algebraic because it is a solution to x2-2=0. If we raise this irrational to itself, rasied to we get a number that is not only irrational, but also transcendental. But what if we raise this transcendental power to the irrational power of ? When we raise a power to a power, we multiply the powers so [(a^b)^c)=a^(b*c)] (the quantity of the square root of two raised to the square root of two) raised to the power of the square root of two is just , squared, which brings us back to 2.... A transcendental number raised to an irrational power may give a rational integer.

The trans is from the Latin for "over" or "across". The second part is from the Latin root scandere, to climb. Literally then, the transcendental numbers climb over the algebraic boundary that held the other irrationals. The more ancient root skand gives rise to interesting words such as scandal. The Greek skandalon meant a snare and then more generally anything you could trip on. And if you trip over a moral snare, you may become involved in a scandal. The Latin scalae, from the same root meant ladder or steps to climb on, and worked its way into the modern words echelon, escalate, and scale.

Here are parts of two posts from the Historia Matematica list which amplify the development of the term. The first is by Daniel Burckhardt :
leibniz introduces in the 1670s both the concept of transcendental curves and quantities.
for curves: "Ich nenne also die Kurven, die Herr Descartes zugelassen hat, algebraische, weil sie von einem bestimmten Grad einer algebraischen Gleichung sind, und die anderen transzendente, [...]. Und wenn ich so sagen darf, habe ich damit die Absicht, die Analysis über die Säulen des Herkules hinauszubringen." [Leibniz to Arnauld, 14. 7. 1686]
he can prove in his quadratura from 1676 that both sine and logarithms must be transcendental curves.
on transcendental quantities: "So wie die Unmöglichkeit der Wurzelziehung im Rationalen zu irrationalen Grössen führt, so führt die Unmöglichkeit der Integration algebraischer Ausdrücke zur transzendenten Grössen." [LMG V, 377, Übersetzung nach Breger 1986, 128]
leibniz suspects that the ratio between the circumference and the diameter of a circle is transcendental, even if he could not find a proof for this yet: "Ich sehe noch nicht, was daran hindert, dass der Kreisumfang selbst oder irgendein festgelegter Teil von ihm gemessen wird, und das Verhältnis eines gewissen Bogens zu seinem Sinus durch eine Gleichung eines bestimmten Grades ausgedrückt wird. Aber die Beziehung zum Sinus im allgemeinen durch eine Gleichung eines bestimmten Grades zu erklären, ist unmöglich, [...]" [LMG V, 97]
LMG: Leibniz, G. W. Mathematische Schriften. Hrsg. von C. I. Gerhardt. 7 Bde. Berlin [später: Halle] 1849-63. [Nachdruck Hildesheim 1971: Olms] can be found at gallica:
for more information on leibniz' use of transcendentals, see Breger, Herbert. 1986. Leibniz’ Einführung des Transzendenten. In: Albert Heinekamp (Hrsg.), 300 Jahre "Nova methodus" von G. W. Leibniz (1684-1984): Symposium d. Leibniz-Ges. im Congresszentrum “Leewenhorst” in Noordwijkerhout (Niederlande), 28.-30. August 1984 (= Studia Leibnitiana Sonderheft 14), 119-132. best
and a second by James A Landau
Jeff Miller's Web Page "Earliest Known Uses of Some of the Words of Mathematics" at URL http://members.aol.com/jeff570/t.html gives the following:
Leibniz coined the word transcendental in mathematics, using transcendens in the fall of 1673 in Progressio figurae segmentorum circuli aut ei sygnotae.
Leibniz used curvae transcendentes and figurae transcendentes in December 1674 in De progressionibus et geometria arcana et methodo tangentium inversa.
Leibniz used aequatio transcendens in 1676 in Series convergentes duae. [these Leibniz papers can be viewed at an archive of publications by Leibniz at ].
According to Paulo Ribenboim in My Numbers, My Friends, "LEIBNIZ seems to be the first mathematician who employed the expression 'transcendental number' (1704)."
Euler used transcendental in his 1733 article in Nova Acta Eruditorum titled "Constructio aequationum quarundam differentialium quae indeterminatarum separationem non admittunt":
Now there are kinds of constructions, which can be called transcendental, which arise in solving differential equations and cannot be transformed into algebraic equations.
Euler used a phrase which is translated transcendental quantities in 1745 in Introductio in analysin infinitorum. Euler wrote that these numbers "transcend the power of algebraic methods" (Burton, p. 603). He also used the term in the title "De plurimis quantitatibus transcendentibus, quas nullo modo per formulas integrales exprimere licet," which was presented in 1780 and published in 1784 in Acta Academiae Scientarum Imperialis Petropolitinae.
An English translation of the 1745 Euler quote, if my notes can be trusted, can be found in D. J. Struik, A Source Book in Mathematics, 1200-1800 Princeton NJ: Princeton University Press, 1986, pages 346-350. "Burton" is Burton, David M. The History of Mathematics - An Introduction. Dubuque, Iowa: William C. Brown, 1988.


Vertex/Vertices Vertex is from the Latin vertere, to turn, and had meanings related to highest or foremost. We still use vertex for the area on the top of the head which would be the highest point of the body from the ground. Vertex had uses related to circular turning, and it seems that vortex, fluid flow rotating about an axis such as whirlpools, came from an alternate spelling of vertex. Vertigo, a disorientation which makes you feel as if you are spinning about, is from the same root as are verse, vertebra, and wreath. The use of Vertical for a line, or other object, perpendicular to the horizon also comes from the same root. (See also versine)



Yard In ancient times before paved roads and public services a staff or rod had many uses. It could help you keep your balance on uneven cobblestone paths. It could be used to push uncooperative livestock out of your way or defend an attack by rogues and bandits. It could also be used as a rough measure of distance by laying them end to end and counting the number of lengths to cover a distance. In the old Germanic languages gazdaz was the name for a staff or stick carried for such purposes. The English changed this to gierd and eventually to yard. Variations and remnants of the early meaning still persist. One example is "goad", to push with the staff, and sailors know that a tapered spar (long straight piece of wood or metal) is used to support a square sail, and this is also called a yard.

In France, the equivalent unit is called a verge, from the Latin root virga for a stick or rod. Verge remains to day as the name for the staff carried as an emblem of authority, often in religious processions. I think this use of a verge as a guide for those following is responsible for the use of verge in the sense of leaning toward or going toward, as in "I'm on the verge of a breakdown."

The area around your house where you have grass and a garden is also called a yard, but came from a non-related root.



Symbol for Zero Positional number systems seem to have been invented around 2000 BC, but the idea of using something, a symbol, to represent nothing, the absence of a quantity, did not arrive until the time of Alexander, around 300 BC. It may be that the Mayan civilization of Central America had a symbol for zero even earlier. The Mayans had a sort of place value system that used a closed shell as the place holder for complete sets, much as we use a zero. More about Mayan mathematics can be found here
Babylonians tried several different zero-symbols. Usually these constituted counting symbols slanted to the left or right. Sometime around 100 AD the Greek astronomer Ptolemy left one of the first records of the use of an open circle as a zero marker.
Robert Kaplan suggests in The Nothing That Is, The Natural History of Zero that the merchant class of Greece may have used the open circle as far back as Alexander for commerce, but this is not noted in the writings of that time because of the Greek intellectuals' scorn for the commercial applications of math. He suggests that perhaps the symbol spread to India with the movement of Alexander's army.
Whether they obtained it from the Greek's or invented it on their own, by the 8th century AD the mathematicians of India were using a dot, bindu and the open circle shunya-bindu to represent the empty space left by a marker on a sand counting table. Our present day ellipsis (...) which shows something missing is from this same source.
The Arabs conquest of Western Asia and North Africa spread the "nine Arabic numbers and the single 'cipher' and by the 1200's they had been translated into Latin to begin to revolutionize European mathematics. For the next six-hundred or so years the "0" symbol, under several names, would emerge from the status as representative of something missing to gain full and equal stature as a number