The three words above are a frequent source of confusion. The symbol ^ used as an exponent in mathematics comes from the same symbol used in proofreading to indicate something is missing. The word for this symbol is caret and it is drawn from the Latin root carere which literally means to cut off or be without, but came to mean something is missing. The basic root refers to cutting, or cutting off, and is the same root that gives us castrate (to cut off), castle (a building cut off from the surrounding danger), caste (a people cut off from the rest of society), and chaste (cut off from certain pleasures).

Part of the confusion is because the caret symbol, ^, looks like a little horn, and the root of carat is the Greek keration which means a little horn according to my dictionary. Some other possible origins are given below. Perhaps at some time animal horns were used as a reference for weight, but now the carat represents a weight equal to 200 milligrams. The standard is used for precious stones such as diamonds. The web page How Many? A Dictionary of Units of Measurement, offers two alternatives to the "little horn" meaning of carat. One suggestion is that the word comes from an Arabic word for the seed of a coral tree. Another suggestion is that the Greek word was for the name of a carob bean. Perhaps the carob bean or the coral tree seed or both look like a little horn.

Karat also relates to precious materials, in this case gold. A karat is a unit of purity for gold representing 1/24 gold. 12 karet gold is 50% gold and 50% other material. It is drawn from the same root as carat. To add to the confusion, it seems that the British spell both words with a c, Germans spell both with a k, and in the US we use one of each.

The familiar garden vegetable called a carrot is from the same root as keration. The ancient root has left its mark with words ranging from migraine to cervix, and the English "hart" for a stag.

The process of casting out nines has been around for a long time. It seems to have first appeared in the the tenth century and quickly became the common method of checking arithmetic. Heinz Lueneburg wrote that "Fibonacci used the remainders modulo 7, 9, 11, 13 (and others?) to check addition, subtraction, multiplication, and division with remainder. He knew and proved that each number is congruent to the sum of its digits modulo 9. Checking addition using the number 9 was known before Fibonacci. I think it is already in the latin versions of Al-Hwarizmi's arithmetic."

The remainder on division by nine is also known as the digital root of the original number. [Professor Rickey at USMA credits Henry E. Dudeney with first use of the term digital root.] The digital root refers to the repetitive act of taking the sum of all the digits of a number, and if the sum has more than a single digit, repeating the process on the new value until a one digit number is derived. The remaining number, the digital root, is the remainder on division by nine (a root of nine is reduced to zero). For example to find the digital root of the number 4587, first we add the digits ... 4+5+8+7 = 24 and since 24 has more than one digit we apply the same rule to its digits to get 2+4 = 6 so the digital root is 6.

The "Casting out nines" name for the number comes from the fact that we can discard any digits of nine in the number, or any groups of numbers that add up to nine or a multiple of nine. In the case of 4587, we can drop the first 4 and 5 since 4+5 = 9 and just find the sum of 8 + 7 = 15, and 1+5 = 6 to get the same result.

The method also appears in the Treviso Arithmetic which D E Smith calls "the first book on arithmetic to appear from the newly established presses of the Renaissance period.". The book was published in 1478 and takes its name from the Italian city where it was published. The author and publisher are both unknown. The book gives the method to add 59 + 38 =97 and then adds "If you wish to check the sum by casting out nines, add the units, paying no attention to 9 or 0, but always considering each as nothing."

In answer to a question I asked on Historia Matematica, Albrecht Heeffer responded

"Casting out nines is believed to be of Indian origin, but it does not occur before 950. Maximus Planudes called it "Arithmetic after the Indian method". Avicenna "the method of the Hindus" (Swets 1987, p. 189). For a history see:
- M. Cantor (1907) Geschichte der Mathematik, ed. 3, I, 756-757
- Dixon (1952) History of the Theory of Numbers, I, 337-346
- J. Tropfke (1980) Geschichte der Elementar Mathematik, p 166: "erstmals bezeugt in den Mahasiddhanta von Aryabhata II" (c. 950)
- D. Singmaster (2000) Sources in Recreational Mathematics, ed. 7, � 7.K.1, p. 445

It was the most common test of addition in 16th century arithmetic books. Clavius calls it "Probatio additiones per 9" in Epitome Arithmeticae Practicae, 1607, K�ln, p. 16-17.

The above quotes a 950 date for the creation of casting out nines, and includes David Singmaster as a reference, so I found it strange to find the following on Singmaster's website: "325(AD) Iamblichus: On Nicomachus's Introduction to Arithmetic - first mention of Casting Out Nines; first description of the Bloom of Thymarides; first Amicable Numbers."

Kim Plofker added in another response,

The Sanskrit work where this method first appears is, as the Tropfke reference stated, the (probably) tenth-century _Mahasiddhanta_ of Aryabhata (II). The method is described in the last four verses of the work (verses 67--70 of Chapter XVIII, p. 245 of the edition by Sudhakara Dvivedi, Benares Sanskrit Series (ed. G. Thibaut) 148--150, Benares 1910). The verses are translated and the method discussed in Datta and Singh, _History of Hindu Mathematics_, vol. I, pp. 180--183. (There's a 1966 edition with English translation of the first part of the _Mahasiddhanta_ by S. R. Sarma, but it doesn't include the casting-out-nines bit.)

Michel Ballieu sent the following which includes the pharase in Latin with a translation

In Fibonacci liber abaci (1202, revised 1228) (Scritti di Leonardo Pisano pubblicati da Baldassare Boncompagni, Roma, 1857 (Vol. 1)) we find, for instance on page 34, after the division of 13976 by 23 whose solution is 13976 : 23 = 607 and 15/23

Uerum si prescriptam diuisionem per pensam nouenarii probare uoluerit accipiat pensam de 13976 que sunt 8 et seruet eam ex parte. Et iterum accipiat pensam exeuntis numeri, scilicet de 607, que sunt 4 et multiplicet eam per pensam de 23, que sunt 5, erunt 20; de quibus accipiat pensam, que sunt 2 et addat eam cum 15 que sunt super uirgulam de 23, erunt 17, quorum pensa sunt 8, sicuti superius ex parte seruauimus.

We may quickly translate this by :
In fact if you want to verify the preceding division by casting out nines take "pensa(m)" (a word with the idea of weight) of 13976 which are 8 and keep them aside. And again take pensa(m) of the outgoing number, i.e. of 607, which are 4 and multiply them by pensa(m) of 23, which are 5, they will be 20; take pensa(m) of these 20 which are 2 and add to them 15 which are upon the bar of 23, they will be 17, whose pensa are 8, as higher in what we kept aside.

Although it was commonly used in textbooks, it was sometimes dismissed with little comment. Note the comment on checking multiplication from an early (1772)arithmetick book by John Hill,

You could use the remainder of division by any prime to check math operations. Fibonacci points this out in the Liber Abaci (1202) and gives examples for seven and eleven. The big advantage of nine is the simplicity of finding the remainder by just adding the digits. The big disadvantage is that it doesn't catch one of the most common errors, a digit reversal. Casting Out Elevens will find digit reversals, and the method is almost as easy. To find the remainder on division by eleven, just start at the ones digit and alternatly add and subtract as you move to the left. For example, 3124 divided by 11 would have a reaminder of 4 - 2 + 1 - 3 or zero, so it is divisible by 11. Switching hte middle two digits by mistake would give 3214, but on division by 11 we get a remainder of 4-1+2-3 = 2, thus catching the switch of two adjacent digits which the casting out nines would not do. Augustus de Morgan included both methods in his "Elements of Arithmetic" in 1861, and points out the reason the method works; "Observe that 10+1, 100-1, and 1000+1 are all divisible by 11." In spite of the fact it was well known to the math community, there seems to be little evidence that this was widely included in early arithmetic texts (at least that I can find) but I hope if some one finds it in old arithmetics, they will send me a photo. It is mentioned by Here is an explanation of why both methods workfrom the wonderful folks at Dr. Math... ...

The words circumscribe and inscribe are often used to describe putting a circle around, or inside a figure, and students often become confused when the words are used to describe one polygon inscribed in, or circumscribing, another. Today the general definition of circumscribe means to enclose (a figure or solid) within a configuration of lines, curves, or surfaces so that every vertex of the enclosed object is touching the enclosing configuration. The words circumscribe and inscribe are used in conjuction with each other; if A circumscribes B then B is inscribed in A. For example in the images below, the triangle on the left is inscribed in a circle and the circle circumscribes the triangle. In the second image the circle is inscribed in the triangle, and the triangle circumscribes the circle. The third figure shows one triangle inscribed in another. We might also say that the inside triangle is circumscribed by the other. .

Solids may be circumscribed by other solids. For example we may speak of a cone inscribed in a cylinder, which is the same as a cylinder circumscribing a cone. The root of Circum is from the same Latin root as circle, circus. Scribe is from the Latin scriba, to write, mark or draw.

The creation of the this term, and the use of the symbol r is generally credited to Sir Francis Galton. Here are some comments from a discussion on the AP Statistics discussion group about the origin of the terms. The First is from Michael Bulmer who has written a (apparently no-longer) online book on Galton, "Francis Galton: Pioneer of Heredity and Biometry". The book is now available from Amazon.

In his original paper on correlation (Co-relations and their measurement, chiefly from anthropometric data, Proc R. Soc. 1888, 45, 135-145) Galton used r for the "index of co-relation" that we now call the correlation coefficient. He calculated it from the regression of y on x or of x on y after standardizing the variables to have the same probable error, and it is fairly clear from the text that he used r for (standardized) regression.

Dan Yates, a mathematics teacher and author of the frequently used "YMM" Statistics Textbook posted the message from Michael above and had also added

We do know that Sir Francis Galton invented the words 'regression' and 'correlation' and he also gave a definition for r. Although Legendre gets the credit for developing the least-squares method in about 1805, it was Galton who turned regression into a general method for understanding relationships. The term "regression" likely suggested the symbol r for correlation. See page 188 in "The Practice of Statistics," 2nd edition, page 118.
A good reference for Galton and his contributions can be found at Michael Bulmer's online book, "Francis Galton: Pioneer of Heredity and Biometry" which can be found at http://www.francisgalton.com Check out Chapter 5. [Note, it seems that the site has gone somewhere new and it is not clear how you get access now]

In Stephen Stigler's Statistics on the Table he says that it was the work of Adolph Bertillon, perhaps the first "criminoligist" in history and the inventor of the idea of mug-shots for arrested criminals, that led Galton to the idea of statistical correlation. Indeed in his 1888 aritcle that introduced the notion we find :

In the same book Stigler states that Galton was lead to the choice of "correlation" by the reading of a book by economist Willam Stanley Jevons.

Jevon's Principles of Science did make one contribution to statistics that is worth noting; it helped give us 'correlation.' The book does not present the concept of correlation ; rather it seems to have been Galton's source for the word. The word was then in common use, for example, by Darwin in the Origin of Species (1859) and by W. R. Grove in The Correlation of Physical Forces (1865), but it was Jevon's use that we know caught Galton's eye at a crucial time. In his personal copy of Jevon's book, Francis Galton marked the explanation of the term and wrote in the margin:'nice wd." [Statistics on the Table, Note 1, Pg 89]

Here are links to the etymology of the words correlation and coefficient

The word half comes into English from the Old English healf which is almost surely related to the Germanic halbaz for divided or cut in half. I am assured that the modern German language still has a word anderhalben for one and a half. Danish has a similar word halvanden for one and one-half that sounds even more like the Modern English word and similar words exist in Norway and several other countries. The Greek root for one half is hemi, and in Latin the root is semi.

I recently read a question on a newsgroup that must be echoed by students in classrooms all over the English speaking world, "Why do we say hemisphere and semicircle and never the other way around?" The excellent answer from David W Cantrell was

"Because, when forming words, it is considered "good form" if all parts come from the same language. In "hemisphere" both parts are Greek; in "semicircle" both are Latin.
Mixing parts from different languages in a single word is "bad form", although this does happen. E.g., "television" is a Greek-Latin hybrid. There is even an example of such a hybrid in mathematics, and it has an amusing (though possibly apocryphal) story: The engineers at IBM wanted a word to describe the base-16 arithmetic which they were using. Even though they realized that "sexadecimal" would be the better form, they thought it sounded risque! That is how the hybrid "hexadecimal" came to be used.

The use of "half of" versus "halves of" was the subject of the "Word for the Wise" column from the Merriam Webster dictionary people. Here is how they explained the double use:

We are now officially halfway through the year. Does that mean July second is considered the day the year 2007 is cut in half? Or is it more proper to describe the year as cut in halves? Believe it or not, one can find usage writers willing to quibble about this.
The argument for dividing something in halves is (of course) twofold: first, whatever verb results in a thing being in two parts results in two, not one, halves, so the thing divided is in halves, not half. Second (and relatedly), one doesn't say "divided into third" or "cut into quarter," so why should this fraction be different? Nonetheless, the in half construction is completely defensible as an established idiom, and that's not the half of all the half talk, either. Not the half is another established idiom meaning "significant part; crux;" or "whole." Then there's by halves, of course, which means "halfheartedly; inadequately or incompletely;" and the slangy and a half, used to suggest something of "great size, importance, difficulty, or perplexity." Do you have half a mind to forget all this fussing and just say what comes naturally, cut in half or cut in halves? We'd say to go right ahead; you're bound to find at least half the usage commentators agree with whatever you choose.

When James Watt formed a partnership with Matthew Boulton to capitalize on the improved steam process Wat had designed, the major use of steam engines was for the pumping of water, water from the mines around cornwall, or to refill mill-ponds in dry summers, or to remove water from fields and canals.

The machines were sold under a strange (by todays standards) business agreement. Boulton and Watt would design the machines and sent the parts out to be manufactured and delivered to the customer where the customer would hire a crew under the direction of one of the designers engineers to assemble the engine. Boulton and Watt would take a regular payment equivalent to 1/3 the cost savings on coal over the old Newcomen engines. But when they went to make steam engines to run the grinding process for two London brewers, Godwyn and Whitbread, the scheme ran into a difficulty. The brewers had used horses to drive the mills, "so there was no savings of coal, and Watt had to work out a new way of calculating royalties. Eventually he estimated that a horse could raise 33,000 lbs of water one foot per minute. 'Horsepower' has been the unit of power for engines ever since". Quotes from The Lunar Men by Jenny Uglow

The word isoperimetric literally means "same perimeter". It is usually used in mathematics to refer to to figures having the same perimeter but different shapes. Mathematicians use the term Isoperimetric Problem to describe problems relating to finding which of two figures with equal perimeter have the greatest area. The problem dates back at least to Zenodorus , a 2nd century BC Greek mathematician who wrote On Isometric Figures. In it he showed that of all isoperimetric polygons having the same number of sides, the regular polygon had the greatest area. [For example of all quadrilaterals having a perimeter of 16, the 4x4 square has the greatest area.] He also showed that the area of a circle was greater than any regular polygon with the same perimeter as the circumference as the circle.

A related isoperimetric problem is called Dido's problem. Dido's problem is to find the maximum area for a figure with a given perimeter and bounded against a strait line. For example, if you had 100 feet of fence and wanted to enclose the maximum area with one side of the property along a straight river, or the side of a barn; what shape would enclose the maximum area.

Dido was a Phoenician princess in Virgil's Epic tale, THE AENEID. The story tells of her founding of Carthage. When she fled from her brother, Pygmalion, who was trying to kill her (sibling problems seemed pretty dramatic in the Greek classics) she landed along the coast of Northern Africa and tried to buy land from the local king, King Jarbas. She was told she could have as much land as could be enclosed by a bull's hide. Dido had her followers cut the hide into tiny strips which were then strung together to make a great length, and enclosed a great area against the sea. That's where the strait line on one side comes from in the problem. Dido's solution was a semicircle.

Students may wish to pursue minor variations of Dido's problem, which produce an interesting pattern. Find the shape of the boundary with N sides for N=2, 3, etc that produces the maximum area for a given perimeter. The solutions turn out to be one half of a regular polygon with 2N sides divided by a diameter.

Dido's problem is sometime called the Problem of Hengist and Horsa for a similar conclusion to an English story about two wizards/lords who come from Germany to defend against a Saxon invasion in the fifth century. The two turn out to be the antagonists of the young King Arthur and Merlin. Even Dido's story is related to England in a round-about way. Her brother Pygmalion would create a beautiful statue and then fall in love with it. Aphrodite brought it to life for him. This transformation of a stone statue found its way through a George B. Shaw play into the famous English musical, My Fair Lady

Perhaps one more small side-note to bring all this full circle to my interest in language, the model for Professor Henry Higgins in the play was a real live linguist named Henry Sweet who was instrumental in the development of hte International Phonetic Alphabet in 1897.

A conic surface may be (mentally) divided at its vertex into two infinite, identical sheets. Each of these are called a nappe of the surface. Nappe is from a the same word in French which means a sheet of water. Imagine water rushing over a waterfall and the sheeting effect is what brings the term. The French word is actually drawn from an older Latin word nappa which meant a towel or tablecloth. The words origins remain today in the use of napkin, or "nappy" for a baby's diaper.

The foundation of the Imperial Stone weight is rooted in the ancient practice of comparing goods by hanging them in balance pans. We know from archeological finds that the Babylonians began to keep specific stones to use as a reference weight. At first, probably every shop or stall would have his/her own stones to measure with. In the British Isles eventually the stones became more or less standardized in a single area of trade, so that a STONE represented about sixteen pounds of wool, but only eight pounds of meat from the butcher, and fourteen pounds of horseweight to the horseman. Maybe it reflects the favor in which the British hold their animals, but eventually the official Imperial Stone was set at fourteen pounds. The origin of the word stone seems to be very old, as similar sounds for the word appeared in ancient English, German, and Norse languages.

A tree is a planer graph in which any two vertices are connected by exactly one path. Notice in the graph of a tree, at right, for example that if you pick any two vertices, there will be a path connecting them, but only one such path can exist. If there is more than one such path, or if there is no such path, then it is not a tree.

In the world of planer graphs, when you cut a tree you get more trees. If you remove one of the edges joining two adjacent vertices the figure will no longer be a tree (since some vertices are not connected to some others) but instead will form two trees. A newish name for two or more trees considered as a single graph is called a forest Another way to say the same thing is to say that a forest is a disjoint (not connected) union of trees.

A good furthur resource on trees is at Mathworld

According to John Allen Paulos, the Tree diagram was created by Christian Huygens, the Dutch Mathematician, during the 17th century. I was asked recently to cite the origin of this attribution, and was unsure, but think it came from Paulos' "Once Upon a Number." In addition, I came across a similar statement by A. Philip Dawid, Professor of Statistics, University College London; in a review of a book on probability by Glen Shafer.

The "probability tree" is a seemingly elementary device, first introduced by Huygens in 1676 to organise probability calculations.

Professor A. W. F. Edwards of Cambridge University recently (2006) answed my request for information on Huygen's use and confirmed the 1676 date. His post said,

According to me (Int Statist Review 51, 73-9, 1983) Huygens's manuscript is dated August 1676 and is headed: The last problem contained in De ratiociniis; proposed some time ago by Pascal. (emphasis added)
I commented 'It is interesting to note the use of an "event-tree" to describe the possible events, perhaps the first time such a diagram had been used'. Volume XIV of the Oeuvres, pp 151-5. My paper is reprinted in both editions of 'Pascal's Arithmetical Triangle'. A.W.F.Edwards

The first use of the word for a mathematical structure may have been by Cayley, and may have been prompted by the earlier use of "tree" for the similar structure used in genealogical records. Certainly Cayley was one of the first to discuss the objects mathematically in his "On the Theory of the analytical Forms called Trees." (1857)

After the 1880 census, C. W. Seaton, chief clerk of the U. S. Census Office, computed apportionments for all House sizes between 275 and 350, and discovered that Alabama would get 8 seats with a House size of 299 but only 7 with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. They also show a nice example (with small numbers) so you might check their site.

Hamilton's Method Hamilton's method may have been popular because it was so simple. It began with the selection of an Ideal Ratio for the population needed for each seat. This was found by taking the total population from the census and dividing by the number of representatives to be approtioned. For example the US Population is (according to piece of software on my computer) 281,424,177. The number of Representatives is 435, so the Ideal ratio for any state should be that they get one representative for each 646,592 people. So what about my home state of Michigan which has a population of 9955829. Dividing the population by the Ideal Quotient should give the number of representatives or quota for each state. Trouble is, of course, it doesn't come up with an integer answer. Michigan's 3.54% of the population should give 3.54% of the 435 Representatives, but 9955829/646592 = 15.3973.. A similar result happens in every states apportionment. If we truncate down to the whole number, we won't have 435 Representatives. If we round everyone to the nearest integer, we may get 435 by chance, but not for sure (algorithms and law both want "for sure".)

Hamilton's solution was to give each state the truncated number. Then to fill in the remaining available seats, they awarded them one at a time to the states with the largest fractional parts in their quota until they were gone. Here is an interactive java-script of Hamilton's method from Alexander Bogomolny at Cut-the-knot. For another example look here.

By contrast, Jefferson's method used a modification of the Ideal Quotient. Jefferson suggested a trial and error lowering of the Ideal quotient. Then each state would be given their truncated quota. For the Ideal Quotient, this would not produce enough representatives, but as the quotient was lowered bit by bit the sum of the truncated values would go up until, at some point it would be exactly the required number. The Cut-the-Knot site also has an applet on Jefferson's method. And here is another resource for an example (again using small values).

Webster chose to use the traditional rounding method right from the start. After we divide each state population by the ideal quotient, we round up or down depending on whether the fractional part is more or less than 1/2. This can lead to either too few or too many repersentatives being assigned, and so he added a method of determining which state to remove or add seats. If the number of seats was too few, then each states population was divided by the number of their number of assigned seats + 1/2. The state with the highest adjusted ratio would get an additional seat. This was repeated until the correct number of seats had been reached. If the number assigned was too high, they would subtract 1/2 from the alloted number and then divide; and eliminate one representative from the state with the lowest adjusted ratio.

The rebirth of mathematics which had stirred across the continent of Europe from the 12th through the 15th century had made little impact on England. By the time of Henry VIII, and the birth of Robert Recorde in Pembrookshire, Wales, in the year 1510, England lagged well behind the continent in mathematics and mathematicians. There were copies of the great mathematics texts from the continet, but as these were written in Latin few of the English tradesmen, builders, and sailors could read them. Whetstone of Wit was printed in English and presented in the form of a dialogue between a master and a student. The Whetstone of Wit is a shortened, and modernized interpretation of the rather imposing original title, The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers which he published in 1557. The title is actually a play on words. Algebra was referred to as the "cossicke practice" and the word "cos" was used as an unknown in equations. Cos was derived from the Latin cosa meaning "a thing". The word cos in Latin, however, meant a grindstone. The sharpenng of intelligence or wit, cos ingenii, became the English Whetstone of Wit.

The book contains the earliest record of the use of parallel horizontal lines, =, to represent equality, and was also the first English text to use the symbols "+" and "-" for addition and subtraction. Actually Recorde's equal signs were very long indeed.

The first section of the book deals with whole numbers, ratios, and the extrction of roots; in fact, over fifty pages are devoted to the topic of extraction of roots. The second part of the book is devoted to the solution of equations, and much of the work is a direct translation of the 1551 Algebra by Johannes Scheubel. The third part of the book is devoted to irrational numbers, which Recorde calls "surds".

A photocopy of the cover page to the book is here. A physical copy of the book is in the Turner Collection at Keele University.

In 1755, when Samual Johnson published his English Dictionary, he described width as a "low word". The word came into being in the early 1600's as a shortened form of "wideness", the more common, and, according to Johnson, more acceptable term. Today width is used in mathematics to represent a distance across something from side to side. The "th" ending in words like length (from long), width (from wide), truth (from true), etc is derived from a long line of Indo-european suffixes which were used on abstract nouns of state.