Math Words, pg 18


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Astronomical Unit

An astronomical unit, usually abbreviated as AU, is the mean distance from the Earth to the Sun, or very nearly 149,600,000,000 meters. The AU is also used in the definition of a parsec, short for "parallax second". A parsec is the radius of a circle for which a central angle of 1 second (1/3600 of a degree) would cut an arclength of one AU. This makes a parsec equal to about 206265 AU, or about 3 1/4 light years. If you like big numbers, the unit of distance called the Hubble, named for American Astronomer Edwin Powell Hubble, is a billion light years, or about 1/12 the radius of the Universe.



Atlas

One of the earliest (see note below regarding Waldseemuller) people ever to bind a group of seperate maps together into a single book was a Belgian-German cartographer named Gerhard Kremer. Kremer is better known for the Latinized version of his name, Mercator . The name Kremer in German, and Cremer in Dutch both mean merchant. The Latin name for a merchant was Mercator, and the word was commonly used for the travelling book merchants of Germany and the lowlands. When Kremer followed the fashion of educated German men of his period to Latinize his name, he chose Gerardus Mercator de Rupelmonde. Rupelmonde was his birthplace. Mercator gave the name Atlas to these volumes of maps to suggest that they "held" the whole world. He also included a graphic of Atlas holding the sphere of the world on the title page. Mercator explained his choice with the words, "I have set this man Atlas, so notable for his erudation, humaneness, and wisdom as a model for my imitation."

Mercator's Atlas was not the first publication of a systematic collection of modern maps. That honor would go to Martin Waldseemuller in 1507 (Mercator's complete Atlas was not published until 1595, after his death) when he published a Supplementum modernior seperate from his Ptolemaic geography, Cosmographiae Introducto. Among his other unique firsts, Waldseemuller should be remembered for championing the name of "America" for the "new" continent across the Atlantic. Waldseemuller wrote in his geograhy, "Since another fourth part (of the world, which until then had consisted of three parts, Europe, Asia, and Africa) has been discovered by Americus Vesputius, I do not see why anyone should object to its being called after Americus the Discoverer, a man of natural wisdom, Land of Americus or America...". He did not completly ignore Columbus, placing a note near the Caribbean Islands, he wrote, "These islands were discovered by Columbus, an admiral of Genoa..". In maps a few years later Waldseemuller gave Columbus credit for the discovery of the "new world", but after his death in 1518, the name "America" was used again on his maps. Waldseemuller used America only as a name for the South American continent. Mercator may have been the first, in his maps of mid-century, to include the name for North America. Waldseemuller also was the first to draw a world map that covered 360 degrees of latitude, and the first to publish a globe, released with his Cosmographiae Introducto with America on it. He was also the first to print a map with more that two colors. Mercator's Atlas was not the first collection of maps, but it was a step beyond Waldseemuller's collection in many ways. Mercator's maps were all original and to the same scale.

The earliest modern Atlas, in many minds, is the one by Abraham Ortelius, : Teatrum Orbis Terrarum. You can see the Cover page here

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Bayes' Theorem

Thomas Bayes was an English Presbyterian minister and mathematician who lived from 1702 to 1761 in and near London. He only published two things while he was alive: a paper that tried to prove that God wants everyone to be happy, and another defending the accuracy of the calculus. {in which Bayes defended Sir Isaac Newton's position...[my note]}

After he died, though, is when Bayes's most important work was published. Called "Essay towards solving a problem in the doctrine of chances," it was discovered by an acquaintance (one Richard Price, for those of you who like the details) who sent it to the Royal Society which published it in its Philosophical Transactions in 1763.

Out of that paper came what's called Bayes's Rule or Bayes's Theorem. It's a formula for guessing how likely it is that something will happen based on two or more independent events.

[From an article by Andrew Kantor, CyberSpeak, Sept 17, 2004; USA Today]

Bayes theorem has emerged recently as a news topic because of its use in two of the big headaches of modern life; internet spam and terrrorism. Links to news articles die quickly, but here is a link to one about Bayes' thm and Spam, which has been partially quoted above; and here is another about Bayes' Thm and terrorism. Hope they are there when you come to search.

Here is how the folks at Wikipedia explain Bayes' Theorem, in part:

"The main result (Proposition 9 in the essay) derived by Bayes is the following: assuming a uniform distribution for the prior distribution of the binomial parameter p, the probability that p is between two values a and b is


where m is the number of observed successes and n the number of observed failures. His preliminary results, in particular Propositions 3, 4, and 5, imply the result now called Bayes' Theorem (as described below), but it does not appear that Bayes himself emphasized or focused on that result.

What is "Bayesian" about Proposition 9 is that Bayes presented it as a probability for the parameter p. That is, not only can one compute probabilities for experimental outcomes, but also for the parameter which governs them, and the same algebra is used to make inferences of either kind. Interestingly, Bayes actually states his question in a way that might make the idea of assigning a probability distribution to a parameter palatable to a frequentist. He supposes that a billiard ball is thrown at random onto a billiard table, and that the probabilities p and q are the probabilities that subsequent billiard balls will fall above or below the first ball. By making the binomial parameter p depend on a random event, he cleverly escapes a philosophical quagmire that he most likely was not even aware was an issue.

Bayes is buried in Bunhill Fields in the heart of the City of London. Nearby in the same cemetery is his friend Richard Price who read Bayes'paper to the Royal Society after Bayes death. The cemetery seemed to specialize in non-conformists in the 18th century, but is now a park maintained by the City of London. Devlin's article gives directions to the site, "Stroll 100 yards south along City Road from Old Street tube station in London and on your right, you will find Bunhill cemetery." An image of Rev. Bayes tomb is here . An inscription on the top reads, "In recognition of Thomas Bayes's important works in probability this vault was restored in 1960 with contributions received from statisticians throughout the world." According to a site by the Bayesian organization (honest); "There are many notable graves, including John Bunyan, William Blake, Daniel Defoe, many of the Cromwell family and Susanna Wesley (mother of John Wesley, the founder of Methodism, who is buried across the City Road where his chapel still stands)."


The Big Bang

In 1929,Edwin Hubble discovered that stars which were farther away had a greater redshift, meaning they were moving away with a greater velocity than nearer stars. The obvious implication was that the universe was growing, getting bigger and farther apart every minute.

Strangly, this fact had been computed mathematically by Albert Einstein over ten years before, but in what he would call "the biggest blunder of my life" Einstein inserted an adjustment in his equation, a "cosmological constant", that would remove the expansion of the universe.

By the early 1950's, George Gamow looked backward in time to a moment when all the mass in the Universe was a bundle of energy the size of a softball. From that moment on, he felt, the universe had "blown up" like a balloon and continued to grow forever. A contemptuous phrase by English Astronomer Fred Hoyle described the Gamow model as starting with a big bang, and the term has resisted every effort to remove it.



Coordinates

The use of a graphing system based on two measures or ordinates from a common origin dates back to before Apolonius, but the use was not in any way like the association we make in modern analytic geometry. John H Conway points out that the term is of recent origin in a post to Historia Matematica, " Let me remark here that in all its early uses (in fact until quite recently), this word was almost always either hyphenated ("co-ordinate") or written with a dieresis. This, and the form of the phrases containing it, make it quite clear that through the 18th and most of the 19th centuries one thought of each of x and y as an "ordinate" rather than a "coordinate" - it's only together that they were "co-ordinates", or, as it were, "complementary ordinates"."

It seems that Leibnitz was the first to use the term COORDINATE (in Acta Eruditorum [1692], p. 170). Gabriel Cramer was the first to make formal use of two axes and to define two coordinates simultaneously with reference to the axes (in Introduction a l'analyse des lignes courbes [1750])



Derivative

The term derivative seems to have come from the pen of Lagrange in hisTheorie des fonctiones analytiques. In the same paper he introduced the prime symbols, f', f", etc for the cascade of derivatives of a function

The root of derivative, like that of derive, is from the word river



The word Galaxy is now used for any remote cluster of stars, but was orignally intended to describe our galaxy, the Milky Way. The word galaxy, in fact, comes from the Greek root for milk, gala. It is related to the Latin root lac from which we get words like lactate,to give milk, and similar words.

Harmonic Series

The Harmonic Series is the sum of a sequence formed by the reciprocals of the counting numbers. . The name dates back to the early Greeks and is related to the association between proportions and musical harmony as indicated by the following clipped from a response to a question at the Historia Matematica discussion list.

Randy Schwartz wrote "The harmonic sequence/series was so named for its connection to harmony in music. The Pythagoreans explored the notes produced by plucking the taut chords of the lyre, a small harp-like instrument. They discovered precisely how the length of a chord determines its pitch, and they found that when the length is divided by any whole number, and the resulting shortened chord is plucked, the note that it produces is harmonious with the original one. Thus, chords of lengths 1, 1/2, 1/3, 1/4, ... make pleasing harmonies and were said to be in "harmonic ratio" (see Van der Waerden, _Science Awakening_, pp. 93-6, 157). Many scholars believe that the Pythagoreans' mathematical study of ratio and proportion originated with this interest in musical harmony. "

The harmonic series (the sum of the terms of the harmonic sequence) is unique in that it was the first series discovered whose nth term went to zero, but whose sum diverged to infinity. The proof dates back to Nicholas Oresme in the 14th century. He showed that the sum diverged by grouping partial sums as below:
(1 + 1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + .... >
(1/2+ 1/2) + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + ...
and in general 1+1/2 + 1/3+...+1/2^n > (n+1)/2

Although the series does diverge to infinity, it does so very slowly. For example, the sum of the first one thousand terms adds up to 7.485, and after a million terms the sum still has only reached 14.357. Oresme was the first person to use fractional exponents, and developed a precursor of coordinate geometry long before des Cartes.

Jakob Bernoulli explored this and other infinite series in his Tractatus de Seriebus Infinitis and gave another proof of the divergence of the Harmonic series that he credited to his brother, Johann. The idea of small quantities adding to an infinite sum seemed to be the motivation to close the article with the following poem.

So the soul of immensity dwells in minutia.
And in the narrowest of limits no limits inhere.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!

As slowly as the Harmonic sequence grows, and as small as the fractions become, it is also somewhat surprising that the sum is never an integer after the first term of one. In fact, there is no consecutive subset of the series that is integral. The proof is based on the fact that any partial sum after n=1 will have a denominator that is even, and a numerator that is odd when expressed in its simplest form. In fact, beyond the sixth partial sum, (1+1/2 + 1/3 + ... + 1/6)= 2.45, there is no partial sum that can be expressed as a terminating decimal.



Induction and Inductive Proof

The OED gives a good definition of induction from a famous historical text."1875 TODHUNTER Algebra (ed. 7) xxxiii. §484 The method of mathematical induction may be thus described: We prove that if a theorem is true in one case, whatever that case may be, it is true in another case which we may call the next case; we prove by trial that the theorem is true in a certain case; hence it is true in the next case, and hence in the next to that, and so on; hence it must be true in every case after that with which we began." You can find more explanation, and worked examples at "Purplemath, Your Algebra Resource"

From the web page of Jeff Miller on the first use of some mathematical terms I found, "The term INDUCTION was first used in the phrase per modum induction is by John Wallis in 1656 in Arithmetica Infinitorum. Wallis was the first person to designate a name for this process; Maurolico and Pascal used no term for it (Burton, page 440). " Note that others had used the mathematical method of induction prior to Wallis' giving it its current name. I found a nice history note about Induction in a response to a Question at the Ask Dr Math site at the Math Forum. Dr Rob writes, "It is believed that the first person to explicitly state and use the Principle of Mathematical Induction was the French mathematician Joseph Louis Lagrange (1736-1813). He used it to prove the Binomial Theorem, which had been known earlier to Sir Isaac Newton.
It was formalized and further developed by Augustus DeMorgan (1806- 1871). At the St. Andrews University web siteDeMorgan is credited with the formalization of the term, "In 1838 he defined and introduced the term 'mathematical induction' putting a process that had been used without clarity on a rigorous basis. The term first appears in De Morgan's article Induction (Mathematics) in the Penny Cyclopedia. (Over the years he was to write 712 articles for the Penny Cyclopedia.) The Penny Cyclopedia was published by the Society for the Diffusion of Useful Knowledge, set up by the same reformers who founded London University..."
There are many equivalent forms of this idea. Probably the first to use an equivalent form was Pierre de Fermat (1601-1665), who used what he called the Method of Descent.
The idea was to show that if a value of n exists for which a statement is false, then a smaller value also exists for which it is false. Using this, a still smaller false value could be found, and so on. This would produce an infinite chain of descending values of n for which the statement is false. This situation is impossible, however, since there are only a finite number of whole numbers between the starting value and zero. Thus no such value of n could possibly be found, and the statement must be true for all n.
Do you see why this is equivalent to the Principle of Mathematical Induction?
Fermat used his method of descent to prove that x^4 + y^4 = n^2 has no solution in positive integer values of x, y, and n, by showing that if there were a solution, there would be another with a smaller value of n."

The history is pushed back a little more in this interesting note from Ed Sandifer to the Math History newsgroup, "Victor Katz in his _A History of Mathematics_, pp 278 ff, describes induction as it was used by Levi ben Gerson in 1321, well before Maurolycus. Shortly after Katz's book came out, I talked to Dirk Struik about it. Struik had also cited Maurolycus as the first use of induction, but he readily conceded precedence to ben Gerson.
That was kind of a thrill for me, because you don't get to see 99 year old guys change their minds on long-held beliefs very often.
Ed Sandifer
sandifer@wcsu.ctstateu.edu "



League (distance measure)

When the "Light Brigade" charge "half a league" into the cannonfire in the classic poem by Alfred, Lord Tennyson, they would have ridden about a mile and a half. The original unit of a league, on land, was the distance a person could walk in an hour. Now it is defined as exactly three miles.

But for Jules Verne, half a league would not have been the same, as a league on water is considered to be three nautical miles, or about half a mile farther than the "land league".



Mil

A mil (sometimes mill) is a general substitute for one thousandth of something. It is shortened from the Latin millesimum which meant the thousandth part of something. It seems to have first appeared as a unit of money when the US Federal Money system was set up. The mill represented 1/10 of a dollar (see dollar for more detail). The mil is also used in many states as a unit of taxation for property taxes where the tax is assessed at some number of mils (thousandths of a cent) per dollar of property value. Even though created as part of the federal money system, there has never been a one mill coin minted by the US. (several states,{maybe Oklahoma and Missouri for example} have minted tokens worth one or five mills used in conjunction with taxes and change up to about the second world war.) The smallest coin the US has ever minted was the half-cent, and these were made from 1792 until 1857. Today mils are also used as a measure of the thickness of sheets of paper, plastic, cloth etc where the term stands for .001 inch (in the US). There is even a symbol for promil corresponding to the idea of percent. (see percent)



Monge's Theorem on the Tangents of circles

With a moments thought, most students of geometry would realize that if the two external tangents to a pair of circles is drawn the tangents would intersect in a point (if the circles are congruent then the point of intersection might be at the point at infinity). Monge proved a conjecture by d'Alembert that if the mutaul pairs of tangents were drawn for three circles, the three intersections of the mutual tangents would lie on a line. An nice interactive sketch by Antonio Gutierrez can be viewed here. Be sure to see both of the dynamic sketches, as it makes clear that the theorem is correct for pairs of internal or external tangents. Students of geometry can learn much from browsing through the other demonstrations on Antonio's site.... just an excellent place to learn about geometry. The theorem is named for Gaspard Monge of France.






Nine Point Circle


The Nine Point Circle is so named because it was first shown to pass through the three medians, the three feet of the orthocenter, and the three midpoints of the segments of the perpendiculars joining the orthocenter to a vertex. Although the circle was known before, the first proof was published in a joint paper of Brianchon and Poncelet which appeared in Gergonne's Annales de Mathematiques, Vol II in 1820. They were also the first to use the name "Nine Point Circle". The radius of the Nine Point Circle is 1/2 the radius of the circumcircle. The center of the Nine Point Circle is the midpoint of the segment from the Orthocenter to the Circumcenter.

The circle is also called Feuerbach's Circle after Karl Wilhelm Feuerbach who showed in an 1822 paper that the circle "touches all four of the circles which are tangent to the three sides of the triangle, and specifically, it touches the inscribed circle internally and the escribed circles externally."

We now know the circle contains even more than these nine points, but also contains the midpoint of the two Fermat Points, and the center of the Kiepert Hyperbola.



Perigon

The Perigon was used once commonly used as the name for a 360o angle, sometimes called a "circular angle". Today the term is seldom used.


Period

The word period is used in several ways in mathematics. The first use is as a name for the groupings when large numbers are written and seperated by a space, or a comma (or a point in some countries). For example, if we write a large number such as 23, 456, 789 the three digits 789 are in the units period, the digits 456 are in the thousands period and represent four hundred fifty-six thousands. In a similar way each period of three digits continue to represent sequential powers of 1000. In a similar way some old methods of teaching root extraction would begin by dividing off the argument in periods of length equal to the root to be extracted. To take a square root the digits were grouped in periods of two numbers, for example.
added noteI assumed that period had always referred to a set of three digits in a numeral, but recently came across a note in Charles Huttons A Course of Mathematics,1836, that the term first referred to a cluster of six digits, and the three digit groups were called half-perids.

The second use of the word is to describe the length of the repeating section of rational numbers expressed as decimal fractions. For example the fraction 3/11 expressed as a decimal fraction is .2727272727.... repeating forever. Since the length of the repeating part is 2 digits long, we say that the period of the decimal fraction is 2. All fractions with 11 as a denominator will have period two, but not the same two digits. 1/7 has a period of six since it repeats 142857 and 1/3 has a period of one since it repeats the single digit 3. The OED indicates that the first use of period for the length of the repeating cycle in a decimal was by B. Smith in 1869 in Arithmetic and Algebra(edition 6).

A third use of the word is to describe certian functions which repeat the same group of range, or y, values after some distance along the domain, or x, value. The simple example y=sin(x) where x is in degrees gives an easy to see example. As the values of x go from 0 degrees to 360 degrees the value of y goes up from zero to one, down to -1, and then back toward zero. Between 360 and 720 degrees the function repeats the same behavior, and in fact will continue to repeat the behavior over any range of degree measures. We say the Period of the function is 360 degrees. In general the period of a periodic function is d if d is the smallest non-zero number so that f(x+d) = f(x) for any x in the domain of the function.The first use of period for this idea seems to have been by Arthur Cayley in 1879.

The word period is from the Greek periodos which combines the roots for around peri with the root for way or path hodos. This last root shows up in words like Exodus (the way out) and odometer (the gauge in the car that tells how many miles the car has travelled).



Prosthaphaeresis is a combination of the Greek words for addition, prosth and subtraction aphaeresis. Prior to the discovery of logarithms it was very difficult to solve spherical triangle equations because it required several multiplications of sines and/or cosines to solve fir a single unknown. Since most of these were represented as chords of a radius of 10^7, it involved the equivalent of multipliying two seven digit numbers together by hand for each multiplication. In 1582, a Jesuit Priest named Christopher Clavius found a better way. He showed how to employ the trigonometric identity, Cos(A) Cos(B) = [Cos(a + b) + Cos(a - b)]/2 to make faster work of such problems.

Here is how it might work for a simple problem. We will multiply 314 x 245. We express 314 as .314 and find that is the cosine of 71.6995 degrees. Doing the same with 245 we find Cos-1(.245) = 75.8182 degrees. Now we simply apply the identity above to show
Cos(A) Cos(B) = [Cos(a + b) + Cos(a - b)]/2
.314 * .215 = [Cos(71.6995 +75.8182) + Cos(71.6995 - 75.8182]/2
.314 * .215 = [Cos(147.5177) + Cos(-4.1187)]/2
.314 * .215 = [-.843557+ 997417]/2
.314 * .215 = [.15386]/2 = .07693 and multiplying by 10^6 to restore the correct magnitude to the original problem we see that this is 76930, and 314 * 215 = 76930.

One of those who made very good use of the method of prosthaphaeresis was the Danish observer Tycho Brahe. Tycho tried to claim credit for the invention of the method, but we know believe he obtained the method from the itinerant mathematician Paul Wittich or the instrument-maker Jost Bürgi who may have been introduced to Clavius method during their travels. Bürgi is also noted for independently discovering the logarithm.


Rolle's Thm

In 1691, Michelle Rolle developed a method for finding roots called the "Method of Cascades" which, in greatly different form and notation, shows up in almost every first semester Calculus book. Rolle use the word "cascade" to describe the sequence of derivatives of a function and used these cascades to set limits on the location of roots.

Today we state his rule in formal notation thus; "If y=f(x) is continuous throughout the interval a < x < b, if it vanishes at x=a and x=b, and if the derivative f'(x) exists at each value of x between a and b, then there is at least one value of x, say X, which lies between a and b and at which f'(X) = 0." (From Moris Kline's Calculus, An Intituitive and Physical Approach.

For many years math historians could not find any record of Rolle's publishing and some even doubted that he had ever published it until it was found in a little known book almost hidden in the explanation of his "Method of Cascades." According to David Eugene Smith, "the first occurance of the name "Rolle's Theorem" appears to be in the writings of the Italian Mathematicain Giusto Bellavitis. He uses the expression theorema del Rolle in 1846."



The Rule of Three

The rule of three was a name given in earlier days to an algorithm for solving proportions. The method required setting up the problem so that the unkown quantity is always last "extreme" in the proportionality. The image below shows the rules as given by a 1827 arithmetic.

The example and the solution from the same page are shown below

This rule is covered in almost all the arithmetics up to the beginning of the 20th century. The images here were taken from a version of the popular Pike's arithmetics re-issued after his death by Dudley Leavitt in 1827 with the imposing title, Pike's System of arithmetick abridged : designed to facilitate the study of the science of numbers, comprehending the most perspicuous and accurate rules, illustrated by useful examples : to which are added appropriate questions, for the examination of scholars, and a short system of book-keeping.

The rule of three was such a common part of arithmetic education that it found its way into common expressions. In his autobiography, Lincoln writes that he that he learned to "read, write, and cipher to the rule of 3." A poem often used in student copy books was:

Multiplication is vexation;
Division is as bad;
The Rule of Three doth puzzle me,
And Practice drives me mad



Triangle

From the Latin for "three-angled", a triangle is the simplest closed plane figure. In keeping with the "gon" ending of other names for geometric plane figures, it has sometimes been suggested that "trigon" would be a better name.

Rather than repeat lots of terminology and formulae that are well done somewhere else, let me offer a link to the Dr. Math FAQ on triangles.



Taxicab Geometry

Taxicab Geometry is a non-Euclidean geometry developed by Hermann Minkowski. The effect is that all points lie along the vertical and horizontal gridlines passing through a rectangular lattice. Distance then is measured as it would be by a taxi driving on the square grids of a city. Minkowski is famous for coupleing together the geometric space with time to create the "space-time" continuum.

Here is a link to a somewhat advanced page about Taxicab Geometry and some applications. An excellent introduction is at this page by Cynthia Lanius which she calls School Bus Geometry



Telescope

Although Galileo did not invent the instrument now called the telescope, he applied his mathematical skill and his studies of optics to create one far superior to the simple instruments created by others at the time. More importantly, in January of 1609, he used the new instrument to shake the foundations of the Aristotelian belief that all the stars and planets moved in orbit around the Earth, he discovered the moons of Jupiter. Galileo called the instrument his perspicillum and sometimes it was simply referred to as "an instrument for seeing far." Then on April 14,1611 at a dinner to honor Galileo's discovery a Greek poet and priest named Demisiani gave the tube with two lenses the Greek name telescope, from the Greek for "to see at a distance."

An excellent discussion of history of the name "telescope" is found at this Renaissance Mathematicus blog, an excellent historical reference site.



Logic Symbol for Therefore
According to Cajori's history of notations, the triangular array of dots was first used with the meaning "therefore" by Johan Heinrich Rahn. Rahn used the symbol with point up (mostly) and point down (less often) versions in his Teutsche Algebra in 1659. The sign was even used occasionally with the point to the right (William Jones, 1706). Cajori points out that, "...with Rahn and many other writers of the eighteenth century, the three dots were used especially in connection with the process of finding the products of means and extremes of a proportion."