# Math Words & Other Words

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Antiparallels

Almost every geometry student learns that a line through two sides of a triangle parallel to the third side (DE in figure) will cut off a triangle, ADE, that is similar to the original triangle ABC. Almost none of them are shown that the reflection of such a parallel line segment in the angle bisector will produce another segment cutting off a triangle AE'D' that is also similar to ABC. Such a segment is called an antiparallel. The antiparallel also creates a quadrilateral of the remaining piece of the triangle which is always cyclic, and therefore can be circumscribed. The opposite sides of any cyclic quadrilateral are thus referred to as antiparallels also, since extending any two opposite sides will create a pair of similar triangles also.

In the famous books on the conic sections by Apollonius of Pergia he refers to this as a triangle lying "subcontrary wise". The only two planes cut through a cone that will produce a circle are the cuts parallel to the base, and the cut subcontrary or anti-parallel to the base. (book I, Proposition 9)

Arbelos was the Greek word for a knife used by a shoemaker to cut and trim leather. The name was applied by Archimedes (about 250 BC) to the region below bounded by three semi-circles.

In his Liber assumptorum (Book of Lemmas), Archimedes wrote of some properties of the arbelos. It is perhaps trivial that the sum of the two smaller arc lenghts are equal to the larger, but the fact that the area of arbelos is $$\frac{\pi}{4}$$ times the product of the two smaller diameters (AC and CB) is certainly less trivial. Archimedes also showed, in lemma 4, that the area of the arbelos was equal to the area of a circle whose diameter was the length of a perpendicualr segment from the tangent point of the two smaller semi-circles (Point C) up the the larger semi-circle. This is easy to derive from the previous result since the vertical segment is the geometric mean of AC and CB.

Pappus (320 AD) wrote on the relations of chains of circles mutually tangent to the two larges semi-circles and each other. You can find some interactive java-scripts and excellent commentary on the arbelos at Alexander Bogomolny's wonderful web site.

The Bride's Chair In book one of The Elements in proposition 47, Euclid proves the Pythagorean Theorem by a method often called the Bride's chair, and sometimes the Franciscan Cowl. The suggestion is that the construction lines resemble a drape that was once commenly used to drape the ceremonial chair of the bride at a wedding reception, and sometimes still is. Franciscan monks wore hoods of corse material that formed an A-frame appearance about their face, and that seems to be the reason for the other term. You can see the proof and step through it with a java-script from Texas A&M university.

Equilateral joins contractions of the Latin aequus, the root of equal, and lateralis for side. The term is used to describe a polygon with all sides congruent, or a polyhedra with all faces congruent. The prefix equi- is commonly used in math and science terms to express the idea of things that are equal, level, or unchanged. The geometric term equilangular applies in the same way to show that all angles of a polygon are congruent. The equinox occurs when the sun is at the celestian equator, and day and night are approximately equal in length. The -nox- stem is related to the Latin word for night.

The Roman surveyor Junius Nipsus (150 AD) introduced the symbol L (latus) for the root (or side) of a square. Ramus' arithmetic of 1592 includes the use of "lc" for the cube root. In the Clavis Matematicae (1647 and later editions) William Oughtred used L for the side of a square, and as the square root of a quantity. The invention of logarithms and the use of l for logs may have influenced the choice of mathematicians to adopt the modern radical over this simpler symbol, although Briggs used the l in the manner of Ramus.

The Greek for equal sided, isopleure, also is sometimes used, although more often in science books than those on geometry. Isopleuron is sometimes used for the triangle with all sides of the same length.

Halmos symbol It is common in mathematics now to use the symbol to indicate the end of a proof. It is named for mathematician Paul Halmos who seems to have first applied it to mathematics. Halmos has stated that he got the symbol from popular magazines where it was used to indicate the end of an article. It is also frequently called the tombstone.

Halmos had (has) a sharp wit and sense of humor according to Peter Ross, who shared the following Halmos anectdote with a discussion group. G. P. Hochschild worked with Halmos at the University of Chicago, and on hearing that Halmos was writing a book asked if his name would be in the book. When the book, Finite Dimensional Vector Spaces", was published, an entry on page 198 of the index read, "Hochschild, G. P. .... 198". The reference was to the index page itself, a genuine self-referential index.

IFF The use of IFF to represent the phrase "If and only if.." in mathematics is the creation of Paul Halmos. Here is the story in his own words from his autobiography, I Want to be a Mathematician.

My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented "iff", for "if and only if" -- but I could never believe that I was really its first inventor. I am quite prepared to believe that it existed before me, but I don't know that it did, and my invention (re-invention?) of it is what spread it through the mathematical world.

Infinity is the negated form of the Latin word finire for limited, bounded, or finished. Infinite then would literally mean that which is unlimited or unbounded. Originally only used to apply to things that were unmeasureably large, the origin of theories of limits led to the need for a word that expressed the idea of things growing smaller and smaller without bound. Mathematicians added the Latin suffix "esimus" to create infinitesimal for a word to describe the dimenishing quantities.

Near the beginning of "One, Two, Three... Infinity" George Gamow has the following little limerick which seems perfect as part of this definition:

There was a young fellow from Trininty
Who took the square root of infinity
but the number of digits
gave him the fidgets;
He dropped Math and took up Divinity.

Infinity symbol

The symbol we now use for infinity (left in figure), was first used by John Wallis (1616-1703) in 1655 in his treatise On Conic Sections. Why he used it seems lost to history. The two most popular suggestions are these. The Late Romans used a symbol like two hooked together zeros, 00, for the number 1000. Since Wallis was a classical scholar, this may have been one source. A second theory is that he used a variant of the lowercase symbol for Omega (right in figure), the last letter in the Greek alphabet, to symbolize the "final number" in a sense.

IODINE --- An ugly black solid, a beautiful flower, a mythical river nymph, and a colorful flame all come together in the story of Iodine.  According to the Greek myths, when Zeus saw, Io, the beautiful daughter of the river god Inachus, he fell instantly in love.  He tried to cloak the world in clouds, but Hera, Zeus' wife, found out about his liason from a giant named Argus who had a thousand eyes, and was thus a pretty good lookout.  Zeus then tried to hide her by disguising her as a white ox, but Hera was not fooled and sent a pestering fly to torture her.  Tortured, Io traveled across the globe trying to elude the tormenting flys.  The spot where she supposedly crossed from Europe into Asia at the point where Istanbul is now located is called the bosporus which literally means "ox crossing" (Oxford England is named for a less mythical spot where cattle crossed the river).  Io finally swam to Italy (across the Ionian Sea) where she found rest, but no food. Zeus was moved to pity when he saw her crying, so he turned each tear into a beautiful violet, which the Greeks called Iode.
Much, much, later, around the year 1811,  French Chemist Bernard Courtois was studying with the ashes from burning seaweed (Chemist did that kind of thing) and discovered a strange blackish gray solid he could not identify.  Some say it was Gay-Lussac, others suggest  it was Sir Humphrey Davy, but in any case, someone thought to heat the solid and  it gave off a gas with a beautiful violet blue vapor.   The Greek name of the flower was chosen, and the substance was named Iodine.  Here is a table of  the properties of Iodine.

IRIDIUM--- Hermes (Mercury) was not the only messenger of the mythical gods, in fact they had a special messenger to run messages to the mortels, her name was Iris. Since she went back and forth to the heavens so often she would glide up and down on the rainbow.  Today iris is still the Greek word for a rainbow. The beauty of a rainbow is in its bright colors, and so other bright colorful things were given the same name, .  The Danish naturalist who studied the colored portion of the eye choose to call it an iris, and a plant that is often called the flag plant in some cultures, is called the iris in the US because of its bright blossoms.  The film of rainbows on bubbles or in the lining of shells is called iridescence from irid, the plural of iris. So in 1803 when Tennant found an element which formed many different colors when combined with other substances, he called it iridium, the element of rainbows.

Kaprekar Numbers

Take a number (I'll use 45 as an example), square it (45^2 = 2025), and add the left half (20) to the right half(25) and if you get the number you started with, the number is a Kaprekar number. If the squared number has 2N+1 digits, take N on the left and N+1 on the right to add together. The first few Kaprekar numbers are 1, 9, 45, 55, 99,297, 703, 999, 2223, 2728, 7272, 7777... It seems that for every Kaprekar number K of N digits, 10^n-K is also a Kaprekar number. Notice 1+9=10, and 45+55=100, etc. It is also should be easy to convince yourself that a string of nines will always be a Kaprekar number, 9 or 99 or 999, etc..

If you form sequences of the numbers that each number maps to, you quickly notice that some numbers map to a Kaprekar number and stop; for example 8--> 10-->1. Other numbers map into loops such as 70--> 49-->25-->31-->70. The numbers 2, 4, 5, 7 and 13 all map into this loop. Other numbers just seem to eventually get into a path that continues to grow bigger and bigger. They may have moments where they drop down for a bit, but then they continue to grow. The question, of course, is will they ever either hit a Kaprekar number, or a loop, or do they continue to grow infinitely

I have noticed some interesting patterns in playing with these; for example, among two digit numbers where the digits differ by one, both permutations go to the same number. That is, 21^2 = 441 and 4+41 =45; while 12^2 = 144 and 1 + 44 =45. A similar thing happens with 23 and 32 which both map to 34; 34 and 43 which both map to 67, etc...and all the two digit numbers that are mappings also have digits that differ by one. More generally, if both two digit numbers of the form xy and yx map to a two digit number, then they map to the same two digit number. If you look at three digit numbers that have two digits alike, and one that differs by one, 112 for example, we see another curious pattern... 112 maps to 556, 121 maps to 655, and 211 maps to 565. With 221 we get ,221--> 889, 212-->988, and 122-->898. Any permutation of the digits one, two and three will produce a number with two fours and a one, 123^2 = 15,129 and 15 + 129 = 144.

The numbers are named for D. R. Kaprekar who came up with the idea of searching for them. He is also known for discovering that if you take a four digit number, write the digits in ascending and descending order and subtract, and then repeat the process with the new result, eventually you will end up with 6174. A similar method works for numbers with two or more digits. This process is called the Karprekar process. For example if we start with 3245 we write the digits in order both ways 5432 - 2345 and get 3087. Now we repeat the process with 3087 to get 8730-0378 = 8352. Continuing the process we get 6174 for the next result. It is an interesting exercise for elementary students to try to find the length of the Kaprekar sequence for three and four digit numbers (and they never realize they are practicing subtraction skills).

Lucas Sequence

The Lucas sequence is similar to the Fibonacci sequence. The Lucas sequence is given by {1, 3, 4, 7, 11, 18, ...} . Each term is the sum of the two previous numbers, as in the Fibonacci sequence. Just as in the Fibonacci sequence, the limit of the ratio of consecutive terms is the Golden Ratio. The Lucas numbers can also be constructed from the Fibonacci numbers by the function Ln = Fn-1 + Fn+1, thus the fifth Lucas number, 11, is the sum of the fourth and sixth fibonacci numbers (3+8).

The sequence is named for Edouard Lucas, a French mathematician of the later half of the nineteenth century. He used his sequence and the Fibonacci sequence to develop techniques for testing for prime numbers. Lucas is also remembered for his unusual death, caused by a waiter dropping a plate which shattered sending a piece of plate into his neck. Lucas died several days later from a deadly inflamation of the skin and subcutaneous tissue caused by streptococcus. The disease, officially listed as erysipelas (from the Greek for "red skin") was more commonly known as "Saint Anthony's Fire".

Lucas was also the creator of a popular puzzle called The Tower of Hanoi in 1883. You can see the original box cover here. Note that the author on the box cover is Professor N. Claus de Siam, an anagram of Lucas d' Amiens (his home). The professors college, Li-Sou-Stian, is also an anagram for "Lycee Saint-Louis" where Lucas worked.

France was building an Empire in Indochina (the peninsula stretching from Burma to Viet Nam and Malaysia) and the "mysterious East" was a very fashionable topic. Lucas created a legend (some say he embellished an existing one, but I can find no earlier record of one) of monks working to move 64 gold disks from one of three diamond points to another after which the world would end. The solution for a tower of n disks taks 2n -1 moves, so the game often had less than the 64 disks of the legend. Solving the 64 disks at one move a second would require 18,446,744,073,709,551,615 seconds, which at 31,536,000 seconds a year would take 584 Billion years. (and you thought Monopoly took a long time to finish).

Here is a link to a java applet of the game that allows you to adjust the number of disks for play. Students/teachers interested in furthur explorations of the history and math of the famous game should visit the work of Paul K Stockmeyer who maintains the page with the cover illustration mentioned above, and his Papers and bibliography on the Tower of Hanoi problem.

Another recreational game first published by Lucas is the well known Dots-and-boxes game in which players connect dots with lines to claim completed squares. The game was first published in Lucas' Recrational Mathematics, vol II in 1889.

Pedal Triangle

For any triangle ABC and a point P, not necessarily inside the triangle, we can create three new points called the "feet" of P on ABC by drawing a perpendicular line from P to each of the lines AB, BC, and CA. The point where each perpendicular intersects the side is called the "foot" of the perpendicular to that side. If we connect the three "feet" of P on ABC, we get a triangle called the pedal triangle. The word pedal is drawn from the Latin ped, and the earlier Greek pod, for foot. The pedal triangles of some well known points have special names. The pedal triangle of the circumcenter of ABC is called the medial triangle.

If the incenter is chosen as the pedal point, the vertices of the pedal triangle will occur at the points where the incircle is tangent to the sides of the triangle.

The orthocenter H, has a pedal triangle that has the feet of the altitudes from A, B, and C as its vertices, and is often called the Orthic triangle.

If the point P falls on the circumcircle of ABC, then the three feet of the perpendiculars will lie in a straight line, known as the Simson line.

Perimeter The origins of perimeter are the Greek roots peri (around) + metron(measure). Peri shows up in other "around" words like periphery, the word the Greeks used for the circumference of a circle. The pher in the word means "to carry", and is the same as the ending in circumference. The first use in English seems to have been in the late 1400's.

The peri root also remains today in many biology terms. The pericardium is a membrane that is "around" the cardi "heart", and periderm is used for the outer bark or covering of a plant , literally the dermus , skin, around" the plant.

Something that was around you was often close to you, and so the word picked up an association related to nearness, and that is how perigee (near the geos, earth) came to mean the point where the moon is closest to the Earth, and perihelion (near the helios sun) became the word for the point in a planets orbit that is nearest the Sun.

Sometimes pimorials are indicated by the number sign, #, which also goes by the names "pound sign", "octothorpe", "crosshatch", "tic-tac-toe", and in England and some international standards, "square" . So 3# would indicate the product of the first three primes, which I showed above was 5*3*2=30.

One of the interesting features of the primorials is that the limit as n goes to infinity of the nth-prime root of the nth primorial approaches the natural base, e. [for example when n=3, the third prime is 5 and the third primorial is 30, so the third term of the sequence would be the fifth root of 30.

Although he never used the word, Euclid used the idea of a primorial to prove that there was no largest prime number. Here is a link to Euclid's proof.

The Pythagorean Theorem is one of the most famous theorems in all of math. Because so much has been written, and so well, I will content myself to giving you some links that I think are excellent tools for finding out out about the famous theorem. A good introduction to the History of the theorem is at the Geometry Center Page. You can also find more indepth historical information about the life of Pythagorus at The Math History link at St Andrews University in Scotland. This link is to a map showing the island of Samos, off the coast of modern day Turkey. Samos is just north of Rhodes and south of the island of Lesbos.

One of the nicest proofs of the Pythagorean Thm that I've ever seen is a step-by-step Java script from Texas A&M. It lets you move back and forth through each step of an area preserving transformation until it is perfectly clear. The perfect level for HS Students.

The Dr. Math FAQ page on the Pythagorean Thm at the Mathforum has answers to the most common questions, and several links to sites for more exploration.

Here is a proof that I often use with my students that is based on a figure from a Chinese block print dating around 40 AD, and reputed by oral tradition to predate the life of Pythagorus in Chinese History. In the first figure we see a large square with two small squares of Area a2 and b2 in the white space. By simply moving the right triangles with sides of a, b, and c, we redistribute the white space to show that it is also equal to c2. The final white space is still a square because at each of its vertices two complimentary angles of the red triangles meet.

This proof is also sometimes attributed to the fifth century Indian mathematician, Arya Bahta, who demonstrated it in his writing. Steve Phelps, a teacher at Madeira HS in Ohio posted an interactive Geogebra applet of this at his blog. This link has an animated Java script of the proof, and a dissection proof credited to Thabit ibn Qurra of Mesopotamia (now Turkey). You can find the same proof demonstrated using simple algebra at Issac Reed's Famous Problems web sight.

It is common to hear statements suggesting that either the Babylonians or the Egyptians had knowledge of the Theorem. Otto Neugebauer has suggested on the basis of what we call Pythagorean number triples which appear on Babylonian tables that some Babylonian people knew the theorem as applicable to right triangles. Another common story is that the Egyptian rope-stretchers harpedonaptai used 3-4-5 triangles to produce right angles. Both these seem to be disputed by modern Math Historians. A recent post to Historia Matematica by Bob Stein, says that a "recent article by Eleanor Robson on Plimpton 322 in Historia Mathematica makes a very convincing case that Plimpton 322 does not indicate a knowledge by its creator either of the Pythagorean theorem or the systematic generation of Pythagorean triples."
In another post in the same string, Roger cooke writes, "Alas, there is really no good evidence that the Egyptians did this at all! It was CONJECTURED by Moritz Cantor in the first volume (1881) of Geschichte der Mathematik. Then, as Van der Warden remarked, frequent repetition turned the conjecture into an "established fact." ... to my knowledge the earliest Egyptian document with the Pythagorean theorem in it dates to the time of Euclid."
In Mathematics in the Time of the Pharaohs Richard Gillings includes an appendix with several quotes by famous math historians rejecting the idea that the Eqyptians knew that a 3-4-5 triangle made a right angle. One of them, T. L. Heath, states,"There seems to be no evidence that they knew that triangle (3,4,5) is right-angled; indeed, according to the latest authority (T. Eric Peet, The Rhind Mathematical Papyrus, 1923) nothing in Egyptian mathematics suggests that the Egyptians were acquainted with this or any special cases of the Pythagorean theorem."

President Garfield's proof of the Pythagorean theorem may be a subject of as much misinformation in classrooms as information. I trust the following version which comes from Howard Eves' Great Moments in Mathematics.

...James Abram Garfield (1831-1881), the country's twentieth president, who in his student days developed a keen interest and fair ability in elementary mathematics. It was in 1886, while he was a member of the House of Representatives, and five years before he became President of the United States, that he independently discovered a very pretty proof of the Pythagorean Theorem. He hit upon the proof in a discussion with some other members of congress, and the proof was subsequently printed in the "New England Journal of Education".
Garfield actually was a professor of mathematics at Hiram College in Ohio for several years before being elected to the Senate in 1859.
A side note about other mathematical Presidents may be of interest to students. Ulysses S Grant wrote that he had hoped to take a position as a math instructor at West Point before the Civil War changed his plans. "Stonewall" Jackson, the Confederate General, was a Professor of Mathematics at VMI. Washington was an accomplished surveyor, and Lincoln took time off from his study of law to learn the proofs of all the propositions in the first six books of Euclid so that he could truly understand the meaning of "demonstrate".
The Garfield proof used the figure at the right and found the area of the trapezoid by two approaches, one using the traditional formula with the two bases, the other using the area of the three right triangles. The trapezoid has bases a and b, and a height a+b, so the area = 1/2 (a+b)(a+b)= 1/2 (a2 + 2 ab + b 2). The two congruent triangles with legs of a and b have area = 1/2 (ab) so together their area is ab. The isosceles triangle with legs of c has an area of 1/2 c2. The total area of the three triangles add up to ab+ 1/2 c2. Setting this equal to 1/2 (a2 + 2 ab + b 2) and doubling both sides gives us the equation
a2 + 2 ab + b 2 = 2ab + c2
and subtracting 2ab from both sides gives us the desired result.

I only recently came across a neat proof of the Pythagorean theorem that uses some similar triangles. The proof was listed as a "proof without words", so I will say no more. You can see it here.

The 19th Century British mathematician Henry Perigal ( 1801 - 1898) invented a disection proof of the Pythagorean thm that he was so proud of he used it on his tombstone. You can see the picture of the tombstone and a discussion of his proof at this link.

Gougu Rule A rule equivalent to the Phythagorean Theorem appeared in the Chinese classic Zhoubi suanjing, "Zhou Shadow Gauge Manual", and was called the Gougu Rule . A "shadow gauge" is the Chinese equivalent of the Greek gnomen. The work was written sometime between 100 BC and 100 AD. In it the author claims that the Emperor, ".. quells floods, deepens rivers and streams, surveys high places and low places by using the Gougu rule." This is the earliest written evidence of the Pythagorean Theorem in China, but legends suggest it may have been known in China prior to the life of Pythaogorus.

Generalizations of the Pythagorean Theorem. The most well known generalization of the Pythagorean theorem is the well known Law of cosines. Less well known is a 3rd century AD generalization credited to Pappus. Using the figure at right to help, I will explain. On sides AB and AC of triangle ABC, construct parallelograms ABFG and ACDE. Extend the sides FG and DE until they intersect at A'. Construct BB' and CC' parallel and cogruent to AA'. The parallelogram BB'C'C has area equal to the sum of the areas of ABFG and ACDE.

de Gua's Theorem is a 3-D analogue of the pythagorean theorem. If a tetrahedron has all right triangles for the three faces meeting at one vertex, de Gua showed that if the areas of the three right angled triangles are A, B, and C, and the triangle on the base has area D, then A2+B2+C2=D2. The theorem is named for J. P. de Gua de Malvex but it seems it had been previously known to both Descartes and Faulhaber.

Pythagorean Thm for Spherical Right triangles

Just as the Pythagorean theorem is a simplification of the Law of Cosines, there is a simplification of the spherical Law of Cosines that produces a relationship between the sides of a spherical triangle. In spherical triangles both sides and angles are usually treated by their angle measure since sides are arc lengths of a great circle. Using capital letters to represent angles, and lower case to represent the opposite sides, the law for sides is given as cos c = cos a cos b + sin a sin b cos C . When angle C is a right angle, then Cos(C) = 0, and the "Pythagorean relation for spherical right triangles" is given by cos c = cos a cos b.

Quincunx is the name for an arrangement of five things about a square with one located at each corner and one in the center. Originally it was a method of planting fruit trees, and probably a decorative latticework; but it worked its way into mathematics when Sir Frances Galton used it to demonstrate binomial probability. You can find reproductions of some of Galton's illustrations at this site at Kyoto University

An early illustration of a quincunx appears in the frontispiece to Sir Thomas Browne's The Garden of Cyrus in 1658. The subtitle is "The Quincunciall, Lozenge, or Net-work Plantations of the Ancients, Artificially, Naturally, Mystically Considered".

Although many dictionaries suggest the origin is related to quinque + uncia and means five ounces or five-twelfths (See Merriam-Webster's Collegiate Dictionary, for example) it seems more likely to me that the origin was related to the fifth point of a die, as suggested in "Now though in some ancient and modern practice the area or decussated plot, might be a perfect square, answerable to a Tuscan Pedestall, and the Quinquernio or Cinque-point of a dye" from the first chapter of The Garden of Cyrus. I was relieved to find a post from John Bibby which supported my view on this matter, and also gave some nice references to Galton's use of the Quincunx in statistics, and with his permission, I have copied it here:

Incidentally, Galton's original quincunx was in the Pearson Building at University College, London in the 1960's, but now it has gone. Does anyone know where it is now? Steven Stigler gives a marvellous discussion if Galton's Q in his "History of Statistics". This includes a fine demonstration of how conditional arguments can be used to derive complex results such as "A normal mixture of normal distributions is itself normal". Galton is also discussed in Theodore Porter's "The Rise of Statistical Thinking". PS for etymologists: Ther origin of the term "Quinc-unx" is "five ounces", where an ounce is a dot or small object: "An arrangement or disposition of five objects arranged so that four occupy the corners, and the fifth the centre of a square or rectangle" (OED). In Galton's Q, the "objects" are pins. The Quincunx also features in Florentine or Cosmati marble-work, as discussed in the marvellous book "Patterns of thought: The hidden meaning of the Great Pavement in Westminster Abbey", and the eponymous poster produced by The Mathematical Association and available like all other items mentioned from QED =============================================== John Bibby

You can find a java script of a (simulated) quincunx in action with a nice explanation of the probabilities involved created by Chris Stevenson and Goeff Coates.

Teachers (or others) looking for a physical model can purchase one online from this site.

Trigonometry

The root of trigonometry comes from the union of the Greek trigonon for triangle, and metron for measure. Although the roots are nestled in ancient Greek, the word seems to have been the creation of Bartholomaus Pitiscus, who used it in the title of a book, Trigonometriae sive de dimensions triangulorum libri cinque in 1595. Among other things the book includes a demonstration of the law of sines and the law of cosines.

The mathematical ideas we now call trigonometry have been studied in connection with astronomy at least as far back as 140 BC when Hipparchus produced the first table of "chords". Early work in trigonometry was often more concerned with the triangles on spheres, like the Earth, than they were with those on idealized flat planes. A brief history of the development of trigonometry is available from the Math History site at the University of St. Andrews in Scotland.

It may be of interest to students to know that the use of the unit circle was "unknown much before 1800" according to an article on "Benjamin Banneker's Trigonometry Puzzle " by Florence Fasanelli, Graham Jagger, and Bea Lumpkin that appeared in the MAA online magazine Convergence. [The magazine is free for the present but will be subscription at some future time, so the link to the article may disappear without notice] The method of constructing tables in the early years is given in another article about Frans van Schooten (1615-1660) .

During that career, van Schooten seems to have published four books. The first, and most forgettable, was a collection of trigonometric tables, published in 1645. This book, with the typically long 17th Century title, Table de sinus, tangentes et secantes ad radium 10,000,000 : Avec une methode de resoudre très-facilement par leur moyen tous les triangles rectilignes & spheriques, is a little unusual in that the title changes language changes from Latin to French half way through. At the time, decimal fractions were not in wide use, so van Schooten’s tables were in the old style. They gave the measurements for the sine, tangent and secant on a circle of radius 10,000,000 rather than on a circle of radius 1, as we do today. Thus, the sin 90°, also called the “total sine” was given as 10,000,000, and the sine of 45° was 707,107 and not 0.707107, as we would use today. Anyone using these tables would use rules of proportion to make any necessary conversions. Schooten probably wrote only the text that described how to use these tables, and the tables were probably just a reprint of tables that had already been published before.

The British Society for the History of Mathematics has a note that Thomas Blundeville a student at Cambridge, possibly non-collegiate, (c1545), later wrote the first English book using trigonometry.

According to Bruce Burdick in a post to Historia Matematica, "The first printed mention of a trignometric function by name in the New World was in the Reportorio de los Tiempos by Enrico Martinez (Mexico, 1606)"