Math Words, pg 13

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Aliquot parts of a number are proper divisors of the number that are smaller than the number. The aliquot parts of six are one, two, and three. The word joins two unlikely partners, the Latin ali for "other" and quot for how many. Together they came to mean a part of something, in this case, a part of the number of which it is a factor. The "other" meaning of ali remains today in words like alius, alibi, and alien. The quot root remains in quotient.

Aliquot chains, sometimes called sociable chains, are formed by taking the sum of the aliquot parts and adding them to form a new number, then repeating this process on the next number. For some numbers, the result will bring you directly back to the original number. In that case the two numbers are called amicable numbers. For example, 220 and 284 are amicable numbers. The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 and if you add all these numbers together, you can see they sum to 284. The aliquot parts of 284 are 1, 2, 4, 71, 142 and these sum to 220.

The relationship between 220 and 284 was known at least as far back as Pythagorus (500 BC). Sam Kutler has written to tell me that the first use of a term like "friend" for the pair was in a commentary on the work of Nicomachus by Iamblichus, around 300 AD. He also thought the Greek term was arithmoi philos, literally, friendly numbers. The numbers were inscribed on "magic charms" in the middle ages which were sold to insure the fidelity of ones lover. Other stories suggest that the gift of 220 Goats from Jacob to Esau in the Biblical story was an expression of love made significant by the use of one of the pair. It is likely that the ancients assumed that this pair was unique and there were no others. Western mathematics knew only this set until 1636 when Fermat discovered a second pair; 17,296 and 18,416 (we leave the proof that they are amicable as an exercise for the reader, :-} ). After his discovery, the search for amicable numbers became quite a trend among mathematicians of the time. Descartes discovered a third pair and Euler added over sixty more pairs to the known list. The second smallest pair,1,184 and 1,210, was overlooked by all of these people, and found by a sixteen year old Italian student, Nicolo Paganini, in 1866.

Arabic mathematicians had preceeded their western counterparts by many hundreds of years. Thabit ibn Qurra discovered, and proved, a rule for creating amicable pairs in the ninth century. St Andrews web page tells us that
Al-Farisi (born 1260) gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler(I have only seen this creidited to Fermat, as I have above. Not sure if they have made a mistake (shudder) or know something I don't (more likely))., but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself.

One chain of sociable numbers is given by 12496, 14288, 15472, 14536, and 14264. When the sum of the aliquot parts of each number are added they produce the next number in the series. Amicable comes from the Latin word amicus for friend and the related word amare for to love. Some other words from the same root include amigo, amateur, and amour.

Angstrom Electro-magnetic radiation beyond the visible spectrum has such a short wave length that it becomes difficult to express in traditional units of length. For that reason, scientists who work with x-rays and other radiation with very short wave lenth have adopted the use of a measure equal to 1/10,000,000,000 or 10-10 meters. The unit is called the Angstrom after the Swedish physicist Anders Jon Ångström (1814-1874), a pioneer in the field of spectroscopic analysis.

BaseThe word base is so ancient and its meaning so generalized that it is used in many ways both in and out of mathematics. My on-line dictionary list forty-five definitions. The word comes from the Greek word for step, or beginning. What most of these definitions share is the idea of the fundamental or first part of something. This is true in mathematics as well. When Euclid used the word basis in the Elements he referred to the base of a triangle, parallelogram or trapezoid to mean a side drawn horizontally, or in the words of Proclus, "level to the eye." It is probable that then, as now, this is the common orientation for most drawings of these objects. As the objects are drawn in different ways, or as we generalized the term "base" to be used in equations like A=bh, we have to widen out interpretation, and today any side of a triangle or parallelogram may be considered a base, but in the trapezoid the term is usually only used for the two parallel sides.

The word base is also used in exponential relations. In 23=8 the two is called the base, the 3 is the exponent, and 8 is the power, although it seems that misuse of this term is leading to many people referring to the 3 as the power. This comes, I think, from misunderstanding the statement "eight is the third power of two." We also refer to 3 as the logarithm of 8 base 2. Base is also sometimes used to refer to the amount which is the reference for 100% in a percent sentence. In the sentence 25% of 80 is 20, 80 is called the base of the percent. This refers to the fact that 80 is the base, or foundation, from which the percentage is taken. Number systems are also describe by the number of characters in the system. Our decimal, or base ten system, is so named because it has ten characters, 0, 1, 2, 3, 4, .... A system which uses only zero and one is called a base 2 system. Many computers use a base 8 or base 16 system.

Gnomen is used today almost unchanged from its Greek origins. The literal meaning of the word in Greek was a marker, something which enabled something else to be known. But from antiquity, the gnomen represented the vertical bar or style of a sundial which enabled one to know the time. When Oenopidea of Chios investigated the problem of drawing a perpendicular from a point to a line, he described the line as "kata gnomen", like a gnomen.

By the time of Euclid, a device for marking and checking right angles which looked and was used much like the modern carpenter's square was also called a gnomen. In Euclid's definitions for book II of the "Elements", he expands the word to a similar angle cut off any parallelogram.

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In a similar way, gnomen was applied to the odd numbers because they are added around two sides of a square number to make the next square. If a square of side three was increased to a square of side four, it would need 7 points arranged in a gnomen like shape around the two sides.. Theon of Smyrna extends the idea to the difference between any two figurate numbers. The difference between the pentagonal numbers, for example, are the numbers 1, 4, 7, 10, ....

The gno root exists today in many English words, most obviously in know. Other words related to the same root include gnome, notice, prognosis and narrate.

The most common use of the word today is for the marker that rises from the face of a sundial.  Man has attempted to tell time from the shadows of the sun since antiquity, and great monuments like stonehenge, built as early as 2000 BC, may have been built to use the sun to mark out the seasons,   but as much as 1500 years earlier the Egyptians had erected huge stone monoliths to cast a shadow for the purpose of telling time.

In the Bible at 2 Kings, chapter 20, verses 9-11, there is mention of a sundial, the "dial of Ahaz", that is thought to have been built around 700 BC. When Isaiah brought God's promise to deliver King Hezekiah from the jaws of death and of 15 years extended life and to deliver for his people from the Assyrians, he also had the shadow on the sundial move backward, a miracle to confirm it was God's will. "And Isaiah, the prophet, called upon the Lord, and he brought the shadow ten degrees backward, by which it had gone down in the dial of Ahaz". A scientific explanation for how this could have occured by cloud effect is provided at this Popular Mechanics site, although calling up a cloud on command is a little of a miracle in its own right.

By 600 BC, The Egyptians had come up with the merkhat,  a device for measuring time at night by lining them up with the pole star and observing when certain stars cross this line..  This is usually recognized as the first astronomical instrument.
The problem with ancient shadow clocks were that they did not move through the same angle per hour each day.  In the summer the angle distance between noon and 3 pm was different than in the winter.  To resolve this difference the ancients resorted to cutting hemi-spherical depressions in the face of the sun-dial and making markings for time in several different seasons.  It seems that Ptolemy may have known that this could be corrected by pointing the gnomen in the correct angle (the angle of latitude for the dial) but this idea was not put into common practice until the Islamic mathematicians of the Middle Ages made some advances in spherical geometry.
Sundials can also be mounted vertically as seen in this 1749 dial I photographed in the Cathedral Close at Salisbury Cathedral.  The sign below makes referrence the Gregorian calendar adjustment three years after the clock was erected.  The lines drawn across the face mark the sun's path at different seasons of the year.  One of the characteristics of vertically mounted sundials (in the northern hemishphere) is that the shadow moves counter-clockwise or in a mathematically positive rotation. In the Duomo (cathedral) in Florence there is an unusual clock dating to the 1400's by Paulo Uccello. The clock not only rotates in a positive (anti-clockwise) direction, it is a 24 hour clock. You can see a rather dark image of the clock taken by my wife on our visit there. Here is a brief note from an on-line art site. "In 1436 the administrators of the Opera del Duomo in Florence commissioned Paolo Uccello to paint a fresco in the Cathedral, a monument commemorating the English soldier of fortune Sir John Hawkwood. Not long after the Monument to Sir John Hawkwood, the Opera del Duomo commissioned from Paolo the designs for three stained-glass windows (the Resurrection, the Birth of Christ and the Annunciation destroyed in 1828) for the oculi of the drum of the dome, as well as the decoration for the clockface on the inner façade of the Cathedral." The site also has some excellent images of the works done there, and other places by Uccello.
Many sundials have a figure eight looking image called the analemma  It is also called the "Equation of Time" since it describes the amount by which the sun exceeds or falls behind the "clock" time at the clocks local noon.  In a recent discussion group Herbert Prinz has suggested that the word is a misnomer. In response to a statment that ""Going back to ancient times, specifically in astronomy, one finds reference to the so-called analema, which is the path the sun-at-noon
traces out in the sky at latitudes between equator and poles.  Here is Mr. Prinz's remarks
"In the English speaking world, the lemniscate (figure 8 shaped curve, distorted hippopede) produced by the sun in the way described above is indeed often referred to as "analemma". I consider this a misnomer.  In German we call it "Zeitgleichungskurve", which means "curve representing the equation of time". The Equation of Time is the difference between the True and the Mean Sun. I have tried in vain to track down an ancient reference to "analemma" in the above meaning and would therefore be grateful for a citation.  From looking at Vitruvii "Ten books on architecture", Book 9, Ch. 7, and some notes that I have on what Neugebauer says on Ptolemy's "On the Analemma" I get the understanding that in antiquity analemma refers exclusively to a projective method used for constructing horizontal sundials, particularly to find latitude from shadow length or vice versa."
He also posted a brief note about the reason for the odd path of the sun,
"The figure 8 shape is solely due to the fact that the earth's orbit (ecliptic) is inclined with respect to the pole of rotation. The only effect of Kepler's 2nd law is to DISTORT the curve's otherwise perfect symmetry. If the orbit were elliptical but not inclined,  The movement of the noon shadow tip would be in a straight line east-west. If the orbit were perfectly circular, but inclined, the curve would be a "hippopede". The first astronomer to study (and name?) this curve was Eudoxus, who based his planetary theory of homocentric spheres on it.
The hippopede has the remarkable property that it is the intersection of  4 surfaces:  a sphere, right cylinder, double cone and parabolic cylinder!!!".
The Link above to Eudoxus also has a nice illustration of the hippopede on a sphere as visualized by Eudoxus for his sytem.  More about the advances in time keeping from ancient time to the present can be found at this NIST web site

Grad Many students and teachers still have calculators with a key that is labeled "DRG". A large percentage of them who know that the D stands for degree and the R for radian have no idea what the G is for. If you have gotten to here, you probably have figured out that the G is for Grad, a unit of angle measurement in which 100 grads = a right angle. Here is the definition as given by How Many? A Dictionary of Units of Measure.

a unit of angle measurement equal to 1/400 circle, 0.01 right angle, 0.9°, or 54'. This unit was introduced in France, where it is called the grade, in the early years of the metric system. The grad is the English version, apparently introduced by engineers around 1900. The name gon is used for this unit in German, Swedish, and other northern European languages in which the word grad means degree. Although many calculators will display angle measurements in grads as well as degrees or radians, it is difficult to find actual applications of the grad today.
In a second definition he explains the use of the term "grade" as still practice by civil engineers in the USA, which may lead to some of the confusion.
a measure of the steepness of a slope, such as the slope of a road or a ramp. Usually stated as a percentage, the grade is the same quantity known as the slope in mathematics: the amount of (vertical) change in elevation per unit distance horizontally ("rise over run"). Thus a 5% grade has an elevation gain of 0.05 meter for each meter of horizontal distance, or 0.05 foot for each foot of horizontal distance. The angle of inclination, in grads or grades [1], is not equal to the percentage grade in this sense: for a 5% grade the angle of inclination is about 2.86° or 3.18 grads.

The use of a decimalized right angle would require the creation of new trigonometric and log tables for the surveyors and astronomers using the new system and the French savants did not shy away from the task. The complex formulas were broken into a series of easy tasks and the ready supply of out-of-work wig makers were drafted into computational training to form a human computer. This employment may have been the greatest benefit of the proposed system of angular measure. It was not just money, meters and angles that the French sought to decimalize, they also proposed and created clocks that divided the day into ten hours of 100 minutes, each divided into 100 seconds.

I found a description of the French "Grade" in Plane Trigonometry by the Right Reverend J W Colenso printed in 1859 by Longmans Green & co. of London.

It is interesting to note that the French unit suffered the same fate as the later American creation according to another note from Rev. Colenso's text.

Abandoned or not by the French, nine years after the date of Colenso's text (1868), the 15th edition of James B Thompson's Higher Arithmetic still informed readers of the French grad, minute, and second and their English conversions

Knots and Knot theory The best mathematical definition I could find comes from a web page created by students, The Knot Theory Home Page. It states

"Knot Theory is a branch of topology that deals with knots and links. In topology, a sphere is the same as a cube, and a doughnut is the same as a coffee cup. It does not deal with the rigid properties of objects, such as length and angles, but instead the properties that no amount of bending, twisting, stretching, or shrinking can change.
A knot is a closed, one dimensional, and non-intersecting curve in three-dimensional space. From a more mathematical and set-theoretic standpoint, a knot is a homeomorphism that maps a circle into three-dimensional space and cannot be reduced to the unknot by an ambient isotopy."
The word knot itself is derived through the Old English "cnotta" and the earlier Greek term "gon" for knee. Gon is the same Greek root that means angle in words like polygon.

Jozef Przytycki recently contributed a note to a Math History news group regarding the early origin of knot theory from which the comments below are clipped .

The first mathematical paper which mentions knots was written by A.~T.~Vandermonde in 1771: "Remarques sur les problemes de situation"... ... In 1771 Alexandre-Theophile Vandermonde (1735-1796) wrote the paper: {"Remarques sur les probl`emes de situation "} (Remarks on problems of positions) where he specifically places braids and knots as a subject of the geometry of position . In the first paragraph of the paper Vandermonde wrote:
"Whatever the twists and turns of a system of threads in space, one can always obtain an expression for the calculation of its dimensions, but this expression will be of little use in practice. The craftsman who fashions a braid, a net, or some knots will be concerned, not with questions of measurement, but with those of position: what he sees there is the manner in which the threads are interlaced."
Yours
Jozef Przytycki

The image below is from the gates at the Center for Mathematical Sciences at Cambridge which house the Isaac Newton Institute

Notes on the history of pencil

From http://www.berol.co.uk/education/cont/writing/content_writing.html I found much of the following with some additional comments I added from some other sources:

The ancient Egyptians, and the Greeks and Romans, too, used a small lead disc for ruling guide line on the papyrus to keep the lettering even. The Romans called it a plumbum - Latin for lead.
{added content...Philip of Thesalonica, the Greek poet, in about 20 B.C., mentions the use of writing tools fashioned of lead in disc shaped pieces. The small lead disc used by the Romans to rule guidelines on papyrus may have been responsible for the error that led to identifying our family core as lead.}
Some additional input from a column by Zora Sweet Pinney:
The purest graphite discovered was revealed in 1564, when an Oak tree fell during a storm near Borrowdale, England. The shepherds in the area found the rough chunks to be useful to mark their flocks, but the raw material was also very dirty and messy to handle. That problem was addressed by cutting the material into square pieces and encasing them with wood. The material discovered was called "plumbago" (imitation lead). In 1779 K. W. Scheele, a Swedish chemist, found "plumbago" to be a form of carbon and suggested that it be called "graphite" from the Greek word for writing. The first hand made pencils, in the form that we know today are the "Crayons d'Angleterre", made from Borrowdale graphite. One year after the discovery in Borrowdale, Conrad Gesner of Zurich, wrote the earliest surviving description of a pencil in his Treatise on Fossils, illustrated with a woodcut by the author showing a wooden tube holding a piece of graphite. Some scholars believe this "Gesner pencil" was used by Shakespeare.
It was only logical that someone should eventually think of using a thin rod of lead for scribing fine lines, and equally logical that the invention should be called a 'lead pencil'. Who first invented, or who first named it, is unknown, but such pencils were in use by the fourteenth century, primarily as an artists' tool. Very beautiful, pale grey drawings, done with rods of lead, zinc or silver can be found in museums today, though all of them are now classified as silver point drawings.
Small things in daily use often seem so obvious that no one takes the trouble to write about them; and it was not until 1565 that one Conrad Gesner of Zurich described a pencil. Even then, it was only an aside in his Treatise on Fossils, but the description is sufficently detailed so that we know the writing rod was held in a wooden case.
It took a discovery in 1564 to make the name 'lead pencil' a complete misnomer. In that year, in the reign of Queen Elizabeth, a deposit of graphite (pure black carbon) was found at Borrowdale in Cumbria, in a form so solid and uniform that it could be sawn into sheets and then cut into thin square sticks. (it is interesting to note that these were still made square by tradition as late as 1860, though the reason for this shape had long since vanished). Little chemistry was known in 1564, so the material was called plumbago, or that which acts(writes) like lead. The pure graphite of the Borrowdale mines was the only such deposit ever found{added in England they must mean, for the Chinese also made wood pencils in the 1800's}, and its value was fantastic. It was mined only six weeks a year; armed guards escorted the wagons to London; and export of the ore was prohibited. The English Guild of Pencilmakers hand-carved wooden cases for the writing sticks, and enjoyed a world monopoly on the sale of the finished product.
From http://www.pencils.com/history.html I found :
The first mass-produced pencils were made in Nuremberg, Germany in 1662. Until the war with England cut off imports, pencils used in America came from overseas. (William Monroe, a cabinetmaker in Concord, Massachusetts, made the first American wood pencils in 1812.) Benjamin Franklin advertised pencils for sale in his Pennsylvania Gazette in 1729. George Washington used a three-inch pencil when he surveyed the Ohio Territory in 1762.
. At the same sight, I found the answer to that pressing question, "Why are Pencils Yellow?" (at least they almost ALL were in my youth,when dinosauers roamed the Earth). There is more about pencils, including a picture of The oldest pencil known to exist so check it out.

Simpson's ParadoxSimpson's Paradox is the name for a paradox that occurs when a reversal of results occurs if several groups are combined together to form a single group. This explanation will give you an idea of why the idea can lead to confusion

The Simpson in the paradox is E H Simpson who discussed the paradox in a 1951 paper, "The Interpretation of Interaction in Contingency Tables," Journal of the Royal Statistical Society, Ser. B, 13, 238-241.". In a note to the AP Statistcs discussion group Lee Creighton points out:

"he wasn't the first person to discover the paradox. The "Aggregation Paradox" had been known for a long time before that. The earliest citation I have handy is Yule, G.U., 1903, "Notes on the theory of association of attrbiutes in statistics," Biometrika 2: 121-134. "

Jeff Miller's website on the first use of math terms cites an even earlier referrence to the effect. Pearson had written on the topic as early as 1899,

The possibility of a conflict between total and partial analyses was first noticed in 1899 by Pearson, Lee & Bramley-Moore "Mathematical Contributions to the Theory of Evolution. - VI. Genetic (Reproductive) Selection: Inheritance of Fertility in Man, and of Fecundity in Thoroughbred Racehorses," Philosophical Transactions of the Royal Society A, 192, (1899), p. 278): "We are thus forced to the conclusion that a mixture of heterogeneous groups, each of which exhibits no organic correlation, will exhibit a greater or less amount of correlation. This correlation may properly be called spurious . . ."

Miller also credits the creation of the name "Simpson's Paradox" to a paper by C. R. Blyth, ("On Simpson's Paradox and the Sure-Thing Principle", Journal of the American Statistical Association, 67, (1972), p. 364.)

Slide Rule

The slide rule appeared shortly after the invention of the logarithm.  The invention of the slide rule seems to have been by William Oughtred who created and used them as early as 1621, some six or seven years after the first publication of logarithms by Napier.    The slide rule is made by marking off two logarithmic scales on seperate straight (or sometimes circular) pieces of material that could be positioned to "add" the logarithmic lengths in order to multiply the reperesentative numbers.

In spite of a few early users (Newnton for example), the Slide rule seems not to have been a early hit. Cajori's history of the slide rule includes :

"De Morgan says that slide rules were little used and little known till the end of the 17th century. He bases this conclusion on the fact that Leybourn, himself a fancier of instruments, and an improver (as he supposed) of the sector, has 30 folio pages of what he calls instrumental arithmetic in his ‘Cursus Mathematicus’ (1690), but not one word of any sliding-rule, though he puts fixed lines of squares and cubes against his line of numbers in his version of Gunter’s scale.”

On the European continent such an instrument was hardly known to exist. The only German writer to be mentioned is Biler, who in 1696 brought out a publication under the title: Descriptio instrumenti mathematici universalis, quo mediante omnes proportiones sine circino atque calculo methodo facillima inveniuntur. He called his instrument the instrumentum mathematicum universale. The instrument is semicircular in form and differs from that of Oughtred also in dispensing with the sliding indices and using instead the sliding concentric semi-circles. Biler does not state the source whence he obtained his idea of the instrument. A few years later (1699), Michael Scheffelt brought out in Ulm a book, describing an instrument called by him pes mechanicus, which was not a slide rule, but employed logarithmic lines of numbers, together with a pair of compasses, as in Gunter’s scale.

While in France, Gunter’s line was made known by Edmund Wingate as early as 1324,[This date is a misprint in Cajori's text, and should be 1624, I think] and was again described by Henrion in a work, Logocanon ou règle proportionnelle (Paris, 1626), we have not been able to secure evidence that would show familiarity with the slide rule before about 1700.
From the Oughtred Society web page (collectors of slide rules) I found that, "In 1675 Sir Isaac Newton solves cubic equations using three parallel logarithmic scales and makes the first suggestion toward the use of the cursor (earlier called a 'runner'). " Their history notes also add that, "Early in the 19th century the first slide rules come into use in the United States. Ex- president Thomas Jefferson has one...". Charles Hutton's A Mathematical and Philosophical Dictionary from 1795 describes Gunter's scale, and several types of sliding rules in use by that date.

I have not found any evidence that Newton actually used a cursor, or even suggested the use of one except in a quote from Cajori's The Slide Rule in which he writes about Newton's method of solving Cubic and Biquadratic equations on logartihmic rules:

The practical operation of this scheme would call for the use of a device to enable one to read corresponding numbers on scales that are not contiguous. Such a device would fulfil some of the functions of what is now called the “runner.” We must therefore look upon Newton as the first to have thought of such an attachment to the slide rule. Sixty-eight years later, Newton’s mode of solving equations mechanically is explained more fully and with some restrictions, rendering the process more practical, by E. Stone in the second edition of his Dictionary (1743).

Newton's "suggested" cursor did not much find favor until "In 1851 a French artillery officer named Amedee Mannheim standardizes a set of four scales for the most common calculation problems. The four scales include two double length, named A & B, for squares and square roots … and two single length, C & D, for multiplication and division. This scale set becomes the basis of slide rule design for the next 100 years and bears his name today. His design and use of a cursor hastens the eventual widespread acceptance of this feature." [Also from the Oughtred Society page ]

Another nice page about the history of slide rules, their use, and lots of pictures from his personal collection is maintained by Eric Marcotte.

Given the early date of invention, I was surprised to find that the slide rule did not become commonly used in some colleges until much later. For example, at a US Naval Academy web site I found, "In 1929 the slide rule became a formal part of the mathematics program and served as the midshipmen's primary calculator until 1976.” (to be replaced, I assume, by electronic calculators) [From A Brief History of the Department of Mathematics by Professor T. J. Benac ]

Teachers wishing to introduce students to the slide rule might use the onlineJava Pickett Slide rule by Derek Ross; or another java slide rule here as a classroom model by projecting it on a white board (it is very effective on a smart-board). Here is another very good site for information about slide rules, including the picture below.

For notes related to the history and use of the slide rule in math education, look in the 1900 - 1950 section of My notes on Early American math books and teaching

While at the Museum of the History of Science in Oxford, UK, I came across a cylindrical slide rule shown here.

A precursor to the slide rule is the Gunter Scale which was used with a pair of dividers to do the operations.

Tangents, Law of

The Law of Tangents is also sometimes called the Theorem of Tangents. Given a triangle with sides a, b, and c and opposite angels A, B, and C respectively, the Law of Tangents states that  . Since the naming is arbitrary, the formula is cyclic for any two sides and angles. Carl Boyer's "A History of Mathematics" suggests that the theorem may have been first discovered by Francois Viete,("variorum de rebus mathematicas" 1593) but it seems to have been published first by Thomas Fincke, in the "Geometria rotunda", the same book in which he introduces the terms tangent and secant. The theorems are often derived from a set of trigonometric identities known as the Mollweide Equations after Karl Brandan Mollweide (1774-1825) which state, using the same side/angle names as above:

and

Mollweide is also well known as an astronomer, and as an influential teacher of Moebius, as well as for a map projection that still bears his name. An explanation of the Mollweide "elliptical equal area" projection can be found