**Volume N - P**

**N- Dimensional**The earliest mention of dimensions in mathematics are related to algebra, much like we would today speak of the "degree of a polynomial", more than geometry. Still the association between the dimension of an expression was associated with spacial dimension as early as Robert Recorde's Whetstone of Witte in 1570, "The nomber that doeth amount thereof (3x3x3) hath gotten 3 dimensions, whiche properly belong to a bodie, or sound forme, and therefore it is called a cube, or cubike nomber." This usage persisted in to the middle of the 18th century. In 1843, Arthur Cayley wrote,

"Chapters in the Analytic Geometry of (n) Dimensions," but despite the title, it seems the work was more about algebra than geometry. Reviews of this (I haven't found the paper yet for a personal reading) suggest that he was still using the term in the way Recorde had used it.

Certainly though, by 1878, the idea of a fourth spacial dimension was being discussed and written about. In that year, B Stewart and P G Tait wrote Unseen Universe, and included, "suppose our (essentially three dimensional) matter to be the mere skin or boundary of an Unseen whose matter has four dimensions. C H Hinton, in the same year(1880) he became the son-in-law of the then Late George Boole, wrote a paper in entitled, What is the Fourth Dimension, he suggested that three dimensional objects might be merely cross-sections of four dimensional objects. He would also began teaching that year in the Upland School in Rutland where Howard Chandler also taught, and was a friend of Edwin Abbot, who would write the classic, Flatland, in 1884, describing a similar relationship between some two dimensional creatures and their three dimensional intersections.

Hinton also tutored his sister-in-law Alice (Alicia) Boole, who went on to independently rediscover many facts about the fourth dimension, such as there were only six regular polytopes (Like Platonic solids for higher dimensions) and demonstrated incredible ability to produce careful illustrations of three-dimensional intersections of these objects. For her work,The University of Groningen honoured her by inviting her to attend the tercentenary celebrations of the university and awarding her an honorary doctorate in 1914. Through her nephew, Geoffrey Taylor, a major figure in physics and wave theory, she met HSM Coxeter in 1930 and they began working together and produced a joint paper. To his friends at Cambridge Coxeter referred to her as his "Aunt Alice".

I have written much more about the incredible progeny of George and Mary Boole, Those Amazing Boole Girls, that you might enjoy.

**N - Gon**The earliest reference to N-gon I can find dates to 1806 in George Washington Hull's Elements of Geometry: Including Plane, Solid, and Spherical Geometry, where he writes, "What fractional part of each interior angle does each exterior angle of a regular N-gon equal?" The term seems to have been little used until the end of the 19th century.

I would have anticipated that number gons like 6-gon or 13 -gon would have been earlier, but the earliest example I found was in Thomas Tates Practical Geometry in 1860 where he writes, "calculate the angle at the center for each of the regular polygons:(i)5-gon,(ii)8-gon, (iii) 9-gon, (iv) triangle, (v) 10-gon, (vi) 16-gon"

A Google N-gram viewer search showed the popularity of N-gon rising quickly after 1900 and then decreasing around 1930 (depression effect on book publishing?) and then beginning to rebound around 1960 (New Math?) until the present.

**Napoleon's Point/Theorem**says that if you construct three

*Wikipedia |

The credit to Napoleon is thought to be misplaced, as the earliest attribution to him occurred in Faifofer's 17th Edition of Elementi di Geometria in 1911. However, Chambers Cylopedia includes a earlier reference to the name being applied, "Another remarkable property of triangles, known as Napoleon's problem"

*.**Wikipedia

The problem first appeared as a Gold Medal problem set at the University of Dublin in 1820. The problem consisted of three parts, to prove the inside triangles and outside triangles formed an equilateral triangle at their centers. The third question asked for the relation between the two formed equilateral triangles and the original. This part of the relationship is seldom mentioned now, it seems.

The problem ( was posted again in the "Ladies Diary" problems in 1829, already without the third part of the Dublin Competition (the magazine was a good source of a wide range of mathematical problems, and attracted many professional mathematicians. ) After numerous readers responded with solutions, there seems not to have been a single solution mentioning the Emperor.

**Napoleon's Point**is related to the same equilateral construction used in Napoleon's Triangles, but does not seem to be mentioned in any of the early problems. The same construction of equilateral triangles as before, but join the centroids of the equilateral triangles each to the vertex opposite the base they were built upon. These segments will intersect in a point called the Napoleon Points (one for inside, one for outside facing equilateral triangles. Such naming, I suspect, came from the name of the previous problem and the similarity of the preconditions of the result.

**Natural Logarithm**Logarithms are the inverse of exponentiation. A simple example of the relation is that 10^3 = 1000 and the base ten logarithm or 1000 is 3. A natural logarithm is a logarithm to the base e, so that if e^a = b, then the Natural logarithm of b is a.

The OED cites Natural Logarithm as first used in post-classical Latin by P Mengoli in 1659. This term was applied in French as "logarithme naturel" , and made it's way into English in 1746 by Ab raham De Moivre in the Philosophical Transactions, "The quantity that is here called a , is that which some call the hyperbolic logarithms;others, the natural logarithms."

These refer to logarithms with a base equal to the constant e (2.7182818284590...).

**Common Logarithms**are used to describe base ten logarithms which were created by Henry Briggs with the encouragement of Napier himself.

They are still sometimes referred to as hyperbolic logarithms, a term seemingly created by Euler in 1748 in his Introductio, since they are related to an area under the rectangle hyperbola y= 1/x. A logarithmic relation between areas swept out under the hyperbola had been recognized as early as 1649 by Gregoire de St.-Vincent.

You may also read or hear them referred to as Napierian Logarithms, but this is a misnomer; as Napier didn't actually use a number base, but were related to ratios of distances rather than exponents. They are however transformable to the natural logarithms with a function .

**Natural Number**The idea of the positive integers (and sometimes including zero as "natural" numbers seems to have emerged in the 15th Century. Nicolas Chuquet used the term "progression naturelle", from the French derivation of the Latin root, Naturalis, (by birth, according to nature). The word came into English in 1763 in Emerson's Method of Increments, "To find the product of all natural numbers from 1 to 100."

The usual mathematical symbol for the group is the bold N. Due to the frequency with which applications choose to include, or not include the number zero, it has become common for writers to adopt a different symbol for the two "natural" sets. The set including zero is sometimes written as N ∪ 0 (the union of N with zero) , \$ N_0\$ and \$ N^0 \$ are all used. For the set without zero, common symbols include \$ N^+, n^*, N_1 \$ and probably many more I haven't seen.

**Nebula**comes to us from the same word in Latin, which meant a cloud. The Greek root seems to be nephos, and is related to both clouds and rain. By the time of the Roman Empire, the word

*nimbus*was used for rain, or specific types of clouds, which we now call nimbostratus, those low dark spreading clouds which deliver rain or snow. The stratus root is also Latin, and gives the "spreading" meaning to the term. Nimbus is still used to describe the aura or cloudy luminescence that appears in pictures of saints and the Diety.

Somewhere in the late 1600's, or early 1700's astronomers began to describe the faint patches of diffused light in the night sky as nebulae, probably they resembled clouds. Nebula was also a medical term from the 1400's on to describe cloudy spots on the cornea of the eye.

Around the 1700's the term nebulizer was used to describe what we now call an atomizer in English, becasue the spray comes out in a cloud like mist. If neither term is familiar to you, it's the little rubber bulb spray that you might have seen on your Grandmothers dressing table for spraying perfume. Lister used such an item to spray ether antiseptic throughout his operating rooms in the 1860's as part of his pioneering work in antiseptic. On one of the Connections videos on Lister, James Burke made the clever pun that, "before every operation, Lister would say, "Let us spray."

**Negative**Negative numbers, and the equivalent word for negative were introduced by Brahamagupta, a Hindu mathematician around 600 AD. The Latin root of today's word is

*negare*, to deny. The negative numbers, in this sense, denying or invalidating an equivalent positive quantity.

The negative numbers were themselves denied for a long part of mathematical history, and only slowly came to be accepted. The first operational rules for what we today call positive and negative numbers came to us from the quill of Diophoantus(around 250 AD) who referred to them as "forthcomings" and "wantings". (I see this as similar to an accountant's debits and credits then the current use of positive and negative numbers. His work probably came from proposition five in Book II of Euclid's Elements in which he demonstrates with geometric figures what we would write in modern algebra as (a + b)(a-b)+b

^{2}= a

^{2}. This is of course easily recast as (a + b) (a - b) = a

^{2}- b

^{2 }

Al-Khwarizmi (850 AD), whose writings brought Arabic numbers (and Algebra) to the west, used a similar approach with negatives allowed in-process, but not as a final result.

Descartes, around 1636, used the French

*fausse*, for false, for negative solutions.

Thomas Harriot used a clever twist of logic to sort of accept the idea of negative roots. His method was to explain that the appearance of -c as a root of f(x) was only to be understood that c is a root of f(-x).

Franz Lemmermeyer wrote in a posting to the Historia Matematica Newsgroup that "Gleanings from the History of the Negative Number" by PGJ Vrendenduin suggests that a number line with both positive and negative numbers could be found in the work of Wallis (1657). Another posting to the same group quoted Kline's "Mathematical Thought from Ancient to Modern Times" :

Though Wallis was advanced for his times and accepted negative numbers, he thought they were larger than infinity but not less than zero. In his 'Arithmetica Infinitorum (1665), he argued that since the ratio a/0, when a is positive, is infinite, then, when the denominator is changed to a negative number, as in a/b with b negative, the ratio must be greater than infinity. (For me the logic he seems to apply is that if b is greater than c, then a/b is greater than a/c, but in applying that when b=0 is greater than the negative c, disagrees with his result suggesting c, the negative number, is greater than infinity.)Even as late as 1831, De Morgan would still write that one must recollect that the signs + and - are not quantities, but directions to add and subtract.

But along the way, there were pioneers. William Jones (who first coined pi for its present ratio) wrote in 1706, "To add a negative, is to take away a positive."

By 1901 the idea was less in flux, and the Transactions of the American Mathematical Society had an article stating, " The inverse of an element a, is called the inverse, and is denoted by -a."

shaped" .The prefix is from the Greek

The term was once used to describe a number of plane curves, but was appropriated by Richard Proctor in

an 1878 paper on cycloids for the epicycloid of two cusps, the figure made by a fixed point on a circle rolling around the outside of another with twice the diameter.

Another shape called a nephroid is now usually called Freeth's Nephroid after the writer he wrote about it on May 8 of the same year in the Proceedings of the London Mathematical Society.

derivative, or slope, at that point (x, f(x)). By taking the ratio of f(x) and f'(x) you know how far it is from the x value to the intersection of the slope with the x-axis. Subtracting this ratio from the value of the first guess of x, gives you a (usually) better guess for the root. Take that value of x, and repeat the process, or as the mathy types like to say, iterate. It doesn't work with all functions, but for most problems its a pretty effective tool.

Newton was the first to write about it in his Analysis of Equations by Infinite Series (but in Latin) in 1669, but seemed to use it only for polynomials, saw it as a strictly algebraic process and glossed over any mention of calculus. In Japan Seki Kawa independently seems to have discovered the same method, again without calculus, although he had used differential calculus in his astrology work also.

That left it to Joseph Raphson in 1890 to publish a method involving calculus that is the iteration method mentioned above. Maybe his name should be first!

The first citation of the OED, but not for the graphic nodes, was in 1864 discussing the points of intersection of two parallel lines on an infinite plane. The earliest use of the word in graphs is from 1941 in the proceedings of the Cambridge Philosophical Society. Earlier, and related citations include botanical use in 1835 for the nodes on a plant, where branching emerges, and in 1830 for a stationary point in a standing wave (where wave forms neet) and as early as 1572 for a knot, a clump formed by intersections of rope.

The Latin

For Hyperbolic geometry the earliest work was done by the Janos Bolyai,(1832) and Nicolai Lobachevsky (1830). Gauss in a 1824 letter claimed credit for having discovered it but had not published it.

According to Felix Klein, it was Gauss who first named it "non-Euclidean Geometry." The term Hyperbolic geometry had been introduced by Klein himself, in 1871. In English the term is cited in the OED in 1872 in discussing Klein's paper, and in 1878 Simon Newcomb described the "general appellation of non-Euclidean Geometry.

The first use in Statistics of Normal was Francis Galton 1877, "Perfectly Normal in shape."

By 1915 it was used as an adjective for a subgroup in which all the elements of group G, turn elements of the subgroup H into themselves, the H is called the Normal subgroup of G.

Closely after in 1915, the term normalized came to represent a base of a vector system in which all vectors were perpendicular and had a "unit" length.

As early as 1931 the name was used to refer to a number whose decimal expansion had equal probability for which all ten digits are equally likely, and all sequences of n digits are equally probable in the number. For example the phrase, "It is not known if pi is a normal number" by J. H. Caldwell. was used in 1966.

Many credit Gauss for the first use of the normal curve, and he not only figured out the function for the density function, but also the application to the method of least squares, and the maximum liklihood estimator. For this reason, many people call the Normal distribution the

The OED gives the earliest citation of the name, Normal Distribution, in 1877 to Francis Galton. Jeff Miller's page on earliest word use gives Charles S Peirce in 1873. In 1885 Galton extends his name to the "normal curve of distribution of error" a term that is also commonly applied, or shortened to normal curve of error.

"NTH is found in 1756 in Philosophical transactions of the Royal Society of London 49/84: “If the given series..be raised to the nth power, the terms of the series will truly exhibit all the different chances in all the proposed (n) observations.” [OED]

Nth is found in 1820 in Functional Equations by Babbage: "To find periodic functions of the nth order...."

**Nephroid**is a planer curve of degree six whose name means "kidney*Wikipedia |

*nephros*, for kidney, and the

*eidos*for resemblence, so it looks like a kidney. "Nephro" appears in many medical terms related to the Kidney, nephritis, and inflammation of the small filters in the kidney. The "oid" shows up in many math words like cardiod (heart shaped) and ovoid (egg shaped).

The term was once used to describe a number of plane curves, but was appropriated by Richard Proctor in

*Wolfram Mathworld |

Another shape called a nephroid is now usually called Freeth's Nephroid after the writer he wrote about it on May 8 of the same year in the Proceedings of the London Mathematical Society.

**Newton's Method (Newton-Raphson)**Is a method of numerical analysis to use differential calculus to approximate roots (zeros) of real-valued functions. The process is essentially to guess an initial x-v alue, and then find both the function at that value of x, and the

*Wikipedia |

Newton was the first to write about it in his Analysis of Equations by Infinite Series (but in Latin) in 1669, but seemed to use it only for polynomials, saw it as a strictly algebraic process and glossed over any mention of calculus. In Japan Seki Kawa independently seems to have discovered the same method, again without calculus, although he had used differential calculus in his astrology work also.

That left it to Joseph Raphson in 1890 to publish a method involving calculus that is the iteration method mentioned above. Maybe his name should be first!

If you find the midpoints of all three sides of a triangle, then draw the altitudes and mark the points where the altitudes intersect the opposite side, and the point on the altitudes half way between the vertex and the pint where the altitudes intersect (the Orthocenter), then all nine of those points will be concentric (on the same circle). There are actually lots of other important points on the circle, and it has been called the 6 point circle, the 12 point circle, and named for various people, including, most often, the German mathematician Karl Feuerbach, who first found six points. Poncelet and Brianchon (1820)had previously found and proved it, and later French mathematician Oiry Terquem (for whom the circle is sometimes named also), discovered the other three of the nine points above and the first to call it the nine point circle. [Kudos to Terquem for founding the first journal on the history of mathematics.] In 1882 Feuerbach would find four more points where the nine-point circle was tangent to the three ex-circles of the triangle and also where it was tangent to the incircle.

The OED cites the first appearance of nine-point circle in an 1865 Dictionary which refers to a circle passing through the midpoints of the sides of a triangle "referred to by Continental Writers as the nine-points Circle." The earliest mention of "Feuerbach's circle in English is in the Philosophical Society of Washington (DC), which refers to Feuerbach's theorem, "...and the circle is known to the Germans as Feuerbach's circle."

The configureation is rich with other interesting geometric relationships, so go, explore!

The OED cites the first appearance of nine-point circle in an 1865 Dictionary which refers to a circle passing through the midpoints of the sides of a triangle "referred to by Continental Writers as the nine-points Circle." The earliest mention of "Feuerbach's circle in English is in the Philosophical Society of Washington (DC), which refers to Feuerbach's theorem, "...and the circle is known to the Germans as Feuerbach's circle."

The configureation is rich with other interesting geometric relationships, so go, explore!

**Node**In mathematics, and more particularly in graphs, a node is a point on a graph that may be connected to other nodes by an edge. The term vertex is often used for these nodes, but they should not be confused with the vertices of a polygon, which have fixed positions not associated with a node of a graph.

The first citation of the OED, but not for the graphic nodes, was in 1864 discussing the points of intersection of two parallel lines on an infinite plane. The earliest use of the word in graphs is from 1941 in the proceedings of the Cambridge Philosophical Society. Earlier, and related citations include botanical use in 1835 for the nodes on a plant, where branching emerges, and in 1830 for a stationary point in a standing wave (where wave forms neet) and as early as 1572 for a knot, a clump formed by intersections of rope.

The Latin

*nodus*for knot is the root of the English word, and the use for a point of intersection in the 1660's, originally for the intersection point of planetary orbits. Bernoulli is reported to use the Latin word in 1753, but I do not know the exact context.

**Nonagon See Noon**

**Non-Euclidean Geometry**is a geometry constructed without the use of one or more of Euclid's postulates. The most commonly omitted one is the principal that through any given point not on a line, only one one line parallel to the given line can exist. Hyperbolic geometry and spherical geometry are the two most common. In this geometry, where geodesic great circles are the "lines", there are no parallel lines, and all triangles contain more than 180 degrees. The earliest studies of the geometry of the surface of the sphere date back to before Euclid. And the theorem of Menelaus was used in his famous

*Spherica*to prove a three dimensional analogy of the theorem. By 1463 the Arabic extended studies of these early Greek works had made their way to the west and Regiomontanus wrote his On Triangles with much information on the trigonometry of spheres.

For Hyperbolic geometry the earliest work was done by the Janos Bolyai,(1832) and Nicolai Lobachevsky (1830). Gauss in a 1824 letter claimed credit for having discovered it but had not published it.

According to Felix Klein, it was Gauss who first named it "non-Euclidean Geometry." The term Hyperbolic geometry had been introduced by Klein himself, in 1871. In English the term is cited in the OED in 1872 in discussing Klein's paper, and in 1878 Simon Newcomb described the "general appellation of non-Euclidean Geometry.

**Noon**Noon is not related to the number 12 in origin, but to the number nine. It is derived from the Latin*nona*(ninth). It is also the name of a Roman Goddess, who was the Goddess of pregnancy, "the spinner of life", and represented the nine months of pregnancy. The name originally refered to the ninth hour after sunrise, which was closer to the present 3PM.**Nonagon**, a polygon of nine sides, is derived from the same root with the addition of the suffix*gon*, bend or angle, which gives us knee, and the related genuine, and gnaw.**Norm / Normal,**The basic idea behind all uses of normal, and norm, are based on the old Etruscan derivation of the Greek*noman*, for a carpenter's square. Because it was used to indicate if walls were erect, or "the right way" this right angled instruments name came to represent "standard, acceptable, or the 'right' way". This shows up in the names of teaching colleges in the 1800's which were called "Normal Schools", a combined term that appears in English as early as 1797. This usage of "the usual way" is in English as early as 1598. By 1704 the mathematical use for a perpendicular line, "Normal is the same with perpendicular, or at right angles."The first use in Statistics of Normal was Francis Galton 1877, "Perfectly Normal in shape."

By 1915 it was used as an adjective for a subgroup in which all the elements of group G, turn elements of the subgroup H into themselves, the H is called the Normal subgroup of G.

Closely after in 1915, the term normalized came to represent a base of a vector system in which all vectors were perpendicular and had a "unit" length.

As early as 1931 the name was used to refer to a number whose decimal expansion had equal probability for which all ten digits are equally likely, and all sequences of n digits are equally probable in the number. For example the phrase, "It is not known if pi is a normal number" by J. H. Caldwell. was used in 1966.

**NORM**Norma was used by Gauss in 1832 to describe the size of the Gaussian integers (a+bi), asserting that the

*norma*was the sum of the squares of a and b. In 1856 W. R. Hamilton generalized the name to all complex numbers as the norm. Today students studying vectors will call the square root of the norm, the magnitude, or length of a vector drawn in an Argand Daigram. This "norm" was first used by Albert A Bennett in 1921.

**Normal Distribution**in statistics was first studied by Abraham de Moivre in 1733, but it was not published until he included it in the 1738 third edition of his Doctrine of Chances. This paper was the first study of his approximation to the binomial distribution which we now call the normal distribution. Statistical historian Stephen Stigler suggests that de Moivre never uses his law beyond approximating binomials and may not have had any understanding of the density function.

Many credit Gauss for the first use of the normal curve, and he not only figured out the function for the density function, but also the application to the method of least squares, and the maximum liklihood estimator. For this reason, many people call the Normal distribution the

*Gaussian function*.

The OED gives the earliest citation of the name, Normal Distribution, in 1877 to Francis Galton. Jeff Miller's page on earliest word use gives Charles S Peirce in 1873. In 1885 Galton extends his name to the "normal curve of distribution of error" a term that is also commonly applied, or shortened to normal curve of error.

**Nth**The use of "nth" to indicate some unspecific term or element seems to have been a necessary creation of the growth of mathematical language in the 17th and 18th centuries. Jeff Miller's web site on the Earliest use of some Math terms gives two early uses:

"NTH is found in 1756 in Philosophical transactions of the Royal Society of London 49/84: “If the given series..be raised to the nth power, the terms of the series will truly exhibit all the different chances in all the proposed (n) observations.” [OED]

Nth is found in 1820 in Functional Equations by Babbage: "To find periodic functions of the nth order...."

**Null/ Null Set**Null is adopted from the Latin

*nullus*for something not important or nonexistent. It was in the "not important" meaning that the word was first used in English by Francis Bacon. He applied the term to a cipher for blank spaces, or just to throw off the code breaker. By 1648 the term was applied to the "cifer, or null" for the digit 0. The use of "

**null hypothesis**" in statistics is credited to Hill and Kerber in Models, Methods and Analysis in 1967.

**Null Set**is the meaning but not the actual wording of OED's first citation with the meaning of "null set." The 1905 listing gives, "a null, or empty, class." After 1889, when Peano defined class theory as an extension of set theory the idea of null classes and null spaces were often mentioned in the same sentence was not unusual; such as this example from Phillips Course of Analysis, in 1939, "Each bundle is a set whose members are classes< thus each is a set of classes ... By the general definition of number, the number of terms in the null class is the set of all classes which are similar to the null class"

**Nullcline**– null meaning “none” or “not any” and cline meaning “to slope”. Given a set of differential equations, the “x-nullcline” is when x’=0. The definition appeared in a 1967 article by Endre Simonyi although it was called the “directivity curve”. *Derek Orr

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

The word came into English before the 1400's as the sum of quantities, and as a plural for the same use, as well as something graphically or symbolically representing a quantity, even a non-specif one.

**Number Line**While the OED first citation for "number line" is in 1964, Dave Renfro sent me a much earlier source, an article that first appeared in the American Mathematical Monthly in March of 1928 in the "Undergraduate Mathematics Clubs" section by Herbert Ellsworth Slaught, "The evolution of numbers — An historical drama in two acts," in which the term appears three times. The first is delivered by the herald before the closed curtain in the prologue to the play. His remarks include this passage showing to whom he credits the creation of the idea of a number line, "Well might Descartes be awarded martial honors for his services as "general in charge of personnel" when he stationed his positive and negative integers and zero as sentinels along his

**number line**and then filled in the intervening spaces with the lowly fractions..."

prior to this usage, the most common term for the idea was called a number scale, or scale of numbers, a term that is still commonly used in humanities books, such as the 2002 Data Analysis for Behavioral Sciences, "In order for us to make meaningful multiplicative or ratio comparisons of this type between points on our number scale ..."

Woods Analytic Geometry and Calculus in 1917 on page 3 includes, "Number scale: On any straight line assume a fixed point O as the zero point , or origin , and lay off positive numbers in one direction and negative numbers in the other ." And a "number scale" for integers aligned with one of powers of two, are found in the journal Power, on page 384 in 1924.

In Luke Heaton's 2017 Short History of Mathematical Thought, he credits Richard Dedekind for the first formal definition of a number line, but does not cite the source.

An early innovation of the idea of number line was present in the work of 9th Century Moroccan mathematician who used both positive and negative numbers in constructing a number table of integers and their powers, using it to show how adding and subtracting exponents yields the exponent of the product or quotient.

Certainly another candidate for using a number line was John Wallis, in his Algebra, where he uses a number line as a guide to show that negative numbers were not impossible by constructing a number scale/line with both the positive and negative numbers on it, but masked as four points labeled A, B, C, and D. He then presents a story of a walk beginning at A(0) walking forward five units to B(4), then retreating 3 steps to C(2). Then he repeats the imagined walk by having the walk forward to B, then retreating 8 steps, and marks off point D, where he was 3 below zero steps.

**Number theory**Number theory in practice existed in the Pythagoreans, Euclid, and Diophantus, and throughout the history of the study of mathematics. Blaise Pascal used the term in a letter to Fermat on July 29, 1654,"The Chevalier de Mèré said to me that he found a falsehood in the theory of numbers for the following reason."

The first English citation in the OED credits Peter Barlow in 1811 for An Elementary Investigation of the Theory of Numbers. I think by then the word had been in somewhat common use, before it would appear in the title of a book.

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

My current OED gives the definition of innumerate as "ability with or knowledge of numbers."

**Numerator**The top number in a common rational fraction serves to tell how many of the fractional parts described by the denominator, there are. Is is drawn from the Latin "enumerate", to count out. Both denominator and numerator were used in Liber Abbaci by Leonardo de Pisa, which brought Arabic numbers to the Western World.

The OED gives the first citation as 1539 in an unknown authors publication on "learning to reckon."

**O**

**Obelus**The symbol ÷ which is sometimes used to indicate the operation of division is called an obelus. The word comes from the Greek word

*obelos*, for a spit, or spike, a pointed stick used for cooking. Perhaps because it is both sharp and used for piercing meat, the word is sometimes used for a type of stabbing knife called a dagger. The root also gives rise to the word obelisk for a pointed pillar of stone. The symbol was used as an editing notation in early manuscripts, sometimes only as a line without the two dots, to indicate material which the editor thought might need to be "cut out".

The symbol was used for a division symbol as early as 1650, although why it was chosen over some other for the purpose is not clear. It may be it became so popular in England because it was John Pell who seems to have introduced it in Rahn's Algebra. I have written here that I do not believe Pell meant it as a division operator ( it was never used the way in Rahn's book), but instead as an instruction for how to proceed along an procedure in mathematics, like drawing a vincula, or parenthetical enclosure, around everything that came before, in order to divide the whole by some quantity. Cajori says Rahn's Teutsche Algebra in 1649 is the first use of the obelus in print for division. As late as 1923 the National Committee on Mathematical Requirements wrote, "Since neither ÷ nor : , as signs of division, plays any part in business life, it seems proper to consider only the needs of algebra, and to make more use of the fractional form and of the symbol /, and to drop the symbol ÷ in writing algebraic expressions. "

**Oblique**Comes to English as a borrowing from the Latin,

*obliquus*, for indirect, or slanting. The word is still preserved today in military drill commands for a turn of 45°. It came into English first in the general idea of slanted or indirect. The first citation in science was in astronomical term for the horizon. in 1503, and spelled oblyk. By 1560 Leonard Digges geometry, Pantometria, had "of solids called prismata, there are two kindes, the one direct, or upright...the other oblique, or declining. In 1570 Henry More described "Rhomboides, is a parallelogrammicall figure with unequall sides and oblique angles.

*Wikipedia |

**skew.**Both oblique and skew are also used for cylinders and pyramids in which the axis is not perpendicular to the base.

**Note that skew, or oblique, prisms (or cylinders) still have their bases in parallel planes, but they may be shifted so that the line joining their centers, the axis of the body, is no longer perpendicular to the bases. In the case of polygonal faces,**

*Wikipedia |

**Obtuse (angle)**Obtuse is from the Latin ob, against, and tundere, to beat with the literal meaning, to beat against. An object thus beaten against will become blunt or dull, and the application to the obtuse angle is in this sense, not sharp(acute) but dull, (obtuse). The word ambligon or amblygon was used sometimes up into the 19th century. Billingsley, in his 1570 translation of Euclid's Elements, used both terms in a single sentence, "An ambligonium or an obtuse angled triangle". W R Hamilton in his Metaphysics of 1860 gave the three types of triangles as "the rectanghular, ... into amblygon, ... and into Oxygon.

**Occam's Razor**William of Ockham was an English philosopher of the Early 14th Century. He is most remembered today for the quotation, "

*Entia non sunt multiplicanda praeter necessitatm.*" The direct English translation is close to, "Entities ought not to be multiplied except from necessity." Occam's Razor has become the scientific rule for deciding between two theories to explain a single phenomenon; "Given two otherwise equal theorems, the more simple one is the better."

I remember reading somewhere that Ada Lovelace's husband, William King-Noel, first Earl of Lovelace and Viscount of Lovelace, was a descendant of William of Ocham, and that some of her love of math may have been acquired through her mother, Anna Noel Byron, whose intense love of mathematics led her poet husband to declare her the "Princess of Parallelograms."

*Wikipedia |

**Octahedron,**The Octahedron is one of five Platonic or regular convex 3D solids. It has 8 equilateral triangular faces, 12 edges, and 6 vertices. The five solids are called Platonic because Plato itemized them in Timaeus around 360 BC. The octahedron is the combination of two square based pyramids adjoined at their square bases.

The word is almost a direct borrowing from the Greek,

*octaedros*, through the Latin,

*octaedrum*, and was first used in English in Billingsley's 1570 translation of Euclid's Elements, "An octahedron is deuided into two equall..."

For students interested in information on the octahedra volume, surface area, duels, and other relationships, the Wikipedia page is full of good information.

**Odd Function**(See also- Function) An odd function is a function of one variable, so that changing the sign of the argument reverses the sign of the function value, or symbolically, f(-x)=-f(x). The first OED citation is in the Philosophical Transactions of the Royal Society in 1812, "..supposing P to be any odd function..." .

**Odds/Odds Ratio**The earliest mention of odd in England was in relation to something irregularly done, T. Morley used the term in relation to music, "... in two crochets and a mimime, but odded by a rest, so it never comes even till the close." This 1598 meaning shows the irregular and regular event cast alongside the odd (impaired) and even (paired) quantities.

The current idea of odds as a ratio is demonstrated clearly a year earlier in the Language of Shakespeare in Henry IV in part II, act 1, scene 1 "Knew that we ventured on such dangerous seas, That if we wrought out life 'twas ten to one'; and then the word, "I will lay odds that err this yeere expire...".

The idea of odd numbers as unpaired or uneven entered into the use of the plural odds for contests and wagers, when the phrase, "to make odds even," or "to mak our oddis evyne" in 1513.

Again speaking in terms of differences between quantities in 1548 writing," Whiche is by a great oddes higher." The earliest OED mention of odds used as a ratio is from T. Churchyard in 1560, "I durst lay oddes who trust you long...".

Be aware, however, like so many differences between British English and American English, odds are constructed differently in the two countries, so 3 to 1 here is 4 to 1 there (I hope I got that right).

**Ogee/Ogive**The words came into English very early from the French

*ogive*, and perhaps from the Latin,

*obviateata*, but that source is not fully agreed upon. Ogee seems to have been a shortening of the other, but both have been used in architecture since about 1300. By 1800, Chambers Cyclopedia gives the definition of Ogive under the entry for ogee. The shape is a curved arc blending a concave and a convex curve at points of equal tangent, called an

**inflection point**. The two different shapes are termed

**Cyma recta**for the curve that goes from positive to negative concavity, and

**Cyma reversa**for the opposite order. Cyma is from Greek

*kuma*, for a swell or a wave. Recta and reversa are from the same roots that gave us straight and rectangle on the first, and reverse, obverse, from the later. The church entrance shown below is cyma reversa.

The 12th century gothic cathedrals frequently used a pair of ogives as arched ribs for support, and also decorative entries. England has many beautiful examples in little tiny communities almost everywhere.

The statistical term came from Francis Galton in 1875 who used it to describe a cumulative frequency distribution, "When the objects are marshaled in the order of tier magnitude along a level

the ordinates... will form a curve of double curvature... such a curve is called, in the phraseology of architects, an 'ogive'. " I have used both ogive and ogee in instructing my students on cubic and tangent curves. I like ogee especially because I would pun that it was named for people looking at the curve for the inflection point and saying, "Oh, Gee! There it is." as I point out the inflection point.

**Optimization/Optimisation**– late 19th century: from Latin, neuter (used as a noun) of optimus ‘best’. OED gives earliest citation as 1857, but adds that term was used rarely before mid 20th Century.

The study of extremal values (or best values) of functions over some domain given some constraints. Sometimes spelled “optimization” and can also be titled “math programming”. There were essentially two ways to find optimal values: analytically through calculus techniques and iteratively through sequential techniques. The former was studied by Fermat and Lagrange while the latter was due to Newton and Gauss. There are many subfields of this including optimal control, calculus of variations, and linear/quadratic/integer programming. *Derek ORR

**Order of Operations**has been an issue for as long as people have tried to write expressions for combinations of mathematical operations. You still see frequent simple expressions like 8 ÷ 2 x 2 in those "Can you get this correct?" queries on the internet. My answer is always to say "Yes, cross out your obsolete symbol for division." Cajori around 1930 writes that different authors instruct different orders of use, and closes with the suggestion of a British committee on teaching arithmetic that "recommends the use of brackets to avoid confusion...". Advice that is still being regularly issued by teachers today.

However some authors were presenting much the same guidance that has become standard in most places today, In 1912, First Year Algebra by Webster Wells and Walter W. Hart has: "Indicated operations are to be performed in the following order: first, all multiplications and divisions in their order from left to right; then all additions and subtractions from left to right."

Even when students all over the problem are sure they have everything figured out because they memorized Pemdas in US, or Bedmas in Canada, Bodmas in the UK, India and more, or Yes, there's more, Bidmas in some parts of Africa, beware children. If you open Excell spreadsheet or Matlab and enter 2^3^4 you will get 4096, (2^3)^4) (I get the same on my Ti-30XIIS, but if you enter it into Google search, or Wolfram alpha... you get a 25 digit number that is 2^(81), (both happened just now, although kudos to Google, they did write out the bracketed 2^(3^4) above the answer).

**Ordinate/Ordinal**The ordinate of a point is its difference from the x-coordinate, in modern terminology, it is its y coordinate in an x-y coordinate system. The x-coordinate often referred to as the abscissa. The origin is from the Latin,

*ordianare*, to put in order. All the words you might expect, order, ordinary, orderly; are from the same root. The ordinal numbers are the numbers used for indication position, (first, second, third, etc.) The first use of the word for the y-coordinate of a points seems to have been by Leibniz around 1694, although it had been used in a more general sense for distances by others.Jeff Miller offers, "Cajori (1906, page 185) writes: The Latin term for 'ordinate,' used by Descartes comes from the expression

*lineae ordinatae,*employed by Roman surveyors for parallel lines" but no date is given. The first OED citation is 1706 by H. Ditton in a calculus paper.

**Orthocenter/Orthic triangle**While the term altitude is rooted in old Latin terms, the name now most commonly used for the point of intersection of the three altitudes of a triangle is totally native English. The origin, as told by John Satterly (and repeated to me by John H. Conway) was published in a 1962 article in the Mathematical Gazette.

The OED's first citation of word is a year later, 1866 when N. M. Ferrers used the word in an book on Trilinear Co-ordinates.

**Orthic triangle/Pedal Triangle**The Orthic triangle, or the Pedal

*Wikipedia (I think?) |

*ped*)) of the altitudes as it's vertices.

David Wells' Curious and Interesting Geometry offers "The only closed path of one circuit [for a billiard ball bouncing around inside an acute angled triangle] is the pedal triangle.."

The first citation for "pedal triangle" is 1862 by Gorge Salmon in a book on three dimensional analytic geometry. The first citation of OED for an "orthic triangle" is in 1920 for the American Mathematical Monthly, "... is frequently called the orthic triangle". I found a book by Stanley Rabinowitz that is titled Problems and Solutions from the Mathematical Visitor, 1877-1896, on page 22 is a problem by Christine Ladd to find the radius of the "radius of the circle inscribed in the orthic triangle." I would like to imagine that this is the same American Psychologist and mathematician named Christine Ladd-Franklin who was a student of J J Sylvester. I am now certain of this being the same mathematician. I found an note from the editor of the publication, Artemis Martin, who stated much earlier that she was a contributor, and adding that "It should have been stated that Miss Christine Ladd was the first American lady member elected to the London Mathematical Society, she is now Mrs. Fabian Franklin.

**Orthogonal**is used in several ways in different areas of mathematics, but most are related to perpendicularity. The OED cites a Latin,

*orthogonalis*, used by a British source as early as 1109, and the adoption of the French orthogonal in 1560 by Leonard Digges in his geometry, Pantometria, "Of straight lined angles there are three kindes, the Orhtogonall, the Obtuse, and the Acute Angle."

A 1694 citation for "Orthogonal Line", and in 1878 in a crystallography paper, "..planes intersect at right angles.... axis of orthogonal symmetry."

In 1852 J J Sylvester in the Cambridge and Dublin Math Journal used the term "orthogonal transformation" for a transformation on a vector space preserving lengths and angles and inner products. A similar term was used for Matrices in which the rows and columns were orthonormal when treated as vectors.

A very different use occurred in 1933 when the Journal of Agriculture used for the name of an experimental technique using blocked sampling as ,"arranged so as to be orthogonal".

**Oscillation**- early 18th century: from Latin oscillat- ‘swung’, from the verb oscillare. The second earliest citation in the OED is by Welsh mathematician William Jones, famous for the creation of pi as the ratio of the circumference to the diamter, and it is in the same book, his Synopsis Palmariorum Matheseos in 1706 that he is cited for, "in a medium that does not resist, the shorter oscillations in a cycloid are nearly isochroal."

The “oscillation” of a function is a number that quantifies how much the function varies between its extreme values. In general, “oscillation” is a function property with multiple changes in its slope. There are many types of oscillations such as “harmonic oscillations”, “coupled oscillations”, “damped/driven oscillations” etc. *Derek Orr

;

**Osculate/Osulating/Osculum**The terms all came as borrowings from the Latin,

*osculat-, osculari,*to kiss. Mathematically it usually refers to a point where two mathematical objects share a single point of contact, most often with one or more of them being curved. Osculating was used by Leibniz in his

*Acta Eruditorum*of 1686 "Et Gêner alia de natura linearum, anguloque contactus, et osculi, provolutionibus , aliisque cognatis, et eorum usibus nonnullis. C'est-à-

*osculi*refers to curvature, and the second,

*osculation*, to kissing.

Morris Kline credits John Bernoulli for the creation of the term "Osculating Plane" in his Mathematical Thought from Ancient to Modern Times, V2, without giving a date.

The OED has a 1706 citation for

**osculum**, but with no relation to geometry. An OED citation for Chambers Cyclopedia in 1728 uses

**Osculum**as the tangent point where the circle

**osculates**the evolute, and credits the creation to Huygens. This is also the earliest mathematical mention of osculate. Osculating of curves or surfaces is first cited in 1816 in Lacroix's Calculus, "This circle, called the osculating circle..

**Oxygon**is a (mostly) unused name for an acute angled triangle. The roots are from the Greek

*oxy*, for sharp, which is the root of oxygen, and

*gon*for angle, literally, sharp angled. The OED has a citation for the American Mathematical Monthly in 1967. The adjectives oxygonal and oxygonous appeared as late as the 19th Century.

W R Hamilton in his Metaphysics of 1860 gave the three types of triangles as "the rectangular, ... into amblygon, ... and into Oxygon."

A google search was useless as it couldn't distinguish the use of oxygon from oxygen.

**P**

**Palindrome**A Palindrome is a word, or number, that reads exactly the same forward or backward. A famous language palindrome is Madam I am Adam. The word comes almost directly from the Greek,

*palindromus*. Although the word appears in the OED as early as 1636 in regards to language, although the earliest definition is for forward and back syllables, rather than letters. Numerical palindromes are first mentioned in 1972, in regards to the date, 27,7,72 for July 27 of 1972. Obviously the British Times. Even musical palindromes, in which a passage was repeated in reverse appeared earlier, in 1947.

**Parallelepiped**This word for a solid made by intersecting pairs of

*Wikipedia |

A rectangular or orthogonal parallelepiped is the shape of a room or a shoe box. The word is condensed from the Greek word

*parallelepipedon*for the same shape. (I actually love this word, it breaks down as parallel beside the base, or upon the base, and really, pronounce it as that, parallel Epi, pedon, with a mild accent on the epi, just do it for me, please.) Parallelepipedon was the word used by Billingsley in his English translation of Euclid's Elements in 1570. According to John H Conway, this was hte common term in use until around 1870 to 1900 when it gave way to parallelepiped. This newer term now seems destined to demise due to changes in school curriculums and reduction of the coverage of 3D geometry (audible sob!) , although one correspondent suggests that the (beautful) parallelepipedon would be recognized by most Spanish language students. It was a nice surprise to find the parallelepiped was recognized by my spell check some years ago, but alas, that is not longer so.

The

*para*is common in math words and is related to other words like parlour, paragraph, and parable.

*Allei*became our alter, for other and gives us alternative and alternate. The

*epi*root shows up in epidermis (on the skin) and epitath (on the grave) and epicycle (on the circle), and now (writing in the time of corona virus) for epidemic (upon the people).

*Pedon*is from the Latin

*ped,*for foot, and was also generalized to base, or plane.

**Parameter**– In modern Latin, from Greek para- ‘beside’ + metron ‘measure’. It usually is a fixed element of a system which, when changed, creates new qualitative information about the system. For example, in the equation f(x) = ax^2 + bx + c, the “x” is a variable while a,b,c are parameters. Depending on what a,b,c equal, this equation might have 0 real roots, 1 real root, or 2 real roots.

Also related is “parametrization”, which can also be spelled “parameterization”, “parametrisation”, or “parameterisation”. It is the process of finding parametric equations to classify some mathematical object, usually a curve or surface. It would be given as “x=f(t), y=g(t), z=h(t)” for example and “t” would be the parameter.

**Pascal's Triangle ----- See Arithmetic Triangle**

**Pell's Equation**/

**Pell-Fermat Equation**is a Diophantine equation of the form \(x^2 - Ay^2 = 1\), or frequently written with the roles of x and y reversed to be \( y^2 = Ax^2 + 1\) . The name Pell comes from the English mathematician John Pell (1611-1685). This is one more of the many cases in math where the wrong guy gets the credit. It seems that in summarizing the history of work on the problem, Euler gave credit to Pell when it was Fermat who should be cited. This was not a judgement call, but apparently a mistake by the great algebraist. In fact, Pell had done no more than copy it in his papaers from some of Fermat's letters. Fermat had been the first to state that, where A is any integer not a square, the equation always has an unlimited number of solutions in integers (x, y)

The equation was studied in special cases centuries before either of the men named above were born. The records of both Greek and Indian mathematicians who studied the problem where A=2 in the 4th century BC because of the relation of the general equation to the square root of 2. For any solution, the ratio of x/y gives an approximation of the square root of A. Using the first two positive solutions, (3,2) and (17,12) the sequence of approximations is 1.5, 1.41666.. coming very close quickly.

Younger students might want to use a method derived from Brahmagupta mentioned later, to use these solutions to generate others for the A=2 solutions. Taking any (x,y) solution, the new \(x_1, y_1\) is given by \( x_1 = 3x+4y \) and \(x_2 = 2x+3y\) .

Archimedies approximated the square root of three to \(\frac{1351}{780}\) . His method of approximation is unknown, but this is exactly the 6th number that the Pell-Fermat approximation sequence would yield.

Diophantus gave a more general equation in the 3rd century forcing A to be a square, and replacing 1 with any integer.

In the 7th century the great Indian mathematician/astronomer Brahamagupta derived a method of deriving another solution from an existing one.

In the 17th century, Fermat rediscovered the equation, and claimed he had solved the smallest solution for all numbers up to 150, and taunted the English, naming out John Wallis in particular, to solve the next 150. It was one of these methods of solution by Wallis/Brouncker, that Pell wrote into a translation of Rahn's Algebra that caused Euler's mis-crediting the solution to him.

By 1769 Lagrange had come up with a method commonly used today, with continued fractions.

**Pedal Triangle**For any triangle, and any point, P, not on the sides of the triangle, the triangle connecting the three points where the

*Wikipedia |

**incenter**of the triangle, then the pedal triangle is called the intouch triangle, as it strikes the three points where the incenter is tangent to the triangle. If P is the

**circumcenter**then the pedal triangle is the medial triangle, connecting the midpoints of each side of the triangle.

If the point P falls on the

**circumcircle**of the triangle, then the pedal triangle will degenerate into a straight line, known as the

**Simson line**.

If A', B' and C' are the feet opposite angles A, B and C respectiely, then the sum of the squares of (AC'), (BA'), and CB') is equal to the sum of the squares of (C'B), (A'C), and (B'A).

*Wolfram Alpha |

**Anti-pedal triangle**. On each point of ABC, construct the perpendicular to P(A, B or C) and the three points where these perpendiculars intersect, it the anti-pedal triangle. Each pedal, and anti-pedal pair, and the area of ABC is the geometric mean of the areas of the pedal, and antipedal pair. In simple algebra, if P is area of the pedal triangle, and P' is the area of the anti-pedal triangle, and T is the area of original triangle, then \( P * (P') = T^2\)

Pedal triangle seemed to emerge in the early 20th century.

There are, of course,

**Pedal Tetrahedrons**and of course,

**Anti-pedal Tetrahedrons**and as you can imagine, someone has extended these to multi-dimensions. The earliest record (if you can beat it, share) was in the Amerian Mathematical monthly in 1894.

**Pentomino**The word domino was first used to refer to the hooded black cape worn by priests, and later to black masks (if you remember the Lone Ranger, that's the type) worn a masquerade balls. Although people sometimes try to attribute the name to a short for of dues-omino, or some such word, it seems much more likely it was from the Latin root for master, from which we also get dominate, and domain. Around the beginning of the 20th century there were several poplar puzzles using shapes made with more than two squares, stuck together along their edges. This was just after the time the Tangram puzzles, an assembly puzzle of seven geometric shapes, came to the united states and may have inspired some of the others. Because there are few possible shapes with four or less, and so many with six or more, interest centered on twelve unique shapes that can be formed from five squares joined at their edges. The earliest use of what we now call pentominos in a mathematical recreation was posed Henry E. Dudney as the problem of the broken chessboard in The Canterbury Puzzles published in 1919. The term pentomino was created later by Solomon Golomb, as an extension of the word domino. He first used the term at the Harvard Math Club in 1953. His book on them in 1965 and the puzzle sets of pentominoes became very popular after being covered in Martin Gardner's Scientific American column.

*Wikipedia |

**Percent % / Per mil ‰ Basis Point ‱**Percent for parts out of 100 comes to us almost unchanged from its Latin roots, per for each, and centum for 100. In fact percentum is still found in many dictionaries as an antiquated term for percent. Not obsolete to all, on 19 July of 2000, the Cape times printed an advertisement that stated, "payment 10% (per centum) of the purchase price on the day of sale." The exact same per centum appeared in 1565. The earliest usage of percent in the OED is by T. Gresham in 1568.

David E Smith suggest that the % symbol derived from one used as early as 1425 in an Italian manuscript.

**Per mil**means by the thousand, and 3‰ is the same as .3%. Taxes are often quoted in a rate per mil. The earliest citation in the OED is for 1682 in a work on customs. The spelling is sometimes (than and now) per mille. When you see a note on shooting percentage of a basketball player being .645, it's not a percentage that's a decimal fraction. Often the decimal is ignored and they say it's 645. The same for baseball players batting 300. When used this way, they are expressing the efficiency per mil, and could be written as 645‰ .

**Basis Point**, often abbreviated as BP is a method of expressing parts per ten-thousand. When you hear the Federal Reserve say it's lowering the interest rate by 50 basis points, they mean the interest is lowered by 50 ‱ or 5 mills, or 1/2 percent. Baseball on base percentages are presented as four digit decimal numbers, so without the decimal point, you could write 6453

**‱. Per ten-thousand is a common statistical and epidemiology standard, and appears in use as early as the 1850's. Basis points is used in financial notations as early as 1885.**

**Perfect Numbers**The perfect numbers are whole numbers that are equal to the sum of their proper divisors. Six is a perfect number since 1 + 2 + 3 =6. This was known to the ancients far into early mathematical history. This sum is often called the

**aliquot**sum of the original number. Euclid wrote about perfect numbers in is Elements, and gave a rule for the creation of these ideal or complete numbers that is identical to what we now write out as \$ 2^{p-1}(2^p - 1)\$ for prime p. That means all even perfect numbers are of the form n(2n-1) where n is a power of two and thus in binary are always a string of ones, followed by a one-shorter string of zeros. 110, 11100, and 111110000 are binary for 6, 28, and 496. It also means that every perfect number is the sum of some string of consecutive natural numbers; 6= 1+2+3, 28 = 1+2...+7, etc, and are also called triangular numbers, because they can be arranged in a triangle of dots.

It is not known if there are any odd perfect numbers, but if there are, they will be very large.

Another method extends the idea of perfect numbers to what are called

Another method extends the idea of perfect numbers to what are called

**pluperfect**or k-perfect numbers where k is such an integer. This method adds the number itself to the**aliquot**sum, so the original perfect numbers are 2-perfect, and numbers whose aliquot sum is twice the value of n are 3-perfect. The smallest 3-perfect number is 120.**Perimeter**The origins of perimeter come from the Greek roots,

*peri*(around) and

*metron*(measure). Peri shows up other "around" words like

**periphery**, the words the Greeks used for the

**circumference**of a circle. The

*pher*is the root for words about carrying, and is the same as the fer in circumference. Perimeter is first cited in the OED around 1425. Periphery was first used by Thomas Digges in 1560, but surprisingly, to me, was not cited for Billingsley's translation of Euclid's Elements, the first English translation using circumference, from the Latin, instead.

The peri part of both words remains today in many biological terms, the pericardium is a membrane that wraps around the heart, and periderm is used for the outer bark or covering of a plant, literally the 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 this is how perigee (near

*geos*, earth) came to mean the point where the mood is closest to the Earth, and perihelion (near

*helios*, sun) for the word for a point in a planets orbit where it is closest to the Sun.

**Periogon See**

**Apeirogon**

**Pick's Theorem**Georg Pick was a Jewish Austrian mathematician. In 1899 he published a for finding the area of a polygon if all the vertices are on lattice points(points whose x and y coordinates on the plane are both integers. He presented the formula he created in 1899, in Geometries zur Zahlenlehre in Prague. He was also instrumental in introducing Einstein to Ricci-Curbastro and Levi-Civita which helped him work out the mathematics in the general theory of relativity.

Hugo Steinhaus caused a surge in the popularity (and knowledge) of the theorem when he used it in his popular, Mathematical Snapshots, in 1969. It probably would have been much more popular in 1899 if graph paper had been more in mathematics education, but thirty to fifty years would pass before the paper became popular in geometry and algebra classes. Pick would never be aware of this late surge of popularity, he died after two weeks in the Theresiestadt prison camp in 1942.

*Art of Problem Solving |

There is not a higher dimensional analogy of this theorem, counting points inside and on the boundary, but the Ehrhart Polynomial, created by French high school teacher Eugene Ehrhart in the 60's, describes an expression for the volume in terms of the number of interior points in the polytope and dilations.

**La Pipopipette/ Pigs in a Pen/ Dots and boxes**A game created by mathematicican Edouard Lucas in 1895 in his

*L'arithmétique amusante.*The game involves workiing (originally) on a rectangular

*Wikipedia |

**Plane**– early 17th century: from Latin planum ‘flat surface’, neuter of the adjective planus ‘plain’. The adjective was suggested by French plan(e ) ‘flat’. The word was introduced to differentiate the geometrical senses, previously expressed by plain1, from the latter's other meanings. It is indeed a flat, 2-dimensional surface that extends infinitely far. Plane geometry first began by ancient Greeks 2000 years ago. Euclid made incredible contributions to this field with his book “Elements”. The OED gives its earliest citation of plane in English as 1604. *Derek Orr

**Planet**In the days before every city had public lights that dimmed the heavens to our view, and every family had television to bring us in out of the night, the ancients could look up into the sky that seemed filled with a blanket of stars rotating in lockstep around the heavens. But when they looked up from what most believed was an Earth in the center of the universe, some lights in the sky did not act like stars. The Sun and the Moon seemed to almost race across the day and night sky unlike the slowly creeping motion of the stars. They were certainly different, and over time the ancinets seemed to notice five more bodies that seemed to "wander" across the sky, some even seeming to change direction and go backwards against the background of the stars. These were the planets that even now, are mostly visible in the night sky, Mercury, Venus, Mars, Jupiter, and Saturn. The Greek word planasthai meant, to wander, and so they took the name Planet for these night sky wanderers. The ancient astronomer/astrologer (and they were indistinguishable in early science) would study these seven wandering objects to predict the important events of your life (and death). Many believe they are the source of a seven day week, and also account for the idea of a "Lucky Seven". You were indeed lucky, if you were alive in a period where several of these wanderers aligned in the sky, and still are.

**Plus and Minus**(as in two plus four) comes from the early Latin word of the same spelling, but a long u sound, meaning "more". Extensions of the root were used for related ideas like fill, full, and abundant.

The Egyptian Hyroglyphs in the Rhind Papyrus from around 1550 BC used a pair of walking legs forward for addition and away for the sign of subtraction.

Diophantus around 225 AD used the occasionally no symbol, and sometimes a / for addition, and a curve looking like the right hand side of a valentine heart for subtracting. Most Hindu writers used no sign at all for addition, and but most Hindu writers used a dot to mark negative quantities, but the 4th century scholar Bakhshali use +, for subtraction.

Leonardo of Pisa (Fibonacci) in his 1202 Liber Abaci used the Latin plus and minus in his example on double false position for the excess, or deficit.

The modern signs + and - came into use in Germany in the late 15th century. The Dresden Library has algebra manuscript in German from 1481 in which the minus sign is used, called minnes, and placed before and sometimes after the amount to be subtracted. Addition is express in the same document by "und". University of Leipzig lectuere J. Widman was the first to use the two symbols in print. The Latin documents using the +, short for

*et,*sometimes had the vertical cross somewhat slanted. The - symbol had been used on German (and some Dutch) commercial documents to indicate the tare in weights, the difference between the combined content and container weight with the - setting off the raw container weight. This was often stamped on the top of containers. It seems the + was occasionally used for an overage of the content weight. The same excess an deficit used by Leonardo of Pisa.

In the 15th century the Italian writers continued the practice of using script p and m, for plus and minus, as they had in writing in Latin earlier. In 1608, Christopher Clavius, a German Jesuit living in Rome used the + and - symbols in his algebra. Others slowly followed. Robert Recorde in England had used the signs in 1557

Some writers used the

**obelus**, ÷, perhaps to distinguish it from the - which also was used in proportions and some other uses. This symbol would become the dominant symbol for division in England after Pell translated Rahn's 1659 algebra into English.

The OED cites the first use of the word plus to indicate the addition of quantities in 1537, "...item plus for his fee..". The mathematical symbol "+" is first cited in 1579 in the Stratioticas of Thomas Digges, and his father, Leonard, "..like signes multiplied produce + Plus... diverse signes produce always - Minus" , ignoring the use by Recorde 22 years earlier. It appeared in an equation of his Whetstone of Witte, at one place in an equation using his own creation, the = for equals.

**Plus/Minus**\( \pm \) sign – To denote the next term could be subtracted or added (e.g. quadratic formula). A version of the sign was used in 1626 by Albert Girard but its more modern form took shape in 1631 in William Oughtred’s “Clavis Mathematicae”.

There is a reversal of that which comes in handy when you want two or more places to switch signs in each term to be the opposite of each other. An example is the rule for factoring the sum and difference of two cubes written as a single expression, \( x^3 \pm y^3= (x\pm y) (x^2 \mp xy +y^2) \) Notice the sign in the binomial is always the same as the sign between the cubes, and the sign of the xy term is always the opposite of it. *Derek Orr

**Point**The word point comes from the Latin word

*pungere*, which means to pierce or prick. The word became generalized to things related to both the tip of the sharp object (point of a pencil) and the mark left by its use. Our modern word puncture comes from the same root, as do pungent (a sharp odor) and punctual.

Euclid used the term

*semeion*which refers to a sign or mark to distinguish a person or thing from another. Most translation into Latin used the term

*punctum*. This passed in to many English writers as well, as in 1602 when J Davies wrote, "arithmetike from Vnity flows, en'n as from Punction flowes Geometry." Alongside this some, like Robert Recorde in 1551 wrote, " a poynt or prick is a thing, ... which hath in it no partes..." Punctum continued to be cited into the 18th century, and it was Newton himself who referred to his fluxions with little dots above them as "pricked letters."

**Polar Coordinates**

**Polygon**is from the Greek roots

*poli*(many) and

*gonus*(bent,angle, knee) and interprets literally as many angled. And polygon was used by the Greeks in Euclids time, but he preferred the many sided version, polyplueron.

Leonard Digges in 1560 used both polygona, and pollygonium and restricted the term to those with more than four sides. Henry Billingley in 1570 also confused his sides with angles when he wrote that, " a poligonum is a figure that consists of many sides." The distinction of whether a polygon should have "three or more sides/angles" or "more than four" was still not sorted as late as 1828 when Webster dictionary writes "more sides than four", and Brewster's geometry gives examples of, "the equilateral triangle is one of three sides, the square is one of four", when talking of regular polygons.

The relation between knee and angle relates to the flexed position of the knee. Poly appears in many math words, and other words, but

*gonus*remains mainly in its Latin derivative genus, from which we get genuflect (to bend the knee). According to John H. Conway, terms like gnaw, and jaw, are from the same root, perhaps because the jaw forms the same shape as the bent knee. Another word with the "gen" connection is genuine. In the early days of the Roman Empire, a parent would sit in the squares of the city and place their real or adopted child (and sometimes adults) on their knee, and pronounce, "this is my genuine son/daughter".

**Polygonal Numbers,**sometimes called

**figurate numbers**, was used commonly by the Greeks and others, with examples of triangular numbers, square numbers and pentagonal numbers, and in general polygonal numbers. They did not make their way into English language it appears, until 1842 in the Dictionary of Science, Literature, and Art, "Fugyrate Numbers: first sums, or polygons of the first order." with examples of triangular, 1, 3, 6... square and pentagonal examples. The late emergence of "polygonal numbers" may have been due to the preference for "figurate Numbers" which were used by Wiliam Jones in his 1706 Synopsis Palmariorum, famous for introducing Pi in its modern C/D manner and frequent others before 1842. And Peter Barlow in a book on Number Theory, referred to them as "multi-angular."

**Polyhedron**is the name for a solid with "many faces", the joining of

*poli*(many) and

*hedros*(face or seat). The

*hedros*originally referred to any flat surface. Later, in the Latin, hedra was used for a chair, flat places are good to sit on, and the root is preserved in our words for cathedra (the Bishop's chair) and the Cathedral where it is kept. An excellent source of information and grpahics about polhedra is the Virtual Polyhedra page of George W. Hart. Come on, say rhombicuboctahedron three times really fast.

Polyhedron was used by Euclid without a definite definition, and Henry Billingsley in his 1570 translation used "a solid in many sides(which is called a polyhedron.) where today we would call them "faces". The English often spelled the word polyedron, perhaps because the sound of the h in French was almost a silent exhale, as in the name of L'Hospital, which is correctly pronounced, and often spelled as if the H was not present.

There are

**Polyhedral numbers**too, Tetrahedral numbers, the sum of the first n triangular numbers, and the square pyramidal numbers, the sum of the first n square numbers.

*Wikipedia |

**Pons Asinorum (bridge of asses)**When geometry was first taught in schools, it was taught straight from the translations of Euclid's Elements. In book one, proposition five, Euclid proved that the base angles of an isosceles triangle were congruent by using a figure like the one at right.

Although some claim the name comes from the fact that it looked like a steep bridge, too steep for a horse but not for an ass, it is more likely that it is named because it was a challenging obstacle to the students who were "too stubborn" to learn their proofs. Because Euclid chose to prove this theorem before he had developed many of the congruence relationships, it was much more difficult than it might have been. When given the problem in 1959, a computer AI solved the problem by showing that triangle ABC was congruent to triangle ACB and thus they were congruent. The solution had been solved in the same way by human intelligence by Proclus in the fifth century.

The term was applied to the idea of a logic diagram or figure invented for helping someone find the middle term of a syllogism, "long known as the pons asinorum."

Another uses the term as any problem that will defeat an unskilled person. Both of these, 1641 and 1645 and others, well before the first citation of a reference to Euclid's Elements, 1718.

**Positive**comes from the root word

*posit*which means to place or set... think position. This probabily releates to ancient methods of counting and calculating on sand dishes and abacci, and the checkered counting tables. Postive numbers can be set out one by one.

Positive was used in English as early as 1400 to express the presence of a quality, as opposed to negative for its absence, "somewhat positif affyrmnge". Cocker's Decimal Arithmetic used, "nature of the numbers.. whether it be positive or negative. The word "affirmitive" was used by some, particularly after Newton used it in 1665, "a whole and affirmative number." The use continued at least as late as 1885 when it was used in a book of surveying, "multiplying ... a negative index by an affirmative number, the product will be negative."

For the symbols + marking a positive number, and - marking a negtive one, See

**Plus and Minus**

**Pound**The unit of weight now common in the US came from a long string of language ideas that, at first, seeem poorly related. The original Indo-European

*spen*, related to spinning or twisting, and gives our current word for spin. Hanging things sometimes spin, and so the Romans attached the word to things that were hung up, and led to words like suspend. Over time the s slipped away and we got pendant and penthouse. The act of hanging things on a balance rod to verify weights led to the use as an amount of weight, and from which we get peso, pound, and pendulum. From the twelve ounces in the weight system the Romans beget both ounce, and inch, both from the original unit called the

*uncia,*for a twelfth part. Eventually it became a term for a coin also, and the weight of 12 of these coins on a balance,

*Libra*, became a Libra pound, and preceded to be the Lb symbol for the British Pound.,

**Power**comes from the French,

*poeir*, and perhaps the earlier Latin word

*potere*from which we get potent. Both words refer to ability or bening able. In mathematics, power refers to a number which is the result of some base number being raised to an exponent. For example we say that 8 is the 3rd power of two. Students, and teachers, often refer to the exponent as the power, but this is not historically correct. It has become so common that I've seen it so used in some dictionaries.

The term appears in English as early as Billingsley's 1570 Elements translation, "The power of a line is the square of the same line." The earliest arithmetical citation in the OED is about 1690 when S Jeake, "numbers given by these powers given by these alternate indices..." .

The 1995 New Scientist, however, seems to directly refers to Fermat's Little theorem clearly indicating the exponent as the power, "subtracting n from n raised to the power p always leaves a number divisible by p."

In most early books I have studied (before 1850) the exponent is more often referred to as the index as in Jeake above. Exponent is first cited in the OED in a mathematical usage as G Berkeley's Analyst in 1734.

Power was used by Georg Cantor in 1895 to refer to the cardinality of a set.

**Power**is also used geometrically in the Power of the point. The product of the distances from the point, P, to the two intersections with the circle. It seems to have surfaced around 1872. The term was created by Jacob Steiner in 1826. In geometry it is often known as the

**tangent-secant theorem**, and if the point is inside the circle, the

**intersecting chords theorem**.

And yes, there's more. Power is a statistical term for a quality of a hypothesis test created by Neyman and Pearson in 1933.

**Prime**is from the Latin word for first,

*primus*, and related to the Greek,

*Protos*. They probably came to English through the French as a borrowing of "prime" in that language. The Church used the term for the first service of the morning, and the term was also used for early dawn as early as 1300, and Chaucer used it in reference to these morning bells in the 1410 Pardoner's Tale. Around the same year the OED cites a use of prime for high quality food, and generalize to other "best" things.

In mathematics, the first century scholar, Iamblichus, referred to what we would now call prime numbers as "rectilinear" because they could only be expressed as a length, as opposed to composite numbers which could be shown as rectangles with edges of two factors.

By the end of the fourteenth century, astronomers were able to measure down to the sixtieth part of a degree of arc. They adopted an English simplification for the Latin name for these little graduations between angles, pars minuta primus, (first little parts), calling them minutes. Advancing technology was soon adding a second division between these parts of degrees, which became seconds. By 1392 an astronomy book stated, "...it shall be deuyed in mynutis, in degrees, in nombres." In Chaucer's most famous work when he was alive, A Treatise on the Astrolabe, he wrote, "... degres ...considered of 60 mynutes, and every mynute of 60 seconds. The first was made little, and prime went on to find numbers in a purer field.

In 1570 prime number appeared in Henry Billingsley's translation of Euclid, "A prime (or first) number[L., numerous primus] is that which onely vnite doth measure."

Prime had found a home in math, and all that was left was figuring out a definition, and that took a while. 200 years after Billingsley, Euler decided that one was not a prime, but even his greatness could not stem the tide. Throughout the 1800's and into the early 20th century some writers still persisted with listing the primes as, 1, 2, 3, 5....

In 1895, on page 92 of William J Milne's Standard Arithmetic for Schools and Academies, you would find:

**WHAT about TWO?**

Then a few years later, 5th August 1900 in The Weekly Dispatch, the famous puzzlest, Henry E. Dudeney challenged the readers to find a magic square with all prime numbers that had the lowest sum for rows and columns. He had already solved it, and his example had that "prime one" sitting right there front and center in the top row.

But wait, remember that "what about two" thing I did. Well it seems for some folks, neither one or two was a prime well into the eighteenth century (but Milne was writing in the late 19th century.) So here is how Chris Caldwell explained that some folks, very early on, thought neither one or two was prime:

"Martinus and others such as Nicomachus (c. 100) and Iamblichus (c. 300) [26, p. 73], Boethius (c. 500) [53, pp. 89–95], and Cassiodorus (c. 550) [20, p. 5] make the primes a subset of the odds, excluding both one and two from the primes—so for them, the smallest prime was three. (It is easy to extend this list of those for whom the ﬁrst prime was three well into the 16th century .) Most of the ancient Greeks however, like Euclid, began the sequence of primes with two."

**Primes with a Prime Subscript/Index**A prime number, such as 5, which is indicated for its order in the sequence of primes by \( p_3\) is called a prime with a prime subscript, or prime index. The earliest paper I know to introduce this concept is in February of 1974, and published in October of 1975. R. E. Dressler and S. T. Parker, Primes with a prime subscript, J. ACM 22 (1975) 380-381.,

The 2nd prime, is 3, so it is the first number in the sequence, and it begins 3, 5, 11, 17... and the rest can be found here. In the paper they prove that any integer greater than 96 can be written as the sum of distinct numbers from this sequence. They have recently been called

**super-primes**, or just superprimes, but that name is often used for a prime number that remains prime when its last digit is removed. I found it in the Cambridge IGCSE® Mathematics Core and Extended Coursebook, 2018, and all the way back to 1980 in a computer book for educators promoting the radcial idea of computers in the classroom and gave examples of learning problems, including finding superprimes of the elimination of digts style. The earliest paper I can find that used superprime/super-prime/super prime, for primes with a prime subscript was in 2010. Superprimes has also been used for a type of prime near-ideal ring, and numbers such that, for any prime p, 2p-1 is a superprime if it is prime.*Wikipedia |

**Prism/Prismoid**If two congruent polygons, called the bases, are in parallel planes and similarly aligned and with the axis joining their centroids normal to the bases, it is called a prism. All the non-base faces will be rectangles. In Billingsley's translation of Euclid, he uses the word to apply to the instances where the centroids are not normal to the bases, and in which case the non-base faces would be parallelograms. This is often referred to today as an

**oblique prism.**If one of the two congruent basis is rotated in the parallel plane, it can be called a

**skew prism**(all the edges in one plane are skew lines to all the edges in the other),

**or anti-prism**, and the faces joining the bases would be triangles. If a prism is cut by a plane

Skew, or Anti-prism *Wikipedia |

**truncated prisms**.

If the two bases are similar but not congruent, in the same orientation in parallel planes, then the object is no longer a prism, but a

**frustum of a pyramid,**and if the axis joining the centroids is not normal, we apply the adjective oblique to the frustum of the pyramid . If the polygon bases are neither congruent or similar, it is called a

**Prismoid,**(Still sometimes called a

**prismatoid, )**a name that is applied to any polyhedron with all the vertices lying in two parallel planes. The prismoid may be normal (or orthogonal) if centroids are aligned, oblique if they are not, and the faces joining the two bases may be rectangles, trapezoids, parallelograms, or triangles. All these terms are essentially borrowings from earlier Greek words.

**Primorial ---- See Factorial**

**Probability**comes to us from the Old English word provable, and from the Latin

*probare*, to prove, and the earlier

*probus*, goodness or value. The word made its way into mathematics in the early 1700's and may have been introduced by De Moivre.

Pascal, a leader in the early development of what we now call probability theory, never used the term. Today in the practice of law, a probated will is that that is proven (tested in the courts). A person on probation is given a period of time to prove themselves of value.

The OED cites uses of probability for general things that are likely to happen as far back as 1443, and probable over 60 years earlier. Chance events were cited before 1300, as in "feble chaunce." Christen Huygens used the Dutch Kant (chance) in the Dutch version of the more famous van Schootin translation into Latin

*De Ratiociniis in Ludo Aleae*in 1657. The English translation of that would be, "Reasoning on the game of dice", but the actual translated book was titled "Reasoning on the games of chance."

The first OED citation of a mathematical use was in 1692 when John Arbuthnot published Of Laws of Chance, ".... a calculation of the Quantity of Probability founded on experience..." . By 1718 De Moivre used the term in the title of his famous The Doctrines of Chances: or, a method of calculating the probability of events in play."

**Product,**the past participle of produce, which is the union of hte Latin

*pro*, for forward, and

*ducere*, which means "to lead". The original meaning was thus, to lead forward. Product, then, referred to that which was lead forward. It has been suggested that the mathematical product of the result of a multiplication may arise from the fact that multiplication of whole numbers leads forward to a larger result. This may be the best guess, given that the mathematical product appeared in post-classical Latin writings in England as Early as the 13th century .(OED) It appeared in the first English language arithmetic, Art of Nombryng by Steele, in 1450, "product, or provenient, of takyng out of one fro another, as twyes 5 is 10."

The OED has no citations for other uses of product before 1600 when it was first used as a thing generated by "The product of the vnion." A the word as a mathematical word first seems to have occurred in France,

*produit*, in 1554; and in German,

*produkt*, in 1522.

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

*exducate*, to lead out.

**Proof**– The word “proof” comes from Latin probare meaning “to test”. The OED cites the word in English as early as 1225, but it may have appeared earlier as a French borrowing.

“Proof by picture” was most likely the first way proofs were introduced. Proofs began with geometric problems (e.g. land measurement/disputes). Thales (624–546 BCE) and Hippocrates of Chios (c. 470–410 BCE) were some of the first mathematicians to give proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove them. Aristotle (384–322 BCE) said definitions should describe the concept being defined in terms of other concepts already known.

Euclid (300 BCE) introduced the axiomatic method still in use today. It starts with undefined terms and axioms, propositions concerning the undefined terms which are assumed to be self-evidently true (from Greek "axios", something worthy). Hence, this method proves theorems using deductive logic. Euclid’s book, the Elements, describes this and contains theorems of geometry, such as the Pythagorean theorem, and number theory, including a proof that the square root of two is irrational and a proof that there are infinitely many prime numbers.

While earlier Greek proofs were largely geometric demonstrations, Islamic mathematicians used arithmetic and algebra to allow more general proofs with no dependence on geometric intuition. In the 10th century, the Iraqi mathematician Al-Hashimi worked with numbers as such, called "lines" but not necessarily considered as measurements of geometric objects, to prove algebraic propositions concerning multiplication, division, etc., including the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal’s Triangle. Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.

Modern proof theory treats proofs as inductively defined data structures, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example axiomatic set theory and Non-Euclidean geometry.

The rigor of a proof is a long debate in the mathematics community and has changed throughout history, it really depends on who the audience is. However, formal proof is well-defined and is written in formal language instead of natural language. Proof theory studies formal proofs and their properties, looking at the logical statements that create a formal proof. Proofs may also be admired for their beauty. Paul Erdos would reference “The Book” when a proof was so spectacular…he believed it was one of God’s proofs.

The most common types of proofs include but are not limited to: direct proof, math induction, proof by contradiction, proof by construction/example, proof by exhaustion, proof by contrapositive, combinatorial proofs, and computer-assisted proofs. *Derek Orr

**Proportion**The word proportion come from the Latin proportio, a tanslation form the Greek word for analogy. The roots

*pro*, for, and

*potrio*, share or part, combine to mean "for (its/his/their) share." The The roots are preserved in words such as portion, parity, compare, and the golfing favorite, par.

Proportions seems to have come into English in the mathematical sense in Chaucer's Treatise on the Astrolabe in 1450, with many different earlier usages regarding poetry, music, land mass and more. Symbols for both arithmetic progressions and geometric progressions were varied but the the 1.3::4.12 used by Oughtred in his Clavis Mathematicae, 1631, found the most common theme. by 1661 Thomas Streete had replaced the single dots with colons as well. By 1700 these two symbols were becoming common in continental Europe, and they continued to be the most common in usage until the early 20th century, when English and Americans began using an equal sign between the two parts, 1:3 = 4:12.

**Protractor**The protractor seems to have originated in land surveying in the early 16th century, and by in late 16th century it was often used for sailing in conjunction with a pegged board called a traverse board which allowed for holding a heading while dead reckoning. Later they proved practical for navigation, and engineering drawing. The ODE lists a citation in 1559 where W. Cuningham's Cosmography contained, "The principal treasure of cosmography, that is to delineat, protract, or set fourth the platforme of th' vniuersall face of th' earth.," The term was still in use in a surveying book in 2001.

By 1602, A citation for Seaman's Kalendar under heading gives, "...the tip of a Trauerse board and a protractor. "

The roots of the word come from the post-classical Latin

*protractor*, a person who summoned another to court. and the earlier

*protract*, to draw out, and was used in such a manner in English for a person, or action which extended or drew out some process.

**Proposition**– The word is drawn from the mixture of borrowings from Latin and French, and is cited in English with the mathematical meaning in 1425 in a work by Goffrey Chaucer on Boethius.

A statement that proposes some idea that could be true or false. Do not confuse with “lemma”, “theorem” or “corollary”, as they are all different. Proposition most often appear in formal logic, where symbols and quantifiers can be meshed to create a proposition. *Derek Orr

**Pseudosphere**from the Greek

*pseudes*for false, a pseudosphere is

*wikipedia |

Christian Huygens found (1693) that the volume and surface of the shape are finite, in spite of its infinit expanse. Just as the sphere has a volume of 4 pi r^2, so too does the pseudospere. The volume of hte pseudospere, however, is only half the volume of a sphere of the same radius, (2/3) pi r^3.

**Pyramid**may come from the Greek root

*pura*, for fire.

*The Greek word for pyramid was peramus, and this word also referred to a type of wheat bread or cake made in the shape we now call a pyramid in geometry. David Eugene Smith suggested the borrowing went the other way. The Egyptians created the name pyramid, and the Greeks adopted the word for the shape, and then named, or renamed, the cake. This seems the much more likely event, since the Ahmes Papyrus, around 1650 BC, in the 2nd book, ends with six problems about the pyramid, using that word. Impressively, the problem on finding the seked (slope) of a pyramid given the sides of the base and the height, solving it by using the ratio of half the side to the height... the cosine of the base angle of the inclined face.*

Rhombic rhight pyramid *Wikipedia |

Even if pyramid may have had nothing to do (most likely) with the Greek word pura, the word has still left its mark in English. The same root has given us pyrite (firestone in Greek), and fire related words like pyromaniac, pyrometer (measures heat)

**Pyramidal Numbers**are first cited in the OED by Ozanam in his 1708 matheamatical recreations book. "1, 4, 10, 20 &c formed by the continual addition of the triangular."

The formula for the square pyramidal numbers was known to Fibonacci in his Liber Abaci in 1202, although I can not show that he used that term, but just added the sum of the first n squares, 1 + 4 + 9 + ... The square pyramidal number is always the sum of two consecutive tetrahedral numbers, just as the square numbers are always the sum of consecutive triangular numbers. For any figurate number, there is a pyramidal number adding the sum of the first n figurate numbers.

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