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70-725: The Mathematical Association is a professional society concerned with mathematics education in the UK. It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1894. It was the first teachers' subject organisation formed in England. In March 1927, it held a three-day meeting in Grantham to commemorate the bicentenary of

140-537: A major subject in its own right, such as partial differential equations , optimization , and numerical analysis . Specific topics are taught within other courses: for example, civil engineers may be required to study fluid mechanics , and "math for computer science" might include graph theory , permutation , probability, and formal mathematical proofs . Pure and applied math degrees often include modules in probability theory or mathematical statistics , as well as stochastic processes . ( Theoretical ) physics

210-563: A board into thirds can be accomplished with a piece of string, instead of measuring the length and using the arithmetic operation of division. The first mathematics textbooks to be written in English and French were published by Robert Recorde , beginning with The Grounde of Artes in 1543. However, there are many different writings on mathematics and mathematics methodology that date back to 1800 BCE. These were mostly located in Mesopotamia, where

280-637: A geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate ( elliptic geometry ). If one takes the fifth postulate as a given, the result is Euclidean geometry . • "To draw a straight line from any point to any point." • "To describe a circle with any center and distance." Euclid, Elements , Book I, Postulates 1 & 3. Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating

350-589: A given method gives the results it does. Such studies cannot conclusively establish that one method is better than another, as randomized trials can, but unless it is understood why treatment X is better than treatment Y, application of results of quantitative studies will often lead to "lethal mutations" of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting new hypotheses , which can eventually be tested by randomized experiments. Both qualitative and quantitative studies, therefore, are considered essential in education—just as in

420-460: A glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield , went home to my father's house, and stayed there till I could give any proposition in

490-534: A line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier. As was common in ancient mathematical texts, when a proposition needed proof in several different cases, Euclid often proved only one of them (often

560-575: A teaching award that was examined was the Diploma of the Mathematical Association , later known as the Diploma in Mathematical Education of the Mathematical Association. It exists to "bring about improvements in the teaching of mathematics and its applications, and to provide a means of communication among students and teachers of mathematics". Since 1894 it has published The Mathematical Gazette . It

630-522: A thousand different editions. Theon's Greek edition was recovered and published in 1533 based on Paris gr. 2343 and Venetus Marcianus 301. In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley . Copies of the Greek text still exist, some of which can be found in the Vatican Library and

700-414: A variety of different concepts, theories and methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education. Elementary mathematics were a core part of education in many ancient civilisations, including ancient Egypt , ancient Babylonia , ancient Greece , ancient Rome , and Vedic India . In most cases, formal education

770-466: Is mathematics-intensive, often overlapping substantively with the pure or applied math degree. Business mathematics is usually limited to introductory calculus and (sometimes) matrix calculations; economics programs additionally cover optimization , often differential equations and linear algebra , and sometimes analysis. Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on

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840-682: Is one of the participating bodies in the quadrennial British Congress of Mathematics Education, organised by the Joint Mathematical Council , and it holds its annual general meeting as part of the Congress. It is based in the south-east of Leicester on London Road ( A6 ), just south of the Charles Frears campus of De Montfort University . Aside from the Council, it has seven other specialist committees. Its branches are sometimes shared with

910-403: Is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then

980-504: Is still an active area of research. Campanus of Novara relied heavily on these Arabic translations to create his edition (sometime before 1260) which ultimately came to dominate Latin editions until the availability of Greek manuscripts in the 16th century. There are more than 100 pre-1482 Campanus manuscripts still available today. The first printed edition appeared in 1482 (based on Campanus's translation), and since then it has been translated into many languages and published in about

1050-623: Is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students. The social status of mathematical study was improving by the seventeenth century, with the University of Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics being established by

1120-510: The Association of Teachers of Mathematics (ATM): Past presidents of The Association for the Improvement of Geometrical Teaching included: Past presidents of The Mathematical Association have included: Mathematics education Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing

1190-701: The Bodleian Library in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available). Ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of

1260-579: The Elements from the Byzantines around 760; this version was translated into Arabic under Harun al-Rashid ( c. 800). The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation. A relatively recent discovery

1330-547: The Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements , encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Furthermore, its logical, axiomatic approach and rigorous proofs remain

1400-406: The Elements is largely a compilation of propositions based on books by earlier Greek mathematicians. Proclus (412–485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements : "Euclid, who put together the Elements , collecting many of Eudoxus ' theorems, perfecting many of Theaetetus ', and also bringing to irrefragable demonstration

1470-452: The Elements , and applied their knowledge of it to their work. Mathematicians and philosophers, such as Thomas Hobbes , Baruch Spinoza , Alfred North Whitehead , and Bertrand Russell , have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced. The austere beauty of Euclidean geometry has been seen by many in western culture as

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1540-466: The Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although Euclid was known to Cicero , for instance, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received

1610-691: The National Council of Teachers of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000 for the United States and Canada, which boosted the trend towards reform mathematics . In 2006, the NCTM released Curriculum Focal Points , which recommend the most important mathematical topics for each grade level through grade 8. However, these standards were guidelines to implement as American states and Canadian provinces chose. In 2010,

1680-491: The University of Cambridge in 1662. In the 18th and 19th centuries, the Industrial Revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money, and carry out simple arithmetic , became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age. By

1750-548: The What Works Clearinghouse (essentially the research arm for the Department of Education ) responded to ongoing controversy by extending its research base to include non-experimental studies, including regression discontinuity designs and single-case studies . Euclid%27s Elements The Elements ( Ancient Greek : Στοιχεῖα Stoikheîa ) is a mathematical treatise consisting of 13 books attributed to

1820-403: The minor or AS in mathematics substantively comprises these courses. Mathematics majors study additional other areas within pure mathematics —and often in applied mathematics—with the requirement of specified advanced courses in analysis and modern algebra . Other topics in pure mathematics include differential geometry , set theory , and topology . Applied mathematics may be taken as

1890-488: The 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation. No indication is given of the method of reasoning that led to the result, although the Data does provide instruction about how to approach

1960-411: The 1300s. Spreading along trade routes, these methods were designed to be used in commerce. They contrasted with Platonic math taught at universities, which was more philosophical and concerned numbers as concepts rather than calculating methods. They also contrasted with mathematical methods learned by artisan apprentices, which were specific to the tasks and tools at hand. For example, the division of

2030-665: The National Governors Association Center for Best Practices and the Council of Chief State School Officers published the Common Core State Standards for US states, which were subsequently adopted by most states. Adoption of the Common Core State Standards in mathematics is at the discretion of each state, and is not mandated by the federal government. "States routinely review their academic standards and may choose to change or add onto

2100-622: The Sumerians were practicing multiplication and division. There are also artifacts demonstrating their methodology for solving equations like the quadratic equation . After the Sumerians, some of the most famous ancient works on mathematics came from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus . The more famous Rhind Papyrus has been dated back to approximately 1650 BCE, but it

2170-552: The United States. During the primary school years, children learn about whole numbers and arithmetic, including addition, subtraction, multiplication, and division. Comparisons and measurement are taught, in both numeric and pictorial form, as well as fractions and proportionality , patterns, and various topics related to geometry. At high school level in most of the US, algebra , geometry , and analysis ( pre-calculus and calculus ) are taught as separate courses in different years. On

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2240-423: The ancient Greek mathematician Euclid c. 300 BC. It is a collection of definitions, postulates , propositions ( theorems and constructions ), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry , elementary number theory , and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in

2310-400: The angles sum to less than two right angles. This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician Nikolai Lobachevsky published a description of acute geometry (or hyperbolic geometry ),

2380-508: The birth of Pythagoras . In Plato 's division of the liberal arts into the trivium and the quadrivium , the quadrivium included the mathematical fields of arithmetic and geometry . This structure was continued in the structure of classical education that was developed in medieval Europe. The teaching of geometry was almost universally based on Euclid's Elements . Apprentices to trades such as masons, merchants, and moneylenders could expect to learn such practical mathematics as

2450-702: The changes in math educational standards. The Programme for International Student Assessment (PISA), created by the Organisation for the Economic Co-operation and Development (OECD), is a global program studying the reading, science, and mathematics abilities of 15-year-old students. The first assessment was conducted in the year 2000 with 43 countries participating. PISA has repeated this assessment every three years to provide comparable data, helping to guide global education to better prepare youth for future economies. There have been many ramifications following

2520-439: The chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being 10 3 ( 5 − 5 ) = 5 + 5 6 . {\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.} The spurious Book XV

2590-435: The continuous and discrete sides of the subject: Similar efforts are also underway to shift more focus to mathematical modeling as well as its relationship to discrete math. At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included: The method or methods used in any particular context are largely determined by

2660-432: The cornerstone of mathematics. One of the most notable influences of Euclid on modern mathematics is the discussion of the parallel postulate . In Book I, Euclid lists five postulates, the fifth of which stipulates If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles , then the two lines, if extended indefinitely, meet on that side on which

2730-410: The current findings in the field of mathematics education. As with other educational research (and the social sciences in general), mathematics education research depends on both quantitative and qualitative studies. Quantitative research includes studies that use inferential statistics to answer specific questions, such as whether a certain teaching method gives significantly better results than

2800-502: The death of Sir Isaac Newton , attended by Sir J. J. Thomson (discoverer of the electron), Sir Frank Watson Dyson – the Astronomer Royal , Sir Horace Lamb , and G. H. Hardy . In 1951, Mary Cartwright became the first female president of the Mathematical Association. In the 1960s, when comprehensive education was being introduced, the Association was in favour of the 11-plus system. For maths teachers training at university,

2870-478: The development of logic and modern science , and its logical rigor was not surpassed until the 19th century. Euclid's Elements has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since

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2940-583: The effects of such treatments are not yet known to be effective, or the difficulty of assuring rigid control of the independent variable in fluid, real school settings. In the United States, the National Mathematics Advisory Panel (NMAP) published a report in 2008 based on studies, some of which used randomized assignment of treatments to experimental units , such as classrooms or students. The NMAP report's preference for randomized experiments received criticism from some scholars. In 2010,

3010-602: The errors include Hilbert's geometry axioms and Tarski's . In 2018, Michael Beeson et al. used computer proof assistants to create a new set of axioms similar to Euclid's and generate proofs that were valid with those axioms. Beeson et al. checked only Book I and found these errors: missing axioms, superfluous axioms, gaps in logic (such as failing to prove points were colinear), missing theorems (such as an angle cannot be less than itself), and outright bad proofs. The bad proofs were in Book I, Proof 7 and Book I, Proposition 9. It

3080-403: The existence of some figure by detailing the steps he used to construct the object using a compass and straightedge . His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct'

3150-457: The figure used as an example to illustrate one given configuration. Euclid's Elements contains errors. Some of the foundational theorems are proved using axioms that Euclid did not state explicitly. A few proofs have errors, by relying on assumptions that are intuitive but not explicitly proven. Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements ]

3220-453: The first printing in 1482, the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read. Scholars believe that

3290-646: The levels of achievement that were relevant to, realistic for, and considered socially appropriate for their pupils. In modern times, there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England , for example, standards for mathematics education are set as part of the National Curriculum for England, while Scotland maintains its own educational system. Many other countries have centralized ministries which set national standards or curricula, and sometimes even textbooks. Ma (2000) summarized

3360-402: The most difficult), leaving the others to the reader. Later editors such as Theon often interpolated their own proofs of these cases. Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles,

3430-441: The number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals . The presentation of each result is given in a stylized form, which, although not invented by Euclid,

3500-507: The objectives that the relevant educational system is trying to achieve. Methods of teaching mathematics include the following: Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries. Sometimes a class may be taught at an earlier age than typical as a special or honors class . Elementary mathematics in most countries is taught similarly, though there are differences. Most countries tend to cover fewer topics in greater depth than in

3570-415: The other hand, in most other countries (and in a few US states), mathematics is taught as an integrated subject, with topics from all branches of mathematics studied every year; students thus undertake a pre-defined course - entailing several topics - rather than choosing courses à la carte as in the United States. Even in these cases, however, several "mathematics" options may be offered, selected based on

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3640-652: The other social sciences. Many studies are “mixed”, simultaneously combining aspects of both quantitative and qualitative research, as appropriate. There has been some controversy over the relative strengths of different types of research. Because of an opinion that randomized trials provide clear, objective evidence on “what works”, policymakers often consider only those studies. Some scholars have pushed for more random experiments in which teaching methods are randomly assigned to classes. In other disciplines concerned with human subjects—like biomedicine , psychology , and policy evaluation—controlled, randomized experiments remain

3710-457: The preferred method of evaluating treatments. Educational statisticians and some mathematics educators have been working to increase the use of randomized experiments to evaluate teaching methods. On the other hand, many scholars in educational schools have argued against increasing the number of randomized experiments, often because of philosophical objections, such as the ethical difficulty of randomly assigning students to various treatments when

3780-443: The research of others who found, based on nationwide data, that students with higher scores on standardized mathematics tests had taken more mathematics courses in high school. This led some states to require three years of mathematics instead of two. But because this requirement was often met by taking another lower-level mathematics course, the additional courses had a “diluted” effect in raising achievement levels. In North America,

3850-485: The results of triennial PISA assessments due to implicit and explicit responses of stakeholders, which have led to education reform and policy change. According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist." However, there are useful theories on how children learn mathematics, and much research has been conducted in recent decades to explore how these theories can be applied to teaching. The following results are examples of some of

3920-494: The six books of Euclid at sight". Edna St. Vincent Millay wrote in her sonnet " Euclid alone has looked on Beauty bare ", "O blinding hour, O holy, terrible day, / When first the shaft into his vision shone / Of light anatomized!". Albert Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book". The success of

3990-475: The standards to best meet the needs of their students." The NCTM has state affiliates that have different education standards at the state level. For example, Missouri has the Missouri Council of Teachers of Mathematics (MCTM) which has its pillars and standards of education listed on its website. The MCTM also offers membership opportunities to teachers and future teachers so that they can stay up to date on

4060-460: The status quo. The best quantitative studies involve randomized trials where students or classes are randomly assigned different methods to test their effects. They depend on large samples to obtain statistically significant results. Qualitative research , such as case studies , action research , discourse analysis , and clinical interviews , depend on small but focused samples in an attempt to understand student learning and to look at how and why

4130-751: The student's intended studies post high school. (In South Africa, for example, the options are Mathematics, Mathematical Literacy and Technical Mathematics.) Thus, a science-oriented curriculum typically overlaps the first year of university mathematics, and includes differential calculus and trigonometry at age 16–17 and integral calculus , complex numbers , analytic geometry , exponential and logarithmic functions , and infinite series in their final year of secondary school; Probability and statistics are similarly often taught. At college and university level, science and engineering students will be required to take multivariable calculus , differential equations , and linear algebra ; at several US colleges,

4200-461: The teaching of mathematics. While previous approach focused on "working with specialized 'problems' in arithmetic ", the emerging structural approach to knowledge had "small children meditating about number theory and ' sets '." Since the 1980s, there have been a number of efforts to reform the traditional curriculum, which focuses on continuous mathematics and relegates even some basic discrete concepts to advanced study, to better balance coverage of

4270-609: The text. Also of importance are the scholia , or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study. The Elements is still considered a masterpiece in the application of logic to mathematics . In historical context, it has proven enormously influential in many areas of science . Scientists Nicolaus Copernicus , Johannes Kepler , Galileo Galilei , Albert Einstein and Sir Isaac Newton were all influenced by

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4340-525: The things which were only somewhat loosely proved by his predecessors". Pythagoras ( c. 570–495 BC) was probably the source for most of books I and II, Hippocrates of Chios ( c. 470–410 BC, not the better known Hippocrates of Kos ) for book III, and Eudoxus of Cnidus ( c. 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians. The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated

4410-400: The twentieth century, mathematics was part of the core curriculum in all developed countries . During the twentieth century, mathematics education was established as an independent field of research. Main events in this development include the following: Midway through the twentieth century, the cultural impact of the " electronic age " (McLuhan) was also taken up by educational theory and

4480-419: The types of problems encountered in the first four books of the Elements . Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition, is general, valid, and does not depend on

4550-474: The use of letters to refer to figures. Other similar works are also reported to have been written by Theudius of Magnesia , Leon , and Hermotimus of Colophon. In the 4th century AD, Theon of Alexandria produced an edition of Euclid which was so widely used that it became the only surviving source until François Peyrard 's 1808 discovery at the Vatican of a manuscript not derived from Theon's. This manuscript,

4620-934: Was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the Almagest to Latin. The Euclid manuscript is extant and quite complete. After the translation by Adelard of Bath (known as Adelard I), there was a flurry of translations from Arabic. Notable translators in this period include Herman of Carinthia who wrote an edition around 1140, Robert of Chester (his manuscripts are referred to collectively as Adelard II, written on or before 1251), Johannes de Tinemue, possibly also known as John of Tynemouth (his manuscripts are referred to collectively as Adelard III), late 12th century, and Gerard of Cremona (sometime after 1120 but before 1187). The exact details concerning these translations

4690-413: Was not uncommon in ancient times to attribute to celebrated authors works that were not written by them. It is by these means that the apocryphal books XIV and XV of the Elements were sometimes included in the collection. The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by Apollonius . The book continues Euclid's comparison of regular solids inscribed in spheres, with

4760-568: Was only available to male children with sufficiently high status, wealth, or caste . The oldest known mathematics textbook is the Rhind papyrus , dated from circa 1650 BCE. Historians of Mesopotamia have confirmed that use of the Pythagorean rule dates back to the Old Babylonian Empire (20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before

4830-670: Was relevant to their profession. In the Middle Ages , the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian. Although it continued to be taught in European universities , it was seen as subservient to the study of natural , metaphysical , and moral philosophy . The first modern arithmetic curriculum (starting with addition , then subtraction , multiplication , and division ) arose at reckoning schools in Italy in

4900-538: Was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose." Later editors have added Euclid's implicit axiomatic assumptions in their list of formal axioms. For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Known errors in Euclid date to at least 1882, when Pasch published his missing axiom . Early attempts to find all

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