I hope you had a great Christmas for those that celebrated, or a great holiday and some relaxing time. I just realized that I haven’t posted about the event: History Facts in December. I found the time to post daily and I am quite happy by your excitement and appreciation. So here are 5 more days of posts from this event:
11th December: For this day I have to thank Math-Update for finding the time and writing some incredible blog posts for this event. The posts represent and combine wonderfully the history of some higher mathematical concepts such as trigonometric series, Fourier series, integrals and finishing with something about Cantor and infinity. So I totally recommend: Math Story & Creation part 1, part 2 and part 3.
For 2000 years, Euclid’s system was held as the only possible foundation of geometry, thought to be supported by logical reasoning and empirical proof.
In the early 19th century, mathematicians demonstrated that geometries that were just as valid and consistent as the age-old Euclidean geometry could be produced by replacing two of the Euclid’s postulates – the one about infinite continuation of a straight line, and the one about parallel lines – by other postulates that were employed with the remaining Euclidean postulates and the common notions.
The new concepts of geometry were later named hyperbolic geometry and elliptic geometry. These names, founded on the analogies with conic sections, were suggested by the German mathematician Felix Klein in his notable program for classifying geometries (1872); for Euclidean geometry, Klein coined the name parabolic geometry, a term which is, however, rarely used.
I was asked to do some research on why we use “m” for representing the slope or gradient of a line. Here is what I have found about this:
J. Miller has undertaken a detailed study of the origin of the symbol m to denote slope. The consensus seems to be that it is not known why the letter m was chosen. One high school algebra textbook says the reason for m is unknown, but remarks that it is interesting that the French word for “to climb” is “monter.” However, there is no evidence to make any such connection. In fact, Descartes, who was French, did not use m (Miller). Eves (1972) suggests “it just happened.”
The earliest known example of the symbol m appearing in print is O’Brien (1844). Salmon (1960) subsequently used the symbols commonly employed today to give the slope-intercept form of a line
y=mx+b in his famous treatise published in several editions beginning in 1848. Todhunter (1888) also employed the symbol m, writing the slope-intercept form y=mx+c. However, Webster’s New International Dictionary (1909) gives the “slope form” as y=sx+b. (Miller).
Also, the are different ways to represent it depending on the country:
In Swedish textbooks, the slope-intercept equation is usually written as y=kx+m, where k may derive from “koefficient” in the Swedish word for slope, “riktningskoefficient.” In the Netherlands, the equation is commonly written as one of y = ax+b, y = px+q, or y = mx+n. In Austria, k is used for the slope, and d for the y-intercept (Miller).
The earliest occurrence of cubic (third-degree) equations may have been in antiquity in connection with the famous problems of duplicating the cube and trisecting the arbitrary angle, which both would lead to cubic equations if stated analytically. The task was to solve these problems using only an unmarked ruler and a pair of compasses, which we now know is impossible. However, several mathematicians of antiquity solved them geometrically using other curves as tools.
An important advance was made by the mathematician-inventor Heron of Alexandria by reviving and developing old Babylonian and Egyptian practices of extracting roots by successive approximation.
The cubic equation was a cherished topic of study among mathematicians of the Muslim world. Best known is the work of the Persian poet, mathematician, and astronomer Omar Khayyam. Building on the Greek tradition, he obtained solutions to the cubic equations as intersections between conic sections; such studies had, however, been made by Archimedes about 1000 years before Khayyam. In mathematics, the importance of Omar Khayyam and other early Muslim mathematicians lies mainly in their perpetuation of ancient Greek and Hindu knowledge.
The word mathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means “that which is learnt”, “what one gets to know”, hence also “study” and “science”, and in modern Greek just “lesson”. In Greece, the word for “mathematics” came to have the narrower and more technical meaning “mathematical study” even in Classical times.Its adjective is μαθηματικός (mathēmatikós), meaning “related to learning” or “studious”, which likewise further came to mean “mathematical”.
In Latin, and in English until around 1700, the term mathematics more commonly meant “astrology” (or sometimes “astronomy”) rather than “mathematics”; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine’s warning that Christians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), and meaning roughly “all things mathematical”; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek. In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.