60

This device was invented by a Jewish Rabbi, Levi Ben Gershon. It was used to measure the angular distance between two stars or, in general, any pair of celestial bodies. Ptolemy lived 1000 years before the invention, so this is only the product of the artist’s fantasy.


55

It is called "Jacob's staff". It was an old astronomical tool used for trigonometric purposes.


17

It depends on the meaning of "calculate", since $\pi$ is a transcendental number it can not be "calculated" in the usual meaning of the word. The first analytic formula (in the form of an infinite series) that in principle can calculate $\pi$ to any required accuracy is probably due to medieval Indian mathematician Madhava, who was first to conceive of ...


14

And this tool has been known under many other latin names than baculus Jacob (or Jacob's staff): radius astronomicus (astronomic ray), crux geometrica (geometrical cross), revelatorem secretorum (secret decipherer). This one is the hand-held version, a mounted version exists. A collection of studies on Gersonides (or Levi ben Gershon) is found in G. ...


14

For sinus, see : Victor Katz, A History of Mathematics (3rd edition, 2008), apge 253 : The English word “sine” comes from a series of mistranslations of the Sanskrit jya-ardha (chord-half). Aryabhata frequently abbreviated this term to jya or its synonym jiva. When some of the Hindu works were later translated into Arabic, the word was simply transcribed ...


14

This number has no significance. Its origin is historical. Originally meter was defined as 1/40,000,000 part of the Paris meridian. Based on the measurement of this meridian they made a standard rod in Paris. Since it is inconvenient to base the definition on something which is difficult to measure, meter was soon redefined simply as the length of this rod. ...


12

Poincare refers to the Lie's solution of the so-called problem of space, a.k.a. the Helmholtz , or Riemann-Helmholtz, or Helmholtz-Lie problem of space, which amounts to characterizing all manifolds (originally, only 3-dimensional) with free mobility of figures (roughly, homogeneity and isotropy). In modern terms, free mobility amounts to constant Riemannian ...


11

Victor Katz is not a linguist and a lot of what he says in the quoted extract is wrong: for example that “Arabic is written without vowels” and that the word in question is spelt “jb”. In fact it is written jyb جيب (as mobileink has pointed out). But the decisive error from the viewpoint of the history of science is his failure to remark that Sanskrit jyā ...


11

A nice account is found in a note to R. Steiner's Die vierte Dimension (1995; translation): Felix Klein (1845–1925) seems to have been the first mathematician to draw attention to this phenomenon in the early 1870s. According to an account by Zöllner (1878a), Klein spoke with him during a scientific conference on this subject shortly before publishing a ...


10

No additional or more precise approximations to $\pi$ seem to have been found in Babylonian records up till now. Herman C. Schepler, "The Chronology of PI", Mathematics Magazine, Vol. 23, No. 4, Mar. / Apr. 1950, pp. 216-228 (as reprinted in L. Berggren, J. Borwein, and P. Borwein, "Pi: A Source Book, 3rd Edition", Springer 2003) summed ...


10

It is the Minkowski plane, the lightlike lines are “perpendicular to themselves”, see e.g. Stachel, Poincaré and the Origins of Special Relativity. To get this geometry, one needs to expand what “geometry” means. The classical geometries of Euclid, Lobachevsky, and Riemann have what is now called sign-definite metrics, which define ...


9

Problem: classical geometry is not happy with infinitesimals Newton is systematically trying to avoid basing calculus on infinitesimal geometric quantities. We can see this from how he emphasizes that his method is consistent with the "ancient" standard of rigor: To institute an Analysis after this manner in finite Quantities... is consonant to ...


9

What do you mean "correctly or not"? Here is a brief history. The Bible has a sentence which can be interpreted as implying that $\pi=3$. The EXISTENCE of $\pi$ (the ratio of circumference to diameter) was rigorously proved by Archimedes. He also calculated it approximately. For many centuries it was called the "Archimedes number". As centuries passed, ...


9

Newton studied at school and at the university, but he mostly taught himself by reading. (At his secondary school he certainly learned Latin, Greek, the Bible and some arithmetic. In the universities, they mostly studied Aristotle at that time, which has nothing to do with mathematics). Besides textbooks that existed at that time he mastered Euclid, and ...


9

Such terms as “given in species” are defined in Euclid’s Data (Greek, English): III. Rectilineal figures are said to be given in species, which have each of their angles given, and the ratios of their sides given. (English version, R. Simpson, 1810, p. 367) [Species is the translation of eidos, shape or form; see LSJ, εἶδος, def. A.2.b.]


8

"Geometry" literally translates from Greek as "land measurement", and reflected a popular Greek belief that they learned it from Egyptian "rope stretchers", land meausurers, who used ropes to perform what came to be called straightedge and compass constructions. Geometric mean for $n=2$, or mean proportional as Greeks called it, was likely so named because ...


8

The coordinate method may be traced to antiquity, specifically to the works of Apollonius of Perga (c. 262 – c. 190 BC) The following quotation from Carl B. Boyer,"Apollonius of Perga" (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. pp. 156–157. ISBN 0-471-54397-7 (as referenced in the Wikipedia article on Apollonius) explains ...


8

The first sentence of the question is not justified. Accuracy of Erathosphenes measurement was much discussed in the literature, and it is certainly not "incredible". What the actual accuracy was is unknown, because we are not sure what his unit "stadium" exactly was. Considering time measurement, he did not need it. He used two cities on approximately the ...


8

It seems that this attempt made an impression, when one needs to make the point Indiana Pi itself is typically invoked. NMSR Reports modeled their 1998 April Fool's story on it: "NASA engineers and mathematicians in this high-tech city are stunned and infuriated after the Alabama state legislature narrowly passed a law yesterday [March 30, 1998] ...


8

The "staircase paradox" (or "Pythagoras paradox") name appears to be recent, so it is hard to search for it. Wolfram calls it "diagonal paradox", but that may be conflating it with a different paradox due to Leibniz, which he used to argue against the actual existence of indivisibles, see The Philosophical Assumptions Underlying Leibniz's Use of the Diagonal ...


7

See : Detlef Laugwitz, Bernhard Riemann 1826-1866 : Turning Points in the Conception of Mathematics (1996 - German ed.1996), page 182; we have : Riemann's habilitation paper "Uber die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe" (W.227-265) ("On the representability of a function by means of a trigonometric series") was completed ...


7

Since the question specifically says "to any accuracy," I will assume you mean approximations as well. In this case, the first recorded approximation of $\pi$ comes from the Babylonians, who not only had an awareness of it as being a specific constant, had approximated it's value to $3\frac{1}{8}$ or $3.125$. This is recorded in a tablet fond near Susa ...


7

Stillwell gives some details in Mathematics and its History. In modern terms, Newton made a general transformation of axes reducing the general cubic to four equation types, and then classified them into "species" according to the roots of the polynomials on the right hand side. "His paper does not contain detailed proofs; these were supplied by Stirling ...


7

Terrestrial longitudes were derived from the astronomical longitudes given in the "Hakimite Tables" of Ibn Yunus (d. 1009). Essentially, during a lunar eclipse (predicted by the astronomical tables) two observers located some distance apart make measurements of when they first observe the shadow of the earth entering the disk of the moon. Source - ...


7

While you already accepted an answer, it seems not superfluous to add another one, in particular since you are implicitly asking for a better translation/understanding of the passage you quoted. I will give you a translation here, and some comments of mine. Beforehand, though, I think it is fair to say hat you seem to have been misled when you wrote ...


7

We have Archimedes's construction of regular heptagon due to Arab transcription by Thabit Ibn Qurra, see Mendell's translation. Aaboe gives modernized exposition and commentary in Episodes from the Early History of Mathematics, MathWorld's Heptagon has a shorter version along with modern variants. The construction uses neusis of sorts, i.e. marked ...


7

We do not know. First, Syene and Alexandria are not on the same meridian, Alexandria is about 3° to the West, and second, Syene is not on the tropic (where the Sun is straight up on the summer solstice) either, which is another assumption Eratosthenes makes, according to Cleomedes. Our primary source on Eratosthenes's Geographica, where these assumptions ...


7

According to Klein, the first mathematician who considered space-time as a 4-dimensional manifold was Lagrange, but these ideas were not immediately developed by others. Then he mentions Cauchy and Cayley (Cayley published in 1844 "Chapters on the analytic geometry in $n$ dimensions"), and credits Grassmann (1844) with the first systematic exposition. While ...


7

Delaunay (Gallicized version of Russian Delone) did not invent them, they were used long before 1934. Delaunay triangulations, or more generally tesselations, are dual to Voronoi diagrams, the circumcenters of Delaunay triangles are the vertices of the Voronoi diagram. One can infer the motivation from the very title of Delaunay's 1934 paper: Sur la sphere ...


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