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26

The main question is why the Pythagorean theorem for right triangles: $$ a^2+b^2=c^2$$ is such a central tool of Euclidean geometry. There are many different approaches one can take to this; I'll give it a shot. One of the key observations is that the triangle is the most basic 'non-trivial' shape in plane (two-dimensional) geometry. Any three points - one ...


24

The truth is that we do not know. We do know of the person who is credited with the discovery, Menaechmus (c. 350 BC), a student of Eudoxus of Cnidus and a friend of Plato's, one of the most prominent mathematicians of his time. The names ellipse, parabola and hyperbola were given to them by Apollonius of Perga over a century later however. Menaechmus called ...


23

This is one of those questions that is much trickier than it appears, many different people contributed to the formulas as we write them today. The short answer, that doesn't really do justice to history, is that only Euler presented volume formulas in this form in his textbooks after 1737. The principal step was no doubt made by Archimedes in On Sphere and ...


18

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 ...


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 ...


10

This question has been discussed several times on math overflow: https://mathoverflow.net/questions/191909/discovery-and-study-of-conic-sections-in-ancient-greece It also has references. One theory is that they appeared when the Greeks started to think how to make accurate sundial. This theory is developed in several books and articles on the subject, and ...


10

Certainly that the Earth is spherical was a commonplace (among the educated people) at the time of Eratosthenes. Once you start traveling on sea (or climbing mountains) you immediately notice that the Earth is curved. Observations of the sky from different places confirm that and tell you that it is spherical, at least approximately (before that there were ...


10

Archimedes calculated the exact formulas (in the way that the ancient Greeks gave formulas) in his book On the Sphere and Cylinder. This was not "experimental": He gave a full geometric proof, rigorous for its time period. He considered this his greatest work. He asked that a diagram representing his proof be inscribed on his tomb. This was apparently done ...


10

While Elements contains no reference to conic sections it does define angled cones, given as definition 18 of Book XI, and it examines some of their properties in Book XII. History identifies Menaechmus as inventor of conic sections (350-360BC), obtained by considering these angled cones intersecting a fixed plane. In addition to Elements, Euclid also ...


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

The reasons are "simple". All other axioms and postulates appeal to our "everyday experience", at least in principle. The straight lines correspond to light rays in everyday experience. However it was probably already recognized by Euclid, that the parallel postulate is different from the other axioms. Of course, a degree of mathematical sophistication is ...


8

There are too separate issues here. The method of fluxions and fluents, Newton's version of calculus, is amply represented in Newton's extant papers, starting with 1669 On Analysis by Equations with an Infinite Number of Terms sent as a letter to John Collins, and disseminated by him to multiple correspondents, including Leibniz. The dotted shorthand was ...


7

There are several ways to answer this depending on what "Cartesian plane" means. Most literally, Cartesius is the Latinization of Descartes name, so one can look at pictures in Descartes's La Geometrie, which was the first systematic use of coordinates to solve geometric problems. However, coordinate graphs were introduced before Descartes by Oresme, to plot ...


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 ...


6

This is in Rouse Ball's Mathematical recreations and problems (2nd edition, 1892, p. 33), and later editions carry this footnote (6th edition, 1914, p. 45): I believe that this and the fourth of these fallacies were first published in this book.


6

This is a neat observation. I was unable to find historical references for it in particular, but there is a rich history of solving this type of problems. The problem is clearly reminiscent of finding the Pythagorean triples, right triangles with integer sides, for which Elements X.29 gives a solution without a hint of how it was obtained (of course once the ...


6

According to The Hindu Business Line, quoting the scholar TA Sarasvati Amma: It was only in the 18th century, nearly 2,000 years after active contact of Indians with the Greeks, that Euclid’s Elements were translated into Sanskrit and even then perhaps the example of the Arabs provided the inspiration. TL/DR; Concerning Indian mathematics at about the ...


6

A good place to look for such things is Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics, where we read: "Boyer (page 404) seems to attribute the term origin to Philippe de Lahire (1640-1718). The term presumably appears in Sections Coniques by Marquis de l'Hopital, since the OED shows a use of the term in English in a 1723 ...


5

For a broader perspective see How was geometry historically used to solve polynomial equations? For early practical problems that would lead (today) to quadratic equations see e.g. Friberg's discussion of cuneiform tablet YBC 3879 (c. 2000 BC), a judicial document from third Sumerian Ur period, that describes field division problems leading to quadratic ...


5

Pappus in Mathematical Collection, book IV "squared" the Archimedean spiral, the spiral is to the circle enclosing it as 1:3. The curve is obtained by composing uniform linear and circular motions in the plane, Archimedes himself in On Spirals attributes the discovery of it to his friend Conon. Some care is needed with calling this "squaring with ...


5

Not much history to it, I am afraid. It seems that methods of trisection rather obviously (to those who considered them) applied to quintisection as well, so the problem was of little theoretical interest in itself. Without restriction on tools the problem of angle multisection is easily solvable. For instance, the quadratrix of Dinostratus, originally the ...


5

Coolidge (1940, p. 273) gives a proof which ends: (...) Hence any real transformation of the Euclidean plane which keeps distances invariant is either a rotation, a translation, or the product of one of these and a reflection in a line.† † The earliest proof I have seen for this is Chasles (1830, p. 321). I think, however, it must have been ...


5

The contest between synthetic and analytic methods in geometry predates Hilbert and even calculus, one can trace its origins to Vieta's algebraic conversions of geometric problems that streamlined their solution, see Viète's Relevance and his Connection to Euler and their systematization in Descartes's analytic geometry. But the rise of calculus and ...


5

The real reason is probably that straightedge and especially compass are the simplest, most primitive instruments, and also easy to make. At the same time they are quite accurate. (Straightedge is not so easy to make and is not so accurate, but certainly it stands next to the compass on these criteria). A modern high school student tends to think that to ...


5

They were the opposite of close. Geometers and atomists were bitter ideological enemies since before Euclid. The reason for such high passions was that Greek view of geometry was different from the modern relativistic/formalistic view, to them the nature of real space was at stake, not one idealized fiction among others. The indivisibles (atoms) and the ...


5

Ostrogradsky is well-known for his negative reaction to Lobachevsky's work, but in his signed reviews, at least, his complaints were different. Lobachevsky, in contrast to Ostrogradsky, was not a good expositor, and the paper he submitted to the Academy was obscurely written. Apparently, Ostrogradsky was only able to make sense of two integrals computed ...


4

It appears in the demonstration of Pappus's hexagon theorem as a tool in the demonstration. given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points X, Y, Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the Pappus line (illustrations from the wikipedia article) (Note: ...


4

Although this question and the answers now have some age to them, I suggest that it's important not to overlook the mythical character of the assumption that underlies this question. The question explicitly supposes that 'calculus is missing from the Principia'. But that is not true: it is not missing, not only have skilled commentators from the 17th to the ...


4

Regarding II(c): Strogatz/Schroeder does offer a string of connections to Einstein through Shneior Lifson and Erst Straus, one of Einstein's assistants at IAS. But it would be hard to connect Einstein and the young Jashemski, since Einstein came to the US in 1933. There are many instances of people coming up with the same proof independently, especially on ...


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