The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to 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 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 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.

Transmission of the text

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, 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 the Elements from the Byzantines around 760; this version was translated into Arabic under Harun al-Rashid (c. 800). 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.

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 a thousand different editions.

Influence

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 the Elements, and applied their knowledge of it to their work. 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 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 the cornerstone of mathematics.

In modern 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 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), 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.

  • Book 1 contains 5 postulates and 5 common notions, and covers important topics of plane geometry such as the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures.
  • Book 2 contains a number of lemmas concerning the equality of rectangles and squares, sometimes referred to as "geometric algebra", and concludes with a construction of the golden ratio and a way of constructing a square equal in area to any rectilineal plane figure.
  • Book 3 deals with circles and their properties: finding the center, inscribed angles, tangents, the power of a point, Thales' theorem.
  • Book 4 constructs the incircle and circumcircle of a triangle, as well as regular polygons with 4, 5, 6, and 15 sides.
  • Book 5, on proportions of magnitudes, gives the highly sophisticated theory of proportion probably developed by Eudoxus, and proves properties such as "alternation" (if a : b :: c : d, then a : c :: b : d).
  • Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures.
  • Book 7 deals with elementary number theory: divisibility, prime numbers and their relation to composite numbers, Euclid's algorithm for finding the greatest common divisor, finding the least common multiple.
  • Book 8 deals with the construction and existence of geometric sequences of integers.
  • Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers and the construction of all even perfect numbers.
  • Book 10 proves the irrationality of the square roots of non-square integers (e.g.2 {\displaystyle {\sqrt {2}}}) and classifies the square roots of incommensurable lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of irrational numbers. He also gives a formula to produce Pythagorean triples.
  • Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of parallelepipeds.
  • Book 12 studies the volumes of cones, pyramids, and cylinders in detail by using the method of exhaustion, a precursor to integration, and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing that the volume of a sphere is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids.
  • Book 13 constructs the five regular Platonic solids inscribed in a sphere and compares the ratios of their edges to the radius of the sphere.

Euclid's method and style of presentation

Euclid's axiomatic approach and constructive methods were widely influential.

Many of Euclid's propositions were constructive, demonstrating 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' 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.

  • TheDrink [he/him]
    ·
    1 hour ago

    currently setting up my little brother's new Retroid Mini that he's getting for Christmas, damn you can really just download whole ass romsets on archive.org huh.

  • Frank [he/him, he/him]
    ·
    2 hours ago

    "the incoherent of the irrational numbers"

    Mathematicians; everything is maths

    Chemists; everything is medicine

    Political economists! Everything is politics

    Anthropologists (me!); everything is culture

    Martial Artists; everything is martial arts and if you disagree with me I will do cool martial arts punching until you are impressed and agree.

    Everyone; you have opened my eyes, teacher. I see now that everything is martial arts. Let us retire to a remote mountain to practicing punching until we achieve enlightenment.

  • CthulhusIntern [he/him]
    ·
    3 hours ago

    I passed both CompTIA A+ exams! Now time for a break before I hunker down for the Network+!

    • blipblip [she/her, they/them]
      ·
      2 hours ago

      Hell yeah! Pretty sure you posted about passing one of them and I said I should get back on my network+ book. Haven't touched it still lmao

  • UmbraVivi [he/him, she/her]
    ·
    4 hours ago

    Still on my BG3 honour mode run. Having Astarion in my party for the first time and man...

    cw: discussing themes of sexual violence in games

    So in Moonrise Towers there's this NPC name Araj Oblodra who wants Astarion to bite her and drink her blood, she gives you a potion in return that permanently increases your Strength stat by 2. Astarion doesn't wanna do it and you have to persuade him, he immediately throws up afterwards and from all the surrounding dialogue afterwards it's pretty clear that he feels violated by the whole ordeal. His story overall revolves heavily around his history of effectively being a sex trafficking victim and having no agency over his own body, so this hits even harder. The romance plot probably explores that a lot more, but I never romanced him.

    This feels really messed up. The game is not subtle about the whole bloodsucking thing being about sexual gratification for Araj Oblodra (there's even a dialogue option pointing this out) and I imagine if the genders were reversed, it would be a lot more apparent how inappropriate this is.

    Imagine: A girl who was a sex trafficking victim for years, forced to seduce and sleep with hundreds of men against her will and only recently managed to escape. She's very clearly traumatized and struggles with feeling like her body is not her own anymore, since it has exclusively been used as a tool for others to get what they want. This girl travels with you and you run into a man who promises to aid you, but only if the girl will get on all fours and lick his feet. The girl is vehemently opposed to the proposition and the only way to get the man's aid is to tell the girl to suck it up and do it.

    BG3 has a lot of evil dialogue options, but I feel like pimping out a sex trafficking victim to a strange pervert against his will crosses a line and it doesn't sit right with me that this is the only way to get an extremely powerful permanent buff. It's obscured by the fact that the strange pervert is a conventionally attractive woman and the sex trafficking victim is an otherwise arrogant and unkind man, but that's stil the gist of it and it clearly affects Astarion negatively. There was another moment at the start of Act 3 when you talk to Astarion about the Astral-touched Tadpole. He says (paraphrasing) that he doesn't wanna let it transform him because for once, he wants to have agency over his own body. But you can once again convince him to do it anyways.

    Again, BG3 has a lot of moments where you can be mean and cruel. You can torture and kill, you can mock people who lost loved ones, you can let innocents get executed. But there's a reason you never get the option to outrightremoved someone, touch someone inappropriately against their will or even just tell someone to take their clothes off. There are no limits to the amount of physical violence games may depict, but letting players commit sexual violence is a taboo in video games for a variety of very good reasons.

    Am I being too woke about this? The player having the option to deny Astarion agency over his own body irks me given his history of what was effectively sexual slavery. I think the gravity of what you're doing to him is obscured by him being male and an arrogant asshole, but it still feels like there's a line being crossed here.

  • Washburn [she/her]
    ·
    6 hours ago

    Brian Thompson: The difference between you and I is that I rose to power from the working class, while you are from the privileged petty-bourgeois class.

    Luigi Mangione: True. But there is this similarity. Each of us is a traitor to his class.

  • Gosplan14_the_Third [none/use name]
    ·
    7 hours ago

    I watched the Big Lebowski yesterday, to see what Reddit has been raving on about all these years. It's a decent movie, fun even if basic. Very GenX in feel.

    Good? Yeah. A masterpiece? Eh.

    • Frank [he/him, he/him]
      ·
      46 minutes ago

      The GenX thing is key. It really spoke to people who felt trapped in the end of history.

    • Are_Euclidding_Me [e/em/eir]
      ·
      5 hours ago

      I really like it, although I'm not going to sit here and say you're wrong for not liking it more than you did.

      What I will say though is that The Big Lebowski contains the funniest scene I've ever watched in any movie, TV show, what have you. And that's at the very end when they're scattering the ashes. For whatever reason, that scene gets me every time, I can't help but crack up every single time. Even when I'm anticipating it and thinking "it can't possibly be as good as I remember", it always is.

      I also really like the vocabulary it builds up in the form of catchphrases. The obvious one being, of course "the rug really tied the room together", but there are others. I can imagine if your friend group is obsessed with the movie that those catchphrases repeated constantly would get really, really old, but luckily I was never in a friend group who felt the need to do that, so they've stayed pretty fresh for me, and I always enjoy watching the movie and being drawn into its world.

      Also, when you're high as hell (the best way to watch it), the plot seems pretty convoluted. Like sure, objectively it's not that difficult to follow, but when you're super duper high it sure seems complicated, and oddly enough, I think that's part of the draw too. At least it is for me.

    • Moss [they/them]
      ·
      6 hours ago

      Yeah The Big Lebowski is decent but I'll probably never watch it again. I really don't know why its become such an iconic thing in media

      • Frank [he/him, he/him]
        ·
        36 minutes ago

        I think it spoke to people who felt trapped in the miasma of the end of history. Communism was defeated, no future existed, nothing was possible. Gen X was still trapped under the thumb of their boomer parents, going nowhere with nowhere to go.

        The contrast between the incredibly petty plotting and scheming of most of the cast, compared with The Dude and his loser friends just trying to live life and enjoy simple pleasures, spoke to deeply alienated people who couldn't tell themselves a story about their lives and their jobs that made sense to them.

        "The Dude Abides" was kind of the antithesis of the dot com boom, the internet superhighway, clintonism, the meaningless of it all. The Dude didn't need meaning. He had a simple life that was enough for him

        I think a good way to analyze it is to contrast it with the self annihilating violence of Fight Club. Jack/Tyler had the same problem; a life without purpose or meaning. The Dude embraced the void and found a reason to live just enjoying weed and bowling with his dirt bag friends. Jack and Tyler rebelled against the void and tried to construct their own meaning, but having negative one hundred chill they built this machine of self destroying violence that could only seek meaning in destruction.

        Jack and Tyler wanted to prove themselves, probe that they were worth more than the corporate cube they felt trapped in, prove their masculinity and their warriorness and capability.

        The Dude didn't want to prove anything and was openly contemptuous of the whole idea. He just vibes, and his greatest ambition was getting compensated for his rug.

        Its two ways of engaging with the emptiness of the End of History era, a desolate eternal present where no one was worth anything and there was nothing to aspire to except peonage and consumerism forever.

        And the contrast that with The Matrix, which said screw all that, history is not over, the system is vulnerable and can be destroyed by principled people with cool trench coats who do terrorism and shoot cops.

      • dustbunnies [she/her, comrade/them]
        ·
        6 hours ago

        my understanding of its popularity is that the main thing people like is how The Dude's chill attitude is a beautiful antidote to grind culture.

        but also, iirc, it's one of those movies that rewards rewatches because of all the silly little details that don't register the first time. it's been a minute since I've watched it, so I'm blanking on them all, sorry 😔

        another film that does that is "The Forgotten" with Julianne Moore. I just saw it again it last weekend for the first time since I saw it in the theater two decades ago, and hot damn does that one reward a rewatch too! there were so many things that stuck out this time – there's a circle motif happening from the beginning that doesn't pay off until the climax, characters who are in the background in the beginning who become important, etc.

        so yeah, I think part of the fun with "The Big Lebowski" is catching new stuff you didn't catch the first time or ten.

        • Frank [he/him, he/him]
          ·
          34 minutes ago

          I think what you said is a big part of why Corn brothers movie hit and keep hitting. The whole narrative and the characters and the jokes are built up of layers within layers and there's always more to see, connect, and consider.