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.

  • peppersky [he/him, any]
    ·
    45 minutes ago

    Can you have a panic attack in a dream? I think I just had a panic attack in a dream. Everything sucks, I've got one fucking life and even with all the privileges I've had my loser ass hasn't managed to do anything with it, like two decades ago I could have made it work, but with the never ending rise of rent and precarity overall there's no way I could ever do what even my poor-ass parents did. Also no woman wants to date my poor ass, which is smart of them. Also there's a genocide going on and many more to come and I'm losing my mind. I don't want to have to life with all of this shit on my own, I don't have the fucking energy for it. I've even got a good amount of friends, but with the "job market" requiring you to move every few years none of them love anywhere close anymore. Nothing is ever going to get better and nothing is ever going to get easier. I don't want to be on this stupid website but it's the only place I've got left. I don't want to die but I don't want to live like this most of all. I don't want to move back to my parents place, my dad is terribly sick and my mom is a nervous wreck half of the time. The city where I study has (comparably) good mental health facilities, but I can no longer afford the rent there (and I certainly can't when I throw away the job I searched for half a year after the first shift) and where my parents live is a fucking wasteland of all life.

  • TheDrink [he/him]
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    edit-2
    2 hours ago

    Just got banned from another Blackjack table. They said they'd banned me before but I forgot about it lmao - but either way someone at the casino dropped the ball because I sat down at the "show ID" table and showed them my ID and they let me play for like three hours and win $500 before they came out and made me leave.

  • GalaxyBrain [they/them]
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    edit-2
    3 hours ago

    At my parents place cause I have yo be for Christmas and would r3ally like them to chill the fuck out on me smoking weed indoors. They let their friends who smoke cigarettes smoke inside but instead of going to the garage where I used to when I lived here I need to go out into the cold for no reason. I hate staying here I need to smoke a joint in bed. That's how I sleep. Sorry you grew up in the 6os and never fif a dingle crime. I hate staying here over night. Weed is fucking legal and I don't like drinking to compensate. Just let me smell like fuckign pot, the rest of my wasp family tho sint gonna die in a year is fine. I fucking hate Christmas cause I can't not do the thing thst helps my anxiety and get my anxiety triggered. Next year I'm coming over Christmas day. I just worked a gucking 6 day week and the day before dur9mg and after Christmas is all I have off and my next weekend is cut in half cause I have to work a kitchen bee years eve. I don't want to do Christmas unless I can enjoy it and as a kitchen employee who busts my asd for a month prior, let me smoke weed without needing yo go out into freezing temps. You let cigarette smokers smoke indoors. It should be fun for everyone and I hate this. I'll go over Christmas morning next time

  • Lovely_sombrero [he/him]
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    edit-2
    4 hours ago

    Luigi is ranking highly across the board, even with stupid cognitive dissonance-having individuals!

    A guy started aggressively defending Biden when I mentioned Biden's bankruptcy bill and how it made it harder for individuals to discharge medical debt (and especially student debt). I mentioned it because it was relevant to the conversation (someone mentioned how Biden could do the epic thing and pardon Luigi before leaving office). His comments were obviously stupid, but then I looked at his posting history and it was 60% complaining about Trump & defending Biden. But about was 40% equally aggressively defending Luigi. Amazing.

  • Rojo27 [he/him]
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    3 hours ago

    We were originally going to have a large family gathering, but we didn't really have the space for it. It was just my cousin and her son. It was nice to have them over. Kind of wish I had less on my mind though.

    A bit of a rant on some work drama, some social stuff too

    I'm thinking about how one of my co-workers continuing to say they don't want to bother me on my off days. But I'm lead/manager and I have told them several times that I don't mind being contacted if its important.

    So recently there was a job that they kind of screwed up on that really didn't have to be such a problem if they had called me. But even more importantly they didn't call me when they had missing hours. And it was partially my fault because I was trying to give them extra hours since I had gotten the wrong schedule info and I didn't want to keep them longer than they were already scheduled for.

    They kept telling me about the missing hours and how they'd begun the process of getting paid for them (its a really tedious process because HR is really there to make things difficult), but today they let me know the exact issue that there was (they kept talking about missing hours, but didn't specify that it was the extra hours I was going to give them for the scheduling mistake). So I said, hey this is why I tell you to call me to remind me of these things. And in this case I was going to go on PTO the following day. But they tell me, no, I'm not going to bother you for that.

    All I could think was "what do you mean?". There's nothing more important for me than to get my crew paid because its already tough enough with how poor the compensation is in comparison to the work we do. And now they're mad about it. Maybe mad a me, but I don't really know. They also said they'd consider quiting over it in the past few days. I have a couple of alternative solutions that I proposed if HR doesn't come through, but fuck...

    I just hate that everyone also treats me like this. Doesn't matter if its coworkers, friends, family. Every time someone gets in touch with me they lead off with "sorry to bother you". Why the fuck does everyone think they are bothering me if they call me? I get I'm a quite and reserved person, but I've never rebuffed anyone's advances whether its just to have conversation or help with work. Shit fucking sucks because its part of the reason I always feel so isolated. And it just makes me wonder if I annoy people when I reach out to them. Ugh...dennis-stare

  • keepcarrot [she/her]
    ·
    5 hours ago

    I wish people who commit petty theft stole from corporations or the rich instead of from people as poor as them. My biggest gripe is when I would have loaned them the thing, but they just take it and never give it back, leaving me to be surprised when I need a vacuum cleaner or whatever.

  • hexaflexagonbear [he/him]
    ·
    7 hours ago

    Home alone is about a trust fund kid using castle doctrine against the working class.

  • Cowbee [he/him, they/them]
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    edit-2
    6 hours ago

    2025 is going to be a bit of a training arc for me. I'm going to finish my "Need to Read" list for theory (of course still continuing to read theory and reread afterwards, just covering the "essentials"), going to start physically training, and focus more on my language learning. I want to build steady, stable habits, and cultivate myself through consistent training. My understanding of Marxism-Leninism is certainly better than a year ago, but now I am at the point where I can see just how much more there is to learn and grow, and I want to put that to practice as well. zhou-envibe

  • 0__0 [he/him]
    ·
    7 hours ago

    How come every single time I go on the internet, I see a brand new Luigi picture from him like, taking a tab of LSD or some shit lol. How many pictures of him are there on the internet? (pre him doing adventure-time

  • SupFBI [comrade/them]
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    edit-2
    7 hours ago

    Don't get to play Santa and eat the cookies.

    Don't get to fill the stockings

    Don't get to see my kid on Xmas morning.

    Yeah, it's the first Xmas post-divorce.

    Holiday cheer can lick my hairy anus.