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.
Santa just came by
I sucked him off
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.
Home alone is about a trust fund kid using castle doctrine against the working class.
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.
What denomination of Christianity do.you think The Who’s of Whoville are? I’m guessing catholic
PSA: It seems like needs have been going unmet lately in the mutual aid comm. If you have anything to spare, please consider checking it out.
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
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.
What happened here? It was so active for a while with all the new bluesky accounts and then activity plummeted? Was there another struggle session?
say what you want about luigi mangione, he would have made a fantastic live-action luigi mario
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.
My grandpa is in the hospital this Christmas . He was having recurrent heart problems, and had a pacemaker installed yesterday. Situation seems stable now.
Apparently when my father got him to the hospital he had a heart rate of 20 bpm and he was still conscious
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.
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.
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
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.
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.
The GenX thing is key. It really spoke to people who felt trapped in the end of history.
depression
Forgot to take my meds and I’ve been feeling like shit today. Manager noticed and was annoyed about it. Fucking middle manager parasites. Broken down several times today over various shit. Want to rant about capitalism but every time I try to type it just makes me more sad. Hate this shit. My meds don’t even get me to a stable neutral so I’m just always sad, yet it is continually surprising how bad I feel unmedicated.
Merry christmas to logic nerds and vibing birds