Post recent books you've finished

>what do you study?
>mathematics
>OH REALLY I SUCK AT MATH AND I NEEEEVER DO IT HOW CAN YOU THINK ABOUT MATH IT'S SO HARD AND STUPID ??

>what do you study?
>literature
>OH REALLY I SUCK AT READING AND I NEEEEVER DO IT HOW CAN YOU THINK ABOUT BOOKS IT'S SO HARD AND STUPID ??

what are some other classic 'math' books? is principa mathematic interesting?

Geometry and the Imagination by Hilbert and Cohn-Vossen is a good one.

Newton's Principia is decently interesting but you need to have Euclid mastered to follow his proofs. Russell and Whitehead's Principia is a funny curiosity to look at but borderline impossible to read.

thanks, didn't know the first one. will check it

Speak for yourself cunt I love smashing me a cone packed with the finest

>how much math and geometry do i need to understand to read and enjoy it
almost none

>are there annotated versions
yes but they ruin the spell by pointing out all the errors the ancients made

Pine cone!

After On Conic Sections I'm reading The Almagest by Ptolemy

my bad there's only 60 props in book one

Fuck your "spell". The reason to read a historical mathematical document is to learn about the history of mathematics, not to read a fun fiction, or to accept a demonstrably false or confused mathematical idea as true, just for the sake of discussion. This is a large part of the /appeal/ of math: ideas can straightforwardly either be confirmed or banished, given basic assumptions. It's not political. At least, not nearly so political as all other fields of human endeavour.

If a Euclid or an Apollonius makes errors, then you should want to know about them. If you don't, then you really don't have any business reading these sorts of texts in the first place. You're not reading some ancient /myth/, or novel. You're not even reading some weird old philosophical treatise where you have to take a few silly ideas and "just go with it, man", for the sake of reading the text. No, simple statements about points, lines, curves, number etc do not fall into the above purview, and this because the ideas are well understood. You're reading something that makes concrete, /relatively simple/, and testable claims about basic abstract aspects of existence, which must be judged according to such standard of truth as is applicable (which is uncontroversial in mathematics (so there's really no weasal-room for you here even though I appear to have opened some up). That's the difference.

that seems very interesting. if you could throw some more titles you already read or want to read I would appreciate it, I am very interested in works like that and I have never had good recomendations here or in Veeky Forums

already read goddel, escher, bach?

thanks again

m8 you're obviously a high schooler learning geometry for the first time

>testable
confirmed for not knowing what you're talking about

you're correct that you're reading about the history of mathematics - but don't confuse that with real mathematics
if you want to learn math, read a math textbook
if you want to learn the history of math read the ancient books (and a textbook on the subject as well)
fact is most of pre 19th century math is considered unrigorous today and very few works from before that are considered good math today

Euclid makes a ton of unsubstantiated and dubious proofs (most famously the very first proof does not hold) and so do a lot of the other famous historical mathematicians
even most of Newton's Principia is considered poorly substantiated today
there is 0 reason to read anything historical if you want to learn proper rigorous math

t. math major senior

On the contrary, if you believe for example that Euclid's propositions are not testable, then it is you who do not know what you are talking about. I have a math degree with a history minor, and I know what I am talking about.

Obviously physics students are not generally expected to read the principia / math students are not generally expect to read Euclid right out of the box, and so on, so since we both understand that what I've been talking about is the /history of math/, as opposed to math, you're not helping yourself with this detour of yours.

My point goes like this:

1) reading a historical mathematical/science document is not generally useful to learning mathematics today (obviously you're right about this but it's such an obvious banality that I don't credit you for making the observation), but it can be an interesting enrichment activity for your subject. But if you choose to undertake such, then you ought to read such texts in certain particular ways: as historical documents on the one hand (giving clues as to the author's circumstances in some cases, presenting ideas, many outdated, yet some still true), and as scientific documents on the other hand (the quality of the document as a scientific document, apart from its historical importance and character, is to be independently evaluated according to present understanding).

2) If you are reading such a text and complaining about annotation "breaking the spell", you're doing it wrong, because you're coming at the text from a point of view which is unconcerned both with the science involved, and with history. Basically you're trying to read such a dry text like light fiction. This is wrong.

Dude geometry lmao

>if you believe for example that Euclid's propositions are not testable
I don't care if they're testable or not
I care if they're consistent (which they are) - that's a key difference and one that Euclid himself would not have appreciated even if he was a platonist who didn't care much about testability either

>you're coming at the text from a point of view which is unconcerned both with the science involved, and with history
except that's wrong on both counts
you do care about the "science" but you have no illusions that it's a modern rigorous work
and you're reading it because of the history, you're concerned with how math developed, how it was influenced by what came before and how it influenced later mathematicians

then again I don't care that Euclid just assumes a point exists and that there is only a single such point because it's irrelevant to the history

>Obviously physics students are not generally expected to read the principia / math students are not generally expect to read Euclid right out of the box
>right out of the box
more like "at all" because again there is zero reason to do so
the only reason to read it is historical curiosity

your point basically boils down to that you think the work is worthless today because it contains errors in logic and that not correcting all of those makes reading it pointless
I disagree and say the work stands on its own merit as a historical document
nothing wrong with annotations if you prefer them but I like the work like Euclid intended

>(most famously the very first proof does not hold)
What are the fucking earth are you talking about? The first proof is constructing an equilateral triangle, trust me that holds.

What famous untenable axiom you're referring to is, is Postulate 5, that two lines which are crossed by a transverse and make the two interior angles together less than two right angles, must intersect each other if produced.

This has been the thorn in the side of Euclidean geometry, because some aspect of geometric understanding is needed to be understood correctly to identify parallel lines, and this axiom has a decent amount of things wrong with it.

You forgot about the other reason people may want to learn geometry: logic. The level of reasoning behind the propositions is so intense and rigorous, that it provides a thorough groundwork on how to syntactically construct arguments. The reductio ad absurdum, the methodology of Euclid's Book X Prop. I or XII prop. 2, (for instance, are both used rigorously by Archimedes), superposition, the very concept of working back from the ending propositions and deriving the former (the cyclical nature of the propositions themselves), these are all wonderful reasons to read early geometric works.

You need not even have an interest in mathematics.

First I want us all to recognize what you are implying by saying mathematics is 'testable', it is very different than, say, chemistry. The theorem is not a theory, it is proven using mathematics. Ergo, when you say something is 'testable', you really mean the entirety of the foundation of the system itself, or most likely a very important aspect of it.

Euclid doesn't assume a point exists, he says it is something with no part. The absence of something proves it, an important difference for the argument you're making.

Everything else is fine, and I agree. The thirteenth book is obviously my favorite because it mixes everything read in the previous books together.

why are you commenting when you don't know what you're talking about
>The first proof is constructing an equilateral triangle
true and it does not hold
there is no justification that the circles intersect
there is no justification that the circles intersect in one and only one point
there is no justification that the triangle is a plane figure

the proof is good by ancient standards but is not considered adequate today

the fifth postulate is perfectly consistent and independent of the other four even if it's unintuitive and historically controversial
letting go of it yields non-Euclidean geometry as developed by Gauss and Riemann

>You forgot about the other reason people may want to learn geometry
Euclid is not a good source to learn geometry because his work is sloppy by modern standards
that's the point here, read Euclid if you want to read Euclid not if you want to learn math

>Euclid doesn't assume a point exists
I wasn't referring to points in general but like in the example above (the first proof) where he assumes the circles meet without justification - Euclid makes many such errors

and I don't get what you mean by "testable" in a mathematical context? are you just meaning proving that the axioms are consistent

>The theorem is not a theory, it is proven using mathematics. Ergo, when you say something is 'testable', you really mean the entirety of the foundation of the system itself, or most likely a very important aspect of it.

say more.

how does the "testability" of a theorem relate to the "testability" of a hypothesis in chemistry? why would you call the latter "testable" but not the former?--(and obviously this is more than just to implicate the role of experiment--i mean what is it about experiment itself that matters?)

First
>there is no justification that the circles intersect
What justification is necessary?? You are drawing two circles with a protractor, this is a pretty fucking basic QEF. You place the protractor on one end of the line and spin, and you place it on the other end and spin. The circles, by their very definition, have got to intersect exactly twice.
>there is no justification that the circles intersect in one and only one point
It's two. Two points.
>there is no justification that the triangle is a plane figure
What do you mean by this?
>the fifth postulate is perfectly consistent and independent of the other four even if it's unintuitive and historically controversial
The first postulate is a construction postulate related to drawing straight lines, so I'd be tickled if you could show me anything in Euclid which doesn't depend on it. But you're right, it's just the debate has been raging in academia for centuries now concerning it and it's possibility of being formalized in a computerized system. Charles Dodgson (Alice In Wonderland fame) and Nathaniel Miller of Stanford Publications are just two people in Academia who have brought up concerns about this in the last two centuries.

And to your last point, that's exactly what I'm trying to figure out. Someone mentioned 'testing' these mathematical systems, which doesn't make sense considering things are proven under a system of mathematical logic, recursively, so they are continually tested and proven even further by propositions after and before it.

The whole latter part of this reply is a mess. The second reply-section makes some blend of misunderstanding what I had written earlier, and/or just going "nuh uh" and supporting same in a confused way that doesn't say much.

Also the third reply-section is simply false in that there are college courses to read these two texts. You also contradict yourself, I know that you want to dismiss the reason as not being a real reason, but the point is that you mentioned a reason of some kind right after saying that none exists.

Your conclusion is also so confused here that in between contradicting yourself some more and missing the point of what I'd written, you still manage somehow to /make an agreeable statement/, that "the work" (sticking with the Elements, now) stands on its own merit as a historical document.

I'm not sure what he means by saying math statements are testable I asked him above

but I can shed some light on math in general

in science a statement is proven by being tested against the real world and the models are made to fit the experiments

in math a statement is proven by logically deriving it from axioms and axioms are "proven" by proving they're consistent
this in a sense has nothing to do with the real world and indeed you can have multiple contradictory axiomatic systems where certain things are true within one system and false within another
for example the statement that parallel lines never intersect is true within Euclidean geometry but false within Differential geometry (of course when this happens you generally try to treat one as a special case of the other if it's feasible and in this case Euclidean geometry is the special case)

Yes precisely, because propositions are proven. You can't just have a mathematical problem or theorem not be proven within the framework or it wouldn't exist. Theories are theories because they haven't been proven, the very antithetical of what a theorem is.

>I'd be tickled if you could show me anything in Euclid which doesn't depend on it
the first 10 proofs or something of Elements is a good start
I'm not sure what the rest of your point is here though

>What justification is necessary??
saying you see it from drawing it is not a proof
Euclid treats it as such but there is no justification for it
is it an axiom that two circles with the same radius line meet? no, therefore that step needs to be justified
this was pointed out error even in Euclid's time
>It's two. Two points.
yea whatever point still stands
>What do you mean by this?
that the figure necessarily lies in a plane
Euclid mentions this specifically in the beginning of book 11 but that's after using it multiple times

>You also contradict yourself, I know that you want to dismiss the reason as not being a real reason
If it was not obvious from context I can spoonfeed you: there is no reason from the perspective of learning math
this should have been clear considering the reason is the next sentence after claiming there isn't a reason and since it was in reply to a reference to math or physics students

and you may call it confused or "nuh-uh"-ing but there wasn't much substance to reply to other than the two points here , the first of which is incomprehensible and the second one some vague statement about being concerned with the history and science boiling down to the completely unsubstantiated conclusion that you should not read the book without annotations

>the first 10 proofs or something of Elements is a good start
first 28* proofs of Elements do not require the Parallel postulate

Proposition one of Euclid's Elements requires you to draw three finite straight lines which require Postulate one. The circles require postulate three.

I was partially agreeing with you, the fifth postulate is a fairly good way of proving Proposition I. 27. It's just that most mathematicians have a problem with formalizing Euclidean geometry these days.

>saying you see it from drawing it is not a proof
That's because Proposition one isn't a QED, it's a QEF, big difference.
>is it an axiom that two circles with the same radius line meet?
No but it's evident, so unnecessary.
>that the figure necessarily lies in a plane
I haven't heard this argument before. Is there a reason specifically this causes problems?

Sorry you're right, 29 is the prop that utilizes post. 5. 27 seems kind of redundant and even unnecessary viewed under that light.

There is a lot of confusion in this thread, I suppose I shouldn't be surprised.

I am a mathematician (just about) at a top uni and I covered most of classical geometry via Olympiads in middle school or high school. I have never read through the elements (except for the initial postulates), but I had to prove many of the results in competitions or just working through problem books. There is a great beauty in classical geometry but you don't need to go read the original texts to access those ideas.
Math is about problem solving. I remember working through this book:
amazon.com/Challenging-Problems-Geometry-Dover-Mathematics/dp/0486691543
When I was 13ish. I still have it, it is bent, stained and covered with notes, so it evokes that first brush with beautiful mathematics and rigour.
Euclid was important historically for giving the first real formal system, so in that spirit the elements are worth reading, but not to learn geometry. That would be like learning calculus from the principia of newton.

>big difference
not really
it's not a recipe but a proof that the recipe works
>it's evident, so unnecessary
that's not logic
Euclid goes into this (intersecting circles) in book 3 in more detail but even there he falls short
>Is there a reason specifically this causes problems?
yes because you need to postulate or justify that everything ends in a plane
he does this at the start of book 11

there is also the third problem I didn't mention which is that it's never justified that AC and BC (pic related) don't intersect at E before C and for the non-equilateral triangle ABE instead - to fix this you need to postulate that straight lines are either the same or have no common segments or something equivalent to that
this may seem like nitpicking and in a sense it is but that's the job of math, making 100% logically sound arguments with no possible holes

I competed in the IMO for my small European country a few years back and obviously trained for the geometry

I don't see much of your supposed confusion in this thread

it's worth noting though that unlike Principia which is notoriously difficult reading even for trained mathematicians Elements is a very clear and intuitive book about math and if not for its slight errors would be a perfectly adequate textbook and it was used for that purpose for like 2000 years

Which year? Might have been the same.

2014 Cape Town

I was done with Olympiads by then. Did you get a medal? I got a Silver the one year I went. Ironically it was a geometry problem which cost me a gold.
Anyway, by confusion I meant in regard to the nature of rigour and mathematical proof. Rigour comes in degrees, working mathematicians don't write every proof out in set theoretic form.

math.cornell.edu/~mathclub/Media/thurston-proof-and-progress.pdf

A good article on the subject.

There is a huge difference between Quod Erat Demonstradums and Quod Erat Faciendums.

As far as Greek geometry goes, for instance, the last two propositions of Apollonius' book one would make literally no sense at all if they were QEDs. The QEDs presuppose a conditions and then propose something else as necessitated by the condition. If you wanted to construct something, you need to have certain properties necessitated by previous propositions proven by propositions before that one. Which is why every mathematical work starts and ends with QEFs and have the QEDs sandwiched in the middle. Hence why you see a larger proportion of QEFs in Books one and thirteen than anywhere else.

>>it's evident, so unnecessary
that's not logic
But it is. Because if you delved into geometry and proved or even mentioned literally every single causation of every intersection of every line meaninglessly, the book would never end. It would be recursive infinitely. So in other words, the fact that two circles drawn with a given distance being constructed using Post. 3 from different ends of the finite line coincide exactly twice is hardly necessary to mention.
>yes because you need to postulate or justify that everything ends in a plane
he does this at the start of book 11
The only reason he mentions planes in the propositions and definitions at the beginning of book 11 is because you're dealing with three dimensional space. Besides, you're incorrect. Euclid mentions planar geometry in the definitions for Book one. Definitions 7 and 8 define literally everything drawn in the entirety of The Elements as existing in a plane. It's almost like you're asking me to justify or prove definition 23 before it's used in Proposition I. 27, which is ridiculous.

Believe it or not, the methodology behind Postulate one solves your problem, essentially this postulate understands the construction of a straight line to be with a straight edge.

got an honorable mention for the combinatorics problem and a handful of other points here and there but no medal

and you're correct about how math is done in practice but the point is often that there's no reason to reinvent the wheel
Euclid on the other hand is explicitly setting out to inventing the wheel so absolute rigor is mandatory, especially in the first proof

there is a difference between qef and qed but not in the sense that you can assume things as given
you can construct something and prove it meets the definition and call it a qef
often the proof is implicit in construction but not here

>It would be recursive infinitely
no it wouldn't
as soon as you show (or postulate) that two straight lines meet at most at a single point and that the two circles intersect in precisely two points (which I think he does or almost does in book 3) the proof is rigorous

>Definitions 7 and 8 define literally everything drawn in the entirety of The Elements as existing in a plane.
no, if they did there wouldn't be a problem
definition 7 defines a plane as a possible surface which implies not everything happens on a plane
this is a bit nitpicky though I'll give you that since it's implied we're only working on a plane here and that there is no other surface
this implicitly means that everything after definition 7 is in 3d space and needs to be shown to be on the special case of the plane if that's the case

The point being, you're saying that Euclid should have defined the figures and shapes as being in a plane before book XI, but book XI is the first book working in three dimensions. It's no coincidence that's when he decided to start to define what plane things are in. Otherwise, again, it's evident. See my argument regarding definition 23. Please.

You're connecting both ends of the line to a single point. Very few have a problem with the first proposition of Euclid. You're just saying they do because it's the first prop. The problems are typically the fifth postulate and the methodology concerning II. 4.

>you're saying that Euclid should have defined the figures and shapes as being in a plane before book XI
I think I'm rather saying that definitions 7 and 8 should have been postponed until then, or that it should have been otherwise clear that we're only working on a plane

>Very few have a problem with the first proposition of Euclid
but that's not true, the proof suffers from these three holes
it's not justified that the circles intersect at 2 points
it's not justified that the lines meet at only those two points
and it's not justified that the resulting figure is a plane figure

the first two were pointed out in Euclid's time, were valid criticisms then and are valid today - the result simply does not logically derive from the 23 definitions, 5 postulates and the common notions

>The problems are typically the fifth postulate and the methodology concerning II. 4.
the errors here are pretty typical of Euclid
he's implicitly using properties of lines and circles he never justifies but takes as given without postulating or proving them
these errors can be corrected and indeed have been by many over the centuries
it gets better as the work goes on though because he starts relying on theorems proven before without repeating the same errors

fact is though Euclid has too few postulates for what he intends to prove and there is no way to make Elements completely rigorous without adding more postulates

>I think I'm rather saying that definitions 7 and 8 should have been postponed until then, or that it should have been otherwise clear that we're only working on a plane
That's not what you said initially, no. That's not a tremendously valid argument either, what justification do you have that any of these planar lines and angles are, you know, not in the plane of reference???

>it's not justified that the circles intersect at 2 points
Justified by the third postulate and the beginning of the construction. It is self evident, do what is required and you see that they just intersect at two points only.
>it's not justified that the lines meet at only those two points
What do you mean by this? The distance, or radius, for the two circles is given. The shape of the circle is given through post. 3
>and it's not justified that the resulting figure is a plane figure
With plane angles and plane lines as defined in def. 7 and 8? Why not?

does reading math books get you better at math?

>Justified by the third postulate and the beginning of the construction
in fact it isn't
there is a valid (ie. consistent) model of geometry in which the circles do not meet
this hole is impossible to fix without a new postulate
>What do you mean by this?
see pic here without proof or a postulate it's not justified that two lines only meet at a single point
>Why not?
he does not mention anything about this lying in a plane where he should have
just because you define a plane doesn't mean this automatically fits the definition
again this is taken care of (rather shoddily) in book 11 and again this is a bit of a nitpick since he should have just had everything until book 11 be in 2d
>what justification do you have that any of these planar lines and angles are, you know, not in the plane of reference?
I'm not the one making a proof

>It is self evident
this is never the answer
least of all here where it's actually false since there is a consistent system in which it's false (ie. the circles do not intersect) without breaking Euclid's postulates by adding a postulate different from the one Euclid tacitly uses
there is no argument to be made that this is a fine assumption

I think I'm done here though
you keep repeating the same nonsense about things being obvious or self-evident or otherwise fine without much interest in the actual real math

Yes, but you need to be reading modern math books, and not the history of math, which is what's under discussion in this thread (for the most part).

The other user is generally wrong and confused in the detailed points that we've been discussing in this thread (multiple other anons have rightly taken issue with the things he's been writing), but he is correct that reading historical math books is generally a terrible way to learn math. If only he could have distinguished between math and the history of math (as he ignored at one point, in order to make his point, the next moment conveniently remembering it) Oh well.

Anyway, back to you. Depending on where you are, I hear that khan academy is a valid solution to learn things, and pick up ideas for yourself. A starting point.

Reading any mathematical text of any kind tends to be a slow process. If you are not asking your own questions, trying things out for yourself, actually doing the exercises on your own, etc, then you're doing it wrong. You can't just read, you have to write, think, sketch and draw as well.

please stop talking nonsense about my posts just because you can't accept Euclid being inaccurate

You don't get to make that complaint and be taken seriously when you yourself have confused the issue at every turn, except for the only half-way right insistence on the lack of rigor in Euclid's first proof, which misses the larger point about Euclid that he built a pretty-good system that is properly mathematical, /and that for general, everyday purposes of discussing properties of shape, it is not absolutely essential to exhaustively treat of every detail/. But I know how your brain works by now, so you'll just read the above, miss the point that Euclid's overall program is sound (and this despite the fact that /you yourself/ admitted to appreciating the latter stuff because it builds on the earlier stuff, how does your brain work if you contradict yourself all the time?), and accuse me of having abdicated reason, or somesuch, because you are fixated on one deficiency of an ancient mathematical text.

I'm the one debating him, this guy is an idiot. He doesn't seem to understand any of the points I bring up, and if you look ITT he just simply says 'ur wrong' without any justification. Any of the postulates/propositions he is talking about have been raised by me, and he even started saying that one of his arguments was using the definitions I brought up.

He denies academically relevant problems with Euclid and focuses on proposition one, which I've never heard of anyone having a problem with... he is a retard.

>half-way right insistence
it is entirely right
>he built a pretty-good system that is properly mathematical
I never denied or argued the opposite of this and never contradicted myself on this topic
>and accuse me of having abdicated reason
you? I have no idea which poster you are of the multiple I have argued with in this thread
the one I said something of the sort to was the one that denied the holes in Euclid's first proof which is simply an indefensible position

Nice try bud, the jig is up. you lose. Get out of my thread or stop being stupid.

remarkable we woke up at the same time

>he just simply says 'ur wrong' without any justification
all my points are properly justified if you actually read the posts
>he even started saying that one of his arguments was using the definitions I brought up
what
>He denies academically relevant problems with Euclid
I never denied any of those
I said there is nothing wrong with the parallel postulate since it's consistent and independent of the other four which it is
it was proven in the late 19th century that there is nothing fundamentally wrong or different about it compared to the other four (ie. its independence was proven but I'm starting to think you don't know what that means)
>focuses on proposition one
it was one example I brought up which you for some reason sperged out about and derailed the argument by denying for 15 posts
>which I've never heard of anyone having a problem with
it's not a failing on my part that you haven't actually studied the subject matter, the first proposition simply does not follow from the postulates and common notions
you can look this up as you like this is not some controversial statement
themathpage.com/abooki/propI-1.htm
aleph0.clarku.edu/~djoyce/elements/bookI/propI1.html

I'm not the stupid one

I would spend time debating this page you've linked, but I've already listed the main arguments.

>What needs to be shown (or assumed as a postulate) is that two infinitely extended straight lines can meet in at most one point.
Funny enough, this is the one point you didn't mention. Perhaps we could try a little exercise to improve our logical systems. Why is this not valid, if it is justified through a combination of definition 10 and Post. 5. Is it really required to prove that right angles exist and that the triangle's base has less than two right angles? Why? It's already defined. We should assume that these sorts of things exist, which is why Euclid never writes a proposition that reads 'two angles in a triangle are less than two right angles'.

If you actually read my posts i explicitly mentioned it here fourth paragraph
Seeing as you didnt read, understand or accept it there i didnt bother to repeat it

And you cant just define things and assume it exists ftom there
First you define it and then you prove its properties

>And you cant just define things and assume it exists ftom there
But that's how definitions work. You couldn't have termed Proposition one without definition 20.

Thats exactly how they dont work
On the contrary proposition 1 can be seen as a proof that equilateral triangles exist, without which you couldnt assume their existence in the rest of the book

Seriously though, to add on, how in the fuck can you possibly say the two lines don't meet if the three sided figure has to have three different connected lines? How are you saying any of the triangle's two lines don't meet in one point if that's been proven in Definitions 19 and 20?

Definitions do work, and they help define things. How in fuck are you going to work with a logical system if you don't define anything before you construct it? Theoretically before you constructed the triangle in proposition one, you should have been able to tell me if it was equilateral or not if I showed you a triangle. Which brings us back to the huge difference of QEFs and QEDs, the point being Prop. one is simply to construct an equilateral triangle, not to define it.

I will admit this argument is a bit recursive, but if what is to be drawn is a circle, you need to know what the circle is before you design the tool to draw it.

>How are you saying any of the triangle's two lines don't meet in one point if that's been proven in Definitions 19 and 20?
nothing was proven in the definitions
>Definitions do work, and they help define things.
true
>How in fuck are you going to work with a logical system if you don't define anything before you construct it?
you define things first that's true

But take this defintion (working in Euclidean geometry):
>A tribangle is a triangle whose angles add up to 270°
This is a perfectly valid definition, the only problem is that the construct doesn't exist, it doesn't work with the postulates
there is nothing inherent about definitions that make them work

This is why constructing a line and circle are postulated (taken as axioms, postulates 2 and 3 respectively) because just defining what they does not mean they necessarily exist or that they're constructible in the system
Note that equilateral triangles are nowhere postulated to exist so the first propositions proves that they do by showing a method that works using only the postulates (in theory but as discussed before he also uses another implicit postulate)
So he constructs the image and then argues (proves) that it fulfills the definition of an equilateral triangle, before he does it can not be taken as a given that such a construct exists

>Theoretically before you constructed the triangle in proposition one, you should have been able to tell me if it was equilateral or not if I showed you a triangle.
Yes I could have. And an example of an equilateral triangle is proof that one exists. But it has to be made using only the 5 postulates (circle and a straight edge) and that's exactly what Euclid does.