Should I Become a Mathematician?

[mathwonk]

Euclid's elements is the best geometry book. The law of cosines is Props. II.12 and II.13,

if you understand them. There is a beautiful edition of Euclid from Green Lion press in paperback at about $15, as well as free ones online.

I will post some remarks about reading Euclid here in a minute or two.

 

[mathwonk]

Introduction to Euclid:
Philosophy: Euclid does geometry without using real numbers. He uses finite line segments instead of numbers, so he wants to be able to compare them, i.e. to say when they are equal, or whether one is shorter than the other, without assigning a numerical length to them. To do this he uses the concept of a straight line, and the principle of betweenness for points on a line. These concepts are not made quite precise in Euclid, but we can see some of their properties in his language.

About Euclid’s definitions:
Euclid attempts to define all concepts, but without complete success it seems. Indeed some of these ideas were not made clear for centuries after him, but he does make important progress. In particular he tries to distinguish objects of different dimensions, and gives some hint of the modern way of doing this. In definition 1, he calls a “point” something with “no part”, which is an attempt to define a zero dimensional object. We prefer now simply to say we are given certain fundamental objects called points of which all other objects of study will be composed. We don’t define the points, we just say they are given and we give some of their properties.

In definition 2, Euclid defines a “line”, [we would call it a “curve”, allowing it to possibly be straight], as something with only length but no breadth, an attempt to say it has only one dimension. This is not a precise definition, but in definition 3 he says that the extremities of a line are points. This gives a clue to the modern inductive description of dimension. Namely we have some way to recognize the border of an object, and an object should be one dimensional if its border is zero dimensional, i.e. if the border consists of a finite number of points.

The same pattern occurs in definitions 5 and 6, where in 5 a surface is something two dimensional, and in 6, we see that the border of something two dimensional should be one dimensional. This is a general pattern, that a border should have dimension one less than the thing it borders.

Today we focus more on the relationship between our objects than on the nature of those objects. So in different situations, what are called points or lines or surfaces could be different things, but in all situations the points will be related to the lines in the same way. I.e. whatever the points are, they should border the curves, and whatever the curves are they should border the surfaces, etc…

So today mathematicians tend to ignore Euclid’s definition 1, and to consider definitions 2 and 5 to be clarified by definitions 3 and 6. Unfortunately definition 4 of what it means for a curve to be straight, is not clarified by any additional property, and we will need one in Prop. I.4. The usual one taken nowadays as basic for straight lines is that two different lines which are both straight, can only meet in one point. This is related to Euclid’s 1st Postulate, that one can draw a straight line between any two points, but only if that means one and only one straight line, so this is the usual modern postulate. So to guarantee that two different lines can only meet once, we need more or less the converse of Euclid’s 1st postulate. I don’t know the original Greek, so I do not know if the words “a straight line” used in that postulate mean “exactly one straight line”.

Terminology that Euclid used differently from mathematicians today
Euclid seems to mean by “straight line” only a finite portion of an infinite straight line. Today we call such finite pieces of lines, line segments, or finite line segments. When Euclid wants to speak of an infinite straight line, he speaks of a (finite) straight line being extended indefinitely or calls it explicitly an infinite straight line. So what he calls a line today we call a curve, what he calls a straight line today we call a line segment, and what he calls a line segment extended infinitely in both directions, or an infinite straight line, we just call a line.

Definition 8 describes an angle as the “inclination” made by two straight line segments which meet but are not in the same straight line. It is not clear to me whether they meet at an extremity, but apparently in that case he considers only the convex angle they make together. E.g. the outside of a 90 degree angle is not considered by him as a 270 degree angle. (Since he does not consider 270 degree angles, it is harder for him to “add” two 135 degree angles.) He defines a right angle as one of the angles formed by two lines that form equal angles. Presumably in this case the lines do not meet only at extremities, since they form more than one convex angle.

Definition 15 describes a circle, but again not quite completely. He says a circle consists of a point called the center, together with a collection of line segments all having that center as an extremity, and all having the same length. But he does not say whether all segments of that length are included, as presumably they should be. E.g. a semicircle seems to satisfy the description given, since all line segments from the center to any point of the semi circle are equal to one another.

We assume he meant that a circle is the figure formed by a center and a segment with that center as extremity, plus all other segments having the same center as an extremity, and which are equal to the first segment. Thus he includes the entire region on and within the circle, whereas today we mean by “circle” only what he calls the circumference or boundary of his circle. I.e. we take a center point A and a segment XY, and we consider the circle to consist of all those points B such that the segment connecting B to the given center A, is equal to the segment XY. It follows that two circles with the same center have either the same circumference, i.e. are the same circle, or else their circumferences are disjoint, i.e. have no common points at all. He is not quite consistent since later he says a circle cannot cut another circle at more than 2 points, apparently referring to their circumferences.

Euclid’s five postulates:
Here are the postulates Euclid explicitly stated (slightly paraphrased):
1. Given any two points, one can draw a straight line (segment) joining them.
2. Given a finite line segment, one can extend it continuously in a straight line, (presumably infinitely in both directions).
3. Given any point as center, and any other point (distance), one can describe a circle centered at the first point and with the other point on the circumference.
4. All right angles are equal.
5. If two lines cross a third line so as to make interior angles on one side total less than a straight angle (two right angles), then the two lines meet on that same side of the third line.

Note Euclid clearly assumes in postulate 5 that a line has two sides. Also there is nothing here asserting that parallel lines exist - rather this has the opposite flavor, guaranteeing that certain lines are not parallel. So this is not the usual parallel postulate I learned in high school. (Through a point off a line, there passes one and only one line parallel to the given line.)

This postulate will imply there is not more than one line parallel to a given line and containing a given point off that line. In the other direction, Euclid will actually prove there is at least one such parallel, using his “exterior angle” theorem.

The properties that Euclid used most without stating them concern how lines and circles meet each other. In modern mathematics we discuss these in terms of connectivity or separation properties. A set is convex if for every pair of points in the set, the straight line segment joining them is also in the set. E.g. a straight line segment is convex. Then Euclid seems to assume basic facts like the following: removal of a point other than an extremity separates a segment into two convex pieces. Removal of an infinite line from the plane separates the plane into two convex “sides”. Removal of the circumference of a circle from the plane separates the plane into two pieces, one of which: the inside, is convex, and the other: the outside is at least “connected” [in what sense?].

What do we mean by “separates”? We mean the segment joining a point inside to a point outside should meet the border which was removed. So if two points of the plane are on opposite sides of a line, then the segment joining them should meet the line. Thus the line separates the two ides of the plane, and forms the border of both sides. If one point is inside and another point is outside a circle, the segment joining them should meet the circle. We can say something about the shape of a circle if we agree that any (infinite) line containing a point inside a circle should meet the circle exactly twice. And we might be wise to agree that a circle that contains a point inside and a point outside another circle also meets that circle exactly twice.

Some of these facts about how circles and lines meet can be proved, and Euclid does so, but others cannot be proved. In general one can prove that circles and lines cannot meet more than expected, but I do not know how to prove that they do meet as often as expected, without more assumptions than Euclid has made. Today many people assume that lines correspond to real numbers, which does guarantee that lines and circles meet as often as expected, since the axioms for real numbers guarantee this. However most geometry books which make these assumptions about lines do not bother to explain the relevant axioms for real numbers, so to me not much clarity is gained, and perhaps some is lost.

Euclid has one postulate (#5) guaranteeing that two lines do meet under certain conditions, but he was criticized for centuries for including this postulate. It turns out he was right, as this postulate cannot be omitted without broadening the possible geometric worlds he was trying to describe. People were unable to imagine any other geometry than Euclid’s however for a long time where this postulate could fail. A Jesuit priest, Girolamo Saccheri, showed that if we deny Euclid’s 5th postulate then there would not exist any rectangles. This and other consequences seemed so impossible to Saccheri that he concluded Euclid’s axiom must be automatically true, and thus did not need to be stated explicitly. However, there is another plane geometry in which there are no rectangles, called hyperbolic geometry, and unless we assume Euclid’s 5th postulate we cannot be sure we are not in that world instead. Today the results Saccheri correctly deduced , but considered impossible, are regarded as theorems in hyperbolic geometry due to him.

So we regard Euclid’s stated definitions and postulates, plus the ones he used but did not state, as rules for the game we are going to play. They tell us what we can do, and we want to deduce as many consequences from them as possible, without violating the rules.

The problem of congruence
If two triangles have vertices A,B,C and X,Y,Z, a correspondence between their vertices, e.g. A→Y, B→X, C→Z, induces correspondences between the sides: AB→YX, AC→YZ, BC→XZ, and the angles: <ABC→<YXZ, <ACB→<YZX, <BAC→<XYZ.
If a correspondence between the vertices induces correspondences of sides and angles such that every side and every angle equals the one it corresponds to, we call the correspondence a “congruence”.

Notice a congruence must be given by a specific correspondence. It is not sufficient just to say two triangles are congruent, one must say what correspondence induces the congruence. E.g. the triangles ABC and XYZ may be congruent under the correspondence A→Y, B→X, C→Z, but not under the congruence A→X, B→Y, C→Z. Other very symmetrical triangles may be congruent under more than one correspondence, but we should always say what correspondence we mean.

Exercise: Given an example of two triangles that are congruent by more than one correspondence.

The first question we ask in plane geometry is when two triangles are congruent, given only a smaller amount of information. The basic criteria are sometimes called SAS, SSS, ASA, and AAS. E.g. “SSS” is shorthand for the fact that if a correspondence of vertices induces a correspondence of sides such that all corresponding pairs of sides are all equal, then all corresponding pairs of angles are also equal, and hence the triangles are congruent. “SAS” refers to the fact that if two corresponding pairs of sides are equal, as well as the pairs of included angles, then the triangles are congruent. Etc…

Once one knows these basic criteria, most geometry courses proceed in the same way, at least for while. Getting started thus means establishing these basic congruence facts. Some books assume them all, while some assume only a few of them and deduce the others. Euclid proves them all, but only by making some implicit assumptions that he has not included among his axioms. See if you can spot some of those assumptions.

 

[mathwonk]

nano - passion, one of the best introductions to topology is First concepts of topology, by Chinn and Steenrod.

here is one for under $5 from the excellent used book site abebooks:

First Concepts of Topology: The Geometry of Mapping of Segments, Curves, Circles and Disks
Chinn, W.G.; Steenrod, N.E.
Bookseller: Meadowlark Books
(Hawley, MN, U.S.A.)

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[dkotschessaa]

Thanks for this guide to Euclid. I will likely be taking a look at this as soon as my break starts.

 

[Nano-Passion]

nano - passion, one of the best introductions to topology is First concepts of topology, by Chinn and Steenrod.

here is one for under $5 from the excellent used book site abebooks:

First Concepts of Topology: The Geometry of Mapping of Segments, Curves, Circles and Disks
Chinn, W.G.; Steenrod, N.E.
Bookseller: Meadowlark Books
(Hawley, MN, U.S.A.)

Bookseller Rating:
Quantity Available: 1
Book Description: Random House, NY, 1966. Soft cover. Book Condition: Very Good Minus. 1st Edition. 8vo - over 7¾ - 9¾" tall. VG-. Text has a couple underlines in intro; name on ffep; page edges, white covers, have slight fading. Bookseller Inventory # 001453

Bookseller & Payment Information | More Books from this Seller | Ask Bookseller a Question


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I'm confused, I didn't ask for topology I'm only up to calculus. Or should I get this book because its a good introduction to geometry, calc, etc.?

 

[mathwonk]

when you wrote:

"But... whenever I go on the forum I hear many many alien mathematical terms that I have never heard before like topology, etc. "

it seemed to me you were asking to learn about topology.

The book on topology I recommended to you is for the average high school educated adult, with no knowledge of calculus. Topology is more elementary than calculus.

Topology is the study of continuity, whereas calculus adds the concept of differentiability.

 

[Vector Field]

Sorry, to bring up something old, but I read this on the first page. I believe you have given some bad advice. I don't know if it was addressed later and you mentioned someone should ask an Applied Mathematician. That would sort of be me.

You advised someone to stay away from majoring in Mathematics/Economics. This was not entirely good advice. Governments all over the place employ armies of mathematicians to study these things. If a person majors in Applied Mathematics they will also be expected to learn the Pure Math as well, Applied Math doesn't mean you sit around learning mechanical problem solving. You need to analytically solve the problems too.

In looking up a lot of jobs data I have noticed that Applied Mathematicians earn 10,000USD more, on average, than Pure Mathematicians. However, in researching this further it is recommended that you choose a field that best interests you. If the person was quite interested in Economics, then they should certainly major in that. Applied Math without a second discipline is harder to find employment with. It is certainly not impossible, but if there is a particular field you love and find interest with, then you are more likely to find employment in that field as a mathematician if you also do some work in that discipline.

 

[dkotschessaa]

Wow, Mathwonk. You blew my mind. I had assumed (since topology is generally offered some time after differential equations and linear algebra) that topology was necessarily an advanced topic. I am sitting in the math section of my library right now and wasn't 20 feet away from the book you mentioned. And by gawd, no calculus.

I seem to have a weakness in math when it comes to the the graphing side of things. The "thinking geometrically" part of my brain is not developed yet. (My current professor thinks very geometrically so I had a hard time following his thinking at times.) I was going to spend my time off going over basic conics etc., again. But do you think this might be an alternative to strengthen that sort of understanding?

-DaveK

 

[mathwonk]

dear vector field. you are attributing to me it seems, advice given by someone else on page one.

i often give bad advice just not that piece.

 

[mathwonk]

DaveK, well, if you don't know geometry, you might also benefit from studying euclid. i just had a long post erased by this finicky browser where i argued that euclid is the best preparation for calculus.

 

[dkotschessaa]

That was something I was thinking of doing, but then you got me all excited about topology. I should probably stay focused. Topology can wait for me.

-DaveK

 

[Vector Field]

dear vector field. you are attributing to me it seems, advice given by someone else on page one.

i often give bad advice just not that piece.

Oh, it wasn't entirely you. It was an exchange made with another poster. You recommended talking to an Applied Mathematician.

 

[Nano-Passion]

when you wrote:

"But... whenever I go on the forum I hear many many alien mathematical terms that I have never heard before like topology, etc. "

it seemed to me you were asking to learn about topology.

The book on topology I recommended to you is for the average high school educated adult, with no knowledge of calculus. Topology is more elementary than calculus.

Topology is the study of continuity, whereas calculus adds the concept of differentiability.

Wow, I'm surprised. Thank you. Any other book you would recommend? I want to get a general handle around mathematics and spike up my interest.

 

[mathwonk]

it is better, grasshopper, to read one good book than to carry around a long list of unread good books. but one more is: what is mathematics? by courant and robbins.

(i think this thread is recycling itself.)

 

[TeckMod]

Hello, this is my first message and I really don't know if this is the correct place to post this but I just want to see if my thought process is correct. I taught myself how to read and understand proofs but since I never got any feedback now I don't know if I'm in the right path when doing them. I'm working through Spivak's Calculus and got to the first exercise but I'm doubting my thinking. Here's the problem and my proof:

Proof that:
a/b = ac/bc , if b, c don't equal 0.

Proof:
Let b,c not equal 0. So the fraction a/b is a rational number. Let that number be z.

thus: a/b = z
(a/b)c= z(c)
(a/b) ([c][c⁻1) = z(c/c)
(a/b) ( 1 ) = z(c/c)

Now since we let z = a/b, we have that
a/b = ac/bc


I'm taking that the above is correct, but I have no idea, please correct my if I'm wrong. Now, my doubt arises at the beginning, since I didn't have a property ( or I didn't remember one, in the book.) that told me that a = a, I did the first step that way. Can I take "shortcuts" when doing a proof by using other truths other than the initial ones? I'm just talking when doing proofs on my own and for myself only; or is this recommended not to do since it could create bad habits or things like that?

Another thing is that Spivak, in his book (first chapter) does proofs almost "algebraically" and while I can follow and understand him, I'm more comfortable doing proofs in another "format", is this just his style or doing things this way is useful? Also, am I OK in proving something using a proof by contradiction while it's implied that a direct proof is expected in a text?

In the end, this question is just a 'Can I do things that I think are reasonable and true but that are not part of a textbook?'. Stupid, I know, but this is the first time that I'm doing proofs and textbooks aren't very good at being interactive.

 

[mathwonk]

the word "proof" is a misnomer. you can't really prove anything, the correct term should be "deduce". I.e. you start from some rules, and you deduce admissible results. so before you "prove" anything you have to know what you are allowed to assume.

so to prove that ac/bc = a/b, you also need a definition of equals for rational numbers.

lets assume that means that cross products are equal.

so you want to prove that ac(b) = a(bc), which follows from commutativity. or maybe you want to prove that acb-abc = 0, and since acb-abc = a(cb-bc) = (by commutativity) a(bc-bc) = a(0) =0, we are done.

 

[TeckMod]

Oh I see, so we are allowed to assume certain things and then we show how what we want to deduce follows from our assumptions. Your post cleared things out, thanks.

 

[mathwonk]

yes you understand completely. proving is a game. but first you have to agree on the rules. the more you allow, the easier the proof is. like poker is hard, but if you play deuces wild it is easier to get a good hand.

 

[tauon]

how about mordell's conjecture that if the smooth compact complex surface obtained by smoothing out the zero locus defined by a polynomial with integer coefficients is a doughnut with more than one hole, then there are only a finite number of rational roots?

Or that in the set of all prime numbers, the density of the subsets of those ending in 1,3,7,9 are all equal?

or that a prime > 2 is a sum of two squares iff it has form 4K=1?

or that all primes are sums of at most 4 squares?

and i like euclid a lot too. did you know he described tangents to circles as essentially limits of secants? Prop III.16.

thanks a lot for these examples.

after looking into them, from here to there, I eventually found that maybe analytic number theory or something similar is more "suitable" for me. an idiot as I was I didn't think of approaching number theory as from other domains; that would've helped a lot and that's what I'm going to do now. :]

 

[Dougggggg]

http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf

Thought you guys might find this a good read, I enjoyed it.

 

[benjaminxx12]

Hi, New member to this board.
I find mathematics both fascinating and beautiful, and i want to get a PhD in the subject within a decade.
My question is, what does getting a PhD consist of? Do you need to develop your own formula? Derive an unanswered equation? what does it consist of i guess is my main question. Thanks in advance!

I am also planning on going to the U of Alberta for my studies. In Jr. High i really wasn't Diligent with my studies and screwed myself over for any chance at one of the more prestigious schools such as Princeton and Stanford, Is it possible to get into these school for graduate studies after doing well In Undergrad studies at a Decent university?

 

[Number Nine]

Hi, New member to this board.
I find mathematics both fascinating and beautiful, and i want to get a PhD in the subject within a decade.
My question is, what does getting a PhD consist of? Do you need to develop your own formula? Derive an unanswered equation? what does it consist of i guess is my main question. Thanks in advance!

I am also planning on going to the U of Alberta for my studies. In Jr. High i really wasn't Diligent with my studies and screwed myself over for any chance at one of the more prestigious schools such as Princeton and Stanford, Is it possible to get into these school for graduate studies after doing well In Undergrad studies at a Decent university?

A PhD requires an original contribution to human knowledge, which means discovering something about mathematics that no one has discovered before. Math at that level is not necessarily "formulas and equations", and may not involve numbers or calculations at all. If you haven't started at University yet (I'm not sure where you are in your studies), you should know that proof based mathematics ("real math") is very different from anything you've ever encountered before. That said, it's also vastly more interesting.

 

[mathwonk]

In my thesis I looked at an interesting mapping between two 12 dimensional spaces,
both parametrizing geometric objects. A point of the source space corresponded to a pair of complex curves (Riemann surfaces) and a 2:1 map between them. A point of the target space corresponds to a complex compact group. the map takes the pair of curves to the quotient group of the associated pair of Jacobian varieties associated to the curves via the map between them.

The problem was to determine the degree of this map. To do that normally one finds a general point of the target space and just counts the number of pre images. But it is hard to find "general" points in practice, special ones are so much easier to find. But the special point I understood well had an infinite number of preimages.

Indeed the preimage had three connected components, one was a point, one was a curve and one was a surface. By inserting more points into the source space I was able to replace it with a new "blown up" space and map in which my point now had only finite number of points.

By generalizing the implicit function theorem I was able to show that my point and its preimages had now become general enough to just count them to get the degree.

this map had been around since the 1890's before anyone learned its degree. In working on this problem I got a big thrill out of learning to visualize spaces of high dimension. I also got a lot of help from my advisor and my brilliant friends.

 

[Dougggggg]

One of my professors talks frequently about when he had finished the actual mathematical work of his PhD thesis and had solved it. He says something to the extent, "at that moment, I realized I knew something about Mathematics, well the world, that nobody else had ever figured out. It was a great feeling considering the age of the subject dates back before Christ (granted his area of study is a bit younger, Graph Theory)."

I want that moment...

 

[mathwonk]

Here is an example of the technique by B. Segre. the space of cubic surfaces in P^3 and the space of pairs (S,L) where S is a cubic surface and L is a line on S, both have dimension 19. the map (S.L)-->S has degree equal to the number of lines on a general cubic surface, which is in general finite. A reducible cubic made of a plane and a quadric has an infinite number of lines but if we add the data of 6 points on the conic where the plane meets the quadric we enlarge the space of cubic surfaces, adding some new points, but with the space still having dimension 19.

Now the preimage of the triple (S,L,p1,p2,...p6) is the set of lines that lie on the surface S = Q+L and also contain one of the 6 points. There are 12 such lines on the doubly ruled quadric and 15 on the plane, making degree = 27, which is well known to be the right answer. In his book on complex projective varieties Mumford shows this in the traditional way, by proving that every smooth cubic surface is a general enough point of the target, and then counting the 27 lines on the special smooth fermat surface.

 

[Robert1986]

Here's a question for those who are professional mathematicians. When you got your Ph.D. you, of course, wrote a dissertation. I have some questions about this:

1) How and when did you decide that you were going to write your thesis on the topic you picked? Did your advisor help? And if so, how did you pick your advisor?

2) Have you (and ask the same questions about your mathematician friends) stayed in this same general area of research, or have you done something completely different?The reason I ask is that I am graduating in the Fall and I plan (*if* I get in) to go to grad school to get a Ph.D. but one thing I am worried about is finding something to research.

 

[mathwonk]

i had three different advisors, who suggested several different problems. the first couple of problems were solved before me by stronger mathematicians.

the next two I solved, but it turned out they had already ben solved by others, unknown to me and my advisors.

the last one was hard for me but my advisor helped me get the idea. Then one day I heard a famous mathematician was working on it too, but I just tried harder, as hard as I could and I solved it first, by providing a new idea of my own.

I guess the moral of my story is to try to pick a supportive advisor, try to think of a problem that interests you. Best is if you find it yourself, perhaps no one else will be competing with you on it.

And don't give up.

 

[mathwonk]

By the way, these 2 weeks I am a teacher at a camp for brilliant children, called epsilon camp, in Colorado Springs. There are 28 kids aged 8-10 here taking 5 classes a day, and in mine we are going through the first 4 books of Euclid. That is a good chunk of the course I taught to college and grad students in 2009, over a whole semester.

These kids are amazing and I am having a blast. We have already done Euclid's original proof of Pythagoras, and learned to construct a regular pentagon. Tomorrow we will discuss how to do algebra geometrically, and I will try to present a new way to do similar triangles without numbers that I figured out just for this camp.

If you know kids this age, or older, these camps, epsilon camp, and math path, are great for very gifted kids aged 8-18 or so. Look them up on the web. My course notes are there in the student forum, but I guess you cannot access that. Maybe I will ask them to post them publicly, or I will just put them on my web page at UGA later. Of course if you are older and more advanced, and interested, the book by Robin Hartshorne, Geometry: Euclid and beyond, is much better. I got my start trying to emulate his course, and I learned from his book.

 

[qspeechc]

Hi mathwonk, can you recommend any Euclidean (classical?) geometry books that go beyond the stuff we learn in school? I.e. books that assume Euclids Elements and go further? I did note the Hartshorne book you recommend, any others? There is no specific thing I am interested in, I'd just like to know more about classical geometry.

Also, when are you going to start writting and publishing books?

 

[mathwonk]

I don't know what you learned in school, but I recommend starting with Euclid. I myself got a lot more from it than I got in school. A good place to begin is with Hartshorne, as he will refer to Euclid.

Another historical source used by Hartshorne is David Hilbert's Foundations of Geometry.

A nice little paperback that assumes Euclidean geometry and mentions some less well known results is Geometry Revisited, by Coxeter and Greitzer.

But Euclid is the best read for me, then Hartshorne. I recommend reading chapter one of Hartshorne, then Euclid, then continue with Hartshorne as the spirit moves you.

 

[mathwonk]

you will find a few books I have written on my web page at UGA, but they are not published.

 

[Nano-Passion]

you will find a few books I have written on my web page at UGA, but they are not published.

What are your books about? I want to read them if you can supply a link.

 

[qspeechc]

I don't know what you learned in school, but I recommend starting with Euclid. I myself got a lot more from it than I got in school. A good place to begin is with Hartshorne, as he will refer to Euclid.

Another historical source used by Hartshorne is David Hilbert's Foundations of Geometry.

A nice little paperback that assumes Euclidean geometry and mentions some less well known results is Geometry Revisited, by Coxeter and Greitzer.

But Euclid is the best read for me, then Hartshorne. I recommend reading chapter one of Hartshorne, then Euclid, then continue with Hartshorne as the spirit moves you.

I have read Edwin Moise's books Geometry and Elementary Geometry from an Advanced Viewpoint (but when I was an undergraduate).
Thanks for the suggestions.

 

[mathwonk]

moise's second book you mention is one i have seen. i found it more formal and less enjoyable than euclid but it is mathematically excellent. hartshorne (and also euclid) contains much more than moise and should be a lot more fun. but if you mastered moise you know a lot. i still recommend euclid and hartshorne. i think you'll be surprised just how much richer the subject seemed before the modern formalists got hold of it.

 

[qspeechc]

I don't doubt that. The 3-volume edition translated by Heath totals 1502 pages!

https://www.amazon.com/dp/0486600882/?tag=pfamazon01-20
https://www.amazon.com/dp/0486600890/?tag=pfamazon01-20
https://www.amazon.com/dp/0486600904/?tag=pfamazon01-20


Then there is the one volume edition, but edited by some Dana Densmore, at only 527 pages. I don't like the idea of some editor intruding on Sir Heath's work.

https://www.amazon.com/dp/1888009195/?tag=pfamazon01-20