"A straight-line is (any) one which lies evenly with points on itself."
This is how Euclid defines a straight line but I don't know what it really means. Is this saying that any point picked on the straight line will have equal distance from each other?
What does he mean exactly by lies evenly?
In modern standards those definitions don't make any sense other than just an intuitive picture of what is intended. And actually even in this case it's not very clear at first what Euclid refers to with that definition of "straight line" if you don't already know what a straight line is.
Today, when you want to study something, you start with objects, which are said to be undefined. They are, well, just things that satisfy certain axioms. In the case of Geometry, you may consider straight lines to be undefined or maybe you can say that the undefined objects are points and that lines are set of points which satisfy certain axioms, which depend on the book you're reading.
Also, as already pointed in the the comments, Euclid's Elements is a book with lots of "holes" that need to be filled by the specialist. In that case you can say that Euclid's elements is more like a sketch. Knowing this, the easy way to read it is just going through it informally by trusting your intuition and using the postulates, just like he did. Along the way you can takes notes about the facts that bugs you the most and try to guess what is missing.
Once you understand what Euclid's Elements is about, then you can skim very quickly how David Hilbert filled those holes of Euclid's in his "Foundations of Geometry" or if you like you can Greenberg's book "Euclidean and no-Euclidean Geometry".
I'm going to come to Euclid's defense.
The modern day tenet that we only know things based on more fundamental definitions and axioms and at the very bottom we have to accept things axioms and definitions by fiat, which we all take for granted, is a relatively new idea and would have been anathema to classical philosophers who would have believed that at the bottom there is some universal idealistic truth.
Euclid's definition of a point "that which has no part" is often given as a prime example pointless of an evasion and meaningless definition that should have just been by fiat. "A point is a point" it is our basic abstract. But I think the intuitive understanding of Euclid's world is that we are talking about space, shapes existing is space and the measurements of the distance in space. Yes, at the core we must accept that space and distance are the abstract concepts we have to accept without definitions. But a "point is that which has no part" conveys that assumption space is built of fundamental immeasurable indivisible units of which all things in space are composed. Those are the points.
So "a line is a length that has no breadth". That means it's a connected entity of points that has no thickness but measurable or infinite length; i.e. a curve. We do need a fundamental idea of "thickness" and of "distance". A curve never varies in thickness. It's width is always zero with no variation.
A "straight" line "lies evenly with itself". Well, I take that to mean if you look at it sideways it has no warps and bumps. That's me being casual. But if we accept a fundamental idea of measure and variation, a straight line is one with no vertical variation in comparison to its horizontal variation. That's not well defined-- it relies on concepts intuited rather than defined and if we attempted to define them it is circular-- but it is meaningful and significant. The basic idea of a straight line, one that fundamentally built into linear algebra and analytic geometry, is that a line has direction if oriented horizontally has no vertical displacement. If it's aligned "a kilter" any vertical displacement is proportional to the horizontal displacement. i.e. it is straight.
Yes, formally we need abstractions. A point is an abstract idea. A line is a collection of points of which any two have an abstractly defined real number value called distance and a line is points so that a point has only two points (in opposite directions) for each distance. A straightline is that in which the distance between two points are minimal. Abstract and divorced from preconceptions; I get that.
But as for what they "mean", I think Euclid did a good job on getting them to stand on their own with minimum circularity, ambiguity and the least (but still some) basic assumptions.
Within the Euclidean system of geometry that is articulated in Book 1 of the Elements, the three fundamental symmetries for objects in two dimensions are reflection, translation and rotation. The idea that a straight line lies evenly between two points can be understood in terms of reflection. If a line A were to be transformed by reflection along an axis running along the two points to yield line B, then the line A would be straight if the reflection to B yielded a line that, when placed on top of A, every point on B lies evenly on A. If a line C had any curvature or deviation from what is straight, then the reflection would yield a line D that does not lie evenly on line C.
Note that this explanation of what it is for a line to be straight rests on a fundamental kind of symmetry--and so it conforms to modern notions of the kinds of group relations that are taken to be fundamental across different kinds of geometries (see Cayley and Klein). As such, Euclid's attempt to dig down to fundamental assumptions does seem to yield remarkable insights.
Euclidean and Non-Euclidean GeometriesbyMarvin Jay Greenberg. Indeed what we know about lines are from Hilbert's axioms. Look up in that book. – Mikasa Aug 19 '16 at 17:25