In one of the previous chapters, a thought experiment was introduced. It involved walking along a straight line, yet it wasn’t specified what a straight line actually is. It’s about time to rectify that.
Straight segments and lines
Every route in space links together at least two locations. Let’s take a closer look at possible paths between two random points: P0 and P1 in empty space. There is an unlimited number of routes we could take. If we connect P0 and P1 with a long piece of string and move along it, every route covers the same distance, yet the number of possibilities is still unlimited. One route stands out though. If we shorten the string to the absolute minimum, the route marked out is unique – it covers the shortest distance possible. We will call routes like this straight segments. The length of a straight segment between any points A and B will be coded with a shorthand |AB|.
Note that we can extend every segment to infinity on both ends. To do that, we choose a new point P2 in such a way that a straight segment between P0 and P2 coincides with the point P1. Then we find P-1 such that the segment P-1P2 coincides with P0 etc.
We can continue this process to infinity on both ends. An infinitely long straight segment in space will be called a straight line.
At the first glance it all looks fine, but is it? If you have a globe in your house, grab it now. You can pick two points anywhere and stretch a string between them to find the shortest distance along the surface. If you choose the opposite poles, the shortest route is not unique. Every route going along the meridian has the same length and it is not possible to choose the shorter one. Another interesting feature is that meridians are not infinitely long. In other words, when travelling along a “straight” line (meridian in this case), after a while you end up where you started.
Unlike in the thought experiment above, in reality we are not restricted to the Earth’s surface, and we can go above and below, so the globe situation is not applicable. We can’t be certain though that after traveling long enough in any direction along a straight line, we wouldn’t eventually get back where we started.
Demonstrating with a string that there is only one shortest route between two points looks convincing, yet there is no guarantee it’s true everywhere. To solve this conundrum, we will simply claim the following two postulates to be always true:
Postulate 1: There is only one shortest route between two points is empty space.
Postulate 2: Every straight segment can be extended to infinity on both ends.
At this point you may wonder whether we can simply postulate something to be true without knowing it for sure? The problem is that it is not possible to know for sure. The best we can do is to accept a handful of postulates that agree with observations and do not contradict each other. After that, using rules of logic we can predict consequences of our postulates and check whether they also agree with reality. If so, our model is considered to be accurate enough for now.
By this logic, the flat Earth model is valid as long as distances taken into account are short i.e. a tiny spot on the surface of a globe is pretty much flat. The longer distances we consider, the more obvious it is that something is not quite right with this model.
Coordinates
Let’s choose two points in space: P0, P1 and pass through them a straight line. We will also agree that P0 is the origin, the distance from P0 to P1 is equal to some unit length U, and positive direction is running from P0 towards P1. If we have some arbitrary point X anywhere on that line, we can pinpoint its location exactly using a single number x that tells us how far in terms of the unit U and in which direction (positive/negative) our point X is with respect to the origin P0. This can be expressed as:
X=P0+x∙U
where x is called the coordinate of the point X in the system defined by P0 and P1.
Summary
In this short chapter we have defined what a straight line is, and shown that the location of a point on every straight line can be described using a single number.
Is there a way to pinpoint locations on planes with numbers too? This is the subject of the next chapter.