Intuition/Motivation - Distance in Euclidean Space In modern mathematics, euclidean distance is defined using the Pythagorean Theorem, that is, by a formula such as $\sqrt{(x_1 - y_1)^2 + \cdots + (x_n - y_n)^2}$. A priori there is not a reason why this should be the definition of distance, but this definition comes from how distance was used in classical geometry (Euclid's elements). 
Is there a way to attain distance from more primitive ideas/notitions in euclidean space. 
 A: Picture a polygon made of pythagorean triplets:
$$(3,4,5), (5,12,13), (13,84,85), (85,132,157)$$
Now picture each of these triplets being orthogonal to all other triplets because the plane of each extend into higher dimensions. The Pythagorean theorem works for any number of dimension $[>1]$ so:
$$3^2+4^4+12^2+84^2+132^2=157^2$$
or $$f=\sqrt{a^2+b^2+c^2+d^2+e^2}$$
If you draw a crude picture, you will see that every $2^{nd}$ and subsequent side-A [odd] is connected to side-C of the previous triangle. You will also see that every hypotenuse has one end at the origin.
A: I'll give two partial answers to your question, hope you'll find it useful :

*

*Looking through the history, first we have the great Euclid(even mentioned by some mathemticians : "the first modern mathematician"). In his great book, "Elements", he proved some basic theorems of geometry(which we call today euclidean geometry)
Pythagorean formula is one these great theorems, which is proved by axioms in Elements.
After Descarts introduced Cartesian plane in 17th century, using Pythagorean formula, he measured the distance between two points.
Being more exact : he accepted the euclidean geometry and by using his theorems, found the distance. Actually, Pythagorean formula and euclidean distance, are equivalent. So, historic intuition is a possible answer.

*After Maurice Frechet and Felix Hausdorff introduced the co\ncept of "Metric spaces", having a metric space $(X,d)$ you may difine a metric on the set $X×X$ by product metric. If we use $X= \mathbb{R}^1$ and absolute norm, for product space which is $\mathbb{R} × \mathbb{R}$ we will get euclidean norm.(But in fact this is all after cartesian plane and euclidean distance. Its better to say: euclidean distance was an intuition for Felix  Hausdorff to get the product metric, but today, if we forget the history, by looking through the defintions of metric space and product metric, we get an abstract intuition of euclidean distance in $\mathbb{R}^2$ and higher dimensions)

