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I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part. Now, any geometrical figure viz. Line Segment,Triangle,Square etc. can be said to be composed of points. No matter how small we try to make a point, it still has some size/dimension. So,how can these infinitude of points add up to give the length/perimeter of the above mentioned figures,when according to Euclid,these have NO PART?

Señor Pérez
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    Points have no part, in (very) old-fashioned language, roughly speaking means that points have no non-trivial subset. (Here I am being very anachronistic.) Points have no part, but they can certainly be part of larger assemblages, such as lines. – André Nicolas Feb 04 '16 at 18:48
  • Thanks a lot for your precious time André. – Señor Pérez Feb 04 '16 at 18:59
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    Also see http://math.stackexchange.com/questions/1083841/if-point-is-zero-dimensional-how-can-it-form-a-finite-one-dimensional-line – David K Feb 04 '16 at 19:07
  • Thanks a lot for your precious time David. – Señor Pérez Feb 04 '16 at 19:20
  • Euclid did not actually use all of his axioms, and used assumptions that he never stated.For example, how do define "$c$ is between $ a$ and $b$ " when $a,b,c$ are co-linear points? Not so easily. In elliptic geometry you can't. See the chapter on Ordered Geometry in Introduction To Geometry by Coxeter. For all its flaws, the Elements is an amazing work.It has been conjectured that Euclid wrote a sequel on 3-D geometry, which is plausible, but there is no evidence for it. – DanielWainfleet Feb 04 '16 at 20:44

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No matter how small we try to make a point, it still has some size/dimension.

It sounds a bit like you are talking about drawing a point, but that is not what we're doing when we imagine a point in geometry. A point is an idealized, primitive notion. It does not have any physical size to speak of.

how can these infinitude of points add up to give the length/perimeter of the above mentioned figures, when according to Euclid,these have NO PART?

As Andre Nicholas mentioned in the comments, the idea of having "no part" speaks to the indivisibility or atomicness of a point. Lines and planes have many parts: in particular, their points are parts of them.

Why do you feel you can raise an objection? Is there some rule somewhere that says a collection of things without size can't have a size? It sounds a bit like you're thinking of these in terms of measure theory, where the measure of the whole can be the sum of measures of the parts (provided there are not too many parts.) But nobody has established any sort of measure in this discussion. Now, even if one established a conventional measure like length and area, the axioms only provide additivity for countable collections of sets, not uncountable collections like the set of points on a segment. There's just no concrete reason to be disturbed when an enormous collection of dimensionless things can be gathered into something with some dimension to it.

PS: Why spend a lot of time discussing matters in Euclid's terms? In this day and age, it's probably better to take a modern approach first, and then can one appreciate Euclid more fully and maybe not get so tripped up in archaic language.

rschwieb
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    I think you diagnosed my problem pretty well.I was studying some advanced stuff when this 'Too simple a question to be posed' stuck me. I referred to Book I of Euclid's Elements and read its definition. Hilariously, I was just too much preoccupied with the archaic language of Euclid. Actually, I think of Hellenistic mathematicians as pioneers of practical application.So,I thought they weren't the ones who did much abstraction. Thanks for your precious time Schweib. – Señor Pérez Feb 04 '16 at 19:10
  • @SeñorPérez No problem. See you around – rschwieb Feb 04 '16 at 19:13