User:TNSE KARUNANIDHI ATHIPALY NLG/sandbox

Empirical devfinitions of dot, line, plane First we admit empirical procedures as valid. Thus, we do not need invisible points. We work with visible dots, and lines which one can see. We devine a dot ostensively, by pointing to it, the way we devne a dog, by pointing at it. At this stage we do not worry about accuracy (how large can a dot be?), which question will be taken up later on.By stretching the string between two dots, we obtain the straight line segment connecting the two dots. We can measure its length in standardised measurement units of centimeters (or inches) by measuring the length of the string with a measuring tape, or directly use the measuring tape which is exible. (We will later on provide a special school kit with a specialised measuring tape.) We do NOT declare a plane to be undened. A at piece of paper or a level surface is a plane. Figures drawn on such a surface are called plane figures. We can also draw these gures on the ground, but the ground must be levelled arst. How do we know the ground is level? As Aryabhat.a put it, we can test the level by means of water. Levelling the ground and determining the cardinal directions was the arst step towards making astronomical observations with a gnomon or shadow- stick (sanku). We don't really care whether these surfaces are exactly plane, some minute ups and downs may survive. We know the plane surface does NOT extend invnitely, for an exact and invnite plane is a fantasy of formal math which does not exist in reality. What we seek is a good approximation, good enough to accomplish the practical purpose at hand which may be to determine one's latitude from the gnomon. We may need to extend straight line segments, sometimes over considerable distance, but never need to or can extend them invnitely. What is a \considerable" distance? First of all it is limited by the sheet of paper or the levelled ground. Next, it is limited by the curved earth: if we extend the line beyond 10 km it can no longer be considered a straight line.