Final answer:
The scratching method of establishing an arc is not similar to erasing with a pencil eraser. While erasing involves back-and-forth motion to remove marks, drawing an ellipse with the string, pins, and pencil method is a continuous motion keeping the length constant, demonstrating the consistent geometry of an ellipse.
Step-by-step explanation:
The statement that the scratching method of establishing an arc is similar to the motions used to erase a mistake with a pencil eraser is false. When discussing the method of drawing an ellipse using string, pins, and pencil, it is important to understand the geometric principles involved. This method, commonly referred to as the gardener's method, relies on the properties of an ellipse, where the sum of the distances from any point on the ellipse to the two foci is constant.
To draw an ellipse, you place two pins in a board to represent the foci and loop a string around them. You then use a pencil to pull the string taut and trace an arc while keeping the string stretched. The reason this method works is that the length of the two sides of the triangle formed by the pencil and the pins remains constant - the length of the string - as you draw. This is quite different from erasing, where you apply friction back and forth to remove pencil marks. The movement in drawing an ellipse is smooth and continuous, while erasing is more abrupt and reverse-directional.
The motion of erasing entails pushing and pulling the eraser across the paper to physically remove the graphite, whereas the gardener's method requires a consistent, steady pull to form a precise geometric shape. Moreover, the gardener's method results in a smooth curve, whereas erasing can cause the paper to become rough or even tear if done vigorously. It should also be noted that erasing does not follow any geometrical principals whereas the gardener's method is a practical application of the constant sum of the distances from any point along the ellipse which is a specific geometrical principle.