FOOTNOTE

simple example of table generated by sage command:

table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/12]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

+----------+------------+--------------+

| $i$ | $cos(2*i)$ | $rational$ |

+==========+============+==============+

| 0 | 1.000 | 1 |

+----------+------------+--------------+

| 1/12*pi | 0.8660 | 1/2*sqrt(3) |

+----------+------------+--------------+

| 1/6*pi | 0.5000 | 1/2 |

+----------+------------+--------------+

| 1/4*pi | 0.0000 | 0 |

+----------+------------+--------------+

| 1/3*pi | -0.5000 | -1/2 |

+----------+------------+--------------+

| 5/12*pi | -0.8660 | -1/2*sqrt(3) |

+----------+------------+--------------+

| 1/2*pi | -1.000 | -1 |

+----------+------------+--------------+

| 7/12*pi | -0.8660 | -1/2*sqrt(3) |

+----------+------------+--------------+

| 2/3*pi | -0.5000 | -1/2 |

+----------+------------+--------------+

| 3/4*pi | 0.0000 | 0 |

+----------+------------+--------------+

| 5/6*pi | 0.5000 | 1/2 |

+----------+------------+--------------+

| 11/12*pi | 0.8660 | 1/2*sqrt(3) |

+----------+------------+--------------+

| pi | 1.000 | 1 |

+----------+------------+--------------+

| 13/12*pi | 0.8660 | 1/2*sqrt(3) |

+----------+------------+--------------+

| 7/6*pi | 0.5000 | 1/2 |

+----------+------------+--------------+

| 5/4*pi | 0.0000 | 0 |

+----------+------------+--------------+

| 4/3*pi | -0.5000 | -1/2 |

+----------+------------+--------------+

| 17/12*pi | -0.8660 | -1/2*sqrt(3) |

+----------+------------+--------------+

| 3/2*pi | -1.000 | -1 |

+----------+------------+--------------+

| 19/12*pi | -0.8660 | -1/2*sqrt(3) |

+----------+------------+--------------+

| 5/3*pi | -0.5000 | -1/2 |

+----------+------------+--------------+

| 7/4*pi | 0.0000 | 0 |

+----------+------------+--------------+

| 11/6*pi | 0.5000 | 1/2 |

+----------+------------+--------------+

| 23/12*pi | 0.8660 | 1/2*sqrt(3) |

+----------+------------+--------------+

| 2*pi | 1.000 | 1 |

+----------+------------+--------------+

attached is the simple graph generated by the sage command:

t = var('t')

polar_plot(cos(2*t), t, 0, 2*pi)

If I am looking for the area inside two curves, it helps to plot both, and then find the intervals of integration.

Too time consuming? Computers help, but when I am calm, I like to draw the polar coordinates and plot them, coloring the areas of interest.

sage: p1 = polar_plot(cos(2*t), t, 0, 2*pi)

sage: p2 = polar_plot(sin(2*t), t, 0, 2*pi, linestyle="--")

sage: show(p1 + p2)

At the point where I begin the integration, this is what Schopenhauer is referring to when he says, "This necessity for space with its three dimensions to be translated into time with only one dimension, if we wish to have an abstract knowledge (i.e., a rational knowledge, and no mere intuition or perception) of space-relations--this necessity it is that makes mathematics so difficult."

While computing the integral there are lots of opportunities for arithmetic errors. I always check my results with a computer algebra system.

The last thing I want to think about when engrossed in such activity is "what will I be studying in the autumn or by the time winter returns or before I am 50."

While my shelves are filled with a lifetime's worth of material to explore, I can only focus on the task at hand. It does take some discipline to stick with the textbooks and not skip through tedious exercises.

Could you imagine a janitor locking himself in his supervisor's office to "play with graphs" on the computer?

I am the rebel slave who has been rewarded with freedom from my atrocious behavior.

FREEDOM!

PS: The following tables were more helpful. All I did was change the step size to pi/8. I did this after algebraically solving cos(2*t) = sin(2*t). t = pi/8

That's where the two curves intersect. Do you see how the algebraic and numeric are so important in communicating the geometric? I think, no, I know this is exactly the kind of thing Schopenhauer was referring to.

When you look at the graphs and then look at the numbers in the tables representing the positions of the curves, think of Schopenhauer's statements.

sage: table([(i,n(cos(2*i), digits=4), cos(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$cos(2*i)$", "$rational$"], frame=True)

+---------+------------+--------------+

| $i$ | $cos(2*i)$ | $rational$ |

+=========+============+==============+

| 0 | 1.000 | 1 |

+---------+------------+--------------+

| 1/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| 1/4*pi | 0.0000 | 0 |

+---------+------------+--------------+

| 3/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| 1/2*pi | -1.000 | -1 |

+---------+------------+--------------+

| 5/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| 3/4*pi | 0.0000 | 0 |

+---------+------------+--------------+

| 7/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| pi | 1.000 | 1 |

+---------+------------+--------------+

| 9/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| 5/4*pi | 0.0000 | 0 |

+---------+------------+--------------+

| 11/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| 3/2*pi | -1.000 | -1 |

+---------+------------+--------------+

| 13/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| 7/4*pi | 0.0000 | 0 |

+---------+------------+--------------+

| 15/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| 2*pi | 1.000 | 1 |

+---------+------------+--------------+

sage: table([(i,n(sin(2*i), digits=4), sin(2*i)) for i in [0..2*pi+0.1, step=pi/8]], header_row=["$i$", "$sin(2*i)$", "$rational$"], frame=True)

+---------+------------+--------------+

| $i$ | $sin(2*i)$ | $rational$ |

+=========+============+==============+

| 0 | 0.0000 | 0 |

+---------+------------+--------------+

| 1/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| 1/4*pi | 1.000 | 1 |

+---------+------------+--------------+

| 3/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| 1/2*pi | 0.0000 | 0 |

+---------+------------+--------------+

| 5/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| 3/4*pi | -1.000 | -1 |

+---------+------------+--------------+

| 7/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| pi | 0.0000 | 0 |

+---------+------------+--------------+

| 9/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| 5/4*pi | 1.000 | 1 |

+---------+------------+--------------+

| 11/8*pi | 0.7071 | 1/2*sqrt(2) |

+---------+------------+--------------+

| 3/2*pi | 0.0000 | 0 |

+---------+------------+--------------+

| 13/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| 7/4*pi | -1.000 | -1 |

+---------+------------+--------------+

| 15/8*pi | -0.7071 | -1/2*sqrt(2) |

+---------+------------+--------------+

| 2*pi | 0.0000 | 0 |

+---------+------------+--------------+

sage:

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