In my head, and probably in yours too, there is a jumble of memories of ancient Greeks — old men, bald and with beards, who did stupid but essential things. There is one who overflowed his bath and ran naked down the street. One drank hemlock, and another lived in a barrel; one married his mother. They fought a lot, invented gods and wrote unreadable books. And they were good at geometry.
I resurrected one of these weirdos from my memory; they all look the same, dressed in chitons or togas and sandals. I first made the acquaintance of this one when I was 12 years old, Pythagoras by name. He died about 500 years BCE, but he was clever with triangles.
‘Pythagoras, please help me. I need a length of rope to hang from a high branch on a tree to make a swing. How long should it be? I can’t get up there to measure the height.’
‘You can’t work it out, old fellow, unless you can measure angles.’
‘Ah-ha! I have a smartphone that can do anything except receive a decent phone signal. It has an inclinometer, especially for nerds like me.’
I stood under the branch and measured 10 metres which took me to the gazebo, then I measured the angle up to the branch of the tree, 45 degrees, near enough. That means 10 metres of rope plus a bit to go around the branch.
I had enough rope in the garage, but I had to get it over the branch. Another character from ancient times emerged. In Florence, David’s statue is the only ancient one to depict a man who is thinking. How could he clobber Goliath without being smitten? A slingshot was his solution. I could do the same. With a weight tied onto a length of thin cord, I swung the weight and let it fly up and over the branch. Then I attached the rope to the cord and pulled it over, tied a loop in the rope and pulled tight – job done.
Or job not done, only beginning. Now I must spend hours pushing the swing with my granddaughter.
‘Faster, Grandpa, faster.’
It won’t go any faster. It takes about 6 seconds to go out and back. Gallileo knew that. Back in 1620, he was bored by the sermon in Pisa Cathedral and watched a swinging chandelier. He noted that each swing took the same time, whether it was a big swing or small. He did a few experiments back at home and found that the time for each swing depends on the square root of the rope’s length, not the arc of the swing and not the size of the child on the swing.
This is the mathematical gobbledegook. T= time for a complete swing, L is the length of the rope and g the force of gravity. If I need to explain π, then you’re lost. If they have a swing on Perseverance on Mars, it will go at half the speed because gravity on Mars is less than half of ours.
Now she cries, ‘Higher, Grandpa, higher,’ but other factors come in to play. The mathematicians expect me to try to explain how the period increases asymptotically at high amplitudes, but it is gardening that limits things. There is a Bay tree at one end of the swing arc and an Acer at the other.
When we are alone, my wife and I attach a double seat with cushions and swing the afternoon away with a bottle of cold beer on my side and a G & T on hers. The problem now is which book to read?
It is said that Pythagoras never wrote anything down; no original writings of his survive. How do we know about his hypothesis? Several contemporaries and later Greek philosophers included his teachings in their books. His teachings survived the loss of the libraries at Alexandria and Ephesus. Euclid collected his geometric teachings in his publications 200 years after Pythagoras’s death.
It is one thing for us to know that Pythagoras’s theorem applies to a right angles triangle with sides 3, 4, and 5, but does it apply to right-angled triangles of whatever size? That’s the trick of geometric proof.
Probably Pythagoras learned the theory from the Babylonians or even the Indians who knew of it 500 years before Pythagoras.