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.
T=2π√L/g
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.
Doug Clews says
Hi Mike
Good one for a Sunday afternoon in Oz …
Unfortunately, the information given is insufficient to give an accurate answer …
Reason, too many unknown factors …
We don’t know whether the height of 10 metres is to the top of the branch, or to the underside, so do we need to allow for branch thickness or not ,,, branch circumference at each tying off point is also needed, as well as the number of turns of rope round the branch when tying off … (Note: Your diagram indicates the branch slopes up, so rope length on the higher side will be greater) …
We do not know whether the rope goes through one hole in the seat and passes under the seat and up the other hole, or whether you are tying off under each hole ,,, if going under, we need the width of the seat, and in both cases the thickness of the seat … (Note: Going under is not recommended, as the seat will surely tip) …
We also need the height of the underside of the seat off the ground (is it granddaughter’s height or grandparents height for use at other times, so that you can enjoy your drinks without your knees getting dirty) …
Enjoy your drinks, and keep smiling … I am (On both counts)
Doug Clews
Chippy says
A good way to approximate a 45 degree angle is to bend over and look through your knees. Of course, this does depend on agility, but if you move to the position where you can just see the top of the tree (or the branch, in this case), the distance to tree is approximately the height.
Or you could use similar triangles, then you don’t need to worry about angles at all. Lie on the ground and position a stick so that it’s perspective (?) just covers the height of the tree. The proportion “distance from you to stick” to “distance from you to tree” will be same as “height of stick” to “height of tree”.
Mike Sedgwick says
I would only attempt those contortions somewhere where no one could see me. I am not sure I can do the knee manoeuvre.
I’ll stick to measuring the altitude of the cloud base. It’s ground temperature minus dew point (in Celsius) times 400 to give the height in feet. I am amazed that it is very accurate. There is also the advantage that there are not many people in a position to contradict you.
Doug Clews says
That formula would make for a very interesting swing Mike, whose height is going to change with a varying dew point … maybe your next formula needs to be for the design and positioning of a trampoline, so that, with granddaughter (and/or wife) in your arms you would be able to seat them with accuracy. (Just joking, as I am sure they would enjoy doing it themselves)
Doug (and STILL smiling) … have a good one