My question for Cecil is simple. If Pythagoras is supposed to have formulated Pi how on earth did our ancient craftsmen make spoked chariot wheels? I am a master craftsman and as far as I’m concerned all Pythagoras did was attach his name to what was once a closely guarded secret. I can delineate a circle into an infinite number of parts using just a straight edge and compasses, but that takes good geometry skills and I still need Pi to make my rims concentric. What do you think Cecil?
I don’t see why knowing the value of pi would be an important factor in making a wheel. You only need to know the basic geometric tricks for dividing a line into two equal parts, and for drawing a perpendicular to that line. As you said, that can be done with a simple compass and straightedge. The basic techniques allow you to build a wheel with spokes in multiples of four or six, which pretty well covers your needs.
Pi will give you the ratio between the radius (or diameter) and the circumference, but did you really need to know that in order to construct a wheel? If you sketched a full-size plan on paper (or parchment, or in the dirt) with a compass and straightedge, you should be able to transfer your plan into a wheel without knowing (or caring about) the ratio of the parts.
I’m also willing to bet that, even after Pi had been “discovered”, there were a lot of uneducated (but trained) craftsmen who were building wheels but who could not do the arithmetic to link diameters with circumferences. But, so long as the wheel was round and didn’t collapse, who really cared?
BTW: the circumference of a circle can be divided into any N equal segments using a ruled straightedge and compass. Sorry I can’t give an online cite- anyone?
I wouldn’t have thought you’d even need to use a compass construction to make a chariot wheel, let alone pi. You could just arrange your spokes (six of them, say) by eye, corssing in the middle (using a length of rope doubled over to find the middle) and then fashion the rim of the wheel between them. It might not be perfectly round but it would be close enough - a few extra bumps wouldn’t make a lot of difference over rough ground, and you could soon plane off any obvious lumps.
Or -
Attach a rope to a stake in the ground. X distance out, attach another pointy stick to the rope, and walk in a circle around the stake, dragging the pointy stick on the ground. Now you have a template for a round wheel. No formal math needed.
Knowing pi would be of limited value, if any, since the length of the spokes is shorter than the radius of the wheel circumference. There’s a hub, and there are fixing holes in both the hub and the wheel. I would venture it was easier to design and make the wheel from common sense coupled with trial and error than to precisely measure the circumference, the distance from the center point to the inboard end of the spoke, and the distance from the wheel edge to the outboard end of the spoke.
Sorry but I didn't explain the experiment properly.
What we have to do is design and 'Tool-Up' a 1285bc workshop to supply Rameses II with 5000 chariots to attack the Hittites with.
The Wheels are to be 4ft dia x 2" with (optional) a copper tyre. no iron!
the number of spokes can be either;
7- hunting
8- racing
or 5 which is Persian.
The felloes are to be as long as possible in layers of 3 so as our wheel is loaded under tension not compression.
The point is they all have to be standardised and run true & concentric.
There are numerous ways to do this, and I've had no problems with the exercise. what I was trying to say was that during the Tooling process Pi became self evident to the Toolmaker. With the necessary tools I only needed to employ semi-skilled craftsman and two apprentices. I achieved minimum costs. I fail to see why this wasn't the case in 1285bc. Besides Doric pillars have 22 flutes so odd numbers were not uncomon.
Next month I'm doing the same thing with a six cornered Cloister Block. A very curiously wrought stone & critical to the structure.
I must say however I thought the same as you at the start but after putting it into practice, I think the ancients were a lot smarter than History gives credit for.Well, I have no doubt that some math-abled ancients were able to come up with some kind of close approximation of Pi without actually defining it as Pythagoras did. A string and a ruler would have been enough to tell them (especially with a large exemplar to work from) that the circumference of a wheel was 3.14 times the diameter. Whether or not they ever figured out (or cared) that this value had any significance in calculating other things like the area of a circle or the volume of a cylinder is up for speculation, but if they found something like the circumference- diameter ratio was useful, I’m sure they could have figured it out.
However, if I were setting up your “Wheels R Us” factory for Rameses, I (as the master wheelwright) would probably build a model, give it to the craftsmen, and say “Make me 9,999 more just like that.” Carpenters’ shops the world over are full of jigs, models and patterns that were used in just that kind of way to make consistent, uniform copies of a design, without the need to calculate component spacings, find centres, etc.
In coming up with my model, would I use math to calculate the length of the felloes, or would I wrap a length of thin, wet wood around the spoke-ends and say to myself “I’ll make this just a skootch longer to keep the spokes in tension” based on my years of wheel-building expertise? Could be either way, but in my experience a craftsman who has done this kind of thing for years will know by rote what is required, and won’t make the formal calculations.
Finding out that the ration between a diameter and circumference is constant is not necessarily such a great discovery.
All you have to do is to measure the diameter and circumference of two wheels, and that’s it.
(from nova)
What Archimedes did, on the other hand was quite a feat. He deduced from first priciples a formula for calculating that ratio, without doing any hands-on measurement.
Not sure how true this would be in truly ancient times, but more modern wheelwrights would certainly have used a set of patterns - nothing much more than a template for making the hub, another for a spoke and another for a section of the rim - making the patterns would require a fair bit of skill, trial and error, perhaps even calculation (but not necessarily) - making wheels from the patterns is still a skilful job, but is much more a case of endless repetition.