Sorry guys, but I'm gonna dump some probably unwanted advice. First thing I gotta say is I love listening to you guys and I'm so glad that you are doing this. The things I am about to discuss are not meant to bring you down, but hopefully to help improve some stuff for the future. And maybe a few nitpicks (sorry). I might come off as a little harsh, but it's meant with love. I will point out that these are just my opinions and suggestions, don't feel too pressured to take the advice. This is your podcast, you should do it the way you feel is best.
You guys have an issue with history. I think your presentation of it is a little confusing, some of the history you discuss might not have been researched the best, and you leave out an unholy amount of information. All of these issues kind of tie in together, but I'll address them one at a time.
First there were some details that I really liked. I was surprised at the discussion of the battle of Uhud, that was a very good detail and I think was described well enough (as far as I know about it). I think different sources will give different weapons for the one that wounded Muhammad, but I think the sling interpretation is the best and the one I would go with (I might be biased). Another thing that surprised me was Cook's notes on slings, and how it was clear it was an object that he had encountered before. This was new to me, and I thought it was amazing that you actually found the reference in the primary source.
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Now to the not so great. First is the presentation. I like that you do it in chronological order, but the way you present the topic is jumpy. You guys jump centuries and thousands of miles with each mention. It doesn't really create a great sense of continuity or how widespread it is. I understand this was an introductory and covered the whole world, but I think anything you do in the future should better connect each instance you are discussing. My suggestion for future historical episodes would be to break it up by region. There are multiple ways you could do it, but Europe, Asia (Would largely be focused on West Asia. East Asia and South Asia have sources but might not be enough to have their own episode) Africa, North America, South America, and Oceania. Again, you could organize them differently. I'll get to it in a paragraph or two, but there is enough information out there to do that. The last thing I'll mention for this is that I wouldn't separate conflict, hunting, etc. in any historical episodes. It's all connected.
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My second issue is kind of nitpicky, but sometimes it feels you didn't research the best. For example, I think Kick talking about Najera said it was between the English and the French. Not quite. I think whenever you are discussing a historical group using it, make sure to know who it was that was using it. I'm sure you have notes, make sure that stuff is in it. You also said that you could find nothing about there being slingers at Najera except for in the picture. There are actually a few references.
(Scroll to bottom of the page, this was a source I found from a thread here on slinging.org)
http://books.google.com/books?id=tSBEAAAAcAAJ&pg=PA372&lpg=PA372&dq=froissart+sl...(I found this one through google, take some of it with a grain of salt since the guy is not a slinger)
http://warfare.gq/WRG/Middle_Ages_1-89-Castillian_Slinger.htmThe point being that there is a lot more information out there. If you can't find any or even if you think there is nothing there, it's worth asking the community or seeing if it has already been discussed.
This is pretty much irrelevant, but I think the name Vegetius in Classical Latin would be Weh-geh-tee-oos (I don't speak latin, someone correct me if I'm wrong). Not sure when the v sound switched in Vulgar Latin, but it might have been around the 5th century when he lived. I don't know how we should say it in English, maybe just Veh-geh-tee-us (the g is still a hard g). It probably doesn't matter.
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The third issue is probably the biggest, and that is missed information. Although there is no way of knowing that you are even missing something, it might be a good idea to ask the community. As mentioned earlier, I know this is more of an overview, but you missed so much of the available information out there. While you said the earliest evidence of slingers was from Catalhoyuk around 7,000 BCE, there are stones from the Monte Verde site in Chile and biconicals from the West Coast of North America that might date to around 11,000 BCE. The comment on slings being around for as long as humans have been is probably not correct (we have no evidence and the sling was not known in most of Australia), but we can probably guess the sling was probably around when humans crossed into the Americas. This was all info that was floating around on this site or could be asked about. Even info more recently discussed on slinging.org was overlooked. The chalk biconical sling stones from Northern France are one such detail, that would be a cool thing to note.
I think what you are doing wrong is sticking only to the more well known aspects of slinging in slinging circles, besides a couple of the cases I mentioned at the beginning. You definitely need to talk about David and Goliath, the siege of Lachish reliefs, Greek and Roman lead sling bullets, Incan slingers, etc. Those are all very important topics, and I'm not saying in any way to skip over them. I am advising to look for information on the less investigated topics, which is why I suggested earlier to tackle history by region, so you don't have to hit only the most notable topics.
History doesn't even stop at just discussing it's use by region. You could probably do an entire episode on the surviving historical sling designs, like the Peruvian sling (2500 BCE), Lovelock cave sling (1200 BCE), Cortaillod sling (circa 1000 BCE), Egyptian slings (circa 1900 BCE - 800 BCE), etc. and all the ones that have survived through continued use and ethnographic collection. You could probably do a whole episode on historical sling bullets, perhaps two. The history of the sling might seem sparse, but when you take into account the whole world and a lot of searching, there is actually an absurd amount of content to cover.
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Again, sorry if this critique was too much. I am hoping to be constructive. That brings me to the final thing I want to say. This is pretty much the biggest piece of advice, and what I already mentioned. Use the community! If you are ever wondering if there is more information is out there, ask everyone else. There's of course slinging.org, but we also have discord and stuff like that. There's lots of places to ask. I know I would be willing to help you find whatever you need, and I'm pretty sure that many others would love to help and have information they could contribute. Despite all my criticism, I still learned something new from this podcast. That's why the community is important. You're just 2 people, and this is a massive and spread out topic. With help from the community, there is so much cool stuff you could dig up and include. All you have to do is ask.