Now, for contrast, let's look at the Indo-European words for five that I mentioned.
First of all, the words all have the exact same meaning, so we don't have to worry about that. Secondly, we already have a good idea that some of the words within families are related (five and fünf, pãch and panj—even some of the others look somewhat similar).
But once you have the idea that they may be connected, you have to dig into the history and prove how. Lots of those words start with a p (and sometimes end with p), but the Germanic words have f. A comparison of Germanic and other Indo-European words shows that there's a sound correspondence: father and pater, fish and pisces, for and por, and so on. So we know that when other languages have p, Germanic has f.
But then there's the Romance words beginning with an s or ch sound. However, we know from records that the Latin was quinque. It seems that the original p must have become a qu from the influence of the second qu and probably the influence of the preceeding number, quattuor. (Actually, the change from p to qu or k isn't unheard of, and neither is the opposite, qu to p, but I'll get to that in a minute.)
So at some point the Latin quinque must have lost the first u, becoming *kinque. In Vulgate Latin, /ki/ became either /si/ or /chi/, so that explains the Romance languages today.
Now back to the problem of kw and p. The original Indo-European sound wasn't really like the qu sound today. It was more like a k pronounced with your lips close together, and in some languages, it changed to just being pronounced with the lips (p).
This happened in all the Brythonic languages (like Welsh) some dead Italic languages (cousins of Latin that are long dead), and in proto-Germanic only when kw was followed by e. So for those languages, the original word becomes pempe. Then the Germanic p > f rules applies and we get femf and so on.
In the Gaelic languages (the other half of the Celtic branch), kw became simply k, and all ps became k, too. This explains the Scottish cóig.
In some other languages, kw became t. I don't understand this sound change as well, but I believe it only occurred when followed by e, much like the Germanic change. So Greek and Russian get a t instead of a q at the end.
The Indo-Iranian changes are simple. Much like the Romance languages described earlier, the w drops out, and then ke becomes che. In some languages, ch then became j (this is simple voicing of the consonant).
A couple common changes that run through most of these words is the loss of the final vowel (nothing surprising—look at all the silent final vowels in English and French) and the disappearance of the n. Nasal consonants have a tendency to disappear over time. First the preceding vowel becomes nasalized (and the actual consonant disappears), and then the vowel denasalizes. Two of these examples (cinq and pãch) are currently in the nasalized vowel stage.
So that, in a nutshell, is what a decent etymological analysis looks like. See the difference? You start with a wide selection of words and then trace them back step by step until you arrive at the common ancestor. There's no picking and choosing words to support your hypothesis while ignoring others that don't. There are no huge leaps when it comes to semantic shifts or sound shifts. Even the weird changes involving kw have shown up in other language families, so there's corroborating evidence. Browsing through a dictionary for 30 seconds disproves that book's assertions, whereas this analysis is watertight.
