Everyone seems to be taking this article as gospel, but that is not the intent at all.
It is an exploration into the idea of basing typographical proportions on the golden ratio; it is NOT saying that the results are the only (or even the best!) way to set type.
Personally, I found the results to be compelling—paragraphs set with the Golden Ratio Typography Calculator look good to my eye (and in fact, trying to understand why certain text looks good and other text looks bad is the reason why I started this research in the first place).
In the comments of the article, I've expounded upon some of the questions posed here, and I hope you'll take the time to explore those if you're so inclined.
For example, some of you have deftly pointed out the lack of clear support for the w = l^2 relationship. I'm going to cover my reasoning behind this in a future post, but I didn't want a heavy focus on mathematics to take away from the impact of the article.
In the meantime, check out the first image on this page (http://dropshado.ws/post/12971305087/webkit-zoomed-out-font-...); the ends of each line of text combine to form a classic x^2 curve. This supports the claim that w = l^2 (but it does not prove it true, obviously).
Most people simply won't read (or even be able to follow) a barrage of mathematical reasoning, modeling, and graphs. That's why I left most of those things out of this article. However, these things are the hallmarks of the research I've done thus far.
Finally, there is a huge benefit to this approach to typography that is worth expounding upon here.
By establishing a mathematical basis to relate typographical variables, you can predict and control type with algorithms instead of relying upon arbitrary selection from designers.
This opens up the doors for sophisticated tools (like the Golden Ratio Typography Calculator) or even cooler stuff, like design controls that allow users to experiment with different fonts/sizes while adjusting all typographical and spatial values based on the resulting line height of the primary text (this is the vertical baseline grid on steroids).
For me, the bottom line is this: Designers rely on arbitrary selection for things like line heights and line widths, and I am convinced there's a better way to go about this.
Research like this is only the tip of the iceberg, obviously, but at the very least, I will continue to ask the tough questions and to challenge the baseless status quo.