I’ve had a spreadsheet for font-size scaling that I’ve used for years. Depending on the physical size of the product, it provides a consistent set of type sizes across different printed formats.
18 pt might look appropriate on an A3 product but disproportionately large on an A6 version. Having an agreed scale means the type is not resized differently by each person working on the job. Everyone uses the same system the hierarchy remains consistent across the products.
That does not mean the scale needs to follow the Golden Ratio. Increasing in steps such as 1.125, 1.2, 1.25 or 1.333 often produces more practical levels while still giving each level of text a clear distinction.
Starting at 10 pt and repeatedly multiplying by 1.618 gives approximately 10, 16, 26 and 42 pt. That might work for a dramatic poster, but it creates very large gaps for a report or academic document. A smaller ratio gives more useful intermediate levels.
The same applies to using multiples of three. There is nothing inherently wrong with 9, 12, 15 and 18 pt if those sizes suit the typeface, content and format. The system provides consistency, but it should not become a rule that overrides what the text actually needs.
In fact, one client insists on a specific tab stop for all their documents, so people don’t use different indent levels.
My own approach is to establish the body-text size first, taking the typeface and its x-height into account. The rest of the hierarchy is then built from that
Body text
– Bullet level 1
– Bullet level 2
– Additional bullet levels
|
Main heading
– Subheadings
– Additional heading levels
So in the above, Main Heading is based off Body Text - and only Subheadings are set to be based off the Main Heading
So everything is based off the body text, then only change font size of the headings or other settings, like rule above or rule below etc.
Ratios provide a framework, but they don’t decide whether the text is readable or whether the hierarchy suits the content.
On the historical point, the ratio was known in ancient Greece and their mathematics. I love Greece, I go almost every year, I’ve been to Athens many times, I’ve stood on the Acropolis and touched the columns, the steps, and it’s fascinating.
All of their stuff for building came naturally in basic geometry of regular pentagons and pentagrams, where the diagonal and side are related by the Golden Ratio. Euclid later formally described the division of a line in “extreme and mean ratio.” That establishes its mathematical use, but it does not establish that ancient architects treated 1.618 as a universal rule of visual beauty.
They did make deliberate optical adjustments rather than relying blindly on mathematical uniformity. Greek architects varied column profiles, spacing and alignment so that buildings appeared visually balanced when seen from the ground.
A related Roman example is Trajan’s Column elements intended to be viewed from below were adjusted for perspective, and the letters in its base inscription are slightly larger toward the top than at the bottom so that they appear more even to the viewer.
I know we all love Trajan font 
I think that’s more relevant to typography than insisting on one supposedly perfect ratio mathematical consistency can establish the system, but optical correction is still needed to make it look right.
The same kinda applies for other design things. You do not always place something at the exact mathematical centre of a page you often position it at the optical centre instead. A rough starting point around 70% of the distance from the top of the page to its mathematical centre, rather than at 50% of the full page height. Of course it still depends on the shape, visual weight and content.
