This went up today. I've been wanting to write about this
particular topic ever since I entered the world of science writing.
The original topic I pitched was to actually show examples of
Ancient-Greek and Medieval-Islamic mathematics, but after some back-and-forth
with my editor we finally cut them. I don't know who reads this blog, but if
you're here you'll probably find these examples interesting.
The Nautilus article mentions the Golden Ratio because it's an
irrational value/length that Ancient Greek scholars were able to determine geometrically. Because of the phobia surrounding irrational numbers an algebraic determination
didn't come until more than a thousand years later. For the algebraic solution, we can thank Medieval Islamic
scholars.
Ancient Greek Geometry:
A rectangle with the golden ratio has the following
property:
When you cut a square off a Golden Rectangle, the
rectangle remaining has the same aspect ratio as the original, rotated at a
right angle, like so:
So how do we construct a
Golden Rectangle? And how can we be
sure the construction actually works?
The following construction procedure is found in Eudlid's Elements (3rd-century
BCE). The cult of the Pythagoreans (6th-century BCE) probably proved
this particular construction produced a Golden Rectangle; Euclid just included
it in his famous geometry textbook:
1.
Start with a square (shown in red).
2.
Draw a line from the midpoint of the base of the
red square to its upper right corner.
3.
Swing this length down to the red square's base.
4.
Using the end point of the swung-down line,
complete the rectangle.
To prove this construction produces a Golden Rectangle, certain areas within the rectangle had to be proven equal. To determine which areas, the Pythagoreans started with the one thing they knew about a Golden Rectangle: matching aspect ratios. While what follows is today called "cross multiplying", Euclid refers to it as Book 6 Prop. 17.
Thus, to show the construction produces a Golden Rectangle, the red and blue areas have to be shown to be equal. If the aspect ratio is off, one of these ares will be
bigger:
Book 2 Prop. 11 shows that for the construction described above, the red and blue areas are indeed equal:
The first part on the left uses a rule about tacking on
lengths to a bisected line (Book 2 Prop. 6). The second part on the right
uses the Pythagorean Theorem (Book 1 Prop. 47).
This shows that Ancient Greek scholars figured out how
to solve this problem geometrically. The algebraic solution came more than a
millennium later in Medieval Baghdad.
Medieval Islamic Algebra:
Algebra began in Ancient Babylonia. The conquests of Alexander the Great in the 4th-century BCE, which stretched from Greece and Egypt to India, likely brought this knowledge to Hellenistic Greece. While Greek scholars did measurably advance the topic, Medieval Islamic scholars took it much further.
In 19th-century BCE Babylonia, a problem like “x² = x + 870” looked like this:
In 19th-century BCE Babylonia, a problem like “x² = x + 870” looked like this:
 |
| British Museum Tablet 13901 |
Here's a translation from Babylonian into English (and sexagesimal into decimal).
Problem: I added 870 to
the side of my square to get its area. What is the side length of my square?
Solution: You divide 1
(multiplied by the side length) by two, it gives ½. You multiply it by itself,
it gives ¼. You add it to 870, it gives 870+¼. It is the square of 29+½. You
add ½ (which you multiplied) to 29+½, it gives 30 (the side length). (Adapted
from this book)
And here it is again translated into modern notation:
Problem: x² = x + 870
Solution: 
More generally, all problems of this form can be solved
using the following formula:
Problem: x² = px + q
Solution:
This "rhetorical form" of algebra was pretty much constant across its advancement under the Ancient Babylonians, Hellenistic Greeks, Classical Indians, and Medieval Muslims. What 9th-century CE Islamic scholars did that nobody else had ever thought to do before was apply these procedures to the quadratic irrational lengths found in Greek
Geometry.
Since the aspect ratios are equal, the problem can be
represented algebraically. Keep in mind, Medieval Islamic scholars would have written their equations out as sentences, but the steps taken would have been pretty much the same.
x/1 = 1/(x-1)
Cross multiply:
x·(x-1) = 1²
Distribute x across (x-1):
x² - x = 1
Add x to both sides:
x² = x + 1
Since the problem is now in the form of x² = px + q, we can apply the ancient Babylonian solution procedure.
Problem: x² = x + 1
Solution: 
A commentary on Euclid’s Elements by the 9th-century Islamic mathematician Al-Mahani is the first known work that algebraically explores quadratic (and cubic) irrational numbers. Thus Medieval Islamic Scholars were able to algebraically solve problems that Ancient Greek Scholars had geometrically solved more than a millennium earlier.
