A Nasty Divorce Has Occurred!
"There is no royal road to geometry." Supposedly, Menaechmus shared this caution with Alexander the Great. Or, was it Euclid's reply to King Ptolemy, struggling to learn geometry?
Then in 1942, mathematician Gaylord Merriman claimed "Algebra is geometry's royal road." His claim was based on his interpretation of "Descartes who performed (1637) a very simple marriage ceremony which united the two." (p. 155)
Now, this grabs my interest. Given current societal conditions, how could algebra and geometry wed?
The essence: Descartes creation of analytic geometry allows us to solve problems by the Transform-Solve-Invert process. For example, we can find the intersection of two geometric lines by TRANSLATING to their corresponding algebraic equations, SOLVING for the intersection algebraically, and then INVERTING back to the geometric diagram of the two lines.
Be that as it may....I claim that sometime during the past 30 years, algebra and geometry have divorced...due to external forces, as the marriage was once happy. All one has to do is look at modern curriculum materials!
Algebra thrives pompously in the public's eye, while geometry has retreated silently to an unknown hiding place.
If only we could find that royal road again...!
Source: Adapted from G. Merriman's To Discover Mathematics, 1942, p. 155