some years ago i discovered that for a long time i had misunderstood how the greek though of mathematics. what today we think of “squaring a number” wasn’t an abstract operation to them. it wasn’t even an operation for them, but a figure, a shape. a square, more precisely. they didn’t really have the concept of “multiplication” but of “area”, they didn’t think of square roots but in terms of “sides of a square” and they didn’t even add numbers really. numbers were not just quantities for them, but content, like length, area or volume. lacking a symbolic language to describe operations, they couldn’t really do algebra. instead, problems that today we reduce to a single line of compact mathematical expression, would require long paragraphs of text and a few drawings. the disadvantage was clearly the lack of a mechanical methodology to problem solving. the advantage, however, the need of a very visual understanding of the mathematics. it’s not a coincidence that geometry was, until the times of Newton, the only “true” mathematics.
this morning i was getting out of my long hot water shower. as usual during my showers in the mathroom
i had been distracted with stupid stuff. today i was computing how much 17² is (the reason has to do with raytracing, micropolygons and the way shading is done by Pixar’s Renderman). i had tried a couple of times to do the long multiplication in my head, you know, the way they teach you at school: multiplying 17 by 7, remembering that quantity, then adding 170 to it, etc. the problem is that at those times in the morning my brain is more parasympathetic than conscious and, basically, it can only respond to basic stimulus, so i miserably failed to perform my computation.
so i gave up on it, closed the water and shifted the curtains in order to leave the bath. i started drying my right foot before stepping outside, while my eyes posed on the tiles in the floor outside the bath. and then, withing the same tiles that i had been seen daily for the last two years, i saw a figure that i had not seen before there, but that i had seen somewhere else. in a trice, i remembered Euclid’s Elements books on geometry, 2400 years old now, and the square made of four different rectangles.
eureka! suddenly, i performed this computation 17²: 256 + 32 + 1 = 289. piece of cake!
the big square tile became of side 16 (area 256, as any coder knows), the small squared tile became of size 1, and the two lateral rectangular tiles were therefore of size 16 each, which all together made an imaginary square of side 17 and size 256 + 32 + 1 = 289. and yes indeed, 17 times 17 is 289. no long multiplications performed nor injured.
this is how the greek though of maths really. i’m not sure they knew how to multiply two numbers or how to square one, or if they were interested in that really. i do know that, by lacking a symbolic language, what we usually write as
(a+b)² = a² + 2ab + b²
they did actually write with these literal words instead:
“if a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments”
which means that a line (a+b) cut in segments a and b, extrudes into a square of size equal to the square obtained by extruding a, plus the square resulting from extruding b, plus the two rectangles resulting extruding a by b and b by a.