George Packard began his little lecture:

The history of humankind and of modern civilization can be described by the history of mathematical discoveries. Each new mathematical understanding has almost immediately led to significant technological advances and “progress”.

First, let’s consider the history of numbers. It’s a long story. From the simple counting numbers – from markings inscribed on pieces of bone dated by archeologists at 15,000 to 20,000 years old – to fractions, to negative numbers, squares, square roots, exponential numbers, irrational numbers, imaginary numbers, and yes, the transcendental numbers (pi, phi, and the rest) – these “discoveries” all plot out a fine map of man’s history. Each “discovery” enabled a leap forward in the advance of technology and science.

The “discovery” of pi, the constant ratio between the diameter and the circumference of a circle, was apparently”discovered” or “revealed” to the Egyptian builders of the Great Pyramid. At about the same time the “discovery”of the golden proportion phi. This was almost 5,000 years ago. Knowlege of pi and phi enabled the Egyptians to build the Great Pyramid of Cheops, the Pyramids that still stand at Giza, on the 30th parallel, in Egypt.

The number zero, “discovered” in India around 500 B.C. and expressed in the Hindi-Arabic numbers around 500 AD, enabled all computations, as well as the so-called algebra, to blossum. The Olmecs in Central America were already using symbols for zero in their mathematics about 500 B.C. The Mesopotamians used a zero in their sexagesimal number system around 2,000 BC; they also used the Pythagorean theorem.

Other mathematical landmarks would be:

Euclid’s Elements—500 B.C. Euclid’s system of logic and proof was the foundation of science.

Archimede’s mathematics : the lever and the principles of mechanics around 250 BC

Cartesian geometry and the mathematics and science of Isaac Newton circa 1640

Binary Numbers, developed by Leibniz in 1679 – which later enabled computers to enter our world.

Non-Euclidian geometry



George Packard would stop at this point, and look out over his class, the assemblage of teen-aged heads and hair-do’s that sat before him, several of them leaning on their propped elbows in listening poses..

As he was about to resume his little lecture a hand rose up from the back of the room. Young Pola was raising her hand.

“Yes Pola”

“Mr. Packard, I mean like why did you put all the parenthesisis marks each time on the word discovery? Huh?”

“Hey Pola that’s a good question. Does anybody out there know why?

Packard scanned the bodies in the room and noticed they were were now paying attention. They were stirring in their seats, sort of roused from their slumber.

“Well, I did that because each of these “discoveries” was present and operative in the world before man “discovered” them. The plants were already growing in phi proportions, the planets revolving in great ellipses, the stars retreating and the universe expanding according to these mathematical underpinnings long before man came along.”

“That’s true, Mr. Packard!” yelled out someone from the back.

George Packard loved to present these interesting things to the kids each year. Maybe some of them might be encouraged and brightened. Here and there anyway.

So thought George Packard as he closed his notebook and packed it in after another day at L.A. High.