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He lived from 1646 to 1716.

Most geniuses shine in one area above all others.  Leibniz was master of many - mathematics, law, religion, philosophy, history, and literature among them..

In mathematics, he has two contributions of amazing import: Calculus, and Combinatorial Analysis.

And even more amazing, these contributions occur in the two separate branches of mathematics: Calculus is the mathematics of the continuous, and combinatorial analysis explains the discrete.

The sun is fierce, the breezes still, the days are lazy.

What better time than now to discover.....

..............................................Calculus.

Okay, I'll say it out loud. Calculus.

And why not?  It's a glorious, glorious subject.  Your kids have heard its name, and they have wondered about it.  Why not give them a sneak preview?

Do it in honor of Gottfried Wilhelm Leibniz, who invented it, independently of the other great inventor of the calculus: Sir Isaac Newton.  Newton's birthday is Christmas Day, when we tend to be a bit too busy to wander into calculus, but Leibniz's birthday is a great time.

Calculus has been called "the mathematics of change."  In our world, things change.  Moons orbit, cars speed up, profits fluctuate.  Calculus is the part of mathematics that lets us understand and work with these changes.

There are two branches of calculus: differential calculus and integral calculus.  The genius of Newton and Leibniz is that they connected the two.   While most of this is too complex to be discussed here, it can be fun to show your children the two basic questions that form the basis of these two branches of calculus.

I. What is the instantaneous velocity of a moving object? (Differential calculus)

II. What is the area of a region under a curved line? (Integral calculus.)

In coming issues we'll introduce you to a key idea of calculus: the limit.   But in the meantime, as you go about your summer, try some of these excursions with your children.

Excursions........

1. Summer vacations usually involve some transport - getting there!  Explore the idea of speed - the rate at which we change distance over time (this is actually the 'derivative,' in calculus).   As you travel, have your child write down your speed at various points during the journey. Talk about the idea of speed -the rate of change. Adjust your discussions to the age of your child.

2. Explore the rates of change of other things: you'll be moving a lot this summer - on the highway, in the water, at amusement parks.  Observe and record rates of change in your activities.

3. Graph the information you gather or observe: graph distance along the vertical axis, time on the horizontal.  On another graph plot speed on the vertical, time on the horizontal. 

4. At the beach, draw shapes in the sand.  Begin with a rectangular shape.   Ask your child how she might figure the area of it.  Then draw a curved shape.   Ask her how you might figure the area of that.  (In calculus, we use the idea of drawing little rectangles inside the curved shape - rectangular areas are easy to calculate, and then we add them up.  The smaller the rectangle, the more we can fit in, and the better our approximation.) 

5. At a used book sale, pick up an old calculus textbook and give it to your child.   (Read the story in the adjoining column about David).  Your child doesn't need to understand it to be thrilled - and proud -to own it.  Someday he will understand it.

Several years ago, I taught math at a psychiatric hospital.  My students faced more than the normal struggles, and their path through school mathematics had involved more than the normal challenges.  Many of them had to work hard to bring their skills up to the level of other kids their age.

Fourteen year-old David was beginning algebra.  His mind was sharp, but his path had been a tough one.  Discarded by his parents, he had grown up in a world of violence and drugs.

The young men he had looked up to had not done math.

Except for Michael.  Michael's hospitalization had overlapped David's long enough for David to watch him study trigonometry and calculus.

"Teach me trig," David asked, after Michael left.

So we digressed a bit, and I taught him the rudiments of trigonometry, which begin with ratios - (fractions by another name.)

"To go further," I told him, "we have to learn algebra."

One day, while I settled other students in their assignments, I looked across the room and saw that David had taken my calculus book off the shelf, and was drawing - in painstaking and beautiful detail - reproductions of the mathematical drawings inside the text.

He saw me watching him, and as I was about to speak, he said only one word:  "Please?"

I nodded. 

David didn't understand the drawings, but he revered them.  In the world he'd come from, the "big guys" did drugs and guns, but he had glimpsed a world where the big guys did calculus and trigonometry, and he saw himself in that world.  The pictures hung proudly on the wall, long after he left.