"If I have seen further than others, it is by standing upon the shoulders of giants."

-- Isaac Newton

Basing an education on experimentation alone at the expense of collaboration and reading is like nipping at the heels of a giant rather than standing on his shoulders. The 10,000 hours required to become an expert will be fruitless if performed in a vacuum. If when we look for greatness, we also hunt for the giants whose shoulders the great stand upon, their genius and talents largely demystify to us and can become our own.

The reason for this is that discovery is inherently difficult. This is often true even if the discovery to be made seems like common sense in retrospect. In fact, as I will demonstrate, just because something is easy to understand, it does not mean that it is easy to discover.

A lesson in art and discovery
To demonstrate the difficulty of discovery, I'm going to teach you a quick art lesson covering some basic principles artists who attempt to draw realistically must know. You will learn a little about linear perspective. Then, we'll take a closer look at who discovered some of the principles of perspective and who used it. More importantly, we'll examine how some of the principles that I'll teach you in just a few paragraphs were missed by great minds like Rafael and Leonardo da Vinci.

As an aside, one thing I love about using art to teach about learning is even those without art training can evaluate art pretty well. While a person with no art training may not be able to create the illusion of a three-dimensional space on a two-dimensional surface the way a master illustrator does, she can still evaluate whether or not a three-dimensional drawing is convincing. A non-expert can't look at algebraic equations side by side and intuitively decide which one is correct in the same way. To prove the point, let's get started. See if you can decide which of the pictures below is the more accurate painting.

Pretty easy, right? Something is off on the picture on the left. Even if you don't understand why, you know what is correct when you see it.

There is a reason the picture on the left looks odd though. It is because the artist painted the picture without the knowledge of the rules of linear perspective. Linear perspective is the theory and methods an artist uses to create a convincing three-dimensional illusion on a two-dimensional surface.

While there is a lot of detail to the theory to help with measuring, proportion, handling circles in space and the like, the basics are pretty easy to grasp. I'm going to give you the basics of one-point and two-point perspective for the purpose of this discussion.

The main principle that governs parallel lines in perspective is the principle of convergence. Essentially, parallel lines that go away from the viewer will always converge at a single point, called the vanishing point. In one-point and two-point perspective, any lines going away from the viewer will converge to a point on the horizon line.

In one-point perspective, this is usually demonstrated with the example of railroad tracks. If we use a transparent box, as in the example below, you will see that there are three line systems. One set of lines runs vertical, one set runs horizontal, and one set of lines converge at the vanishing point on the horizon. Because there is only one point of convergence -- only one vanishing point -- this is called one-point perspective.

Two-point perspective, as you would surmise, has two vanishing points. There are still three line systems, but if the box is tilted at an angle to the viewer, two sets of lines move away from the viewer. Each set of lines still converges to their own vanishing points, and the vanishing points are still located on the horizon line.

I hope you agree that this is relatively easy to understand. Even with my love of art, though, I didn't know this until I was an adult. I wasn't taught perspective in school, much to my artistic detriment, but when I was 23 I finally got a book on the subject and fixed the problem. It took three days of studying and experimenting to change my art ability forever. While I didn't understand the finer details at the time, particularly with handling circles in perspective, I had the basics down pretty well.

It didn't matter that I drew for years prior to reading the perspective book. Experimentation alone wasn't enough. I'm confident that I would never have discovered the principles governing perspective on my own. But, you may think, "you're just not that smart." From me, you will get no argument on this point; however, even if I were that smart, I would have to be smarter than early Renaissance artists to figure perspective out on my own.

Getting from point 1 to point 2
Prior to the Renaissance, artists understood that objects appeared smaller the farther they were away from the viewer. Artists would often draw lines that converged, they just wouldn't converge where and how they were supposed to. They would guess at the angles and would invariably be wrong, much like in the picture at the beginning of the lesson. Then an architect, Filippo Brunelleschi, in Florence, Italy noticed that receding, parallel lines of buildings from a certain angle would converge into one point. The word began to spread to other artists in Florence who started using one-point perspective in their paintings. It wasn't spread through writing at first, but through collaboration.

Leon Battista Alberti was the first to write down the principles discovered by Brunelleschi in his book On Painting. It's still in print today. Once in print, the idea of one-point perspective became common practice among artists everywhere forevermore.

Here is how this pertains to the difficulty of discovery: Even though one-point perspective was discovered, the principles of two-point perspective were not immediately obvious to artists of the Renaissance. As such, if you look at Renaissance art you will see the artists always working around the limitations of their knowledge. They did lots of paintings and drawings using one-point perspective, but never two-point perspective. Whenever they attempted something pictorially that SHOULD have used two-point perspective, they did it wrong.

Point in case: Below is Rafael's painting, School of Athens. All the one-point perspective is very accurate, but look at how he couldn't seem to handle the elements that should have been in two-point perspective. It seems that he didn't realize that all the vanishing points should sit on a horizon line.

This says a lot about talent, how we learn, and the importance learning method. I'm not exactly sure when artists starting using two-point perspective, but from the paintings I've studied, it looks like it took nearly half a century. I've always taken it for granted that once one-point perspective was understood, it was immediately extrapolated out into two-point perspective. That didn't happen. In thousands of pages of drawings, this discovery eluded no less than Leonardo da Vinci, who is considered by many to be one of the greatest geniuses of all time. It is important to note that he saw further, much for the same reasons Newton did nearly two centuries later. They both had their giants.

Whose shoulders will hoist you? You get to choose. As I say often, 10,000 hours of effort is required to become an expert. The time requirement alone isn't enough; it is critical how you spend your time and with whom. Experimentation, collaboration and reading are all required. In retrospect, this is one of those things that seem like common sense. For a long time, however, I didn't understand how it all fit together. Now I do --” and so do you. Thanks for reading.