[In the last episode, Alexandria Jones discovered a mysterious treasure: three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly spaced knots. Her father explained that it was an ancient Egyptian surveyor’s tool, used to mark right angles.]
Back at the camp, Fibonacci Jones stacked multi-layer sandwiches while Alexandria poured milk and set the table for supper.
“Geometry,” Fibonacci said.
“What?”
“Geo means earth, and metry means to measure. So geometry means to measure the earth. That is what the Egyptian rope stretches did.”
Alex thought for a moment. “So in the beginning, math was just surveying?”
“And taxes…”
Measuring the Earth
Professor Jones sliced a sandwich diagonally into two triangles. “Every farmer in Egypt had to pay a property tax to the Pharaoh each year, based on the size of his land, so the rope stretchers spent most of their time measuring farm land. The scribes could easily calculate the area of a rectangular plot of land:
Area = length x width.”
He waved the knife as he talked, drawing imaginary diagrams in the air.
“Dad, would you please put the knife down?” Alex said. “You’re making Rammy nervous.”
“Oh, yes, of course. And since a right triangle made exactly half of a rectangle, the area of a right triangle was simple, too.”
“I know that one,” Alex said. “Area = (1/2)(base x height), where the base and height are the two legs of the triangle—the two sides that form the right angle. But what if the farmer’s property had some really weird shape?”
Her father began to lay the pieces of sandwiches on the tabletop in an asymmetrical design. “The Egyptians discovered that they could divide any piece of property with straight borders into right triangles.”
“They could cut any property into triangles?”
“If it had straight sides.” Fibonacci picked up the last sandwich and took a big bite.
Alex grabbed a sandwich half and held it up, tracing its edges with her finger. “Then when they measured the sides of each triangle, they could calculate its area. What a neat system!”
Her father nodded. “Adding the areas of all the triangles together gave them the area of the entire property—”
“And the proper amount for the farmer’s taxes!” Alex laughed.
Try It for Yourself
Of course, Egyptian rope stretchers, like modern surveyors, laid out land in rectangles whenever possible. But some properties came out with irregular shapes no matter what the rope stretchers did, and the fact that the Nile flooded every year made their work particularly difficult.
On a blank sheet of paper, draw a few large polygons: closed shapes made of straight line segments. Start with relatively simple shapes, using only four or five lines, then work your way up to a complex shape that fills half the page. Use a ruler to keep your lines straight.
Try to divide your shape into right triangles. If you don’t have a drafting triangle, you can use the corner of a sheet of paper to help you draw right angles. The more complicated your original shape, the more lines you will need to cut it up, but try to find the fewest lines you can. Like everyone else, Egyptian rope stretchers tried to make their work as easy as possible.
Finally, try this challenge from my old newsletter:
Egyptian farm puzzle [pdf 415KB]
Can you find the area of this farmer’s property, so that he will know how much of his crop to send to Pharaoh for taxes?
A Puzzle for Older Students
Given the equation for the area of a rectangle:
Prove that Alexandria Jones’s equation for the area of a right triangle is true:
And then prove the area formula for any general triangle:
Make sure your proofs contain enough explanation that a reader can follow your reasoning. The toughest part of any geometry proof is to make sure your logic will stand up to scrutiny. How do you know that everything you said is true?
Hint: You may find the following tips from Euclid useful.
Book 1, proposition 29
What happens when you draw a diagonal through a rectangle?Book 1, proposition 4
How can you show that two triangles are congruent?Book 1, common notions
If you know that this equals that, what else can you do with the information?
And if you get this far, then can you show where geometry’s other area formulas come from? Try to prove the area of a parallelogram or trapezoid, or to approximate the area of a circle.
[The end. Watch for answers to these puzzles and historical notes about the Pharaoh’s Treasure in an upcoming post.]
To Be Continued…
Read all the posts from the May/June 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.
Denise Gaskins' Let's Play Math 
Completely Cool! Thanks for this great post!!
We will be tackling this one tomorrow! What a coincidence, we were just going over area of triangles today!
Jenny, thanks for the encouragement!
Theresa, isn’t it fun when things line up that way? Of course, triangles are only one part of the story problem with the farmer’s field. Another important part (and perhaps more difficult, depending on when in the process you do it) is figuring out how to convert the units from centimeters to cubits—and then from square cubits to acres, if you want to know how the farm compares to a modern farmer’s spread.
He got an answer in square cubits. Now, I don’t want to post it here and give it away for those who are still doing it. So, where can he find out if he is right?
Of course, I could just check it myself, but he seems to think it more fun to check his answers online…
My plan is to post the answers and some related history next time (I hope it will be this week, but we are going on a family trip, which may mess up my plans), and then some more general math history (related to each character’s name) the time after that. In my old newsletter, the history tidbits were sidebars around the stories, but on the Web I can go into a little more depth.
To give Superboy an idea whether he is in the right ballpark: the answer I got for the area of the farm was in the low millions of square cubits. Since there will be round-off error and slight differences in the way people measure (do you read the inside of the dark line or the outside, or try to estimate the center?), I don’t expect his answer will match mine exactly, but the order of magnitude should be the same.
Ok. I will have to check and see where he went wrong, as he is off by an order of magnitude. His answer is in the low hundred-thousands.*sigh*
Well, I will recheck my answer as well. I have been known to make order-of-magnitude errors before! (For example, here. See the comments.)
Oops! Theresa, I just found the error, and it is mine. In my old newsletter, I used this puzzle twice, but I changed the numbers. The page I scanned for this post was from a sample issue that I used to send out, but the answer I looked at was from the original puzzle (where one cm was equal to 200 cubits). It makes a big difference.
Superboy’s answer in the low hundred-thousands is correct. Tell him I’m sorry!—and that he should never trust the “answer in the back of the book” until he has double-checked it on his own.
Hi Denise! Thanks! And don’t worry about Superboy. I went ahead and checked his answer myself and found it correct, so I never even told him it might be wrong.
He did find it fascinating that any shape can be divided into right triangles, as did I! Very cool!
Answer and a few historical tidbits are now posted here. More history coming soon.
thanx