Sadly, most folks don't want to hear about A square plus B square equals C square. I even spell the numbers to try to keep you from skipping this page. The good news: we can now determine them without squaring or taking square roots. Actually, my friend Durwood, the retired carpenter, figured this out many years ago. He could quickly determine the needed values for a fairly large triangle to test whether the framework of a new building was truly "Square". He would mark certain distances from a corner, then he and a co-worker would measure the hypotenuse. He wanted it exact.

Sometimes he looked for a seventeen foot hypotenuse for an eight-by-fifteen foot corner. Other times he might want a forty-one foot hypotenuse in a long room for a nine-by-forty foot corner. He calculated them on the spot according to the room size he was checking for "Square". In his head. Without square roots, with his own, reasoned-out explanation for the patterns he saw in Pythagorean triangles. Can you see any patterns in the numbers below? The ODD series looks easier than the EVEN, but Durwood's method works on both, and for multiples and fractions of basic triangles. (The table covers more than what he used.)

DURWOOD'S TRIANGLES

EVEN Series           ODD Series
 2 -   1.5 -   2.5     3 -   4 -   5
 4 -   7.5 -   8.5     5 -  12 -  13
 6 -  17.5 -  18.5     7 -  24 -  25
 8 -  15   -  17       9 -  40 -  41
10 -  49.5 -  50.5    11 -  60 -  61
12 -  35   -  37      13 -  84 -  85
14 -  97.5 -  98.5    15 - 112 - 113
16 -  63   -  65      17 - 144 - 145
18 - 161.5 - 162.5    19 - 180 - 181
20 -  99   - 101      21 - 220 - 221
22 - 241.5 - 242.5    23 - 264 - 265
24 - 143   - 145      25 - 312 - 313
26 - 337.5 - 338.5    27 - 364 - 365
28 - 195   - 197      29 - 420 - 421
30 - 449.5 - 450.5    31 - 480 - 481
32 - 255   - 257      33 - 544 - 555
34 - 577.5 - 578.5    35 - 612 - 613
36 - 323   - 325      37 - 684 - 685
38 - 721.5 - 722.5    39 - 760 - 761
40 - 399   - 401

The same patterns continue on to infinity, supplemented by many multiples and fractions of the basic units. Why did I include the halves? Because Durwood did. Sometimes he needed a triangle that size, and he could calculate them just as easy as the others. How?? What's the method??

The smallest side limited the accuracy of his work, and he measured it first. I fibbed a little. He did square the smaller side, from memory. Then he halved the square, and added and subtracted one-half (0.5) to get the long sides. For example, a short side measuring 10 feet would be squared to give 100, divided by two to get 50, which then gave 49.5 and 50.5 for the long sides. (50.5 feet is the hypotenuse.) All the ODD Series numbers in the table are also calculated this way.

But notice what happens when you double the 10 foot value to 20 feet. The decimals disappear; all the units are double those with the 10 foot base to 20, 99, 101. Any factor of the base number can be used to calculate.

20 squared = 400; half is 200. You can use the factor One , add and subtract 0.5 to get 199.5 and 200.5, and 20 - 199.5 - 200.5 is a valid Pythagorean triangle.

To use the factor (of 20) Two , 200
2 = 100, and 1 gives the table values of 20 - 99 - 101.

To use the factor (of 20) Five , 200
5 = 40, and 2.5 gives 37.5, 42.5. 20 - 37.5 - 42.5 is also a valid Pythagorean triangle.

To use the factor (of 20) Ten , 200
10 = 20, and 5 gives 15 and 25. The resulting 15, 20, 25 triamgle makes a multiple of the 3, 4, 5 triangle. The difference between the calculated sides is equal to the base number's factor used. You add and subtract half the factor after dividing the square by two. (Reread this paragraph, try a few, then call me.)

How did such a brilliant person become a carpenter? He was bored in school, and liked the work. Lots of Mensans lived similarly. His daughter is a philosophy professor. You figure out how he did it without square roots!