True, but I think he's failing to make his case. Maybe it becomes more apparent if you read his whole book. I'm sure not impressed with his comparison of classical trig and his rational method is section 1.5. Classical trig solves the problem such that d = (5 * sin 41.4096 deg) / sin 93.5904 deg = 3. 3137. The other method requires applying the Quadratic equation to find r in the equation: (7/16 + 1/2 + r)2 = 2(49/256 + 1/4 +r2) + 4 * 7/16 * 1/2 * r.
He's not showing me that his method is easier or more straightforward!
The point is that square roots CAN be calculated by hand, and it's easy and fast. There is no way to calculate trig functions by hand that is remotely practical.
It's called the "divide and average" method. Say we want sqrt(7). The closest integer is 3, so start with 3. the average of 3 and 7/3 is 8/3. The average of 8/3 and 7/(8/3) is 127/48. This is correct to 4 decimal places already (2.6458).
If you do 1 more step and average 127/48 and 7/(127/48) you get 32257/12192 which is correct to 8 decimal places, but the high school textbooks always require at most 4-place accuracy because that's what the trig tables went up to.
When you do the math this way, you don't need a calculator.