Math Discoveries

1. Hargreave’s offhead squaring formula
For numbers between 25-75 you can use it as follows:
x2 = (|x-50|)2 + (x-25)100
For numbers between 75-125 you can use it as follows:
x2=(|x-100|)2 +(x-50)200
Note: you must know all squares offhead of all numbers below
25 to use it.
The 2 stands for square
2. ( The 7 wonders of )theHargreaves wheel

The Hargreaves' wheel

Wonder1:
The smallest circle consists of numbers 1-5 arranged such that no adjacent numbers on the number scale are adjacent to each other.
Wonder2:
There is only one way of arranging them.ie, the way they have been arranged
Wonder3:
On adding each 2 numbers to form the third wheel, you get numbers that are adjacent to each other on the number scale.
The same applies to the 3rd wheel, if every 3 numbers on the first wheel are added to each other.

Wonder4:
On the fourth wheel, when every four numbers are added to each other from the first wheel, we get a rather random pattern. Or is it?
On the first wheel, the differences betweet adjacent numbers forms is 3-2-3-2
On the fourth wheel, this is the same pattern!
Wonder5:
The two greatest numbers on each wheel are the least two numbers in the next wheel!
Wonder6:
The sum of the numbers in each wheel is 15n, where n is the number of the wheel!
3. Hargreaves’ Tan89
tan(89 + 9(1/10^0 +1/10^1 +1/10^2 +1/10^3+…..1/10^(n-1)) = tan89.9(10^n)