Can anyone explain this?

The difference of the square roots for 1 & 2 is 0.414214 (to six places). This number is also the square root of the difference between the the square roots of 8 & 9.

I did a spread sheet on square roots and their differences and there is a pattern of like numbers like this. The interval at which they appear increases by 8 every occurrence. For example, the difference of the square roots for 2 & 3 is 0.317837 This number appears next as the square root of the difference between the square roots of 24 & 25. The lines on which these matches appear are 9, 25, 49, 81, 121, 169, ... increasing by 8 every time.

Is there anything to this?

I can confirm that √2 - √1 = √(√9 - √8)
simplify: √2-1 = √(3-2√2)
square both sides 2-2√2+1 = 3-2√2
and the equality is evident.

The same method will also show that:
√3 - √2 = √(√25 - √24).

You seem to be saying that these occur when the 3rd number (the largest) is the square of an odd integer. But I'm not sure what you mean by "increasing by 8". I haven't seen this before, but it's plausible that there's a general formula there, just begging for an inductive proof, where you show that if it the formula works for the number n then it must work for n+1 as well. I'd have to fool around with it for a while.

Even if there is "anything to this", such integer identities are pretty well-plowed-over ground, where new discoveries are few are far between -- not that I want to burst any bubbles... Well done, Kendor! -- I especially like the spreadsheet approach.

OK, here it is:
For n=0,1,2... you can form odd integers 2n+1 = 1,3,5...

The formula is:
√(n+1) - √n = √(√(2n+1)² - √((2n+1)²-1 )
For example, when n=2 then the formula gives:
√3 - √2 = √(√25 - √24)

Induction is not required -- just straightforward algebra as earlier to prove the identity: Square both sides of the boldface equation, simplify, collect terms, etc. It falls right out.

So for n=100, the formula predicts that:
√101 - √100 = √(√201² - √(201²-1)

Thanks for the input professor. What I mean by the interval increasing by 8 everytime is that on my spreadsheet, the first match appears on line 9, the next on line 25, next 49, next 81, next 121, etc. The number of lines between each match increases by 8 each time,

25-9=16
49-25=24
81-49=32
121-81=40
etc.

The number 8 has got to play into this somewhere. Thanks again! Numbers are so fun to play with. And you are very very good with them.

Things happen on line numbers which are squares of odd numbers, right? In such a sequence, the differences between successive terms yields an arithmetic series, so the differences between successive terms of that series will be constant.

This behavior is related to the basic concept of calculus (derivatives of powers): The rate of change of a quadratic is a straight line, and the rate of change of a straight line is constant. Here, using a discrete integer series, the "rate of change" is represented by the difference between successive terms. Does that make sense?

Thanks for your kind words, Kendor. Through 6th grade, I found -- like most students -- that long division, fractions, and other "math" was joyless drudgery. Then, starting in 7th grade, I was taught mathematics in its much wider sense (algebra, etc). I discovered a talent and affection for math that has remained with me for life. Unfortuantely it's of no use to me in my daily life.