A Beautiful Manuscript of Square Roots, 1830's

JF Ptak Science Books   Post 2082

 

(Note: we'll deal with square root of  3 at another time...)

The author of this manuscript, Charles Fisher, took a solitary pleasure in calculating the square roots of numbers from 2  to 589, not bothering to write down the 24 perfect squares to 576.  (The sqrt being r² = x for every non-negative real number x.) From the few bits that I have checked the man seems to have done a good job back there in the 1830’s. 

I cannot determine where this book was written or who Mr. Fisher was, though it is possible that he was a (Baptist) Minister from the few short textual notes that pop up here and there. 

[This 140pp manuscript is being offered for sale at our blog bookstore, here.]

His work is pretty elegant.  Take for example his solution (and proof) for the square root (hereafter sqrt(x)) 309,

which the calculator living under this page says is 17.578395831246947    Mr. Fisher’s answer is 17 10/17 = 799/17=89401/989=309 100/989, and after some more involved arithmetic comes t the lovely proof number of   

4121989960986322995025 /13339773336525317136 or

309.00000000000000000007496379247

Which is getting pretty close. 

The only note that Mr. Fisher makes on his calculations is for the sqrt(193), which he notes as “the hardest number to find the approximate root of any between 1 and 200. I have found it after repeated trials and have this evening wrote it in as above. March 1^st, 1833. CF.”

Comments

[Ray Girvan]

This is a bit of a puzzle; I can't see what algorithm he's using. For instance, for sqrt(309), I see how he gets 17 + 10/17 as a first approximation:

309 = 289 + 20 = 17(17 + 20/17)
That gives two terms: 17 is an underestimate, and 17 + 20/17 an overestimate, so their mean 17 + 10/17 is a better approximation than either.

But where does that 6 come from, that he uses to derive the next iteration?