Actually the linked site has a very good entry on both "algorism" and "algorithm".    From Wordnik.com.  [languagehat.com: MATHEMATICAL TERMS.] Reference

This is what the OED has under the etymology of "algorism" slightly modified for easier posting.    From Wordnik.com.  [languagehat.com: MATHEMATICAL TERMS.] Reference

Algorisme being popularly reduced in OFr. to augorime, English also shows two forms, the popular augrime, ending in agrim, agrum, and the learned algorism which passed through many pseudo-etymological perversions, including a recent algorithm in which it is learnedly confused with Gr. ‘number.'.    From Wordnik.com.  [languagehat.com: MATHEMATICAL TERMS.] Reference

In arithmetic he was influential in spreading the ideas of algorism.    From Wordnik.com.  [The Hindu-Arabic Numerals] Reference

So the abacus held the field for a long time, even against the new algorism employing the new numerals.    From Wordnik.com.  [The Hindu-Arabic Numerals] Reference

I mortgage loan bad credit the poulet was permissibly the algorism pseudolarix, and plaguey blip unprecedented at postal the coordinator and hyperhidrosis, puckishly.    From Wordnik.com.  [Rational Review] Reference

Hindu-Arabic numerals: "Over the whiche degrees ther ben noumbres of augrim, that devyden thilke same degrees fro fyve to fyve," and "... the nombres ... ben writen in augrim," meaning in the way of the algorism.    From Wordnik.com.  [The Hindu-Arabic Numerals] Reference

Nevertheless, to understand the game aright, they feared neither the cormorant nor mallard of Savoy, which put the good people of my country in great hope that their children some time should become very skilful in algorism.    From Wordnik.com.  [Gargantua and Pantagruel, Illustrated, Book 2] Reference

Al-Khwarizmi’s own name got twisted into “algorism,” meaning “the art of calculating,” what we now call arithmetic.    From Wordnik.com.  [Intelligent Design offers biblical alternative to Darwinian evolution, Dembski says - The Panda's Thumb] Reference

He set it out in a 120x120 frame he termed a "columbarium" (i.e. a dovecote, which has a similar grid of niches) and records that it took him nine years to calculate, even with the newly imported technique of ‘algorism ', calculation with Arabic numerals rather than abacus.    From Wordnik.com.  [The Mandelbrot Monk] Reference

L. algorism-us cf.    From Wordnik.com.  [languagehat.com: MATHEMATICAL TERMS.] Reference

Super fast search algorism.    From Wordnik.com.  [MacUpdate - Mac OS X] Reference

"Hec algorism 'ars p'sens dicit' in qua.    From Wordnik.com.  [The Hindu-Arabic Numerals] Reference