The review praised the book, "One special feature of the book is the great number of excellent diagrams drawn on

*squared paper*. These cannot fail to be of great instructive value to the pupil."

They went on to describe, "Another excellent feature is the early introduction of what Chrystal [George Chrystal, of whom more later] called the '

*freedom equation*' of a curve, in which x and y, instead of being expressed in terms of one another, are each expressed in terms of a third variable."

Then they conclude, "The method is known in some books as the parametric representation; but there is no doubt that Chrystal's distinction between the

**Freedom**and

**Constraint**equations is one that should be early brought to the mind of the student, and the phraseology is

*particularly happy*(does anybody write that way anymore?). In another review where the author did not use the term, it was lamented that he had, "missed the opportunity of calling them by this picturesque title."

The term parametric was far from an established term at the time. According to "Earliest Known Uses of Some of the Words of Mathematics" by Jeff Miller, a teacher at Gulf High School in New Port Richey, Florida.

PARAMETRIC EQUATION is found in 1894 in "On the Singularities of the Modular Equations and Curves" by John Stephen Smith in the Proceedings of the London Mathematical Society. I could not find the document in which Chrystal first used "Freedom equations", and welcome comments from those with more information. It seems like it may have been well before 1911 since an article about Chrystal in the November 2, 1911, copy of Nature indicates his contribution of the term may be little known.

It does seem the term may still have some usage in modern times as I found a definition of it at Math Resources web site, apparently a Canadian site.

Chrystal was a powerful force in Edinburgh Math Society just before 1900, and I mentioned him earlier as one of the contributors to symbols for the sub-factorial notation.

Addendum: In the Spring of 2015, Dave Renfro (one of my most valuable leaning tools) sent me a note about the Gibson/Pinkerton book and described it as "excellent for its coverage of graphing techniques." The book is available on line here.