> How much of geometry was in fact taught in (say) the mid 19th century
> through classical, synthetic methods?
I ask: in which country and to which students.
My modest l9th century book collection and my work with the Artemes Martin
collection at the American University Archives in Washington, DC has led me
to believe that Euclidean geometry was important in the l9th century for
students that would attend the university. At some point (I once knew), in
the l9th century, Harvard made geometry mandatory for admittance. Other
universities followed.
Many geometry books contain much more than Euclid. We find conics in a
number.
Now, I have not found books other than Euclidean geometry for secondary
students. I do find analytical geometries, descriptive geometries, and
perspective geometries which are obviously college texts.
> He (Salmon) opens the treatise with a discussion of the method of
> coordinates, developed ab initio, and using oblique axes in diagrams etc.,
> unless the result under consideration requires rectangular axes.
I find this only in my analytical geometry and calculus books. Never in
algebras until the l900's.
> (In passing I note, in relation to that infamous question 'Why do we
> use m for slope?', that Salmon gives the equation of the line in
> Cartesian coordinates as 'y = mx + b'. Would Salmon be continuing
> a tradition already established in the textbooks of his day? Would
> it be plausible to suppose that Salmon's textbook would have influenced
> those American textbooks identified by Fred Rickey in earlier postings
> on another list?)
One finds the equation y=mx+b (the convention in the US) in analytical
geometries beginning in the mid l9th century. However, we see the equation
written many ways:
y=ax+b (appears most frequently in my books) and even y=ax+k. I have a
British geometry from about l852 or such (don't have the book in front of me)
which has the equation as y=mx+h. My book is the first I have seen using the
"m" for the slope. Fred Rickey has a little earlier book by another author
which also uses the "m". In my book there is no coordinate geometry.
Fred gave an excellent talk this summer to participants in two NSF History
Institutes, one of which I co-direct with Victor Katz, the other Fred and
Victor direct. He thoroughly went through his research about y=mx+b. I know
of no other historian who has done as much research as Fred has about the
topic. Unfortunately, his information has not reached textbook publishers
who continue to spread the wrong information about the history of "m". And,
they do it so dogmatically.
There is a real joy in old book collecting. I have just finished researching
the content and pedagogy in North American l9th century arithmetics. It is
amazing what is covered! Now I'm getting ready to see where (in North
American books) coordinate geometry appears in elementary algebra books. It
is interesting to me that it appears 100 years or so ago. And then it
disappears. My secondary education in the l950's did not include an
understanding of the relationship of linear and nonlinear equations to a
coordinate system.
Also, of interest is the fact that the first mathematics book printed in the
Americas was in Mexico in the l6th century. In fact, algebra books were
printed in Peru and Mexico 200 years before they were written and printed in
the colonial US.
Cheers!
Karen Dee Michalowicz
Upper School Mathematics Chair, The Langley School, McLean, VA
Adjunct Faculty, George Mason University, Fairfax, VA
Treasurer, Americas Section, HPM