It was geometry without algebra; proving theorems and finding loci based on some axioms about angles, parallel lines and so on. What is this called? Euclidean Geometry?
"Euclidean geometry" would do for a name, but to specifically emphasize the point that algebra is not involved, you can say "synthetic geometry." The geometry of the plane can also be taught from an analytic geometry perspective (using ordered pairs of points on the Cartesian plane) as a first high school course, as in the book Vectors and Transformations in Plane Geometry by Tondeur.[1]
Interesting enough, you can also build your whole geometry on the operation of reflection and a bunch of axioms. See Calculus of Reflections at eg http://rd.springer.com/article/10.1007%2Fs00022-012-0123-5 (surprisingly for Springer, not behind a paywall).
There's algebra and calculating, but with reflection operations here, not with coordinate pairs.
"Euclidean geometry" would do for a name, but to specifically emphasize the point that algebra is not involved, you can say "synthetic geometry." The geometry of the plane can also be taught from an analytic geometry perspective (using ordered pairs of points on the Cartesian plane) as a first high school course, as in the book Vectors and Transformations in Plane Geometry by Tondeur.[1]
[1] http://www.mathpop.com/bookhtms/tondeur.htm