# Old geometry terminology

I was reading Ramsey's 1927 paper "A Contribution to the Theory of Taxation" and came across the following paragraph:

"We have $$\lambda_1 = \mu_1,\ldots,\lambda_m = \mu_m$$, $$m$$ hyperplanes ($$n-1$$ folds) whose intersection is a plane $$n-m$$ fold which we will call $$S$$. $$S$$ will cut the hyper-ellipsoids $$u = constant$$, $$R = constant$$ in hyper-ellipsoids which are similar and similarly situated and whose centres are the points $$P'$$, and $$Q'$$ in which $$S$$ is met by the $$m$$-folds through $$P$$ and $$Q$$ conjugate to $$S$$ in $$u = c$$ or $$R = c$$."

Here $$u$$ is a quadratic function such that $$d^2u$$ is a negative definite form. $$P$$ is the centre of the ellipsoid $$u = constant$$.

Does the term "$$n-1$$ folds" here just mean an $$n-1$$-dimensional subspace? And does "conjugate" here just mean "orthogonal"? These are the only interpretations of the terms that make sense to me; I would like to just confirm whether these interpretations are correct...Thank you!

• Hyperplane is the modern term for "n−1 fold". It does not have to be a subspace, it can be shifted away from the origin. "Conjugate" does not mean orthogonal, he is talking about (multiD) planes conjugate with respect to an ellipsoid. The term is also current, you can ask on Math SE. For lines in 2D see Lewis's Conjugate points and lines. – Conifold Apr 3 at 20:22
• @Conifold that sure looks like an answer -- post it up! – Carl Witthoft Apr 4 at 12:53
• @CarlWitthoft I do not like answering off-topic posts, that only encourages more of them. – Conifold Apr 4 at 16:25