Upon Vail's death on June 28, 1900, a notebook containing numerous geometrical problems was found uncompleted. On the left hand page, Vail details the constraints of a problem, and on the right he solves the question often using meticulous diagrams and proofs. The problems begin as simple geometrical inquiries, clarifying basic concepts and progress towards applications in physics and other sciences. This notebook could possibly be the beginnings of a geometry textbook or workbook, aimed at offering great practice with certain proofs and computations.
The problem shown to the left gives a detailed method for constructing various components of a geometrical proof. Here, Vail goes through the process of first constructing a square, then a circle, then an octagon inscribed within that circle in order to prove a geometrical theorem related to the lengths of different line segments in the drawing.
These problems show the continued interest in mathematics that Vail carried over between careers and his relocation to Santa Barbara. At his death he was no longer teaching, but had moved on to various public service projects and personal endeavours. Despite this, he maintained his passion for mathematical inquiry and continued to pursue the task of sharing his in depth knowledge with students. These sorts of demonstrations are reminiscent of the work he did at both Westtown and Haverford, offering learning for the sake of learning in classic Haverfordian tradition by engrossing oneself within the theoretical principles behind mathematical systems.
All of the questions proposed throughout this notebook demonstrate Vail's meticulous attention to detail, covering a large breadth of material with not only written explanations, but very careful diagrams and constructions. The page shown to the right contains an almost artistic drawing of a volume problem. In this particular example Vail proves various properties concerning the lengths and dimensions of a plane in comparison to certain lengths of the three-dimensional shape in which it is contained.
Shown below are several more pages of Vail's notebook of geometrical problems which show both the original constraints of the problem on the left and the solution on the right. Several of these show very intricate diagrams that Vail drew by hand as a means of visualizing certain geometric properties of shapes. The entirety of the notebook contains approximately 87 straightforward problems regarding specific theorems, and several scientific applications of these properties. These applications include lever and pulley systems, impact, screws, inclined planes, pendulums, and the force of gravity upon various objects. These are of particular interest, since they further demonstrate Vail's general interest in science and the interworkings of systems that affect daily lives.