An Elementary Course in Synthetic Projective Geometry — A Reader’s Guide

Pedagogical Choices in the Preface

Lehmer's preface is unusually candid about instructional design. He notes that he has 'never felt satisfied with the usual treatment' of involution by means of circles and anharmonic ratios, arguing that 'a purely projective notion ought not to be based on metrical foundations.' He therefore develops involution without the customary terms 'hyperbolic,' 'elliptic,' and 'parabolic,' which he finds 'very confusing to the student, who inevitably tries to connect them in some way with the conic sections.' This rejection of metrical crutches is consistent throughout the excerpts.

He also explains his approach to examples: many are 'of sufficient generality to serve as a basis for individual investigation.' For instance, the third example at the end of the first chapter indicates a correspondence between lines in space and circles through a fixed point, and he suggests that tracing consequences of that correspondence will give the student 'no little practice in picturing to himself figures in space.'

Deriving Conic Equations from Projective Principles

The excerpts show Lehmer deriving equations of the hyperbola and parabola using purely projective reasoning, not analytic geometry. For the hyperbola, he starts from a theorem about tangents and asymptotes: 'The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area.' He then uses a parallelogram construction to show that the product of oblique coordinates (with asymptotes as axes) is constant, yielding xy = constant. He notes that this 'identifies the curve with the hyperbola as defined and discussed in works on analytic geometry.'

For the parabola, he defines it as 'a conic which is tangent to the line at infinity.' Using Brianchon's theorem applied to a circumscribed quadrilateral, he derives that 'the segments cut off on any two tangents to a parabola by a variable tangent are proportional.' From this he obtains the relation y² = 2px, again identifying the curve with its analytic counterpart. These derivations demonstrate how projective properties can yield familiar equations without coordinate geometry.

Terminology and Notation as a Window into the Subject

Lehmer's terminological choices reflect his synthetic approach. He uses 'pencil of rays' for lines through a point, and 'point system' for a point considered as all lines and planes through it. He rejects 'foci of an involution' and the classification into hyperbolic, elliptic, and parabolic involutions. These decisions are not merely cosmetic; they aim to keep the development free of metrical or conic-associated ideas.

The excerpts also reveal a consistent notation: points are labeled with capital letters (A, B, C, O, P, Q), lines are named by two points (AB, CD), and figures are referenced by number (Fig. 29, Fig. 30). Theorems are stated in italics, as in 'The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area.' The text includes occasional references to earlier sections (e.g., § 88, § 110), indicating a structured progression. The examples at chapter ends are numbered, and the preface mentions that the third example of the first chapter is 'very fruitful in interesting results.'

Readers should note that the excerpts cover only the preface and portions of later chapters on conics. The book's earlier sections—on fundamental concepts, harmonic division, projectivities, and involution—are not represented here. Lehmer's preface suggests a careful sequence building from axioms to conics, with each step justified synthetically. The examples he mentions may be key to understanding his method; readers are encouraged to work through them as he intended, rather than skipping to the equations.

Lehmer’s old book made me wistful for a time when geometry felt like hand-tooled craft, not a race to equations. Flipping through his careful renaming and quiet rejections, I half-remembered why I’d loved it. That same tenderness surfaces in The Way To Geometry — Edition Insights, almost as if two old friends had met.