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RECTILINEAR DRAWING¹

I. INTRODUCTION

SUPPOSE that one draws by means of a well-sharpened
pencil many straight lines of uniform, narrow breadth
extending across a sheet of white paper. What kinds of
drawings may be made with this extremely restricted and
artificial medium ? Here we must hold the pencil point
against the paper with a single definite pressure, while mov-
ing the point always at the same velocity.

More precisely, in the idealized form of the problem here
considered, we take these straight lines to be of microscopic
width, so as not to be individually discernible, and to be very
numerous. The problem is then to determine whether or not
a given (idealized) wash drawing can be reproduced by such
(idealized) means, and further, just how it is arrived at.

Let us begin with a very simple illustration of such a prob-
lem: The given drawing is to be such that the surface density
of lead deposited—determining the degree of blackness—is
inversely proportional to the distance from a fixed point O.
Can this particular drawing be made by such rectilinear
means ?

The answer is clearly affirmative. For imagine a very
large number of lines drawn through O in an equiangular

¹In a paper “On drawings composed of uniform straight lines” which has just
appeared in Liouville’s Journal de mathématiques pures et appliquées, I have pre-
sented the same topic from a strictly mathematical point of view.

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