1.  Introduction

      ``Mathematics is known in the trade as difficult, or penalty, copy because it is slower, more difficult, and more expensive to set in type than any other kind of copy normally occurring in books and journals.'' [1]

      One difficulty with mathematical text is the multiplicity of characters, sizes, and fonts. An expression such as

requires an intimate mixture of roman, italic and greek letters, in three sizes, and a special character or two. (``Requires'' is perhaps the wrong word, but mathematics has its own typographical conventions which are quite different from those of ordinary text.) Typesetting such an expression by traditional methods is still an essentially manual operation.

      A second difficulty is the two dimensional character of mathematics, which the superscript and limits in the preceding example showed in its simplest form. This is carried further by

and still further by

These examples also show line-drawing, built-up characters like braces and radicals, and a spectrum of positioning problems. (Section 6 shows what a user has to type to produce these on our system.)