On Typophile the technical aspects of the em-square have been repeatedly discussed, but the origin of the term seems to be unclear. I did some research and developed some models, of which I discuss and show a couple here. Because of lack of historical documentation, to a certain extend speculation is in this case unavoidable.
The illustration above shows the relation between the height and the width of the capital N from Jenson’s roman type applied in Epistolae ad Brutum from 1470 using a repetition, i.e. fence, of n’s. The counters in the m are in Jenson’s type identical to the one in the n. The height of the N also fits within the stems of the m and subsequently the N fits in a square. The illustration below shows the same relation for Griffo’s capitals from the Hypnerotomachia Poliphili. On top and rotated left is the m, at the bottom a repetition of n’s.
This ‘m-square’ is definitely something else than what is called ‘em-square’ or ‘em’ in contemporary typography. In digital typography, the em-square is a ‘real’ square based on the body size, normally the distance from the top of the ascender to the bottom of the descender and the ‘em’ equals therefore the body or type size. In foundry type casting on the edges of the body was technically not possible and hence the distance between the top of the ascenders and the bottom of the descenders was somewhat smaller than the body.
Nevertheless Moxon writes in Mechanick Exercises that ‘By Body is meant, in Letter- Cutters, Founders and Printers Language, the Side of the Space contained between the Top and Bottom Line of a Long Letter’, which is annotated by Davis and Carter as being ‘Not a good definition because letters are often cast on a body larger than it need be. It is the dimension of type determined by the body of the mould in which it was cast’ (Joseph Moxon [Herbert Davis, Harry Carter, ed.], Mechanical Exercises [New York, 1978]).
In digital type ascenders and descenders can stick outside the body without any (physical) problem. Parts will probably stick outside the em-square anyway, like for instance the diacritics on capitals. Nevertheless, some designers will basically copy the structure of foundry type, just to prevent clipping when zero line spacing is applied.
In the times of the hot metal and photographic composing machines, the em-square was a rectangle that could be square, depending on the design. Vertically the proportions were defined by the body size and in horizontal direction by the width of the widest character, which could be the M or W, which was divided in a certain number of units.
Although the term em-square is often related with the character width of the capital M, which provided the standard for the (division into units of the) em for composing machines, in for instance Monotype fonts the M is not always the widest letter; of a type family for instance the roman capital M could be placed on 15 units and the italic capital M on 18 units, as shown in the schematic representation of a matrix case above. The capital W seems to have been placed by definition on eighteen units and obviously that was part of the original idea: ‘[…] it was decided that the lower case i, l, full point, etc., could be commonly allotted a thickness of five units, the figures and average letter-thickness nine units, and the capital W, em dash and em quad eighteen units’ (R.C. Elliot, ‘The “Monotype” from infancy to maturity’ the Monotype Recorder, No. 243 Vol. xxxi [London, 1931]). The W of for instance of Monotype Poliphilus is much wider than the M.
On the other hand, in The Monotype System, ‘a book for owners and operators of Monotypes’ from 1912 one can read that: ‘The designer of Monotype faces divides the basic character of the font (the cap M) into eighteen equal parts, using one of these parts as his unit of measurement in determining the width of all the other characters in this font’.
Moxon mentions the ‘m Quadrat’: ‘by m thick is meant m Quadrat thick; which is just so thick as the Body is high’ and mentions ‘n Quadrat’ as ‘half as thick as the body is high’. In The history and art of printing from 1771, m and n Quadrats and related variants as ‘Three to an m’ and ‘Five to an m’ are blanks used for indenting and spacing. In An introduction to the study of bibliography from 1814, the function of the m and n Quadrats is described accordingly and furthermore as ‘the square of the letter to whatever fount it belong […] n quadrat, is half that size’.
If m and n stood and nowadays em and en stand for respectively the full and half size of the body, where does the term come from? In Monotype fonts the M is not always the widest letter, but in Moxon’s engraving in which he ‘exhibited to the World the true Shape of Christoffel Van Dijcks […] Letters’ the width of the capital M equalizes the height of the body. The N, however, has not been drawn of half the width of the M. Moxon notes ‘that some few among the capitals are more than m thick’ and he lists Æ, Œ, Q ‘and most of the Swash Letters’ as examples.
The question remains that if the sizes of m Quadrat, m-square and em-square are based on the width of the capital M, why are they not labelled M Quadrat or M-square or EM-square by Moxon and the other named authors? Do the terms ‘m’ or ‘em’ have a different historical background?
The term ‘m Quadrat’ is surely older than its use in Mechanick Exercises. A hypothesis: let’s assume for a moment that the origin of the (e)m-square lays in the lower case m. The relation with the n-square seems to make much more sense then, because the width of the capital N is never half the width of the M. The proportions of the m seem to have been the measure of all –or at least a many– things in Renaissance type. If a square is based on the outside stems of the m of Adobe Jenson and this ‘m-square’ is used to calculate a golden section rectangle (1:1.618) and the height of this rectangle is used for creating a new square, than the ascenders and descenders of (Adobe) Jenson’s type seem to fit perfectly into the latter, as shown above. This square is an extension of the ‘m-square’: an ‘extended-m’ or em-square, although it is perhaps more likely that ‘em’ originates from the rotated m, which reads like an E, in combination with the normally positioned m. If subsequently a square based on the x-height is made and extended to a golden section rectangle, than the proportions of the descenders can be determined.
Assuming that Jenson treated his roman type as a variant of the Textura, it is quite probable that Gutenberg’s type shows the same relation between the ‘m-square’ and the length of the ascenders and descenders. Gutenberg’s Textura from his 42-line Bible (see illustration) indeed shows the relation as in Jenson’s type. The length of the descenders can in this case be defined using a root 2 rectangle.