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David's correction to my formulation for UPM actually comes very close to defining the em:
The em is a vertical segment of the cartesian grid on which glyphs in a digital font are plotted, whose height is such that it matches the nominal point size of scaled type. From this is derived the em as a unit of proportional measurement in typesetting.
This thread is dated, but because it is linked from Wikipedia I will add my 2 cents as a retired typographer whose career spanned the transition from hot metal to cold type.
This conversation has suffered from the nearly complete omission of the scalable typographic measurements' purposes. With those principles in mind, several of the offered definitions are clearly wrong.
Taking the various spaces in order of their simplicity of explanation, the em quad and en quad are units of horizontal white space measurement left over from the late hot metal days that have no modern usage. They were used to fill an incomplete line of text to justify it. Justification in the hot metal days imported a sense of physical pressure as well as text alignment.
For handset, pre-cast movable type setting, a line of individual characters had to "lift;" that is, when locked up in a steel "chase" frame under pressure for placement on a letterpress, the physical pressure had to be right so the line of type would neither push other lines out of alignment nor fall out of the composition. Filling incomplete lines of text with quads and then progressively thinner spaces to achieve the right pressure was called "quadding." Some "quads" were multiples of an em, some were normal em spaces, then there were the en quad and the various thin spaces. In movable type, there was no em or en quad in reality; they were simply the em and en spaces put to a different use.
When working with hot metal line casting machines such as the Linotype, Intertype, and Monotype, "quadding" likewise used em and sometimes the en space along with expandable space bands to fill incomplete lines. Very important stuff when a "loose line" had the nasty habit of persuading the machine to squirt molten lead on you. Because of physical limitations on the number of character molds (matrices or "mats") that could be supported in the Linotype and Intertype line-casting machines's systems for automatic sorting and recycling of mats, their operators did not commonly use quads that were multiples of the em.
In later years, many of the hot metal line casting machines were equipped with mechanical or hydraulic "quadders" that could achieve flush left, centering, or flush right by expanding one or two steel bars to fill empty space in a line rather than by manual insertion of quad matrices ("mats") and space bands. They were a time saver, particularly for centering and flush right lines, since character mats were automatically added left to right and right and center justification required manual manipulation of the quads. Despite the innovation of mechanical and hydraulic quadders, most line casting machines were not equipped with them by the end of the hot type era.
Bottom line: The em and en *quads* are no longer relevant in the digital age. Their survival in the Unicode standard is a relic of the very short period of time when computerized hot metal typesetting was the norm. The name "quad" was undoubtedly a contraction of the geometry term, "quadrangle"
That was the ultra-simple to understand part. But it brings us to the real reason for the fixed width of the various typographic spaces within a given point size. Quite simply, they are typographic devices for alignment of columnar matter. They were largely deprecated in the early years of digital typesetting, having been replaced by fixed position tabs, tables, and such devices. But they are enjoying a well-deserved resurgence in an era that stresses the ability to recycle data in sizes and faces different from the original using, e.g., the various offspring of IBM's General Markup Language (GML), such as Unicode, XML, XHTML, CSS, and XSL:fo.
The one alignment use of the em and en spaces that was discussed in this conversation is paragraph indentations that will scale to other type sizes without revisiting fixed-width tabs or styles used to achieve indentation. That simple example illustrates one major advantage of the scalable typographic measurements as compared to fixed-position tabs and table cells. However, paragraph indentations do not capture the necessity of maintaining the width of the various typographic spaces in all type faces of a given point size. There is no particularly compelling reason that paragraph indentations not be set at arbitrary widths so long as all faces of the same size -- e.g., Roman, bold, and italic -- used in a composition have the same indentation width.
The real trick to understanding the scalable typographic spaces is to realize that not all characters in a variable-width type face are variable and the scalable typographic spaces are devices for alignment of columnar matter.
The widths of the scalable spaces correspond to various visible characters. E.g., the em space corresponds to the width of the em dash and em leader; the width of the en space corresponds to the width of the plus and minus signs, the slashes, numerals, currency symbols, and the en leader; and the various thin spaces correspond to various marks for punctuation.
There is some variability in the particular width of thin spaces corresponding to punctuation marks, with the newspaper publishing industry largely standardized on common punctuation 1/3 em wide and the book and academic publishing industries largely standardized on punctuation 1/4 em wide, with thinner bodies for some punctuation marks such as the single quotes.
The reason for the scalable typographical system of measurement is best explained by example. Let us assume that you have text followed by one column right aligned and another column flush right. In many typesetting systems, the center column would be right aligned using a tab. The example won't be captured accurately on this page because of the page editor and rendering limitations, so it will take a bit of imagination to understand. (This web application does not recognize the relevant ASCII or HTML space characters and strips multiple spaces.) Consider this example:
[Text A] $ 1,037 E 824
[Text B] 17 15
[Text C] -- --
[Text D] 18,046 14,074
The double hyphens would be em dashes when typeset and the "E" in the TextA line is my quick substitute for the Euro symbol. The right alignment would be achieved in the first numerical column by calculating and adding the required number of em, en, and thin spaces after that column to compensate for the variability in the right-most right-aligned column. Then the line would be quadded to fill the space between the text and the center column to justify the line.
Likewise, the system allows you to combine and align type of different sizes. Consider the formatting that often appears in newspaper advertisements with four lines of 12 point justified type with a 48 point price on their right and the last line of 12-point justified with leaders extending to the price. If the price in 48-point is $49.95, the experienced typographer can quickly calculate mentally that the $49.95 price in 48 point is two ems, on en, and one thin space wide (I'll assume a period equal to a one-third em thin space in the price). That width corresponds in 12 point to eleven ems plus a thin space for the 12 point indent on the right. Add another em in 12 point for horizontal separation from the price, and the 12-point lines can be left and right aligned with the desired separation from the 48 point price.
In the hot type days, that kind of formatting could be achieved in only two ways, using the method described with the 48-point price set on the first 12 point line with an overhang across the three lower 12 point lines or by setting the 48-point and 12 point type on separate lines that were butted together. The butted lines method simply asks for errors in composition when associating prices with their accompanying texts. Moreover, butting of lines in general was considered to be a composition method whose usage was to be avoided if feasible because it often introduced "lift" and alignment problems once the composition was locked up.
Such calculations may seem unwieldy if you are not used to doing them. But for experienced typographers, they are near-instinctive. Also, there at least used to be "em" line gauges in common usage for quickly measuring em widths in different sizes so that corrections could be quickly achieved without experimentation to determine how another typographer aligned the original type.
From the above discussion, it can be divined that there is no permissible variability in the widths of em and en spaces among given type faces in a particular point size. A change in type faces from one line to the next must not introduce horizontal alignment issues. So, for example, it is incorrect to say that the width of the em or en space can validly be affected by the amount of leading (additional line spacing) applied between lines of type. All 8 point ems are the same width; just so with 10 point ems. Were adjustments made to the width of the em or en based on leading between lines, the ability to easily align the visible columnar matter would disappear. The width of the em and en are tied to the width of their corresponding visual characters, not to the distance between lines' type baselines. Adjusting the em size based on linespacing or leading would destroy the system.
Much confusion has emerged from the habit of typographers in the hot era of referring to vertical measurements in "ems." When hot type typographers did so, it was understood that ems applied to vertical measurements were 12-point ems, equal to one pica. The habit of referring to vertical picas as "ems" was because there was no commonly understood markup symbol for vertical measurements in picas. But the depth of a desired box, for example, could be easily indicated as being 12 picas deep simply by drawing a blue pencil square em space symbol, writing the number 12 in it, then drawing arrows to indicate that the desired measurement was vertical. (In the specification of line widths, the pica was implicit in the markup that combined the type size and line length separated by a dash, with any deviations from standard leading indicated by a line drawn beneath the type size and line length numbers and a number below the line indicating the desired leading. So "10-20" over "12" indicated 10 point type 20 picas wide set with two points of vertical leading.)
Much of the confusion in modern typography traces to the near-overnight collapse of the typographical trade with the advent of computerized "cold" typesetting that led to the disappearance of the typographical trade in the span of a decade or so. The typographers had been the proud custodians of an oral tradition passed down for nearly 500 years.
The typographers' overnight disappearance marked the ascendency of the "graphic artists" who for the most part lacked more than rudimentary knowledge of typographical skills. At the time, "graphic arts" had been a fairly minor trade mostly composed of graphic designers from the various ad agencies. They generally had few skills in actually composing type, but farmed out their various typesetting and composition work to commercial typographers, the actual relevant tradesmen. (I owned such a business.)
About the time that the typography trade was dying, academia stepped in with programs in "graphic arts" to provide staff for the rapidly growing demands of ad agencies for graphic artists. But academia itself lacked the skills of the typographic tradesmen. The typographers themselves were nearly all left to find new lines of work; for example the International Typographical Union, then the oldest trade union in North America and an AFL-CIO powerhouse, folded; its few remaining typographers merged with the Communications Workers of America in 1987.
The irony for me? -- That single sentence in the first volume of the union's 1951 edition of Lessons in Printing that boldly claimed typographers need never worry about their jobs being replaced by automation because human languages were too complex to be computerized.
The difficult part for me and for the tens of thousands of other typographers around the world who lost our jobs to automation was watching the accumulated 500-year-old wisdom of the typographer's trade -- that incredible oral tradition -- being chucked in the dumpster. Only a miniscule amount of that wisdom was ever reduced to writing and certainly very little of it was widely automated. Those of us who shared the oral tradition now endure the daily shock of viewing a horde of graphic materials, e.g., web pages, newspapers, magazines, and television screens, that defy fundamental principles of typography. I see perhaps 3 or 4 web pages per year that actually approach the former typographical standards of quality. It is a great sadness. The world lost something very precious. Today's far lower general typographic standards are an embarrassment to humanity. Soon, the last of us will be gone.
So I feel a moral imperative to add to conversations such as this one, explaining a bit of the lore and wisdom of typography. It is gratifying to see the scalable typographic measurements in resurgence. But I shed a mental tear whenever I overcome the sadness enough to preserve yet another shred of the history of typography. It is a bittersweet task.
Buck, there are some books which attempt to pass on the knowledge of typography to the current generation. Robert Bringhurst's 'Elements of Typographic Style', James Felici's 'The Complete Manual of Typography' and Mitchell & Wightman's 'Book Typography' are three excellent books in this line.
And not least, now on Typophile there are some with long experience in typography, including the hot metal era, posting with their wisdom on the topics that come up.
So welcome to Typophile, and I hope you will contribute your insights on the topics that come up here. And if you feel that the books above are seriously lacking in some area, I would love to hear about it, and I'm sure others on Typophile would as well.
[There is a typo in your post, where you wrote '6 points' instead of '12' to a pica. You can correct posts using the 'edit' link.]
Thank you William, for the kind words, the invitation, and the proofreading. I will beyond doubt accept the invitation to post more often.
I also hope you post more!
This issue was raised initially by me to correct the misconception in Wikipediadom that the em square has something to do with typesetting.
I was seeking to clarify what the interface between type designer and composition system is, because most trustworthy typographic intelligence in between the type designer's em and the users capabilities is, as you say, gone.
So, the last solid technical bastion of typography that remains in modern systems, upon which whatever quality is to be built, is the EM SQUARE of type design. It's probably still wrong 'n wikidom, but I don't care as long as there is at least one right em in modern fonts. ;)