This research essay aims to underline the significant role

mathematics play in typography, specifically in the creation of a typeface, by

defining a connection between typography and mathematics (particularly

geometry). They first part of the essay will consist of a brief history of

typography, underlining the key turning points it as had across time especially

around the sixteenth century. Further on, the next part will focus on

mathematics as a basis for the development of type, in particular the geometric

aspects, such as the golden ratio. A questionnaire will be at the base of the primary research, examining

whether there is any justification for optically correcting mathematical

proportions in typography, and whether doing is so is noticeable to the common

observer.

Designers and artists have been long fascinated by geometry. Contemporary

typographers view typography as a craft with a very long history, tracing its

origins back to the first dyes and punches used to make seals and currency in

ancient times. These basic elements of typography are as old as civilization

and the earliest writing systems, however a series of key developments were

eventually drawn together into a singular systematic craft. (Steinberg, 1961)

Renaissance artists’ and scholars’ fascination with mathematics

included the domain of the alphabet, and they meticulously studied the styles

and proportions of engraved capitals still visible on classical Roman

monuments. (Loxley, 2006) The

development of geometric perspective in the 1420s convinced artists of the

period that the whole of reality could be expressed in numerical formulae. At

the same time, the works of Greek mathematicians such as Pythagoras were deemed

to be the high point of ancient civilization (whose perfection arose from the

systematic, rational application of the laws of geometry). (Guthrie, 1988)

The concept of defining letters mathematically is by

no means new, as it goes back to the fifteenth century and it became rather

highly developed in the early part of the sixteenth century. In a time where

Renaissance men combined mathematics with the real world, a particular interest

in constructing majuscules with rulers and compasses arose. There are numerous

influential books published at the beginning of the sixteenth century, authors

varying from mathematicians to calligraphers to typographers. Some of the most

notable include Albrecht Dürer, Francesco Torniello da Novara, Geoffroy Tory.

Luca de Pacioli, an Italian mathematician, studies the proportions of classical

Roman letters in his book De divina

proportione written in Milan in 1496-98. This was the first printed book

known to describe the Roman alphabet with the help of Leonardo da Vinci’s

drawings. (Knuth, 1979)

The years between the mid-15th century and the early 18th

century proved to be a time of many changes and developments in the world of

typography, with the development of the printing press greatly influencing the

development of full typefaces and their production. (Eisenstein, 1996) In the 15th century, Albrecht Dürer and Francesco Cresci tried to

rationalize letterforms in accordance with geometric principles from a

lettering copy-book. It was not until the twentieth century, however, that

geometry began to influence type in a direct way.

Mathematical typesetting has always challenged the printer:

“Gentlemen

authors should be very exact in their Copy, and Compositors as careful in

following it, that no alterations may ensue after it is composed; since

changing and altering work of this nature is more troublesome to a Compositor

than can be imagined by one that has no tolerable knowledge of Printing. Hence

it is, that very few Compositors are fond of Algebra, and rather chuse to be

employedon plain work, tho’ less profitable to them than the former; because it

is disagreeable and injures the habit of an expeditious Compositor.” (Smith, 1998)

In 1967, Edward Wright used geometry for his architectural

lettering for the Metropolitan Police headquarter building in London. Related

in part to the Bauhaus idea, geometry was the basis for Paul Renner’s Futura of

1927-9, as pattern drawings from the Bauer Typefoundry.

Nationality does not easily apply to type and culture, as Yvonne

Schwemer-Scheddin has stated: “The concept of ‘nation’ is political, whereas

script is connected directly to language and its geographic linguistic areas.” Languages,

scripts, and typefaces which have represented centuries were intrinsically

related to each other. Earlier Roman capitals became the typographic system

dedicated to monumental inscriptions during the Roman Empire, and later on the

Carolingian minuscule was adopted as the official script for all of the Charlemagne

Empire. In Ireland, as can be seen in books, the Semi Uncial was in use. During

the early years of civilization, unlike in our globalized world, there was no

easy way of communication between various parts of the world.

The idea of defining letters mathematically is in no way new, it

goes back to the fifteenth century and it became rather highly developed in the

early part of the sixteenth century. In a time where Renaissance men combined

mathematics with the real world, there was a particular interest in

constructing majuscules with rulers and compasses. The first person to do so is

thought to be Felice Feliciano, around 1460, whose handwritten manuscript in

the Vatican Library was published 500 years later. Feliciano was an excellent

calligrapher who wanted to a mathematical foundation applied to the principles

of calligraphy. Several other fifteenth century authors made similar

experiments, however the most notable work of this kind appeared in the early

sixteenth century.

Italian mathematician Luca Pacioli, who had previously written the

most influential book on algebra at the time, also published De Divina Proportione (1509), a book

about geometry and the golden section, which included an appendix about

alphabets. Francesco Torniello da Novara published another notable Italian work

on the subject in 1517 titled Opera del

modo de fare le littere maiuscole antique, which focused on enriching, both

geometrically and calligraphically, previously existent fonts. His works

generally focused on adopting the Latin alphabet inscriptions as originally as

possible, whilst improving their geometric conditions. These fonts were not

designed for the printing press, but as a model for artistic inscriptions. He

was also the first to define the ‘point’ as the measure in typography. Similar work appeared in France and

Germany; The German book was probably the most famous and influential: Albrecht

Dürer’s Underweysung der Messung (1525), an

instructional manual in geometry for Renaissance painters. Another rather

popular book was Geoffroy Tory’s Champfleury

(1529), which set a standard for French publishing that in many ways is

still followed today. Nobody carried this work further to minuscule, numerals,

or italic letters and other symbols, until 100 years later when Joseph Moxon

made a detailed study of some letters designed in Holland. The ultimate

refinement of this mathematical approach took place shortly afterwards when

Louis XIV of France commissioned the creation of a Royal Alphabet. A commission

of typographers and artists worked on Louis XIV’s project for over ten years,

starting around 1690.

Francesco Torniello da Novara was

a Milanese typographer who became known for applying geometric specifications

to Latin majuscule fonts. Torniello’s works concentrated on adopting the Latin

alphabet inscriptions as originally as possible, while also improving their

geometric conditions. These fonts were not conceived for the printing press,

but as a model for artistic inscriptions. He designed an 18×18 which served as

a coordinate system for his geometric fonts, which were designed for printing

press usage. In his book Opera del modo da

fare le littere majuscole antique, published in 1517, Torniello improved,

both geometrically and calligraphically, pre-existing fonts. Geometrical

conditions were added to the letters “M”, “R”, “S”, and “T” – in order to

broaden the usage of his fonts in non-Latin texts, he also added the letter

“Z”. He was the first typographer to define the ‘point’ as unit of measure in

typography.

Thanks to the reverential and

detailed reference Torniello made in his work regarding Luca Pacioli, it is

safe to assume they must have either met, or Torniello had access to his work Divina Proportione. Torniello’s work

contains geometric representations of all Latin majuscules. Pacioli’s typeface

at this time represented the culmination of development in the Renaissance, which

strove to adopt the Latin letters of ancient inscription and tombstones as

faithfully as possible. This development began in 1460 with a Vatican archived

manuscript by Felice Feliciano, and was continued by Damianus Moyllus (or

Damiano da Moyle?) in either 1483 or 1484. As tools for the

design of individual letters, only rulers and compasses were available.

However, up to and including Pacioli’s work, only the finished designs were

shown along with some design elements such as circles used and the surrounding

rectangle or square. Furthermore, a complete written specification matching the

charts was missing. In this regard, Torniello’s work broke new ground. This

begins with Torniello’s grid that served as the coordinate system. The double

side length of a grid was defined as a ‘point’. Thus, this became the first

known definition of a point as a typographic unit of measure. However, it is

unlikely that Pierre Simon Fournier knew the work of Torniello when he also

introduced the point in 1737 as a typographic unit. The point as a unit of

measure and the grid greatly facilitated Torniello’s specifications in writing.

While Torniello largely took over

the designs of Pacioli, he deviated significantly from his model for the

letters “M”. “R”, “S”, and “T”. In particular, the letters “S” and “T” were

enriched by calligraphic details, which were also occupied by later

typographers. For example, the design of the parallel diagonally cut upper

serif of the “T” in the alphabet by Albrecht Dürer, can be found again. Only

the “M” was considered less successful in the comparative analysis of Giovanni

Mardersteig. Furthermore, Torniello added “Z” to his alphabet, which can be

taken as evidence that these fonts have also been increasingly used for

non-Latin texts. By the letters “R” and “S”, other peculiarities of Torniello’s

construction drawings can also be recognized. Thus, he drew for each circular

arc used in each case the complete circle segment together with the indication

of the respective radius, which was measured in points.

Torniello’s work and the alphabets of his predecessors,

especially those of Pacioli and Sigismondo Fanti (1514), significantly

influenced the further development. Giovambaptista Verini published in 1526 in

the third volume of his work Luminario

an alphabet that derived largely from the designs of Pacioli and Fanti, however

Torniello took over the definition of the point as the ninth of the page

length. In contrast to Pacioli and Torniello, the much larger circles in the

serifs are noticeable, so that they appear much heavier.

In 1528 Albrecht Dürer published his designs for his

alphabet, which he had made in 1525, in Nuremberg. Dürer knew his Italian

predecessors quite well and was guided mainly by Pacioli and Torniello. He was

the first to present both variants, the completed letter based on Pacioli and

the construction sketched in the same way as Torniello’s model, side by side.

Dürer took over the screening from Torniello as well. The sizes of the serifs are,

however, more similar to those of Pacioli. Though they are slightly less

graceful than Torniello’s, they are also much smaller than Verini’s. Shortly

afterwards, but in full knowledge of the work of Albrecht Dürer and the Italian

role models, Geoffroy Tory published his work Champfleury in Paris in 1529. He took over from Albrecht Dürer the

dual representation according to Torniello and Pacioli and also used a grid,

which is dimensioned in contrast to that of Torniello to 10×10. For the serifs, however,

much larger circles were used, which are more alike to those of Verini than the

smaller circles of Torniello or Pacioli. The departure of the circle as a tool

of typography was initiated by Ludovico degli Arrighi in 1523 and in particular

by Francesco Cresci in his work Essemplare

di più sorti lettere of

1560. Cresci re-studied the ancient inscriptions and came to the conclusion

that compass-based constructions do not do justice to the originals. For

example, the adjacent “B” is based on the corresponding form in the inscription

on the pedestal of Trajan’s Column in Rome. Cresci concludes that “in drawing

every curve of each letter they make more circles than a sphere for the most

part contains. I have come to the conclusion that Euclid, the prince of

Geometry, returned to this world of ours, he would never find that the curves

of the letters could, by any means of circles made with compasses, be

constructed according to the proportion and style of ancient letters.” (Swetz,

1996)

“The

type designer – or better, let us start calling him the alphabet designer –

will have to seek his task and his responsibility more than before in the

coordination of the tradition in the development of letterforms with the

practical purpose and the needs of the advanced equipment of today…” (Zapf,

1970)

The new challenges have

resurrected the quest for the perfect alphabet with an included dimension of

technological accommodation. Intrigued by this challenge and recognizing “a

good mathematical problem still waiting to be resolved”, in 1977, Donald Knuth

of Stanford University decided to tackle the problem of mathematically

designing the “perfect letter”. Systematically approaching this task, Knuth

defined the problems as one of finding the “most pleasing” closed curve, MPC.

He then postulated a set of six axioms to clarify the concept of “most

pleasing”. (Knuth. 1979)

Knuth states the most pleasing

closed curve is one that goes through any n given points in consecutive order,

and the returns to the first point. He postulates six axioms that the most

pleasing curves should satisfy. The first property is invariance, which states

that if any of the given points are rotated or expanded, the most pleasing

curve is to be rotated or expanded in the same way. The property of symmetry says

that any cyclic permutation of the given points does not change the solution,

meaning that if any two of the given points were to switch positions, the curve

would not seize to be a pleasing curve. The property of extensionality explains

that adding a point that is already on the most pleasing curve does not change

the solution. The fourth property is locality, which states that each segment

of the most pleasing curve between any two of the given points depends solely

on those points and the ones immediately preceding and following. According to

the locality property, changes to one part of the pattern won’t affect the

other parts. Taking this into consideration, the search for the most pleasing

curve is simplified, because the problem only needs to be solved in the case of

four given points. This proves to be a great simplification in designing

letters as well, since individual portions of strokes can be dealt with

separately. A way to satisfy all four of these properties is simply to make the

most pleasing curve consist of straight line segments. However, the property of

smoothness implies that there are no sharp corners in the most pleasing curves,

a unique tangent at every point of the curve. Find a way

to describe the math behind defining the perfect curve (with illustrations).

Guided by these postulates,

Donald Knuth derived his MPC as a piecewise continuous curve, where each

segment is determined by an appropriate cubic spline. Appropriate MPCs have

been devised for each letter of the alphabet and loaded into a computer

program. The resulting system, called METAFONT, is flexible and allows its

user, through a variation of parameters, to devise an infinite variety of

letter forms and fonts. Thus the concept of the “perfect letter” now truly lies

within the eye and control of the METAFONT user. Knuth has completed the task

attributed so long ago to Pythagoras, laboured upon by the artists and

calligraphers of the Renaissance and pondered by the Jaugeon Commission. (Show

visual comparison between METAFONT composition and those proposed by

Renaissance masters) Donald Knuth’s resolution of the situation

while ultimately prompted by a mathematical challenge, was also motivated and

facilitated by the existence of new technologies. The scope and power of

METAFONT and its companion TEX typesetting system has revolutionized the field

of typography and demonstrated how mathematics under the rethinking of an old

problem can be applied to new fields. Interestingly, these new resulting techniques

of typography have not narrowed choices as to the “perfect letter”, rather they

have broadened options and personalized the decision of perfection. (Swetz,

1996)

A typeface has the ability to

evoke a personality, a characteristic or even a feeling, as perceived by the

observer. This particular point stems directly form perception, thus observers

may not perceive unanimously. Typography has the responsibility of rendering

specific characters representing speech sound and instruction, and “A” will

always be an “A”, but the governing rules and subtle beauties defining the

design of each individual typeface are what influence perception. Erik

Speikermann noted that “if letters are the clothes that words wear, then it

surely follows that there must be as many typefaces as there are voices,

languages, and emotions.” (Mawby, 2016)

“Like oratory, music, dance,

calligraphy – like anything that lends its grace to language – typography is an

art that can be deliberately misused. It is a craft by which the meanings of a

text (or its absence of meaning) can be clarified, honored and shared, or

knowingly disguised.” (Bringhurst, 2004).

The perception of the voice of typography comes from a richly varied

combination of formal associations with typography, temporal perception in its

reading, and a sense of visual rhythm. (Bringhurst, 2004, p.45)

In 1847, Oliver Byrne wrote and

designed an illustrated book so far ahead of its time that it could have been published

today. The First Six Books of the

Elements of Euclid prefigures the art and design of the 20th

century avant-garde movements. Although the illustration on the title pages is

tremendously similar to that of a de Stijl and Bauhaus design, the mid-19th-century

publishing date disqualifies it from being “modern”. (Heller, 2010)

Every high school student has

suffered though Euclid’s fundamentals of geometry, which is why Byrne stated:

“The

arts and sciences have become so extensive, that to facilitate their

acquirement is of as importance as to extend their boundaries. Illustration, if

it does not shorten the time of study, will at least make it more agreeable.”

(Byrne, 1847)

The colour symbols – circles,

squares, diamonds etc. – effectively substitute for key works, making

comprehension much simpler. By the time the complicated theorems and formulas

appear, toward the end of the book, the reader is fluent in the visual

language. (Heller, 2010)

“This

work has a greater aim than mere illustration; we do not introduce colors for

the purpose of entertainment, or the amuse by

certain combinations of tint and form, but to assist the mind in its

researches after truth, to increase the facilities of instruction and to

diffuse permanent knowledge. .” (Byrne, 1847)