Here you can see a page from Stephen Hawking’s PhD dissertation Properties of Expanding Universes (1966), where he filled out the equations by hand.  IBM’s Model C and Model D typewriters were popular office machines known for their durability and standard fixed-width fonts (usually 10 or 12 characters per inch). Unlike later IBM Selectrics - which had interchangeable typeballs - these earlier models used traditional mechanical typebars arranged in a basket. Precise typographical layout with these machines was challenging, as spacing and character sets were rigidly limited.  *IBM Model C*

 *IBM Model D* Here is the article: https://computerhistory.org/blog/discovering-dennis-ritchies-lost-dissertation/ An electric typewriter mechanically prints characters directly onto paper using fixed spacing and limited fonts, making precision layout and special symbols difficult. An electronic typesetter, however, digitally arranges text with precise control over fonts, sizes, spacing, and alignment, ideal for mathematical and technical documents - which can later be printed. The IBM 2741 was a computer terminal introduced in 1965 that used the Selectric typewriter mechanism as its printing engine. In this context, a terminal was not a computer itself, but a device used to send and receive text from a mainframe - a large, centralized computer located elsewhere. The 2741 had to be connected to a mainframe to function, typically over phone lines or dedicated data links. It offered much higher print quality than line printers or teletypes of the time and was used in research labs, universities, and places like Bell Labs and MIT’s Project MAC. Because of its precision and support for interchangeable typeballs, it was one of the few devices in the 1960s capable of producing the kind of clean, symbol-rich output seen in Ritchie’s thesis.  *IBM 2741 Terminal* ## troff and eqn **troff** (short for *typesetter roff*) was developed at Bell Labs in the early 1970s by Joe Ossanna and later improved by Brian Kernighan. It was written in C and designed to format text for high-quality typeset output, especially on Unix systems. The name “troff” comes from a chain of earlier tools: RUNOFF (from MIT) → roff → nroff (“new roff”) → troff (“typesetter roff”). Here’s a basic `troff` example: ``` .TL Unix Programmer's Manual .AU Dennis Ritchie .PP This is a paragraph in a troff document. ``` **eqn** was introduced in 1974 by Brian Kernighan and Lorinda Cherry as a `troff` preprocessor for typesetting mathematics. It let users write math expressions in plain text that would be rendered cleanly in print. Example: ``` .EQ e = m c sup 2 .EN ``` This would render: $e = mc^2$ Here you can see the difference in alignment of Roman numeral lists:  Ritchie entered Harvard in 1959 and did finish an undergraduate degree in physics in 1963. Later he joked that his “undergraduate experience convinced me that I was not smart enough to be a physicist, and that computers were quite neat” The monus symbol (∸) is used to represent truncated subtraction. It behaves like regular subtraction, except the result is never allowed to go below zero: $$ a ∸ b = \max(a - b,\ 0) $$ For example: $$5 ∸ 3 = 2$$ $$2 ∸ 5 = 0$$ This operation is common in theoretical computer science, especially when working with natural numbers or defining functions in models where negative values aren’t permitted IBM’s Selectric typewriter (1961) was revolutionary due to its spinning typeball - nicknamed the “golf-ball” - which replaced the conventional basket-and-typebar system. With the Selectric, the paper stayed stationary while the typeball moved across it, dramatically improving typing speed and print clarity. Importantly for mathematical and scientific typing, users could quickly swap typeballs to insert special characters like Greek letters or mathematical symbols, rather than manually inserting separate type sticks. This greatly simplified producing technical documents before digital typesetting became common. 
*IBM Selectric* Y&Y’s MTMS script refers to a digital font collection developed by the company Y&Y, which specialized in high-quality TeX and PostScript font support in the 1990s and early 2000s. MTMS stands for Mathematical Typesetting Math Script, a family of script-style fonts designed for use in mathematical and scientific documents. These fonts mimicked handwritten or calligraphic math script (used for things like ℒ, 𝓕, etc.) and were compatible with typesetting systems like TeX and troff. Harvard in the late 1960s followed a tight three-step routine. First a typed draft, usually from an IBM Selectric, went to the dissertation committee for edits. After fixes, the candidate held a small departmental defense, then a Public Oral Examination advertised two weeks in advance in the Harvard Crimson. Passing the orals unlocked the paperwork stage. The student had to produce a flawless typescript on acid-free paper, margins 1 ½ in. left and 1 in. elsewhere, each page initialed by the advisor. Three hard-bound copies were made at a local bindery: one for University Archives, one for Widener Library, one for the department. A bindery receipt plus the signed degree form had to reach the registrar by the semester deadline; otherwise the degree slipped to the next term. In the late 1960s, when Ritchie wrote his thesis, there were no digital typesetting systems like LaTeX or Word. Mathematical documents were usually typed on electric typewriters—often by hand or by hired typists. Symbols like Greek letters or integrals had to be inserted manually, sometimes using special typeballs on IBM Selectric machines, or drawn in by hand afterward. Subscripts and superscripts were produced by physically adjusting the paper in the typewriter - rolling it up or down by tiny increments. The first widely used computerized formatting tool, RUNOFF, had just been written at MIT in 1964, but it couldn’t handle math. Troff and eqn, which could handle mathematical typesetting, wouldn’t arrive until the 1970s. LaTeX came much later, in the 1980s. So at the time of Ritchie’s thesis in 1968, producing a clean, complex math thesis with precise formatting required a level of mechanical precision, patience, and ingenuity that most people today associate with programming - not typing. I notice that the colons in the upper braced section are misaligned with the subscripts. I notice that the colons in the upper braced section are misaligned with the subscripts. Dennis Ritchie was born in 1941 in Bronxville, New York, and grew up in Summit, New Jersey. His father worked at Bell Labs, which exposed him early to science and engineering. At Harvard, he started in physics but became interested in computing after attending a lecture on the UNIVAC I. He switched to applied mathematics for graduate school, where he began exploring theoretical computer science.  *Dennis Ritchie at Harvard* RUNOFF, created by Jerry Saltzer at MIT in 1964, was one of the earliest text formatting programs. It allowed users to produce formatted documents from plain-text inputs with simple markup commands (for instance, line breaks, paragraphs, or page numbers). However, RUNOFF was relatively advanced for the era and designed specifically for MIT’s CTSS operating system, limiting its portability. When Brian Kernighan needed a formatting tool for his own Princeton PhD thesis in 1968, he developed a simplified version called Roff. Written in Fortran for the IBM 7094, Roff removed many specialized features to fit within the constraints of his environment. Despite its limitations, Roff’s simpler design and greater portability made it easier to adapt to new computing environments. This eventually helped it become a foundational text-processing tool at Bell Labs, evolving into nroff and later troff, which significantly influenced document formatting in Unix systems. Think of a loop program as working with a handful of “registers,” each holding a non-negative integer. You are allowed only three basic instructions: ``` X ← 0 # reset X to 0 X ← X + 1 # add 1 to X X ← Y # copy the value of Y into X ``` And one control construct: ``` LOOP X TIMES … instructions … END ``` The body runs exactly the number of times currently stored in `X`. Because loops can be placed inside other loops, the total number of iterations can grow explosively, letting these tiny building blocks mimic any algorithm. #### Example: Multiplying 3 × 4 Registers: `A = 3, B = 4, R = 0` ``` LOOP B TIMES # outer loop runs 4 times LOOP A TIMES # inner loop adds 3 to R each pass R ← R + 1 END END ``` After the nested loops finish, `R` holds 12. Even with nothing but resets, increments, copies, and fixed-count loops, you can write addition, subtraction by counting down, conditional tricks (store a flag, skip work when it is 0), and eventually simulate a full Turing machine.