#### John Napier

John Napier was born in 1550 in Merchiston Castle, Edinburgh into an influential Protestant land-owning family. Little is known about John Napier’s early years, however, there does exist a letter from Napier’s uncle, the Bishop of Orkney, to Napier’s father, Archibald Napier, urging Archibald that John be sent to school in France or Flanders when John was 11. Nevertheless, Napier began his education at St Andrews University, matriculating in 1563 at the age of 13. He lived in St Salvator’s College and his mother made special arrangements for John Rutherford, the principal of the university, to take care of him personally. It was while Napier was in St Andrews that he first gained his passion for theology, and subsequently went on to publish (what he considered) his most influential work *Plaine Discovery of the Whole Revelation of St John* in 1593. There is no evidence of Napier graduating from St Andrews University, but he moved on to study in Europe (although nobody knows where).

Napier had returned to Scotland by 1571 as he was present for his father’s second marriage. Napier himself married and in 1574 moved into a castle at Gartness. Most of Napier’s work on logarithms was done whilst he was living at Gartness. However, his study of mathematics was only a hobby and he often struggled to find time to perform all the necessary calculations.

Napier is best known for his invention of logarithms. In 1614 he published his first work on logarithms, titled *Mirifici Logarithmorum Canonis Descriptio* which translates to ‘A Description of the Wonderful Table of Logarithms’ (a first edition can be found in the University’s Special Collections) which described how to use logarithms. The word logarithm is derived from the Greek word *logos* meaning word, study, reasoning, or, in Napier’s use, reckoning and *arithmos *meaning number. Napier’s logarithms aren’t exactly how we know them today, but although his logarithms weren’t to any base, it is reasonable to say that they are in base 1/e with a constant of 10^7. He chose this based on how the tables of sines were available to him and needed for the logarithms. The use of Napier’s logarithms could save computational time, which was particularly useful to astronomers (Pierre Laplace said they ‘doubled the life of the astronomer’). A second book on logarithms by Napier was published posthumously in 1619, entitled *Mirifici Logarithmorum Canonis Constructio* which described how to calculate logarithms.

Despite being best known for his logarithms, Napier also worked on finding the exponential expressions for trigonometric functions, the introduction of decimal notation for fractions and ‘Napier’s bones’; an invention used for multiplying numbers together where the user only had to use addition. There were 10 rods (often made of ivory) with multiples of different digits inscribed on them. The user could then select the rods needed for their multiplication, adding digits across diagonals to carry out the calculation.

#### John Maior

Scottish by birth, John Maior was a theologian and mathematician who spent much of his working life between St Andrews and Paris. He first came to the University of St Andrews in 1523 where he taught theology and logic. Maior would eventually go on to become the provost of St Salvator’s College in 1534, a post he would hold until his death in 1550.

John Maior made numerous contributions to mathematics in the area of logic and infinity, most notable in this regard is his 1506 treatise *Propositum de Infinito*. In this text, Maior discusses “infinity” as both a so-called categorematic and as a syncategorematic term.

These terms refer to a widely used classification system for words in the medieval period, where categorematic means a word which has meaning on its own out of context (a noun for example), whereas a syncategorematic word needs surrounding words to have use and meaning (a conjunction for example).

While most words fall either into one category or the other, “infinity” is interesting in that it can be considered either or. Quantifiers, such as “three”, “many” or “four million” are historically classified as syncategorematic words. So, for instance, “three” on its own has no significance, but “three sheep” does. It is for this reason, that during the 14^{th}century, “infinity” was typically considered a syncategorematic word.

Importantly, however, John Maior challenged this notion, arguing for “infinity” as a categorematic term. This would imply that infinity exists not just as an idea, but as an actuality in the physical world – a controversial suggestion not only at the time, but into the present day.

St Salvator’s College and Chapel: https://www.st-andrews.ac.uk/about/history/st-salvators/brief/

John Napier: https://mathshistory.st-andrews.ac.uk/Biographies/Napier/

Hobson, E. W. John Napier and the invention of logarithms, 1614 (1914)

Napier’s Bones: https://mathshistory.st-andrews.ac.uk/Extras/Napier_rods/Napier’s Bones: https://mathworld.wolfram.com/NapiersBones.html

Sara L. Uckelman. (2015). The logic of categorematic and syncategorematic infinity. *Synthese,* *192*(8), 2361–2377.

Biard, J. (1986). LA LOGIQUE DE L’INFINI CHEZ JEAN MAIR. *Les Études Philosophiques*, (3), 329-348.