## Ernst Dahlin: *Mathematics in Sweden before *1679 - Introduction

Ernst Mauritz Dahlin's Ph.D. thesis

*Contributions to the history of mathematics in Sweden before 1679*was published as a book. We give below a translation of the Introduction. The translation is by Annette Oldsberg, University of St Andrews. We also give some biographical details of two people that Dahlin mentions in the Introduction.**1. Introduction.**

The traces of mathematical understanding amongst our ancestors are, up until the reformation, few, virtually non-existent. As for the more ancient times, such traces are to be sought in the old Norse literature and mythology. These writings, which are illuminating when studying our forefathers' traditional, religious and daily life, as well as their scientific pursuits, are key sources in such research. They are written by Icelanders and are thus mainly concerned with the conditions on Iceland and in its mother country Norway; but bearing in mind the similarities in language and culture that prevailed in Scandinavia during those times, there are strong reasons to regard these relics as accounts of the civilisation in Sweden also. Content that informs us of our ancestors' insights in mathematics, or more precisely, related topics such as astronomy and chronology, has of course a historical rather than mathematical value. It is primarily for the sake of historical completeness that we have mentioned these themes, as far as they are connected to our topic.

Decrees and prescriptions issued for the educational system have been used when conducting our research, to the extent that they can be obtained. The later part of the time period that this paper encompasses provided lecture notes, specifying the content that was each year covered in the lectures by the mathematics professors. This resource is necessary for gaining a deeper insight about the state of science in general, as well as its various disciplines, to which, at different points in time, we has chosen to restrict attention to.

In order to provide a clear outline, this thesis is split into two main parts, comprised of the time prior to Uppsala University's revival in 1593, and the period 1593-1679. During the former time period, with the exception of the last 11 years, we were unable to find a Swedish establishment for education in mathematics. As for the latter, there have certainly been mathematics professors lecturing prior to 1620, however, we have not found any written evidence of their work dating back to this time. Only after the last-mentioned year did the academic activities become more lively, as a consequence of the measures taken by the university to reorganise and become a more solid institute. Hence the year 1620 was taken as a divide between two subsections in the latter time space.

During the full time span that our report covers, one did never reach - with trigonometry as the only exception - as far as what today constitutes the minimum level of knowledge that is required to pass secondary school. It is not until after 1679, when both of the old mathematics professors at Uppsala passed away, and the more capable scholars Spole and Bilberg simultaneously took their positions, that we could discover in academic works the use of logarithms, exponents, letter notation in algebra, and a robust notation for analogies.

To not burden our paper with biographical notes, we have generally followed the principle that foreign authors will only be stated if they are mentioned in the works of Montucla, Suter or Poggendorff, that are referenced below. ...

Decrees and prescriptions issued for the educational system have been used when conducting our research, to the extent that they can be obtained. The later part of the time period that this paper encompasses provided lecture notes, specifying the content that was each year covered in the lectures by the mathematics professors. This resource is necessary for gaining a deeper insight about the state of science in general, as well as its various disciplines, to which, at different points in time, we has chosen to restrict attention to.

In order to provide a clear outline, this thesis is split into two main parts, comprised of the time prior to Uppsala University's revival in 1593, and the period 1593-1679. During the former time period, with the exception of the last 11 years, we were unable to find a Swedish establishment for education in mathematics. As for the latter, there have certainly been mathematics professors lecturing prior to 1620, however, we have not found any written evidence of their work dating back to this time. Only after the last-mentioned year did the academic activities become more lively, as a consequence of the measures taken by the university to reorganise and become a more solid institute. Hence the year 1620 was taken as a divide between two subsections in the latter time space.

During the full time span that our report covers, one did never reach - with trigonometry as the only exception - as far as what today constitutes the minimum level of knowledge that is required to pass secondary school. It is not until after 1679, when both of the old mathematics professors at Uppsala passed away, and the more capable scholars Spole and Bilberg simultaneously took their positions, that we could discover in academic works the use of logarithms, exponents, letter notation in algebra, and a robust notation for analogies.

To not burden our paper with biographical notes, we have generally followed the principle that foreign authors will only be stated if they are mentioned in the works of Montucla, Suter or Poggendorff, that are referenced below. ...

**2. Anders Spole (1630-1699).**

Dahlin mentions Anders Spole in the Introduction to his thesis. Spole has been described as:

Spole studied at Greifswald University in German, studying subjects such as mathematics, military arts and navigation. During the years 1664-1667 he toured round Europe where he visited leading scientists such as Christiaan Huygens, Robert Hooke, Robert Boyle, Nicolaus Mercator, Ismael Boullieau, Giovanni Cassini and Giovanni Battista Riccioli. He was appointed professor of astronomy at Lund University soon after it was founded in 1666 by Charles XI. He became professor of astronomy at Uppsala University in 1679.The first real astronomer at Uppsala University.

**3. Johannes Bilberg (1646-1717).**

Johannes Bilberg became a student at Uppsala University in 1659 when he was 13 years old. After graduating with a bachelor's degree from Uppsala University, he was employed as a tutor for a young baron Ulf Bonde. He accompanied Ulf Bonde on a foreign trip, visiting foreign courts and prominent universities in Europe. Returning to Sweden in 1677 he was appointed two years later as professor of mathematics at Uppsala University. Bilberg was very much involved in a vigorous debate which went on concerning the new Cartesian philosophy which was strongly opposed by the orthodox theologians. The government appointed a special commission to examine and decide the contentious issue. To complete the victory of Cartesian philosophy, Charles XI moved Bilberg to the faculty of Theology. The rest of his career was in the Church becoming a bishop.