*Contributions to the history of mathematics in Sweden before 1679*was published as a book. We give below an extract from the text. The translation is by Annette Oldsberg, University of St Andrews. We have added some clarification at various points with text enclosed within square brackets.

**II. The period from the re-establishment of Uppsala University in 1593, to 1679.**

A. *From 1593 to 1620*.

However, we have found some disputations [theses] dating back to the beginning of this time period, presented in Germany, where a couple of Swedes were studying mathematics at the time. It can be assumed that these men subsequently moved on to have a significant impact on the studies in their home country, and that their works lay ahead in time of anything equivalent in Sweden. It can therefore be of interest to see what demands one of Europe's foremost Universities for mathematical studies - which Marburg, where the most important theses were examined, undoubtedly was - put on mathematical dissertations. We will thus give an account of these theses, before proceeding to the mathematical studies in our homeland.

*From 1620 to 1679*.

It was not until 1620 that considerable measures were taken for the recovery of its teaching. The clergy, by royal order of 20 March 1620, presented its reform of the academy and schools, whereby it proposed six philosophy professors. The first of these professors would be a professor of astronomy, reading "prima et secunda mobilia" [astronomy was classified as prima mobilia and secunda mobilia or theory of the planet], trigonometry, teaching the youth astronomical calculations and publicly present "de Coelo" [the heavens]. The 6th mathematics professor was to read arithmetic, geometry, optics and mechanics, as well as publicly present "in opticis och de Meteoris". In a resolution dated 13th April 1620, the king let it be known that he intended to found an academy in Uppsala, a gymnasium in Abo and one in Linköping, as well as additional 'half-gymnasiums' or grammar schools. It was decided, by a proposition by the clergy, that each gymnasium should consist of five teachers. The first of these teachers, the principal, should read physics and astronomy and the 4th, a colleague, mathematics and logic. There were to be four teachers in the grammar school, where the principal should read physics and mathematics. This proposal concerning the professors was approved, with the only alteration that the astronomy professor is to be 3rd, and the mathematics professor 4th in order, and the clergy was to give the professors their names.

All that is mentioned above reveals that the beginning of the 1620s brought with it a greater attention to mathematics, as well as academia in general. The fact that mathematics had a higher status than most other disciplines within the academic community, should certainly to a large extent be credited to Skytte. Moreover, we presume that the - in their time - prominent scientists Gestrinius and his brother-in-law Martinus Olai Nycopensis contributed with their keenness, as professors at Uppsala, to a more scientific approach to the study of mathematics. It is evident that the giant stride that our country took at this time in its political structure, also had an influence on scientific progress.

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As we return to the account of the mathematical studies at Uppsala University, the first name we come across is the industrious professor of mathematics Martinus Erici Gestrinus, who was active 1620-48.

**10. Martinus Erici Gestrinius.**

Out of all the academic discourses in a wide range of topics, that were issued during Gestrinius presidium 1622-46, only a relatively small number were in fact mathematical. These are however, to the best of our knowledge, the first of their kind that were published in Sweden. In their capacity as such, they raise a certain interest.

The oldest thereof is a *Disputatio physic-mathematica de Stellis* from the year 1622. Respondent Er. Ivari Wengensis.

The author wishes to examine the nature of the stars. To begin with he defines a star as a simple, ethereal, shining and essentially spherical body, which, through an intrinsic divine force rotates around the axis mundi, completes its orbit in a fixed time and has been created by God for the sake of mankind. He then proceeds to elaborate on this definition with reference to Scaliger, the Bible and Luther, but also Galileo, Tycho Brahe, Plato, Aristotle and Blebelius [Thomas Blebelius (1539-1596), schoolmaster and author of famous astronomy book]. Furthermore, he strives to mathematically prove that the quantity of fixed stars cannot possibly be infinite. He assumes the diameter of the smallest star to be 1 second and calculates the 8th or the "outer area" of the fixed celestial sphere by first reducing 360∞ to seconds, which he divides by 22/7 and thus obtains the diameter of the sphere; half of the diameter times half of the perimeter gives the "area of the sky", which he finds to be 534423744000", and hence as many of the smallest stars could exist in the sky. Since the smallest star that has been observed is 20" in diameter, he concludes that by dividing the specified convex surface by 20 (!) one can give a feasible estimate of the number of stars in the firmament. This twentieth is however regarded as too large, and he finally settles on the number 1022 of 6 magnitudes stated by the elders. Moreover, he describes all planets by adhering to the Egyptian system, as well as Jupiter's 4 moons, which were discovered by Galileo only twelve years earlier. Lastly a few "corollaries" follow, which are solely comprised of questions such as: Is it possible that the new star in Cassiopaea was formed by cumulated dust in the ether? Is it possible that the star that appeared in the east at the birth of Christ, was an actual and natural star? Does the sun dance over Easter? Etc.

Curiously enough, *Disputatio de Philosophiae origine, natura et sobrio* in S S. Theologia usu, which was held under Gestrinius in Uppsala in 1625, contained a systematic classification of mathematics. It was split into two classes: pure or simple and impure or mixed. The former included arithmetic and geometry, the latter music, stereometry, "Optica vel perspectiva", astronomy, astrology, geography and geodesy. Astronomy is further classified as primum mobile and secunda mobilia or planet theory, and geodesy as cosmography or chorography.

As for Gestrinius' work: In *Aristotelis mechanica, Argumenta et Notae, in quibus ... etiam problemata singular e fundamentis partim geometricis, partim physicis resolvuntur ac explicantur* (Uppsala 1627) 97 sid. 8:o, it only contains solutions and explanations of numerous trivial questions, such as why teeth are easier to pull out with a pair of pliers than with the bare hand, why beds are constructed in such a way that the length is to the width what 6 is to 2 or 4 to 2, and why the strings under the beds are stretched aslant, and not diagonally, etc. The last question is "solved" in three ways: 1) since in this way the sides are held closer together and are thus not as strained; 2) since the more the strings are tied across the bed, the more easily they can carry the weight; 3) since this technique requires less string. To strengthen the last proposition, he brings forward a lengthy but incomprehensible proof, about which, all the same, he makes the remark, that almost everything therein is false.

Three *Disputationes Geometricae* that were given during his presidency ought to be of actual mathematical content; but as we soon shall see, few traces of mathematics can be found in them.

*De primo ac simplicissimo Geometricae principio Puncto*, 28 pages, defended 1627 by Petrus A. Schomerus.

*De geometriae Constitutione*, 36 pages, defended by "Johannes Jonae

Tornaeus Both. Math. Studiosus" in 1631.

*De prima Magnitudinis specie Linea*, 36 pages, defended 1633 by Daniel Danielis Dalinus from Linköping

Out of the remaining disputations that Gestrinius presided over, one that undeniably contains more mathematics than any other is *Disputatio inauguralis mathematica de Sole*, 26 sid. 4:o, written and defended 1632 by Simon Kexlerus from Närke, who later on become professor of mathematics in Abo. This thesis reveals what astronomical calculations looked like at the time. The author explains, by using deferent and epicycles and with the aid of a very complex figure, the shifting movement of the sun in its course, different distances between the sun and earth, etc. He adopts certain assumptions made by Ramus, Pitiscus, Copernicus and others, and aims to calculate the position of the sun in the ecliptic at 12 am on the 7th of January 1632, which is when he intended to dispute, i.e. defend his thesis.

The author remarks that he is using Copernicus's observations, but not his "seemingly absurd and physically unrealistic hypotheses about the earth's course and the sun's position in the centre of the universe, which he ingeniously worked out to avoid the large number of spheres", that is otherwise necessary for the explanation of the movements.

In the same year [1632], *Disputatio Planetaria de Stella Marte* (40 pages 4:o) was given by Johannes Tornaeus Bothniensis under Gestrinius' presidium. Bothniensis attempts to illustrate the movements of the planets using a deferent and epicycles, without performing any calculations. He comments that he had planned to consider the observations made by Kepler and Tycho Brahe, but that he had to refrain from those since it would make the treatise altogether too extensive. At any rate this shows us that one had started to become familiar with the most recent developments abroad.

The sheer amount of work that was produced as a result of the favourable conditions cultivated by Gestrinius, testifies to the enthusiasm and dedication with which this professor served his science. Albeit he did not accomplish any great achievements by our standards, one has to take into consideration the general ignorance that was present when he assumed his post at the academy. Taking this into account, one can only assign a great value to his contributions to the mathematical studies in Sweden. Our research has shown that he introduced the writings of Copernicus, Kepler and Galileo to the academy. We can, with a fair amount of conviction, form the judgement that he, more so than any other mathematics professors in Uppsala during this period, strived to keep up to date with the progress that his science made in other countries. Perhaps it was the case that during the years marked by Gestrinius' mathematical insights, Uppsala lagged less far behind its time than at any point during the hundred years that were to follow.

17. Jonas Laurentii Fornelius.

Celsius was born 1621 in Alfta parish in Helsingland. His father Nicolaus Magni Travillagaeus was the son of a farmer from Ydre in Östergötland, and became a church minister in Alfta and Ovanåker. The son initially called himself Magnus Nicolai Helsingus, then Metagrius, and finally Celsius, derived from the name of the vicarage in Ovanaker: Högen. He arrived in Uppsala 1641, graduated with a doctor's degree in 1649 after having disputed three times: 1642 "de aliquot Theorematibus Miscellaneis", 1646 "de Cerebro Humano" and 1647, pro gradu, "de plantis". Celsius became headmaster of the Uppsala school in 1656, adjunct in philosophy at the academy in 1663, extraordinary professor of astronomy in 1665, deputy judge in the Royal Swedish Academy of Letters in 1666, ordinary mathematics professor in 1668 - in the lecture catalogues he is referred to as inf. ordinary professor after the year of 1672. He was ordained in Strängnäs in 1677, became vicar in Gamla Uppsala and died 1679 due to indigestion. Celsius was the principal of the academy in 1675, and was inspector of the österbottniska, finska and aländska nationerna [student societies in Uppsala are (to this day) called *nations*, and are named after Swedish provinces]. His two sons Nils and Olof both became professors in Uppsala.

Celsius, founder of this renown family, was considered to be an excellent mathematician as well as an accomplished mechanic, in that he himself constructed and completed many instruments and machines, such as astrolabes, moon- and planet-telescopes etc. In addition to this he occupied himself with painting, chalcography [engraving copper plates for making prints] and sculpting, as a matter of fact he was also a popular poet. Celsius was interested in antiquarian research, and became famous for deciphering the Staveless runes [or Hälsingle runes discovered on rune stones in Hälsingland in the 17th century]. Someone made the following comment about his appearance "he had chiselled features, as if he had contracted the consumption". After his passing the family was left in straitened circumstances.

Except for the 23 disputations he presided over 1664-78, he published almanacs for the years 1658-61, as well as essays about the Staveless runes. His posthumous work *Observationes circa Cometam in fine anni 1664* was published in "Acta Literaria Sveciae" vol. II, pp. 223-225.

We do not know of any mathematical work by Celsius other than what can be found in the afore mentioned disputations, of which the former are his 8 disputations during the period 1664-78, making up 223 sid. 8:o.

The 4th disputation, *De Hebdomade*, was defended by his brother Nils Celsius. After having described the art of chronology, how intercalations were done in ancient times, the subdivision of time etc., he moves on to "Corollaria Respondentis". Only one of these corollaries is mathematical, but it is actually the best we have seen from these times. It is stated as follows: Consider a triangle with known base. The triangle is split into required parts by drawing lines that are parallel to one of the other sides. This is followed by the "praxis", as the author puts it; "the square root of the ratio between the square of the basis and the number of parts, determines the distance between the point of intersection on the base and the angle opposite the side with which the division line is parallel".

Of course, no proof was provided for the validity of formula, but it can easily be verified. It is a fact that areas of the same shape are related to each other in the same way as the square of two corresponding sides are related to each other. Nowadays, we would write the formula for the length between the intersection points and the opposite angle as follows, where a denotes the length of the base, and n the number of equal parts;

*a*/√

*n*,

*a*√2/√

*n*,

*a*√3/√

*n*, ...,

*a*√(

*n*-1)/√

*n*.

As we have seen, no significant progress has been made in the field of mathematics since the time of Gestrinius, indeed the interest in mathematics appears to have diminished. The old methods and beliefs were maintained with great persistence, and not until the influential and prominent scientists Spole and Bilberg [brief biographical details are given after the Introduction at THIS LINK.] were appointed mathematics professors in Uppsala was there a turning point. It is in theses written under their support that the use of a more rational notation is to be found, as well as signs that Copernicus', Galileo's and Kepler's model of the universe started to gain ground also in Sweden.