August Adler

Quick Info

24 January 1863
Opava, Austrian Silesia (now Czech Republic)
17 October 1923
Vienna, Austria

August Adler was a mathematician who was born in what was then Austria and who worked on geometric constructions using compasses only.


August Adler was born and brought up in Opava, a city also known in German as Troppau. The city was the capital of Austrian Silesia for almost all of Adler's life, only becoming part of Czechoslovakia in 1919 after the defeat of Germany and Austria-Hungary in World War I. All Adler's publications were in German but he did publish a few in Czech journals.

In 1879 Adler graduated from the secondary school in Opava (known as Troppau at that time) and began his university studies at the University of Vienna and the University of Technology in Vienna where he was taught and greatly influenced by Emil Weyr. After taking undergraduate courses at the two universities, Adler undertook research in descriptive geometry graduating with a doctorate in 1884. He began publishing during these years with papers on ruled surfaces and space curves such as: Striktionslinien der Regelflächen 2. und 3. Grades (1882); Raumkurven vierter Ordnung 2. Art (1883); Weitere Bemerkungen über Raumkurven vierter Ordnung 2. Art (1883); and Spezielle Raumkurven vierter Ordnung 2. Art (1883). He submitted papers to the Academy of Sciences of Vienna, now the Austrian Academy of Sciences, for example the first of these papers was reported by the Secretary of the Academy to the meeting of the Mathematical and Natural Science Class on 19 January 1882. The Academy report contains the following abstract:-
In this paper the theory of the striction lines on a ruled surface of the second degree is studied in a purely geometrical way and the characters of the striction lines on a ruled surface of the third degree are determined. The simplest possible construction methods are sought for the striction lines on a second degree surface. In addition, there are also theorems about general rational space curves.
We note that a striction line is a line on a skew surface that cuts each generator in the point of the surface that is nearest to the succeeding generator.

The second of these papers was reported by the Secretary of the Academy to the meeting of the Mathematical and Natural Science Class on 2 November 1882. The author is described in the report as "Mr August Adler, student at the Technische Hochschule in Vienna", see [6]. We note that the Technische Hochschule in Vienna, founded in 1815, was renamed the Vienna University of Technology in 1975. Emil Weyr sent further papers by Adler to the Academy while he was his student.

Adler was appointed as an assistant in astronomy and geodesy at the Technische Hochschule in Vienna in 1885 holding this position for two years. He submitted further papers to meetings of the Academy during this time, for example Über ein allgemeines Princip des graphischen Rechnens to the meeting of 14 January 1886 and a second, Zur graphischen Auswertung der Funktionen mehrerer Veränderlicher , to the meeting of 8 July 1886 where he is described as an Assistant at the Technische Hochschule in Vienna.

Adler taught at Döll's Realschule in Vienna beginning in 1888. Two years later, however, in 1890, he is listed as a Substitute teacher at the Staats-Oberrealschule in Klagenfurt. While in Klagenfurt he published Graphische Auflösung der Gleichungen . He then taught at the German Realschule in Pilsen from 1891 to 1895 before becoming a professor at the German Oberrealschule in Karolinenthal. By 1899 he is listed as a Professor and an Ordinary Member of the Academy of Sciences in Vienna at Karolinenthal, Kollargasse 13. The town of Karolinenthal, also known as Karlin, was founded in 1817 and named after the wife of Francis I of Austria, Caroline Augusta of Bavaria. It was incorporated into Prague in 1922. When Adler taught in Prague there were both German and Czech speakers and education was provided for pupils in both languages. The situation is described in [3]:-
When you look at the absolute numbers, the year 1880/81 and the following year catches your eye, because initially the number of native Czech speakers at German secondary schools rises by more than 200, but drops almost a year later to their old level back. During the increase primarily with the foundation of the German high school in Smichov and especially the German Oberrealschule in Karolinental, which 122 native speakers still choose in spite of the opening of the Municipal Czech Upper Real School in Karolinental in the same year, the following decline is spread across all institutions with the German language of instruction. The division of the university at this time appears to be the obvious reason for this development, because it makes secondary schools teaching in the Czech language more attractive to the developing Czech circles of the education-oriented middle class, which has so far preferred German secondary education.
In 1901 Adler submitted his habilitation thesis on descriptive geometry to the German Technical University in Prague and became a docent there. He continued, however, to teach at the German Oberrealschule in Karolinenthal. For example, at the meeting of the Academy of Sciences of Vienna on 9 January 1902, to which he submitted a paper, he is described as both a secondary school professor in Karolinenthal and a privatdocent at the German Technical University in Prague. Adler went to Göttingen for the summer semester of 1902, applying to the University to he a guest auditor. He attended David Hilbert's lectures on the Foundations of Geometry and took notes which became the published version of these lectures, see [1]. These notes, written in Adler's own hand, are now in the Reading Room of the Göttingen Mathematical Institute.

In 1906 Adler left Prague and took up an appointment as a teacher at a high school in the 6th district of Vienna and, in addition, he was a Privatdozent at the Technische Hochschule there. One year later, in 1907, he was appointed as Director of the Staatsrealschule in the 7th district of Vienna. He retired on grounds of ill health in 1915.

Let us now look in a little more detail at Adler's mathematical contributions. In 1797 Mascheroni had shown that all plane construction problems which could be made with ruler and compass could in fact be made with compasses alone. His theoretical solution involved giving specific constructions, such as bisecting a circular arc, using only a compass.

In 1906 Adler applied the theory of inversion to solve Mascheroni construction problems in his book Theorie der geometrischen Konstruktionen published in Leipzig. Since he was using inversion Adler now had a symmetry between lines and circles which in some sense showed why the constructions needed only compasses. However Adler did not simplify Mascheroni's proof. On the contrary, his new methods were not as elegant, either in simplicity or length, as the original proof by Mascheroni. Richard Güntsche reviewed this book [5]:-
The book gives a coherent presentation of the methods and theories for solving geometric construction tasks, the development of which the author himself is known to have played a significant part. After a description of the construction methods found by Petersen, the Steiner and the Mascheroni constructions are explored, as well as those with the help of a parallel ruler, one or more movable right angles, a gauge or a bisector. This is followed by theoretical considerations about the classification of the geometrical construction tasks and the proof of the possibility and impossibility of solving a specific geometric task with a compass and ruler. After a section on circle division, in which seventeen different divisions are described, there follow solutions of the Delian problem, angle trisection and, in general, third and fourth degree equations by means of various construction methods, as well as historical remarks on the rectification of the circle. A final section is devoted to Lemoinian geometrography.
This 1906 publication was not the first by Adler studying this problem. He had published a paper on the theory of Mascheroni's constructions in 1890 which was reviewed by H Schubert [8]:-
In the direction of the investigation initiated by Mascheroni (Pavia 1797) and von Steiner (Berlin 1833) about the aids or tools that are unnecessary or necessary to solve geometric construction tasks, a noteworthy work is written here, which shows that any problem that can be solved with a compass and ruler can also be solved, if further aids are not allowed other than either only the ruler (two parallel lines at a constant distance) or only the right angle (for example made of wood) or only a certain acute angle (for example made of wood). Since we know from Steiner's work that using the ruler alone can only solve any second degree problem if a fixed circle can be used in the plane, the use of the ruler in the sense of Mr Adler is of course different understand as with Steiner, namely in such a way that it also allows a parallel to a straight line to be drawn at a fixed distance (width of the ruler). Likewise, the application of the wood angle should also allow, for example, the search in a straight line for the point from which a fixed distance appears at the angle of the wood angle. So it is shown that in "practice" each of the four most important drawing tools (compass, ruler, right angle, acute angle) alone is sufficient to solve all tasks of the second degree. At the end it is proven that by using several right angles, tasks of the third and fourth degree can also be solved. Such tasks were once shown by Kortum that they can only be solved with a ruler if a fixed cone can be used.
In fact Hilbert refers to this paper by Adler in the 1902 course which was attended and recorded by Adler. The authors of [1] write:-
Hilbert [refers to Adler's 1890 paper], which deals with constructions executable by a draughtsman whose only means of construction is a wooden right-angle. He shows that, assuming one has several right-angled triangles at one's disposal, one can solve in principle any cubic equation, and with perfect accuracy, assuming the means used to draw the right-angles are perfect. Hilbert's remark implicitly suggests replacing the use of the draughtsman's right-angle tool with the 'movement' of an angle, that is, with a congruence assumption which allows one to construct right-angles at any point and on any line.
Adler had published another paper on the theory of geometrical constructions in 1895, and one on the theory of drawing instruments in 1902 in which [2]:-
... he showed that both a ruler to be used in a certain manner with two parallel edges and a set square, the edges of which meet in the arbitrarily fixed angle, have the same level of performance as circles and straight edge.
As well as his interest in descriptive geometry, Adler was also interested in mathematical education, particularly in teaching mathematics in secondary schools. His publications on this topic began around 1901 and by the end of his career he was publishing more on mathematical education than on geometry. Most of his papers on mathematical education were directed towards teaching geometry in schools, but in 1907 he wrote on modern methods in mathematical instruction in Austrian middle schools. He produced various teaching materials for teaching geometry in the sixth-form in Austrian schools such as an exercise book which he published in 1908.

David Eugene Smith wrote the Report of Sub-Commission A of the International Commission on the Teaching of Mathematics: Intuition and Experiment in Mathematical Teaching in the Secondary Schools which he delivered to the International Congress of Mathematicians held in Cambridge, England, on 27 August 1912. In his report he refers to Adler's teaching texts when he writes about the teaching of geometry in Austria:-
A subject like descriptive geometry, for example, can hardly be said to occupy any definite position, so rapidly changing are the views concerning its importance, its significance, and the schools in which it should be taught. Austria makes much of it in the Realschule, less of it in the Realgymnasium, and almost nothing of it in the Gymnasium.... In Austria, in the Realschulen, the subject [of geometric drawing and graphic representation] is nominally in the hands of a special teacher examined and appointed for the purpose. ... At present, in about a third of the schools, the teacher of mathematics is also the teacher of descriptive geometry in the three upper classes (V-VII). There is a well-defined line of demarcation between stereometry and descriptive geometry as the work is carried on, although the latter is employed in representing the solids met in the former. It appears, therefore, that certain definite work in descriptive geometry is done in the upper classes of the Realschulen. In the Gymnasien descriptive geometry is not a separate object of study. In the work in stereometry, however, plans and elevations are drawn, and orthogonal projections of parallelepipeds, octahedrons, pyramids, and the like are prepared.
One final work by Adler should be mentioned, namely the five figure logarithm tables Fünfstellige Logarithmen which he published in 1909.

For a list of publications by Adler which includes his school textbooks, see THIS LINK.

References (show)

  1. M Hallett and U Majer, David Hilbert's Lectures on the Foundations of Geometry 1891-1902 (Springer Science & Business Media, 2004).
  2. C J Scriba and P Schreiber, 5000 Years of Geometry: Mathematics in History and Culture (Birkhäuser, 2015).
  3. I Stöhr, Zweisprachigkeit in Böhmen: Deutsche Volksschulen und Gymnasien im Prag der Kafka-Zeit (Böhlau Verlag Köln Weimar, 2010).
  4. R Einhorn, Vertreter der Mathematik und Geometrie an den Wiener Hochschulen 1900-1940 (Vienna, 1985).
  5. R Güntsche, Review: Theorie der geometrischen Konstruktionen, zbMATH, JFM 37.0511.03.
  6. August Adler, Meeting reports of the Academy of Sciences of Vienna mathematical and natural science class, ZOBODAT.
  7. M Toepell, Mitgliedergesamtverzeichnis der Deutschen Mathematiker-Vereinigung 1890-1990 (Munich, 1991).

Additional Resources (show)

Other pages about August Adler:

  1. August Adler's publications

Other websites about August Adler:

  1. zbMATH entry
  2. ERAM Jahrbuch entry

Honours (show)

Honours awarded to August Adler

  1. Lectures to the Mathematical Society of Vienna 1905

Written by J J O'Connor and E F Robertson
Last Update July 2020