**Irving Stringham**'s parents were Henry Stringham (1801-1880) and Eliza Tomlinson (1802-1881). Henry and Eliza Stringham were married on 7 March 1823 in Brookfield, Brookfield County, Connecticut, and they had nine children: Amelia Maria Stringham (born 9 Apr 1824, Fairport, Monroe County, New York); Thomas Henry Stringham (born 29 Mar 1826, Rochester, Monroe County, New York); Ann Eliza Stringham (born 26 Apr 1828, Dansville, Livingston County, New York); Daniel Tomlinson Stringham (born 2 Nov 1830, Geneseo, Livingston County, New York); Cecelia Jane Stringham (born 5 Nov 1832, Franklinville, Cattaraugus, New York); Charles Augustus Stringham (born 22 Jan 1835, Lynden, Cattaraugus, New York); Theodore Lefoy Stringham (born 2 Feb 1839, Lynden, Cattaraugus, New York); Oliver Eugene Stringham (born 31 Jan 1844, Lynden, Cattaraugus, New York); and Washington Irving Stringham, the subject of this biography (born 10 Dec 1847, Delevan, Cattaraugus, New York). Two of Irving Stringham's brothers served in the Northern army during the American Civil War. Oliver Eugene Stringham was one of the two brothers and he was taken prisoner at the Battle of Gettysburg. Although he was released in 1865 he died seven years later as a result of his time in prison.

Stringham was brought up at Yorkshire Centre, in western New York, where he attended the village school. When he was twelve years old he became a member of the Methodist Episcopal Church but eventually he decided that none of the recognised churches fitted his beliefs. In the spring of 1865, after the American Civil War had ended, he moved to Topeka, Kansas, to join one of his brothers and one of his sisters who were living there. He studied for a short while at Topeka High School in 1866 and then, in the same year, began his studies at the preparatory department of Lincoln College in Topeka. Lincoln College was a very new establishment, founded only one year earlier in 1865. After a large donation to the College by Ichabod Washburn, the College changed its name to Washburn College in 1868. He spent a year 1866-67 in the preparatory department and showed great promise, being awarded a prize for the best performance in 'Scientific studies'. Although he gained admission to the first year to begin his studies in the autumn of 1867, he delayed his entry for a year and began his studies in the autumn of 1868. He completed one year of study but, not having the necessary financial support to continue, he worked as a house and sign painter for three years, 1869-1872.

In 1872, Stringham entered the second year of studies at Washburn College (as it was named by this time). He studied there for a year but continued to support himself by working as a bookkeeper and clerk in a drug store run by his brother. However, Stringham now aimed higher and was intent on studying at Harvard College. During the year 1872-73 he applied for admission to Harvard and, despite his unusual educational background, he was accepted. He began his studies at Harvard in September 1873 and lived at 43 College House with Richard Smith Culbreth who became a lawyer. In his final three years he lived at 25 Stoughton with James Byrne who also became a lawyer. During his undergraduate years, Stringham was a member of the Everett Athenaeum, a short-lived society for second year students with literary interests. He was also a member of the Signet Society which was founded around 1870 for those "of merit and character", and a member of Phi Beta Kappa. The:-

Stringham received honours in mathematics in his second year at Harvard College and earned the highest honours in mathematics when he graduated with an A.B. in 1877. He had submitted his thesis... purpose of Phi Beta Kappa is to recognize and encourage scholarship, friendship, and cultural interests.

*Investigations in Quaternions.*for mathematical honours with the A.B. This thesis was presented by Benjamin Peirce for publication in the

**I**. Logarithms of Quaternions.**II**. Applications of Quaternion Analysis to Rectification of Curves. Quadrature of Surfaces, and Cubature of Solids*Proceedings of the American Academy of Arts and Sciences*and it appeared in Volume 13 as a 31-page paper.

After graduating from Harvard College, Stringham was a graduate student at Johns Hopkins University in 1877-78 where he worked towards his doctorate. In his application to the Mathematics Department, he had indicated that in addition to mathematics he wished:-

He was an assistant on the... as far as possible, to pursue the study of Fine Arts.

*American Journal of Mathematics*in 1878-79 and he was appointed as a fellow in mathematics at Johns Hopkins in 1878. He published the paper

*The Quaternion Formulae for Quantification of Curves, Surfaces and Solids, and for Barycentres*in 1879. This paper continued the work of his A.B. thesis. Also in 1879 he published the note

*Some General Formulae for Integrals of Irrational Functions*in the

*American Journal of Mathematics*.

At Johns Hopkins, he took part in the Mathematical Seminary where he delivered the lecture *On Vector Ratios considered as Trigonometric Functions of Angles* on 22 December 1879. He delivered the lecture *A Generalization, for n-fold Space, of Euler's Equation for Polyhedra* on 21 January 1880. He attended the following courses in 1879-80: 'Elliptic Functions' by W Story, 'Higher Plane Curves' by W Story, 'Calculus of Variations' by T Craig, 'Spherical Harmonics' by T Craig, 'Symbolic Logic' by C S Peirce, 'Quaternions' by W Story, and 'Theory of Numbers' by J J Sylvester. At the Johns Hopkins Scientific Association on 7 January 1880, he presented paper on *Geometrical Figures in Space of Four Dimensions*. He began with the following:-

On 17 March he delivered the talkA regular figure in space of four dimensions is defined as one which is completely bounded by equal regular figures of three dimensions and whose vertices lie on the four-fold sphere. Three of such figures are,(1)

the Four-fold Pentoid, bounded by five tetrahedra,(2)

the Four-fold Octoid, bounded by eight cubes,(3)

the Four-fold Hexadekoid, bounded by sixteen tetrahedra.

*On Rotation in Four-dimensional Space*to the Mathematical Seminary which began with the following definition:-

He also addressed the Metaphysical Club on 8 March 1880 giving a resumé of Ernst Schröder'sThe rotation of a particle, or point, about a plane in four-dimensional space may be defined as such a motion of the particle that a radius vector drawn from a fixed point in the plane to the particle shall always remain constant in length and perpendicular to the plane.

*Operationskreis des Logikkalkuls*. Stringham said:-

It is interesting to see in what high esteem Stringham was held. For example, J J Sylvester, in his paperThe main features of Schröder's system are,(1)its dualistic arrangement, by which to every addition equation a multiplication equation is made to correspond(and visa versa),(2)the transformation of every logical equation into a form in which the right hand member is zero, and(3)more especially his announcement of a Haupttheorem by means of which by a single operation two equations are obtained, from one of which any given class symbol is eliminated, while the other gives its value. The method of(2)is identical with that of Jevons.

*On a Geometrical Proof of a Theorem in Numbers*delivered to the Mathematical Seminary in January 1882, said:-

Stringham was awarded a Ph.D. in 1880 for his thesesThose gifted with the powers of a Stringham, a Newcomb, or a Charles S Peirce to feel their way about in supersensible space, may, in like manner, obtain if not an intuitive, at least immediate, or non-mediated proof of the[following]theorem ...

*Regular Figures in n-dimensional Space*. In the same year he was awarded a Parker Fellowship by Harvard and, funded by this, he went to Europe. In particular he spent time at Leipzig where he attended lectures by Felix Klein. He wrote in letters back to Daniel Coit Gilman, the president of Johns Hopkins, about attending weekly seminars:-

Although most of the students at the seminar were German, there were, he wrote:-... with Professor Klein's wonderful critical faculty continually at play.

Klein lectured for two semesters on geometric function theory, in particular giving Riemann's theory of algebraic functions. Stringham gave two lectures in Klein's seminar on the work he had done for his Ph.D. thesis. Discussion with Klein led to Stringham's paper... one Englishman, one Frenchman, one Italian, and one American(myself).

*Determination of the Finite Quaternion Groups*which was published in the

*American Journal of Mathematics*in 1881. Stringham submitted the paper from Schwarzbach, Saxony, in September, 1881.

After spending the two academic years 1880-82 in Europe, Stringham would have liked to have remained for another year. He tried to get either Johns Hopkins or Harvard to award him a position which would allow him to spend another year in Europe but he was not successful. Gilman did not come up with a job at Johns Hopkins but, instead, arranged for Stringham to get a professorship at the University of California at Berkeley. He took up this position in the autumn of 1882 but not only was he a professor but he was also appointed as Head of Mathematics, a post that had been vacant for over a year. On 28 June 1888, Stringham married Martha Sherman Day in New Haven, Connecticut. Martha, born in San Leandro, Stockton, California in 1861, was the daughter of Roger Sherman Day and Harriet Eliza Clark. Martha and Irving Stringham had three children: Harriet Day Stringham (born 21 August 1889); Martha Sherman Stringham (born 5 March 1891); and Roland Irving Stringham (born 24 May 1892). Stringham made several trips abroad, both before and after his marriage. For example he studied in Spain in 1887 and in Paris on several occasions between 1889 and 1900.

When Stringham was appointed as a professor and head of mathematics at the University of California at Berkeley he was 35 years old. It was a role he took on reluctantly, however, since he fully realised that his opportunities as a research mathematician would be limited. Indeed, he was soon appointed as Dean of the Faculty giving him an even greater administrative role. As an example we quote from a letter he wrote in 1884, two years after his appointment to Berkeley, to Daniel Coit Gilman, the president of Johns Hopkins. The University of California was governed by a Board of Regents made up of industrialists and politicians. Stringham complained to Gilman that the Board of Regents continued to interfere using:-

In the same letter he wrote:-... an arbitrary exercise of power in matters concerning which the judgement of the Faculty in Berkeley would certainly be more competent.

Tony Robbin writes [4]:-I have not be able to apply myself to my favourite studies.

We mention two books that Stringham published while at Berkeley, namelyThis wonderful mathematician with so many lively interests was swallowed by the morass that is university politics.

*Uniplanar Algebra*, Vol. 1:

*Being of a Propaedeutic to the Higher Mathematical Analysis*(1893) and

*Elementary Algebra for the Use of Schools and Colleges*by Charles Smith, revised and adapted to American schools by Irving Stringham (1895).

We still have not looked at the mathematics for which Stringham was best known. Although today, he is little known among mathematicians, we must point out that in the 1880s he was considered a leading mathematician with great research potential. The results which gave him an international reputation were contained in his doctoral thesis *Regular Figures in n-dimensional Space* and became well-known through the paper he published with the same title. The platonic solids in 4-dimensional space had been found by Ludwig Schläfli in 1852. Stringham's paper [4]:-

Let us give some extracts from Stringham's paper to illustrate how he describes the four-dimensional figures:-... largely duplicated Schläfli's discoveries,[but]for the first time, included illustrations of four-dimensional figures. Long forgotten until its rediscovery by art historian Linda Henderson, this paper swept through Europe when it was written and was cited in every important mathematical text on four-dimensional geometry for the next two decades.

We refer the reader to [4] for an interesting discussion of Stringham's paper and possible sources for the innovative ideas it contains.A regular angle is defined as one all of whose boundaries of any given number of dimensions are the same in form and magnitude. ... the number of regular4-fold angles that can be made up of regular polyhedral angles is five. Out of trihedral angles three regular4-fold angles can be formed, corresponding to the three regular polyhedra which have triangular boundaries. Similarly there is one regular distribution for tetrahedral and one for pentahedral angles. The former consists of a set of six angles corresponding to the faces of the cube, and the latter of twelve angles corresponding to the faces of the dodekahedron. ... A complete n-fold figure is defined as one which is limited on every hand by complete(n -1)-fold figures which are themselves limited by complete(n -2)- fold figures, and so on. ... In particular, the4-fold pentahedroid has5summits,10edges,10triangular and5tetrahedral boundaries. To construct this figure select any one summit of each of four tetrahedra and unite them. Bring the faces, which lie adjacent to each other, two and two into coincidence. There will remain four faces still free; take a fifth tetrahedron, and join each one of its faces to one of these four remaining ones. The resulting, figure will be the complete4-fold pentahedroid.

Stringham gave a number of addresses, both to school teachers and to International Mathematical Conferences. For example he addressed the International Mathematical Congress, held in connection with the World's Columbian Exposition in Chicago in 1893. This Congress was the model for the series of International Congresses of Mathematicians which began in Zürich in 1897. The title of his talk was *Formulary for an Introduction to Elliptic Functions*. In 1900 he addressed the International Congress of Mathematicians in Paris with the talk *Orthogonal Transformations in Elliptic or Hyperbolic Space*. Among his addresses to school teachers we mention *Mathematics in Grammar Schools* delivered to the Alameda Teachers' Institute in October 1885, and T*he Past and Present of Elementary Mathematics* delivered to the California Teachers' Association in December 1891.

Stringham's death is reported in [5]:-

Mellen Woodman Haskell (1863-1948) was an undergraduate at Harvard and then undertook research for his doctorate supervised by Felix Klein at Leipzig and Göttingen. He was awarded his doctorate on 18 June 1889 and appointed as an Instructor in Mathematics at Berkeley. He was promoted to assistant professor at Berkeley in June of the following year. This was the first new professorial appointment in mathematics following the appointment of Stringham. He worked closely with Stringham at Berkeley and wrote these words following his death:-As dean of the, faculties of the university, under the new scheme put into operation shortly before the departure of Doctor Wheeler as Roosevelt exchange professor to deliver lectures at the University of Berlin, Professor Stringham was acting president. During the last two weeks Professor Stringham had been confined to his bed at his home at2250Prospect street and Sunday afternoon was removed to the Alpha Bates local sanatorium for an operation. Doctors Moffitt, Burnham and Terry were called in. An incision was made, but the patient grew so weak that the operation was postponed, but the scholar rapidly sank and last night lost consciousness and died at daybreak this morning. Uraemic poisoning was pronounced the direct cause.

Professor Stringham was one of the really distinguished mathematicians in this country, being the foremost authority on the subject of absolute geometry and of the geometry of more than three dimensions. He was also a man of wide influence in all University affairs, being actively concerned with every forward movement the University has made during the twenty-seven years of his service. But, most of all, he was a man of sincere purpose, of high ideals, and of kindly sympathy.

**Article by:** *J J O'Connor* and *E F Robertson*