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Modern and Pure: Teaching Geometry in Early Twentieth-Century Women’s Colleges

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Women in Mathematics

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 10))

Abstract

In 1913 The American Mathematical Monthly published an article encouraging the inclusion of synthetic projective geometry in the undergraduate curriculum. The author, William Henry Bussey, lamented that American colleges and universities rarely taught the subject despite its potential for industry, teachers, and the liberal arts curriculum. The following year Lao G. Simons responded in “hearty sympathy” describing the potential of synthetic geometry in broadening “the minds of prospective high school teachers.” Both Simons and Bussey particularly remarked on the success of the course at women’s colleges. Simons herself taught at the Normal College of the City of New York and referred to the appreciation “the girls” had of the subject. Bussey observed that “Bryn Mawr, Mount Holyoke, Smith, Vassar, Wellesley, Wells, and Goucher College of Baltimore” regularly featured the course. Drawing on college catalogs, faculty publications, department histories, course notes, and textbooks, this paper will examine the diverse reasons for teaching this modern pure geometry at women’s colleges in the decade following 1913.

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Notes

  1. 1.

    The Conference in Mathematics met in December of 1892 and consisted of Simon Newcomb, William E. Byerly, Arthur H. Cutler, Florian Cajori, Henry B. Fine, W. A. Greeson, Andrew Ingraham, George G. Olds, James L. Patterson, and T. H. Safford.

  2. 2.

    In 1872 Luigi Cremona debated the alternate names of this new form of geometry and came to a different conclusion.

    Various names have been given to this subject of which we are about to develop the fundamental principles. I prefer not to adopt that of Higher Geometry (Géométrie supérieure, höhere Geometrie), because that to which the title ‘higher’ at one time seemed appropriate, may to-day have become very elementary; nor that of Modern Geometry (neuere Geometrie), which in like manner expresses a merely relative idea; and is moreover open to the objection that although the methods may be regarded as modern, yet the matter is to a great extent old. Nor does the title Geometry of position (Geometrie der Lage) as used by STAUDT seem to me a suitable one, since it excludes the consideration of the metrical properties of figures. I have chosen the name of Projective Geometry, as expressing the true nature of the methods, which are based essentially on central projection or perspective. And one reason which has determined this choice is that the great PONCELET, the chief creator of the modern methods, gave to his immortal book the title of Traité des propriétés projectives des figures (1822) [14].

    Nevertheless, even fifty-years after Cremona, the designation among geometries was far from uniform, and most authors seemed to understand any of the above terms as roughly equivalent.

  3. 3.

    We exclude Wells College as we have been unable so far to obtain any course materials from this institution.

  4. 4.

    The recent Cajori Two Project shows that, in contrast to Bussey’s observations, courses classified as projective geometry were taught at diverse institutions including Colorado College, Johns Hopkins, Stanford, Berkeley, and University of Wisconsin at Madison in the year 1915.

  5. 5.

    For the publication and influence of Karl Georg Christian von Staudt’s books, Geometrie der Lage and Beiträge zur Geometrie der Lage, see [25].

  6. 6.

    Additional biographical information on many of the women mathematicians cited here can be found in [17].

  7. 7.

    For instance, other available textbooks in modern pure geometry from this period include [15, 16], and [18].

  8. 8.

    Mathematics departments prized their collections of models bought from German manufacturers or made by students. For instance, a 1919 history of Mount Holyoke described “a collection of models, of plaster and thread, illustrating quadric surfaces, surfaces of the third and fourth orders, Riemann surfaces and surfaces of complex functions” [37, p. 16]. Presumably some of these, like quadric surfaces, would be useful in higher geometry courses.

  9. 9.

    Vassar professor Sophia Richardson’s textbook Solid Geometry also included instances of dual columns and she outlined the concept in a brief note: “In the geometry of the line and plane it happens that so many instances occur of pairs of theorems thus related to each other, that the study of the topic is somewhat simplified by considering in the case of each theorem the theorem derived from it by the interchange of the words line and plane” [27, p. 22].

  10. 10.

    In the translator’s introduction, Holgate quoted H.J.S. Smith with approval as stating “All attempts to construct imaginaries have been wholly abandoned in pure geometry” [26, p. viii].

  11. 11.

    By contrast, some elective courses were dropped or offered only infrequently.

  12. 12.

    For a pessimistic or resigned view on this see, respectively [2] or [7].

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Acknowledgments.

This paper was made possible by the generous assistance of archivists at Mount Holyoke, Smith, Vassar, Wellesley, Goucher, and Hunter Colleges.

Author information

Authors and Affiliations

  1. Pitzer College, 1050 North Mills, Claremont, CA, USA

    Jemma Lorenat

Authors
  1. Jemma Lorenat
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Corresponding author

Correspondence to Jemma Lorenat .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Redlands, Redlands, California, USA

    Janet L. Beery

  2. Department of Mathematics, Appalachian State University, Boone, North Carolina, USA

    Sarah J. Greenwald

  3. Department of Mathematics, Lamar University, Beaumont, Texas, USA

    Jacqueline A. Jensen-Vallin

  4. Department of Mathematics, Fordham University, Bronx, New York, USA

    Maura B. Mast

Appendix: Summaries of Course Catalogs

Appendix: Summaries of Course Catalogs

This appendix was constructed with reference to [44–99].

College

Year

Course

Instructor

Sequence

Credits

Vassar

1913/1914

M. Synthetic Projective Geometry

Richardson

Prerequisite: Course A [Analytic Geometry]

Second semester [3].

 

1914/1915

M. Synthetic Projective Geometry

Richardson

Prerequisite: Course A [Analytic Geometry]

Second semester [3].

 

1915/1916

M. Synthetic Projective Geometry

Richardson

Prerequisite: Course A [Analytic Geometry]

Second semester [3].

 

1916/1917

M. Synthetic Projective Geometry

White

Prerequisite: Course A [Analytic Geometry]. This course is valuable as a supplement to courses I (Curve Tracing) and J (Analytic Geometry of Three Dimensions).

Second semester [3].

 

1917/1918

M. Synthetic Projective Geometry

White

Prerequisite: 11. This course is valuable as a supplement to 1 (plane trigonometry, with logarithms), 11 (analytic geometry) and 15 (analytic geometry of three dimensions).

Second semester [3].

 

1918/1919

26. Synthetic Projective Geometry. Second Semester.

White

Prerequisite: 11. This course is valuable as a supplement to 1 (plane trigonometry, with logarithms), 11 (analytic geometry) and 15 (analytic geometry of three dimensions).

Second semester [3].

 

1919/1920

19 and 20. Introduction to Descriptive Geometry and Mechanical Drawing

Cowley

1. Plane trigonometry with Logarithms and 2. Solid Spherical Geometry

First semester [1], Second semester [1]. Two hours in the classroom, with very little preparation, to be counted as one hour.

  

26. Synthetic Projective Geometry

White

Prerequisite: 11. This course is valuable as a supplement to 1 (plane trigonometry, with logarithms), 11 (analytic geometry) and 15 (analytic geometry of three dimensions).

Second semester [3].

 

1920/1921

19 and 20. Introduction to Descriptive Geometry and Mechanical Drawing

Cowley

1. Plane trigonometry with Logarithms and 2. Solid Spherical Geometry

First semester [3], Second semester [3].

  

26. Synthetic Projective Geometry

White

Prerequisite: 11. This course is valuable as a supplement to 1 (plane trigonometry, with logarithms), 11 (analytic geometry) and 15 (analytic geometry of three dimensions).

Second semester [3].

 

1921/1922

19 and 20. Introduction to Descriptive Geometry and Mechanical Drawing

Cowley

1. Plane trigonometry with Logarithms and 2. Solid Spherical Geometry

First semester [3], Second semester [3].

  

26. Synthetic Projective Geometry

White

Prerequisite: 11. This course is valuable as a supplement to 1 (plane trigonometry, with logarithms), 11 (analytic geometry) and 15 (analytic geometry of three dimensions).

Second semester [3].

 

1922/1923

19 and 20. Introduction to Descriptive Geometry and Mechanical Drawing

Cowley

1. Plane trigonometry with Logarithms and 2. Solid Spherical Geometry

First semester [3], Second semester [3].

  

26. Synthetic Projective Geometry

White

Prerequisite: 11. This course is valuable as a supplement to 1 (plane trigonometry, with logarithms), 11 (analytic geometry) and 15 (analytic geometry of three dimensions).

Second semester [3].

Mount Holyoke

1913/1914

Projective Geometry

Smith

Open to juniors

First and second semester, three hours

 

1914/1915

8 and 9. Projective Geometry

Doak

Open to juniors

First and second semesters, each three hours

 

1915/1916

8 and 9. Projective Geometry

Doak

Open to juniors

First and second semesters, each three hours

 

1916/1917

8 and 9. Projective Geometry

Doak

Open to juniors

First and second semesters, each three hours

 

1917/1918

8 and 9. Projective Geometry

Doak

Open to juniors

First and second semesters, each three hours

 

1918/1919

8 and 9. Projective Geometry

Smith

Open to juniors

First and second semesters, each three hours

 

1918/1919

10. Selected Topics in

Geometry

Martin

Prerequisite: Differential and Integral Calculus, Introduction to the Calculus, Analytical Geometry, Solid and Spherical Geometry or College Algebra and Plane Trigonometry

First semester, three hours

 

1919/1920

8 and 9. Projective Geometry

Smith

Open to juniors

First and second semesters, each three hours

 

1919/1920

10. Selected Topics in

Geometry

Martin

Prerequisite: Differential and Integral Calculus, Introduction to the Calculus, Analytical Geometry, Solid and Spherical Geometry or College Algebra and Plane Trigonometry

First semester, three hours

 

1920/1921

8 and 9. Projective Geometry

Smith

Open to juniors

First and second semesters, each three hours

 

1921/1922

8 and 9. Projective Geometry

Doak

Open to juniors

First and second semesters, each three hours

Smith

1913/1914

3. Descriptive Geometry

Cobb

For Juniors.

Three hours, through the year.

 

1914/1915

3. Descriptive Geometry

Cobb

For Juniors and Seniors

Three hours, through the year.

 

1915/1916

3. Descriptive Geometry

Cobb

For Juniors and Seniors

Three hours, through the year.

 

1916/1917

32b. Descriptive Geometry

Cobb

none

Three hours, second semester.

  

37. Projective Geometry

Cobb

It is recommended that this course be preceded by 32b.

Three hours, through the year.

 

1917/1918

32b. Descriptive Geometry

Cobb

none

Three hours, second semester.

  

37. Projective Geometry

Cobb

none

Three hours, through the year.

  

32a. Descriptive Geometry

Cobb

none

Three hours, first semester

  

37. Projective Geometry

Cobb

none

Three hours, through the year.

  

42. Projective Geometry with Especial Reference to Imaginaries. Beiträge zur Geometrie der Lage by K. von Staudt

Cobb

none

no information

 

1919/1920

32b. Descriptive Geometry

Cobb

none

Three hours, second semester.

  

37. Projective Geometry

Cobb

none

Three hours, through the year.

  

42. Projective Geometry with Especial Reference to Imaginaries. Beiträge zur Geometrie der Lage by K. von Staudt

Cobb

none

no information

 

1920/1921

all courses omitted

   
 

1921/1922

32b. Descriptive Geometry

Cobb

none

Three hours, second semester.

  

37. Projective Geometry

Cobb

none

Three hours, through the year.

  

42. Projective Geometry with Especial Reference to Imaginaries. Beiträge zur Geometrie der Lage by K. von Staudt

Cobb

none

no information

 

1922/1923

32a. Descriptive Geometry

Cobb

none

Three hours, second semester.

  

37. Projective Geometry

Cobb

none

Three hours, through the year.

  

42. Projective Geometry with Especial Reference to Imaginaries. Beiträge zur Geometrie der Lage by K. von Staudt

Cobb

none

no information

Wellesley

1913/1914

6. Modern Synthetic Geometry

Burrell

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

 

1914/1915

6. Modern Synthetic Geometry

Chandler

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

 

1915/1916

none

   
 

1916/1917

6. Modern Synthetic Geometry

Merrill

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

 

1917/1918

6. Modern Synthetic Geometry

Merrill

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

 

1918/1919

17. Descriptive Geometry

Merrill

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

 

1919/1920

6. Modern Synthetic Geometry

Merrill

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

  

17. Descriptive Geometry

Merrill

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

 

1920/1921

206. Descriptive Geometry

Merrill

Open to students who are taking a three-hour elective course in Mathematics.

One hour a week for a year with one laboratory period.

  

306. Modern Synthetic Geometry

Merrill

Open to students who have completed or are taking course 3 [Differential and Integral Calculus].

Three hours a week for a year.

 

1921/1922

206. Descriptive Geometry

Merrill

Open to students who are taking a three-hour elective course in Mathematics.

One hour a week for a year with one laboratory period.

  

306. Modern Synthetic Geometry

Merrill

Open to students who have completed course 202 [Differential and Integral Calculus] or 301 [Calculus and Its Applications]

Three hours a week for a year.

 

1922/1923

206. Descriptive Geometry

Stark

Open to students who are taking a three-hour elective course in Mathematics.

One hour a week for a year with one laboratory period.

  

306. Modern Synthetic Geometry

Young

Open to students who have completed course 202 [Differential and Integral Calculus] or 301 [Calculus and Its Applications]

Three hours a week for a year.

Bryn Mawr

1914/1915

Ib. Lectures on Modern Pure Geometry

Scott

Post-Major

Two hours a week throughout the year

 

1915/1916

none

   
 

1916/1917

Ib. Lectures on Modern Pure Geometry

Scott

Post-Major

Two hours a week throughout the year

 

1917/1918

Ib. Lectures on Modern Pure Geometry

Scott

Post-Major

Two hours a week throughout the year

 

1918/1919

none

   
 

1919/1920

Lectures on Modern Pure Geometry

Scott

Post-Major

Two hours a week throughout the year

 

1920/1921

none

   
 

1921/1922

Lectures on Modern Pure Geometry

Scott

Post-Major

Two hours a week throughout the year

 

1922/1923

none

   

New York Normal School

1913/1914

26. Projective Geometry

Simons

Prerequisite: Course 15 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1914/1915

35. Projective Geometry

Simons

Prerequisite: Course 15 [Plane and Solid Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1915/1916

35. Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1916/1917

35. Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1917/1918

35. Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1918/1919

35. Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1919/1920

35. Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1920/1921

35. Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

 

1921/1922

35. Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

May be given only in the spring term. 3 periods, one half-year; 3 semi-annual credits.

 

1922/1923

35. Synthetic Projective Geometry

none listed

Prerequisite: Course 24 [Plane Analytic Geometry]

3 periods, one half-year; 3 semi-annual credits.

Goucher

1915/1916

8. Pure Projective Geometry

Bacon

Prerequisite, Course 4 [Plane Analytic Geometry]

Three hours, first semester.

 

1918/1919

19. Pure Projective Geometry

Bacon

Prerequisite, Courses 11-12 [Plane Analytic Geometry]

Three hours, first semester.

 

1919/1920

19. Pure Projective Geometry

Bacon

Prerequisite, Courses 11-12 [Plane Analytic Geometry]

Three hours, first semester.

 

1920/1921

19. Pure Projective Geometry

Bacon

Prerequisite, Courses 11-12 [Plane Analytic Geometry]

Three hours, first semester.

 

1921/1922

19. Pure Projective Geometry

Bacon

Prerequisite, Courses 11-12 [Plane Analytic Geometry]

Three hours, first semester.

 

1922/1923

19. Pure Projective Geometry

Bacon

Prerequisite, Courses 13-14 [Differential and Integral Calculus]

Three hours, first semester.

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Lorenat, J. (2017). Modern and Pure: Teaching Geometry in Early Twentieth-Century Women’s Colleges. In: Beery, J., Greenwald, S., Jensen-Vallin, J., Mast, M. (eds) Women in Mathematics. Association for Women in Mathematics Series, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-66694-5_17

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