Mabel Minerva Young (1872 – 1963) was an American mathematician active at Wellesley College.

Life

Young was born July 18, 1872, in

Worcester, Massachusetts Worcester ( , ) is a city and county seat of Worcester County, Massachusetts, United States. Named after Worcester, England, the city's population was 206,518 at the 2020 United States census, 2020 census, making it the second-List of cities i ...

. She began study at Wellesley College in 1894. Going to graduate study at

Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manhatt ...

, she graduated with a

master's degree A master's degree (from Latin ) is an academic degree awarded by universities or colleges upon completion of a course of study demonstrating mastery or a high-order overview of a specific field of study or area of professional practice.

in 1899. First she taught

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...

Northfield Seminary Northfield Mount Hermon School, often called NMH, is a co-educational preparatory school in Gill, Massachusetts, in the United States. It is a member of the Eight Schools Association. Present day NMH offers nearly 200 courses, including AP and ...

. In 1904 she began her long service at Wellesley College, beginning as an assistant in mathematics and becoming a full professor. Taking a leave of absence, she studied for her Ph.D. with

Frank Morley Frank Morley (September 9, 1860 – October 17, 1937) was a leading mathematician, known mostly for his teaching and research in the fields of algebra and geometry. Among his mathematical accomplishments was the discovery and proof of the celebr ...

Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consi ...

. Her thesis was titled "Dupin's cyclide as a self-dual surface". With her doctoral degree, Young was eventually promoted to

professor Professor (commonly abbreviated as Prof.) is an academic rank at universities and other post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who professes". Professors ...

and became Lewis Attenbury Stimson Professor of Mathematics at Wellesley College. In 1933 Young contributed an article to American Mathematical Monthly on a configuration of

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

s associated with a

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...

π. Let π be a parabola, ''p'' and ''q'' fixed

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

s to π that intersect at T. Then a variable tangent to π forms a triangle with ''p'' and ''q''. The variability of this tangent describes the "single infinity of triangles". The corresponding

orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...

s, and centers of the

nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of ea ...

are approached using projective properties of the triangles. Young became

emeritus professor ''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...

in 1941. She died March 4, 1963, at Wellesley.

Solutions of AMM problems

One of the features of American Mathematical Monthly is a section devoted to problems articulated by readers, and eventual solutions of said problems. The published solutions are chosen for their

elegance Elegance is beauty that shows unusual effectiveness and simplicity. Elegance is frequently used as a standard of tastefulness, particularly in visual design, decorative arts, literature, science, and the aesthetics of mathematics. Elegant ...

, and five involving

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

were by Mabel Young. Given a point and a circle, find the locus of second circles where the

radical axis In Euclidean geometry, the radical axis of two non-concentric circles is the set of points whose Power of a point, power with respect to the circles are equal. For this reason the radical axis is also called the power line or power bisector of ...

of the two circles lies on the given point. Young’s

analytical geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineer ...

solution established a condition on the radii. A given segment subtends an angle from a point on another line. As the point moves along its line, find the

envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...

of the bisectors of the angles. Young's solution established the class of the envelope curve using

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...

. Let a point and a pair of intersecting planes be fixed. Then as a variable line lies on the point, find the locus of the midpoint of the segment determined by the planes. Young's solution starts with a line ''p'' through the point and parallel to the intersection of the planes. She identified the locus as a hyperbolic cylinder through use of a third parallel midway between the others that is the

projective harmonic conjugate In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line t ...

of a line at infinity. In a triangle ''ABC'' the feet of the

altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

s and midpoints of the sides are used to define three involutions. The problem was to show that the

double point In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curv ...

s of these involutions are three pairs of opposite vertices of a

complete quadrilateral In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six l ...

. Young's solution used the radical axis of the circumcircle and nine-point circle of the triangle. Young proposed construction of a strophoid: Form triangle ''AOB'' from a fixed point ''A'' and a variable ''B'' on circle centered at ''O''. Then the locus of the

of ''AOB'' is a strophoid. Another problem required the concurrence of three lines determined by a triangle's altitudes and angle bisectors. Young's solution pointed to the

Gergonne point In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

and

Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concu ...

of the triangle to obtain the concurrence.''AMM'' 38(3): 177

References

Boston Globe ''The Boston Globe'' is an American daily newspaper founded and based in Boston, Massachusetts. The newspaper has won a total of 27 Pulitzer Prizes, and has a total circulation of close to 300,000 print and digital subscribers. ''The Boston Glob ...

(March 5, 1963) "Mabel Young 89, headed math department at Wellesley College" * {{DEFAULTSORT:Young, Mabel Minerva 1872 births 1963 deaths Wellesley College faculty Geometers American mathematics educators 20th-century American women scientists 20th-century American mathematicians 20th-century American women mathematicians