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

Wellesley College Wellesley College is a Private university, private Women's colleges in the United States, historically women's Liberal arts colleges in the United States, liberal arts college in Wellesley, Massachusetts, United States. Founded in 1870 by Henr ...

Life

Young was born July 18, 1872, in

Worcester, Massachusetts Worcester ( , ) is the List of municipalities in Massachusetts, second-most populous city in the U.S. state of Massachusetts and the list of United States cities by population, 113th most populous city in the United States. Named after Worcester ...

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

Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...

, she graduated with a

master's degree A master's degree (from Latin ) is a postgraduate 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 prac ...

in 1899. First she taught

English English usually refers to: * English language * English people English may also refer to: Culture, language and peoples * ''English'', an adjective for something of, from, or related to England * ''English'', an Amish ter ...

Northfield Seminary Northfield Mount Hermon School (abbreviated as NMH), is a co-educational College-preparatory school, college-preparatory school in Gill, Massachusetts. It educates boarding and day students in grades 9–12, as well as post-graduate students. It ...

. In 1904 she began her long service at Wellesley College, beginning as an assistant in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

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 celeb ...

Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland, United States. Founded in 1876 based on the European research institution model, J ...

. 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 Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...

and became Lewis Attenbury Stimson Professor of Mathematics at Wellesley College. In 1933 Young contributed an article to

American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...

on a configuration of

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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 exactl ...

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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

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 The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

circumcenter In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumradius. The circumcen ...

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 figure. The same definition extends to any object in n-d ...

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 each s ...

are approached using projective properties of the triangles. Young became

emeritus professor ''Emeritus/Emerita'' () is an honorary title granted to someone who retires from a position of distinction, most commonly an academic faculty position, but is allowed to continue using the previous title, as in "professor emeritus". In some c ...

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

Solutions of AMM problems

One of the features of

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 Taste (sociology), tastefulness, particularly in visual design, decorative arts, literature, science, and Mathematical beauty, the ...

, and five involving

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

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 Analytic or analytical may refer to: Chemistry * Analytical chemistry, the analysis of material samples to learn their chemical composition and structure * Analytical technique, a method that is used to determine the concentration of a chemica ...

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 (message), letter or Greeting card, card. Traditional envelopes are made from sheets of paper cut to one o ...

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 (''p ...

. 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 a point on the real projective line with respect to two other points is defined by the following construction: :Given three collinear points , let be a point not lying on their join and le ...

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

altitude Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...

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 cur ...

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 ...

. Young's solution used the radical axis of the circumcircle and nine-point circle of the triangle. Young proposed construction of a

strophoid In geometry, a strophoid is a curve generated from a given curve and points (the fixed point) and (the pole) as follows: Let be a variable line passing through and intersecting at . Now let and be the two points on whose distance from ...

: 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 concur ...

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

References

Boston Globe ''The Boston Globe,'' also known locally as ''the Globe'', is an American daily newspaper founded and based in Boston, Massachusetts. The newspaper has won a total of 27 Pulitzer Prizes. ''The Boston Globe'' is the oldest and largest daily new ...

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