Logo Homeschool World ® Official Web Site of Practical Homeschooling Magazine Practical Homeschooling Magazine
Practical Homeschooling® :

The History of Geometry Education

By Sam Blumenfeld
Printed in Practical Homeschooling #48, 2002.

Pin It

Sam Blumenfeld


Once a student has mastered arithmetic, including fractions and decimals, the next subject to tackle is Algebra, in which symbols are used to represent unknown quantities. After Algebra comes Geometry. Geometry, of course, is used in many professions involving physics, engineering, architecture, space exploration, and anything that requires measurement. It was used by the ancient Greeks and the Egyptians in building their huge monuments and temples.

What is geometry? It is a branch of mathematics concerned with the properties of space and shapes. According to Yale Prof. Elias Loomis, author of one of the most widely used textbooks on Geometry in the 19th century, the term Geometry is derived from a Greek word, meaning the "science of land-measuring." Ancient writers generally supposed that this science was first cultivated in Egypt. Aristotle attributed the invention to the Egyptian priests, who had abundant leisure for study. Pythagoras, born 580 years B.C., was one of the earliest cultivators of Geometry. He discovered that the circle has a greater area than any other plane figure having an equal perimeter, and that a sphere has a similar property among solids.

The most remarkable epoch in the history of Geometry was the establishment of the school of Alexandria, about 300 B.C. It was here that the celebrated geometer Euclid flourished under the first of the Ptolemies. He is best known for his Elements, a work on Geometry and Arithmetic in thirteen books, under which he collected all the elementary truths of Geometry which were found before his time. It is only the first six books and the eleventh and twelfth books that are much used in schools.

The first four books teach the properties of plane figures; the fifth contains the theory of proportion; and the sixth its application to plane figures; the seventh, eighth, ninth, and tenth relate to Arithmetic and the doctrine of incommensurables; the eleventh and twelfth contain the elements of the geometry of solids; and the thirteenth discusses the five regular solids.

In Euclidean geometry the space corresponds to common ideas of physical space, and the shapes are idealizations of the common shapes that occur in real life. Parallel lines never meet and the angles in a triangle always add up to 180 degrees. The geometry of physical space can be assumed for most purposes to be Euclidean.

There are other branches of Geometry. Non-Euclidean geometry deals with the surface of a sphere to which Euclid's postulates do not apply. In Euclidean geometry two lines at right angles to a third remain parallel and never meet. In hyperbolic or elliptic geometry the two lines eventually diverge or converge.

Then there is Riemannian geometry, which is used in relativity theory. It involves the development of a generalized space of any dimensions in which measurements may vary from point to point. Used by Einstein in his general theory of relativity, it was invented by a German mathematician, Georg Riemann, in the mid-19th century.

There is also Analytic Geometry, in which algebra is used to solve geometrical problems. Geometrical figures are placed in a coordinate system, each point in the figure being represented by its coordinates, which satisfy an algebraic equation. It is also known as coordinate geometry or Cartesian geometry, after its inventor Rene Descartes.

In Cartesian coordinates, a point is located by its distance from intersecting lines called axes. In a two-dimensional plane there are two axes and in three-dimensional space there are three. Usually the axes are at right angles to each other and are known as rectangular axes, but oblique axes are also used in exceptional circumstances.

However, to begin with, you will want to concentrate on elementary Euclidean geometry. Thomas Hill wrote an elementary text back in 1855 entitled in First Lessons in Geometry. He wrote in his Preface:

I have addressed the child's imagination, rather than his reason, because I wished to teach him to conceive of forms. The child's powers of sensation are developed before his powers of conception, and these before his reasoning powers. This is, therefore, the true order of education.

You might consider Hill's views when introducing geometry to your child. He also writes:

Geometry is the most useful of all the sciences. To understand Geometry, will be a great help in learning all the other sciences; and no other science can be learned unless you know something of Geometry. To study it, will make your eye quicker in seeing things, and your hand steadier in doing things. You can draw better, write better; cut out clothes, make boots and shoes, work at any mechanical trade, or learn any art, the better for understanding Geometry.

That was written in 1855. Studying the history of Geometry may get your child more interested in the subject than merely plunging into Euclid's Elements. We believe that 19th century professors wrote better books on Geometry than those writing today. You might search an antiquarian bookshop for a good 19th century Geometry textbook. Prof. Elias Loomis's Elements of Geometry, Conic Sections and Plane Trigonometry, was published in a revised edition in 1888. See if you can find one.

However, if you must buy a modern text, try A Beka Books, Bob Jones University, and other reliable sources of homeschool materials. Their textbooks are usually quite good.


Was this article helpful to you?
Subscribe to Practical Homeschooling today, and you'll get this quality of information and encouragement five times per year, delivered to your door. To start, click on the link below that describes you:

USA Individual
USA Librarian (purchasing for a library)
Outside USA Individual
Outside USA Library

Time4Learning University of Nebraska High School

Articles by Sam Blumenfeld

The Whole-Language Boondoggle

High School for Freedom!

Dyslexia: The Man-Made Disease

Teach Reading to the “Learning Disabled”

Uncle Sam Wants Your Child on his National Database

Why the Internet will Never Replace Books

Teach Reading to the "Learning Disabled"

Homeschooling and Charter Schools

Homeschoolers and Vouchers

The History of Public Education

College At Home

Learning from The "Old Dead Guys"

The Meaning of Educational Freedom

The Importance of Rote Learning

The Exodus Continues

A World Without Public School

The Benefits of Teaching History at Home

How to Tell Real from Phony Phonics?

Getting Started in Arithmetic

Teaching Arithmetic

Teaching the Alphabet

Teaching the Alphabet Sounds

Teaching Blends

Teaching Long Vowels

The History of Geometry Education

Never Bored Again

Learning Greek

How and Why to Teach Shakespeare

How to Get the Most Out of Homeschool Conventions

Forgotten American History: The Barbary Wars

Forgotten American History: God's Providence in the American Revolution

Forgotten American History: The Spanish-American War

Forgotten American History: The Great Awakening

Forgotten American History: Puritan Education

Colonial Education: The Free Market in Action

America Started with Educational Freedom

How Harvard Became Liberal

The Glory of the Alphabet

19th Century Communists & the Origin of American Public Education

The Benefits of Cursive Writing

It Pays to Know Your Legislator

Intelligent by Design

Teaching Kids to Enjoy Classical Music

Before Compulsory Education: The Private Academies

What Schools Teach: Then and Now

The Real Meaning of Easter

The Truth About Independence Day

The Benefits of Reading Biographies

Why We Celebrate Veterans Day

The Purposes of Education

Why Homeschoolers Should be Book Collectors

How History Was Taught Back Then

The American Almanac: A Great Learning Tool

The Fun of Going to an Antiques Auction

Politics and Homeschoolers: A Primer

A Novel Suggestion

Who Wrote Shakespeare?

Why Homeschoolers Should Learn Public Speaking

The Presidency

Party Politics in the United States

The Road to an American Independent Nation

George Washington: Our First President's First Term

George Washington: Our First President's Second Term

Celebrating Flag Day

Popular Articles

The Equal Sign - Symbol, Name, Meaning

Interview with John Taylor Gatto

Getting Started in Homeschooling: The First Ten Steps

Don't Give Up on Your Late Bloomers

University Model Schools

The Benefits of Cursive Writing

Shakespeare Camp

The Gift of a Mentor

Phonics the Montessori Way

Laptop Homeschool

Critical Thinking and Logic

Bears in the House

What We Can Learn from the Homeschooled 2002 National Geography Bee Winners

Teaching Blends

Can Homeschoolers Participate In Public School Programs?

AP Courses At Home

Discover Your Child's Learning Style

Whole-Language Boondoggle

How to Win the Geography Bee

Narration Beats Tests

I Was an Accelerated Child

The Benefits of Debate

Saxon Math: Facts vs. Rumors

Getting Organized Part 3

Montessori Math

Myth of the Teenager

Combining Work and Homeschool

Joyce Swann's Homeschool Tips

Classical Education

Art Appreciation the Charlotte Mason Way

A Homeschooler Wins the Heisman

Why the Internet will Never Replace Books

How to "Bee" a Spelling Success

Top Jobs for the College Graduate

Advanced Math: Trig, PreCalc, and more!

Teach Your Children to Work

Character Matters for Kids

The Charlotte Mason Method

Columbus and the Flat Earth...

Montessori Language Arts at Home, Part 1

What Does My Preschooler Need to Know?

The History of Public Education

A Reason for Reading

Getting Organized Part 1 - Tips & Tricks

Give Yourself a "CLEP Scholarship"

Patriarchy, Meet Matriarchy

Top Tips for Teaching Toddlers

Who Needs the Prom?

Start a Nature Notebook

The Charlote Mason Approach to Poetry