Table Of ContentA CRITICAL AMD HISTORICAL STUD! OP THE ROLE OP RULER AMD COMPASS
CONSTRUCTIONS IN THE TEACHING OF HIGH SCHOOL GEOMETRY
IN THE UNITED STATES
Dissertation
Presented in Partial Fulfillment of tbe Requirements
for the Degree Doctor of Philosophy in the
Graduate School of The Ohio State
University
By
Valter August Albrecht, Jr., B.S., M.A.
I
The Ohio State University
1952
i
contents
Pa^e
1 INTRODUCTION
statement of the Problem 1
Importance of the Problem 3
Scope and Limitations of the Study 4
Definitions of Terms 5
Basic Assumptions 6
II REMARKS ON GEOMETRICAL INSTRUMENTS, POSTULATES, AND
HYPOTHETICAL CONSTRUCTIONS
Geometrical Instruments o
Postulates of Construction 9
Hypothetical Constructions 1r
III A HISTORICAL PICTURE OF THE ROI£ OF CONSTRUCTIONS IN
GEOMETRY
Early Geometry in Egypt 14
Greek Geometry IT
Euclid and the Elements 20
Three Famous Problems 21
Other Sets of Restrictions upon Instruments 27
Geometry During the Middle Ages and the Renaissance 29
Geometry from the l6th to the 19th Centuries 30
Su canary 34
IV GEOMETRICAL CONSTRUCTION IN HIGH SCHOOL TEXTS
Introduction 36
Justification for the Restriction to Straight-edge
and Coiqpasses 37
1* The restriction Is not even mentioned* 37
2. The restriction is specifically mentioned but
is not Justified. 4l
3« Ike restriction is Justified on the basis of
convention or historical usage. 48
4. ike restriction to straight-edge and coqpasses
is Justified on the basis of being demanded by
postulates. 53
5- The restriction is Justified on the basis of
theoretical accuracy. 59
6. The restriction is Justified as being one rule
of a game. 64
7• Miscellaneous other Justifications 66
ii
Treatment of Concepts Closely Related to the Ruler-And-
Compass Constructions 69
1. Carrying distances with the corqpasaes 65
0
c. • frisection of any angle 70
3. Construction of the regular polygons 71
u. The approximation of n 73
5. Discussion of inqpossible constructions 75
6. Improper use of the word ■^neasure* 7b
Summary of Discussions of Justifications Proposed for
the Use of the Ruler and Cotq?ass Restriction 77
V GEOMETRICAL CONSTRUCTIONS AITD JUSTIFICATIONS OR THE RULER AND
COMPASS RESTRICTION APPEARING IH WORKS OTHffl THAN PIANE GE
OMETRY texts
Teaching Manuals 80
Committee Reports 89
Miscellaneous.Standard and Popular Works 98
Articles Appearing in Periodicals 107
Summary ill
VI SUMMARY AND RECOMMENDATIONS
Summary of Earlier Chapters ll6
Sumnary of the Role of the Classical Restriction 119
Recommendations 123
BIBLIOGRAPHY
Part I: A List of the Seventy High School Geometry Text
books Examined in this Study 125
Part IB Bibliography Relating to the Classical Construc
tion Problems and to Constructions With Other
3ets of Restrictions 129
Part His Books Other Than Geometry Textbooks Consulted in
this Study 131
1
CHAPTER I
INTRODUCTION
Statement of the Problem
Whenever geometrical constructions are mentioned, most persons
who are at all familiar with geomstry recall the straight-edge and
compasses, but they remember them as the only tools or Instruments
allowed in geometry, and those with which all the constructions must
be made. For example, the problem "to draw a straight line through
a given point and parallel to a given line" is easily solved by the
use of a pair of triangles or a parallel-ruler, but students are re
quired to use straight-edge and compasses only. Those who study plane
geometry in our high schools come under the Influence of teachers and
textbooks which, almost without exception, continue in the tradition
of universal and unquestioning acceptance of the so-called Tlatonic"
or ’Euclidean" restriction to straight-edge and compasses as an inte
gral part of demonstrative geometry. After a year of high school ge
ometry, students remain unaware of the origin and the great Influence
upon mathematical thinking of the restriction to straight-edge
compasses; they have little or no Idea of what is meant by "impossi
ble" ; they think it strange that so much stress is laid on these two
instruments when all about them they see a great variety of instruments,
such as T-square, triangles, protractor, carpenter'a square, and paral
lel rulers, used in solving practical problems of geometry which con
tinually arise in carpentry, navigation, engineering, architectural
drawing, and other practical occupations and trades.
Educators in general, and writers on the teaching of imthematlcs
In particular, have seldom seriously questioned or given critical at
tention to the role and importance of ruler-and-compass constructions
in the teaching of geometry. It Is possible, of course, that other
phases of geometry are felt to be so much more Important than are con
structions, that the former merit practically all of our attention; on
the other hand, the apparent ignoring of constructions may indicate a
passive acceptance of the traditional restriction to straight-edge and
compasses as an Integral part of plane demonstrative geometry. There
Is little evidence in the literature of mathematics education to Indi
cate whether or not this restriction on the Instruments of geometric
construction Is contributing to generally accepted objectives of geom
etry teaching.
Many textbooks place a rather strong emphasis upon the mechanics
of construction, and because the usual geometry course la governed by
the textbook, much time is devoted to manipulating the instruments. Al
though major stress is purportedly placed upon reasoning and the nature
of proof, few students gain a correct concept of the postulates or as
sumptions which underlie the ruler-and-compass constructions, or of the
logical way in which the constructions follow from the basic assumptions.
Most students and many teachers have never heard that other equally valid
and acceptable sets of restrictions might have been Imposed upon the in
struments of construction, and many do not make the distinction between
drawing and constructing geometric figures.
Construction*, for the most pert, continue to be taught in the
traditional manner, used for two thousand years with little variation,
of memorizing medals presented in the texts, and attempting to prove
that the construction procedure is correct. The pupil has little con
cept of what he is doing or of why it is being done.
The problem, therefore, is this: there is little written evidence,
as will be shown, that much critical thought has been given to the pos
sibility of using constructions effectively in the course in plane ge
ometry. Teachers, on the whole, are not familiar enough with geometric
constructions, the classical restriction to the use of ruler and compasses,
and related ideas and concepts to assign to them their proper role In the
teaching of high school geosmtry.
The purpose of this study will therefore be to Investigate the role
of geometric constructions In the geometry teaching of the past; to in
quire Into their Importance and possible contributions in a modern course
in demonstrative geometry; and to present materials which should help
teachers and prospective teachers to understand the whole problem of ge
ometric constructions and to make good use of them in the classroom.
Importance of the Problem
Comparatively little work of this nature has been done. This study
should serve to focus the attention of high school geometry teachers and
those who train teachers, upon a phase of geometry normally overlooked
when critical examination and analysis of values are brought to bear upon
the teaching of geometry. It should assist teachers and those who train
teachers to determine for themselves what uses can be made in their own
classrooms of geometrical construct Iona in tbs development of important
learning values. Tbs study will bring together in one place information
and materials relating to geometric constructions, which la of value to
teachers, those who train teachers, and textbook writers, and which is
now found in scattered places.
The study should further assist teachers and prospective teachers of
high school geometry to decide how to Justify the ruler-and-compass re
striction to their pupils, to determine the amount and kind of stress to
be placed upon constructions, to stimulate their own and their pupils*
interest In the further implications of the restriction. A thorough
knowledge of ruler-and-compass constructions and an understanding and
appreciation of their role in geometry will result in a much more inter
esting and meaningful geometry course.
Bcope and Limitations of the Study
The study will gather reasons and Justifications that have been pro
posed for the original restriction Imposed by the early Oreeks, and trace
the role and influence of geometrical constructions through to modern
times. Other sets of restrictions will be briefly examined. Tbs various
ways in which modern authors Justify the Platonic" restrictions to stu
dents will be determined and compared with the objectives of geometry
teaching. An attempt will be made to evaluate recent implications that
constructions are over-emphasised, improperly taught, or, as suggested
by Hathan Lasar in classroom lectures and public talks, a hoax foisted
upon the pupils.
Information will be gathered from original and secondary sources
Including:
(1) old and recent high school geometry texts,
(2) works pertaining to the teaching of plane geometry,
(3) reports of Important regional and national councils
and committees on geometry teaching,
(4) articles in periodicals and Journals of mathematics
and mathematics education,
(5) other books which include material on geometry and
constructions, such as works on the history of math*
ematlcs, books on mathematical recreations, math
ematical works of a popular nature, and works devoted
to the study of special problems.
The study will confine itself mainly to materials published end used in
the United States, and to a lesser degree In Canada and England. German
and French works which have had an Important Influence on geometry teach
ing in the United States will also be used; but the great difficulty in
obtaining modern textbooks and manuals from countries outside the United
States has made it necessary to eliminate all such works from consider
ation. The study is further confined to plane, demonstrative geometry,
with little reference to solid geometry or to Informal geometry.
Definitions of Terms
The familiar term "ruler-and-compass constructions" will be used
to designate "geometrical constructions performed with the aid of only
an unmarked straight-edge and a pair of compasses." A geometrical
construction is regarded as a problem situation in which It Is required
that a desired figure be drawn with the aid of specified Instruments
(such as the straight-edge and compasses) and using specific given data.
Textbook will mean any book containing a definite body of systemati
cally arranged subject matter and intended for use as a principal and
primary source of atudy material for a high achool course In plana
geometry (uaually given in the tenth grade). Objectivea of a courae
of lnatructlon are the deaired behavlora or changea in behavior of
the pupil aa a reault of experiencea originating in and arlaing from
the courae. Content of a courae will dealgnate the factual informa
tion, knowledges, and attltudea which are to be imparted to the pupils,
and methods are the proceaaea ueed to impart the mathematical informa
tion and underatandlnga. Authorltlea are thoee lndlviduala and groupa,
who ty virtue of their works, writInga, and pronouncemanta concerning
nathematlca and mathematics education, or poaitlona aa outatandlng
teachera of mathenatlca, are regarded by the profeeelon to be sourcea
of productive thinking in the field of mathenatlca education. Elemen
tary geometry will include the uaual theoreme, principles, construc
tions, and applications found in good textbooks. When It is confined
to points, straight lines, polygons, and circles in a plane, it con
stitutes elementary plane geometry.
Throughout the paper the following distinctions will be made: to
construct a figure means to effect its completion with the straight
edge and compasses only; to draw a figure will mean that any instru
ments deaired may be used, such as ruler, protractor, compasses, tri
angles; to sketch a figure will imply a freehand drawing, made with
out aid of Instruments, which represents the required figure.
Basic Assumptions
A study of this kind must necessarily proceed from a number of
general assumptions, such as: that geometry is of sufficient importance
as a high school subject to warrant a study of one of Its Important
phases; that constructions constitute a large and important phase of
plane demonstrative geometry; that a problem, as stated above, does exist
with regard to constructions; and that the sources of Information ere
authentic, reliable, and adequate.
It will be further assumed that, since most high school courses In
geometry are determined by the textbook used, an examination of text
books will reveal the content of the usual course In plane demonstrative
geometry and, for the purposes of this study, the role assigned to the
classical constructions.
