History of the teaching the elementary geometry by ALVA WALKER STAMPER, 1906

Quintilian's estimate of logical geometry The works of Capella, Boethius, Isidore of Seville, odorus, andtheir use in the schools 43-44 Cassi-The teaching of geometry in the medieval sch

Columbia 1Hniver8it\>

Seacbera College Series

DEGREE OF {DOCTOR OF PHILOSOPHY,

COLUMBIA UNIVERSITY X <A <A <A

BY

B.S., UniTeriity of California 1895 A.M Columbia University, 1905 Graduate Student in Columbia University DuriagTwoYean 1904-1906

"

Geschichte desmathematischen

Amongthe works that treatthe general historyofmathematics,

those of Cantor, Hankel, Gow, and Allman have been of great

miscellaneous articles, standard works on the history of tion, and early texts, the latter constituting, for the most part,

Thematerialthat hasbeen utilizedinthefirstthree chapters

this portionofthe subject-matterand forthe conclusions drawn.The material for the next three chapters has been gleaned very

counsel and criticism, and for rendering much of this work

ALVA WALKER STAMPER.

PAGB

THEBEGINNING OPGEOMETRYAMONGPRIMITIVEPEOPLE . 1-4

classify. Formation of rules Practical use 3-4

Application with logic as a basis 4

Influences thathelpedto develop their geometry 4-7

Utilitarian influence. The overflow of the Nile

Records. The manuscript of Ahmes; inscriptions on the

Thedevelopmentof thesubject-matter ofelementarygeometry . 10-16

The first propositions given to geometry Thepractical not neglected. First problems of construction. Development of a deductive

Development of the geometryof areas. Further

contributions. Nothing known of methods of proof. Geometry becomes an abstract science

Interest ingeometry TheSophists. Theschools

of Plato andAristotle. TheThree Problemsof

of the subject-matter ofgeometry 13Bookswrittenon geometry Euclid not the first.

Summary." Thefoundingof solidgeometryby

Lack of harmony between the historic sequence

andthat given later byEuclid 16 Educational features of the Greek geometry 16-26

In the Ionian and Pythagorean schools 16-17

Neglect of the study of solid geometry . 20

Lettering of figures

Drawing of figures 21Tendency to hold to the special 21-22

Conciseness of proofs 22-23

Methods of attack 23-25

The notion of locus. The methodof

exhaus-tion. Reductio ad absurdum Reduction

Analysis. The diorismus, or discussion

CHAPTER II

SUBSEQUENT TEACHING OF GEOMETRY

The contributions by the earlier Greeks 27-28

Euclid's sequence of subject-matter. Points of parture from the Euclidean system

de-Euclid's sequence not the historic 28-29

The practical eliminated 29

Euclid established a systematic method of proving

Some translations of the "Elements"

The use of Euclid in the medieval universities

CHAPTER III

Hiswritings. Contributions to solidgeometry Value

of TT. Useof methods alreadylaiddown tions of the integral calculus. Geometric and me-chanical proofs. A recognition of the unity of the

Apollonius of Perga 36

teaching ofelementary geometry

Applications ofgeometrytoastronomy and surveying.

ThewritingsofHypsicles, Geminus, Demasciusof

DemascusPtolemy and his interest in geometry

practical writings mentioned that have an

educa-tional significance 38

heights and distances

Land surveying. The CodexArcerianus

Influence of the Alexandrian school

Little influence of Euclid. Quintilian's estimate of logical geometry

The works of Capella, Boethius, Isidore of Seville, odorus, andtheir use in the schools 43-44

Cassi-The teaching of geometry in the medieval schools before

The influence of Gerbert. His geometry 45-47

Leonardoof Pisa. Savasorda Jordanus Nemorarius.

writings

The teaching of Euclid 51-53

AtParis,Oxford,Prague, Vienna, Heidelberg,Cologne,

Leipzig, Bologna, and Pisa. Character of the teaching

THE PRESENT TIME

Paciuolo Fineus. Instruments employed in field work

influence of Albrecht Diirer. Geometry of a single

The early universities 60-61

Place ofgeometryin the curriculum Interest inEuclid

The influence of Comenius The geometries of Schroter and Gruvius

Wolf, Kastner,andSturm,andtheir influence. The

Realschulen

Admis-sion to universities. General natureof the teaching

General resume from the sixteenth century, showing the development oftheteaching of geometryfromthe standpoint

General surveyfrom the sixteenth centuryto the middleof the

Bouvelles. Ramus Errard Mercator. Arnauld

Le Clerc. Theinterest in the teaching ofgeometry

up to the time of the expulsion of the Jesuits

The eighteenth century 78-81

of Rivard, Clairaut, and La Caille. Features with

educational significance. The military schools andtexts used in them

Theteaching ofgeometry inFrancesinceLegendre . 83-86

Founding of the government schools. The lyce"es.Suggestions on teaching by Lacroix. By Busset

The traditions Euclidean

Transition of geometry into the secondary schools 88-89

Before the creation of the Gymnasia 90

in establishing special schools where mathematics

was taught. Methods employed in these

Founding of the Gymnasia 91-92

Attemptstobreakawayfromthedogmatic method

Organization ofpublicschools Schools fortrainingteachers. Gymnasia programs in the early part of the nineteenth century

universitiesupto the timeofentrance requirements

In other schools 98-99

The early academies Colonial grammar schools First course of the Boston English High School in

1821. The accrediting system

Influence oftheEnglishandtheFrench 99

Some references to method 99-100

The place of geometry in the curriculum . 104-105

The place ofgeometry in the curriculum . 106-108

The place of geometryin the curriculum . 112-115

Theteachingofgeometryin theelementaryschools 116-117

Theteachingofgeometryin thetwotypesofnormal

Theteachingofgeometryin thesecondaryschools 118-119

The place of geometry in the curriculum Relation

affecting the teaching of geometry

Schools in which geometry is taught 120-121

The teaching of geometry in the higherelementary

OTHER EUROPEAN COUNTRIES 128-129

The place ofgeometry in the curriculum 130The teaching of geometry in the high school 130-131

Tendencies Relation of the high school to the

university Effects of the "Perry Movement". . 131-132

The practical preceded the logical. The transition into

the logical was not abrupt. Tendencies to hold to the

special have been marked Geometry has been taught

more and moreto younger students 133-135

Correlation of geometry with science. The question of

THE EDUCATIONAL SITUATION TO-DAY 137-138

The balance between the logical and the practical.

the introduction ofphases of the higher work . 146-147

the introduction of modern geometry. Non-euclidean

The question ofmethod Summaryof points 151-154

A HISTORY OF THE TEACHING OF

CHAPTER I

in its widest sense and as restricted to methods of the school

con-cerned with the manner in which man began to formulate thescience, with the additions to the subject-matterin the variousepochs, and with the books written to spread this knowledge

In studying the development of the teaching of geometry1 inany epoch, four factors will be considered: 1. The contribu-

assuchwillbe considered onlyso farasisnecessaryfora

founda-tionfor the present study Naturally this willformalarge part

of our information about the early teaching of the subject.

will be treated first chronologically, bringing it down to thepresent time Certain modern problems will then be con-

discipline. With respect to the general historic development

of the subject-matter of geometry, we shall find: (a) That the

1 Wherecontradictionsdonot arise the term geometry will generally

be employed forelementary geometry

History of Teaching of Elementary Geometry

was developed by the Greeks and reduced to a coherent logical

of Euclid has been closely adhered to; indirectly, where books

logical and practical points of view is fundamental in

Thepupil to-dayisnot readyforhis logical geometryunless

there-forebenecessaryforusto considerthispre-grecian geometry

nature We may postulate that man's first efforts to interpretand adjust himself to nature were intuitive. We are familiar

use of what we term a geometric principle that a straight line

first learning that one side of a triangle is less than the sum of

the other two These facts of nature are used because they are

him to graze in a circle. He knew that the longer the rope, thegreater the area covered, but no exact relation between area

and radius occurred to him Though this early stage of

neces-sarilylead uptothedomainofabstractprinciples, still allscience

com-mon to a higher plane of civilization. The Indian chose the

History of Teaching of Elementary Geometry 3

shapesof irregular form Some of their mounds illustrated thetruncated cone But we cannot draw the conclusion that the

mound builders had any conception of geometric principles If

traced to them But, as far as known, such orientation waslacking

as skillfully as the Esquimau does his. The white ants of

Africa build hills twenty-five feet high, ingeniously

ac-cordingtowhat wecall geometricdesign, heisinsteadmoulding

nature to meet his immediate needs In many ofour so-called

geometric constructions we merely copy what nature hasalready revealed to us. After ages of contact with nature,

man was finally led to an understanding of the laws existing

the first source from which man has drawn his geometric

developed

Primitive man, and like him the child, does not consider

them as necessary constituents of his well-being He observes

and thinks of the plane surface in terms of its serviceability

recognize a difference between a few and many grains of corn,

the exact notion of how many, in all probability, never occurs

physical capacities, as when we speak of stone's-throw,

finger-breadth, span, hand, ell, cubit, fathom, day's journey, and the

like.

History of Teaching of Elementary Geometry

develop a rudimentary science When the mind came to

classify, to define space relations, to summarize the products of

human efforts, a second level inhuman experience wasattained

Onsucha planeworkedthemindsoftheAncient Egyptians, the

Babylonians, and the Chinese The building of the pyramids

of proportion and the simpler facts of plane geometry The

placing of these tombs due north and south led to finding aneast and west line. The effects from the overflow of the Nile

mensuration, thus showing anability to classify theirknowledge;

basing their work on the practical geometry of the Egyptians,

materials of a science Second, the use of these materials in a

that Greece arosewiththemindofthe full-grownman.

of theory to be tested and again expanded.

because our story would not be complete without some

astonishment Though no knowledge of was

History of Teaching of Elementary Geometry 5

required for this work, we naturally query why a people

according to Charles Piazzi Smyth, gives some interesting sults i

re-1

(a) The base is a square. (6) Its height is to twice

Forthe truth of this statement, one can refer only to Smyth's

4' ofthe true north and south line It is reasonable to suppose

that the pyramid builders first located thenorth and south line

perpen-dicular to this line? According to Cantor,3 the philosopher

points out that the name Harpedonaptae is made up of two

pegs soplaced that the threesidesofthetriangleformed were in

fourth dynasty, antedating3000 B C. One has a right to judgethat theabove methodoflayingoffperpendicularswas inuse at

I of the twelfth dynasty, the rope-stretchers plied their art.

to think,however,thatSmythhadextremelygoodluck ingettinghis results.

will be shown from the records.

History of Teaching of Elementary Geometry

that the above method of erecting perpendiculars is in common

pas-sage on the north is a little over 26. (e) The angle of thenorthern airpassage is33 42'.2 Accordingto Colonel Howard-

Vyse's statement3

, in 2400 B.C the lower culmination of thepolar star was 26. So Piazzi Smyth draws the conclusionthat these passage ways pointed to the lower and upper cul-minations ofthe polar star inthe year 2400 B.C The average

due to aesthetic influences In their mural decorations duringthe period of thefifthdynasty immediately following the build-

the isosceles trapezoid One figure shown by Professor Cantor5

represents two squares one over-lapping the other, so placed as

to give the effect of an eight-pointed star. There is also

repre-sented the division of the circle into 4, 8, 6, 12, parts by the

before Greece became interested in geometry The probable

effect of such work on Greece is pointed outby Gow, who says:

"To a Greek, therefore, who had once acquired a taste for

History of Teaching of Elementary Geometry 7

geometrical constructions which, on inspection, suggested new

tells us that Rameses II (cir.

to provide a more convenient means of taxation But on

surveyors tolevy the propertaxin proportion tothe partofthe

land remaining There were certain benefits to be derived from

the overflow of the Nile, and these again necessitated

3

due proportion to each individual, that the lands which were

For this purpose, the necessity of ascertaining the various

canals and dykes, obviously occurred to them . These dykes

of the exact quantity of land irrigated, the depth of the water

necessary practical geometry

an old papyrus,4 which is now in the Rhind collection in the

1

Gow, pp. 132-133.

3

Wilkinson, The AncientEgyptians,Vol IV, pp 7-8.

4This was translated in 1877. See Eisenlohr, Ein Mathematisches

Handbuch der alien Aegypter, pp. 125-150. There is a supposition thatthis is acopyof amuch older work Foracopyof the original, seeFac-

simile of the Rhind Mathematical Papyrus in the British Museum This

a the principalworks

History of Teaching of Elementary Geometty

areas ofsomeofthe plane figures, those forthe areas of isosceles

Figures aregiven for these as well as for squares, rectangles, and circles.

are indicated by means of the hieratic number symbols, which

are copied on the figures just as is done to-day. As this is the

here the earliest known drawing of geometric figures employedfor educative purposes.2

inscrip-tions. The Temple of Hortis at Edfu in Upper Egypt gives

evidence of this fact. Theseinscriptions (cir. 100B.C.) describethe lands "which formed the endowment of the priestly collegeattached to the temple."3 The incorrect formula for theisosceles 1

trapezoid was also applied to any trapezoid The

a + b c + d

the four sides a, b, c, d are in general unequal

circle whosediameteris9unitsandwrites the area 64, usingasa

-D)2= Area This is equivalent to taking

9

TT -=

not accurately describe, and adds some examples on pyramids1

and whoselegis 10 units. Thearea is given as 20,showingthat hisformula

is equivalent to multiplyingthe length ofoneof theequallegsbyone-half the length of the base. He takes an isosceles trapezoid whose parallelbases are respectively 6 and 4 units each long. One of the equal legs is

20 units. The area is given equivalent to the form - X20.

3

claimed, pertains to early Egyptian mathematics. It was sufficiently

examinedtoconveythis impression, but on account ofthe leathertending

8

History of Teaching of Elementary Geometry 9

hypot-enuse and base of right triangles are given to find their ratio

(seqt).1 This ratio determined the cosine of an angle, which,for all the pyramids, gives practically the same slant of thelateral faces. The term seqt was also used to denote the

They allowed the Greeks to come and learn, but just how this

constituted the learned class, undoubtedly it came from them

Perhaps, also, the Greeks got some inspiration, as Gow relates,

by observing the geometric constructions on the walls of thetemples Besides this, the architecture of the various temples

One could well ask why the Egyptians did not develop a

geometrical science But it appears that in Egypt,

land-sur-veying, along with writing, medicine and other useful arts, was

In brief, the Egyptians knew how to calculate the areas

of some of the simple rectilineal figures, using some rules,

barnsby methodsnotclearly defined. Insomeof theirproblems

decora-tions. On the whole, the Egyptians developed a practical

1

Cantor, I, pp 59-60; Gow, pp 128-129. Cantor and Eisenlohr have

worked out these interpretations.

History of Teaching of Elementary Geometry

above, and some temple inscriptions, we have no record of

The Greeks, who were the first to study geometry from alogical viewpoint, brought the subject into a coherent system

during a period of 300 years. This development began when

Thales in the capacity of a merchant visited Egypt1 and there

On his return to Asia Minor, Thales founded the Ionian School

Baby-lonia,

2

the height of its power

to the progress of the science After the time of Aristotle we

again see the study of geometry thrive on Egyptian soil at the

the additions to the subject-matter of geometry during thisperiod of 300 years before Euclid, and also see what contribu-

Thales andhisschool have been creditedwith having added

(1) Acircle isbisectedbyits

diameter; (2) the anglesat the base of an isosceles triangle are

History of the Teaching of Elementary Geometry n

equal; (3) if two straight lines intersectthe vertical angles areequal; (4) an angle inscribedinasemicircleis aright angle; (5)

ThatThales did notproveall the propositions attributedto him

thatEuclid first thought the third worthy of proof.2

geometry, he was primarily an astronomer, and no doubt was

impelled to further study of geometry by recognizing the

Thales is also creditedwith findingthe heights of pyramids by

(Enopides of Chios (cir. 450 B.C.), who seems to have been

associated with the Ionic school, contributed to the

develop-ment of geometry According to Proclus,5 he solved the two

problems: "From a point without a straight line of unlimitedlength to draw a straight line perpendicular to that line," and

"At a given point in a given straight line to make an angle

equaltoa given angle." Concerningthe firstofthese problems,Proclus says that (Enopidesfirstinventedthisproblem, thinking

areasintheir practicalwork, Thalesinhis logicalworkdeveloped

rea-soning applied to geometry

History of Teaching of Elementary Geometry

school of Pythagoras was undoubtedly familiar with many of

the propositions in the first two books of Euclid and with parts

of the fifth and sixth books;1 that is, it was familiar with theordinary theorems in plane geometry concerning equality oflines and of angles and with many of the theorems on equiva-

the anglesumofatriangleequalstworight angles. Hencethey

held a conception of parallellines if we are to suppose that any

proof of the above theorem was accomplished. The theorem

of the three squares (Euclid, I, 47), that in any right triangle

the square on the hypotenuse equals the sumofthe squareson

theother twosides, is known as the Pythagoreantheorem The

truth of this relation in any right triangle.

geometric, and harmonic The Pythagoreans were much

with the construction of the regular plane polygons of 3, 4, 5,

sides. The construction of the regular polygon of five sides

All-man2

quotes from the Eudemian Summary, which attributes thediscoveryof this problem, known as the Golden Section, to theschool of Plato Gow's conclusion admits of less speculation

We must note that in all that is known regarding the

History of Teaching of Elementary Geometry 13

garding the methodsof proof We are, perhaps, safe injudgingthat alargepartofthe workoftheseearly schools wasin findingout geometric truths, and that thesequence ofproved theoremswas not then held in any hard and fast line.

We have no record that the school of Pythagoras was

con-cerned with thepractical, and we mayconcludewithDr Allman1

that thePythagoreans were the first tosevergeometry from theneedsof practical lifeand to treatitas a liberalscience

stageas withthe Egyptians; second, the beginningofthelogical

as a liberal science But we have no proof that the

Mediterranean About this time, fresh from her glories of thePersian wars, Athens, exceedingly wealthy, attracted peopleofall

for hire. Such were the Sophists. To them and the school of

Alexandria We recall that the Pythagoreans developed the

circle.2

titlesof someof the works Euclid is universallycredited withbeing the firstto write a complete textongeometry,but he was

not thefirst to write onparticular portionsofit. Although theschool of Thales is not generally credited with adding a great

History of Teaching of Elementary Geometry

deal to the subject-matter of geometry, to two of its members

FiguresIllustrative ofGeometry,"andPlutarch2(Deexilio c. 17)

(vii. Praef.),

Pythagor-eans published nothing of their work on geometry, although

dis-tinguished geometers; indeed Hippocrates was the first who is

recorded to have written 'Elements.' Plato, who followed him,

make great advances, by reason of his well-known zeal for thestudy, for he filled his writings with mathematical discourses,

of Athens, by whom mathematical inquiries were greatly

ex-tended, and improved into a more scientific system Younger

Ele-ments' more carefully designed, both in the number and theutility of its proofs, and he invented, also, a diorismus (or testfor determining) when the proposed problem is possible and

and a student ofthe Platonic school, first increased the number

of general theorems, addedto the three proportions three more,

Ibid., pp. 135-137(ref. Proclus, ed. Friedlein)

History of Teaching of Elementary Geometry 15

Men-aechmus, made the whole of geometryyet more perfect.

together in the Academy and conducted their investigations in

common Hermotimus of Colophon pursued further the lines

Platonic philosophy."

Besides these, others have been credited with writings on

geometry Diogenes Laertius relates that Democritus wrote

on geometry, on numbers and perspective, and also two

ascribedtohimbears the incomprehensibletitle, "TheDifference

of the Gnomon or the Contact of the Circle and the Sphere."Suidas2

regu-lar solids. Theophrastus,3 a pupil of Aristotle, according to

to-gether with six books of astronomyand one ofarithmetic cording to Hypsicles,4 a book on "The Comparison of the Five

theorem, "The same circle circumscribes the pentagon of the

being inscribed in the same sphere."

As has already been mentioned, during the pre-euclidean

period the subject-matterofelementaryplanegeometrywasticallycompleted In the time ofthe Pythagoreanschool, thefiveregularsolids werestudied, butstereometryas ascience wasnot

History of Teaching of Elementary Geometry

solids.1 A decided advance was made in this study when

par-allels."3 This established a relation between magnitudes of

apparently unlike properties and hence tended to systematizethe logic of solids. And so Eudoxus may be credited with

sequence of Euclid; but maybe illustrated brieflyat thispoint.

The subject-matter of the third book of Euclid, which treats of

circles, was developed in the Athenian period. The theory

of proportion which follows later in the "Elements" was

systematized earlier by the Pythagorean school This school

also studied the properties of the five regular solids beforethe great amount of material of plane geometry had yet been

common logical sequence will be referred to in the last chapter

Educational Features of the Greek Geometry

select body of men, probably few in number, working together

1

Theyhave since been called the Platonic Bodies.

2

text of a newly discovered MS by Archimedes, published in Bibliotheca

article by Professor Charles S Slichter in Bulletin of the American

Mathe-History of Teaching of Elementary Geometry 17

in thesame fieldof investigation, recognizingas their leader the

school of Pythagoras, in Southern Italy. We learn1

that the

his listeners were of two kinds In the first class the lectureswere of a more general nature, and women were allowed to

to judge that only the select who had some skill as

Here the various contributions to the new science were made

The Sophists are credited with having introduced highereducation into Athens There was a demand for the study of

philosophy and mathematics, and so the Sophists came Their

Their method was theSocratic,that ofquestion andanswer So

ap-preciated by minds not yet matured One has good reason

reasoning thanin the subject-matter itself isillustrated in some

ofthe dialogues of Plato.2

eighteen continued his physical and social training. Under the

new education more attention was given to the training of theintellect.3 Welearnfromthe '

from the age of seven to about seventeen was to study musicand gymnastics. Under music was included reading, writing,

1

Ball,A ShortAccountof theHistoryofMathematics, pp. 19-22.

2

See the dialogueon incommensurable linesbetweenSocratesand one

History of Teaching of Elementary Geometry

beno compulsion, it being sufficient that the student enter into

the work as dictated by his interest and his aptitude. From

seventeen to twenty the youth received the military gymnastic

ofdialectics, the trainedman returned to social and politicallife

the old age of Plato, there are the same general

recommenda-tions, only the later years of the man are to be devoted to

music, and astronomy. Later in Rome, Cassiodorus

em-bodied them in the quadrivium, which constituted the

From Prcclus1 we learn that "the Pythagoreans made a

they assigned to each of these parts a two-fold division For

sub-sists by itself, or must be considered with relation to some

other; but that continued quantity, or the how much, is either

con-templates that discrete quantity which subsists by itself, but

nature." Thus we see theorigin of the plan ofkeeping motion

to-day in the study of elementary geometry, was therefore not

ideas. The Pythagoreans founded the theory of proportion,giving a method applicable in the fields of both arithmetic and

geometry, but, notwithstandingthis close relationship, geometry

History of Teaching of Elementary Geometry 19

yet invented, and a great deal of mathematical work, since plified bythe methods of algebra, was laboriously carried on by

. Wherethesubjects are so different as theyarein arithmetic

quan-tities are numbers, which can only happen in certain cases."The theory of geometry was thus isolated by the Greeks from

other branches of mathematics, and mathematical developmentwas thereby retarded many centuries

instrumentsofconstructiondatesfromtheGrecianperiod. Such

in area to a given circle, and duplicating a given cube. Thesewere capable ofsolution by means of conic sections and certain

twogivenstraightlines, Hippocrates havingreduced the

solv-ing the duplication -problem, andsaid that thegoodofgeometrywas spoilt and destroyed thereby,"3 and that owing to this

History of Teaching of Elementary Geometry

the aim of education would lead one to the conclusion thatthestatements attributedtohimarethecorrect ones, andthat at

the use of instruments in elementary geometry Thus we see

the expulsionofmechanics from geometry. Butmechanics was

studied nevertheless, and the science was developed under

geometric problem, while trying to find by means of the

sec-tion of a semi-cylinder two mean proportionals with a view

tothe duplicationofthe cube."2

of solid geometry It has already been mentioned that

Hip-pocrates reduced the problem of the duplication of the cube

solid geometry to one of plane This indicates a tendency of

stereometry went entirely out of fashion Furthermore, we know that by the time of Euclid solid geometry was not de-

veloped in the sense that characterized plane geometry

level, and the gnomen or carpenter's square Theinventionsof

the square and level areattributedbyPliny (Nat Hist VII, 57)

History of Teaching of Elementary Geometry

were placed at the vertices of the pentagram, the Pythagorean

Hippocrates, later, in attempting the quadrature of

of his figures. Thus, he wrote, "the line on which AB is

marked" and "the point on which K stands."2

tion, then A will move twice the distance F in the time A or

that much time and trouble is saved by a general symbolism.3

propensityofthe Sophistsfor verbal disputations, we can judge

to writing which was no longer a subject for argument It is

probable that a board strewn with sand wasin common use for

It will be pointed out later that the tendency to hold to the

should expectto find inthe pre-Euclideangeometrylike

tenden-cies. "Eutocius, at the beginning of his commentary on theConiesof Apollonius (p. 9. Hallev'sedn.), quotesfrom Geminus

it untilhehadfinished hisproblem,thesoldier inapassion drew hissword

andkilled him. Plutarch, Marcellus, trans, Langhorne, p 114,

History of Teaching of Elementary Geometry

right-angled triangle about one of the sides containing the right

sonamed amongtheancients Hencejust astheyconsidered the

and acute-angled cones they exhibited only in such cones

discovered the general theorem that in every cone, whether

cone.'"1

difficulties in reaching the conception of the general, it is the

the race; the tendency to pass from the special tothe general is

of the race also.

As to the conciseness of geometric proofs, it is probable that

previous theorems, and hence a certain amount of tediousness

was avoided Even Euclid, who systematized the existinggeometry, referred to previous propositions in only an indirect

History of Teaching of Elementary Geometry 23

material world there areroads for common people androadsfor

Ptolemy

are standard to-day. We shall referin particular to the notion

of locus, the method of exhaustion, reductio ad absurdum, andanalysis

Thales knew that any angle inscribed in a semi-circle is a right

seem warranted It seems probable though, that the notion

of geometric locus was understood before the time of Archytas

(cir. 400 B.C.), for Archytas not only employed this idea, but

curves ofall kindswere called runningloci, thestraight line and

thecircle werecalled planeloci, andthe conic sectionssolid loci.4

of the circle. This was the process of exhausting the area of acircle by means of inscribed and circumscribed polygons An-

tiphon used only the series ofinscribed polygons, while Bryson

rigid, was adopted by Euclid, and we find it in use to-day in

many of our texts. But the so-called method of exhaustion

cubebythe intersections ofacylinder, cone,and hemisphere.

History of Teaching of Elementary Geometry

ex-haustion, Eudoxus, aswe can see in the propositionscited, made

connec-tion we should mention the use of geometric reduction Thisidea is so common to-day that we do not dignify it with adefinition. Proclus,2

discussing therecasting of the duplication

-method, which he defines as a transition from one problem or

1

I IffromA morethanits halfbetaken,andfromtheremaindermore

than its half, and so on, the remainder will at last become less than B,

A), however small II. Let there be two magnitudes, P and Q, both of the same kind; and let a succession of other magnitudes, called Xlt Xit

X3 , be nearer and nearer to P, so that any one, Xn, shall differ

from P less than half as much as its predecessor differed. Let Ylt

Y2 , Y3, be a succession ofquantities similarly related to

Q; and let the ratiosXl to Ylt X2 to Y2, and so on, be all the samewitheach other, and the samewiththat ofA to B. Then it must bethat P is

toQ asA to B. (It is obvious, fromthe conditions, that ifXt be greaterthan P, Yl is greater than Q,etc.) SupposeXlt X2 , etc., less thanP, andtherefore Ylt Y2t etc.,lessthanQ. Thenif 4 is not toBasPto Q, A is to

B as P to some other quantity 5 greater or less than Q; say, less than

Q. Then(by hyp.andI) wecanfindsomeoneof the series Ylt Y3 ,,

than5. Thensince Xnisto Fuas A to B, or asPto 5,we have Xn is to

yu asPto5, or Xn toPas Fnto 5; from which, sinceXn is less than P,

Fnis lessthan5. But Vnis also greater than S, which isabsurd; fore,Ais not toBasPto lessthanQ. Neither isAtoBasPtomorethan

greater than T. But B is to A as S to P, that is, as Q to less than P,

Conse-quently, A is not to B as P to more than 0, or to less than Q; that is,

A and B the squares ontheir diameters, Xl and Y1 inscribed squares, Xt

and Y2 inscribed regular octagons, X3 and F3 inscribed regular figures ofsixteen sides, etc., the preceding process gives theproof that circles are to

one another as the squares of their diameters See De Morgan's article,

History of Teaching of Elementary Geometry 25

in the discovery of new theorems or problems It sets up achain of steps by which certain conclusions are reached This

we see prepares thewayforthe reductio ad absurdum which in

forthe time being that the desiredtheoremis trueorthe sought

con-ditions. In reductio ad absurdum the contrary theorem is

ex-haustion involves the reductio ad absurdum, and it in turn

1

Plato and

and Diogenes Laertiusgivethecredit to Plato It ismostblethat Plato systematized the method and gaveita definite

proba-form

Greeks We are accustomed to-day to attack a problem by

construction is made, the deductive proof is given, and then is

systematized and practiced in Athens during the time of Plato

elemen-tary geometry, two stand out prominently: (1) It was

essen-tially deductive Undoubtedly this was to a large extent thecause of mechanics being expelled from geometry. (2) The

di-vided the subject-matterof geometry; sothe conic sections did

line andthecircle. Hence in Euclid's "Elements" wefindonly

1

Gow, p 176; also see Cantor, I, p 207.

History of Teaching of Elementary Geometry

someauthorstoputa synthetictreatmentoftheconies attheend

ofthe textafterthegeometryofsolids. Asformethodsofattack

common use to-day. We should specially mention the method

CHAPTER II

SUBSE-QUENT TEACHING OF GEOMETRY1

these words,3 "Those who have written the historyof

Theaete-tus's, and brought to irrefragableproof propositions which hadbeen less strictly proved by his predecessors."

Before considering the special features of this work, let us

opinion of Dr Allman.3 He holds that after the Pythagoreans,Euclid was indebted most to Eudoxus and Theaetetus To the

IV, the doctrineof proportionand.of similar figures "together

of areas the subject matter of the sixth book: the theorems

arrivedat, however, were provedforcommensurablemagnitudes

1Fora readable account of the work of Euclid andhis later influence,

later texts and progress in geometry is set forth by Professor Gino Loria

in his Delia varia fortuna di Euclide. In lighter vein, one may read an

entertainingcontroversy overthe merits of Euclidbyturning to Dodgson,

Euclid and his Modern Rivals.

2

Gow, p 137.

8

Op cit., p 211.

History of Teaching of Elementary Geometry

given a part of Book X and Book XIII Euclid also is given

Euclid contributed to the subject-matter of geometry The

fourteenth book, whichhas been assignedto Euclid, was written

thatthiswaswrittenbyDamasciusof Damascus(cir. 490A.D.).1

lines and areas Book III treats of circles. Book IV, of

reg-ularfigures. BookV, the theory of proportion for all kinds of

figures. Books VI, VIII, and IX treat of the arithmeticaltheory of proportion Book X is on arithmetical characteris-

polyhedra

Book I is introduced by the definitions of point, line, etc.

Here are given five postulates and five common notions (later

for it represents the points of departure of so-called clidean geometries Itis, "Iftwostraightlinesare cutbyathird

samesideofthe transversallessthantworight angles, then these

two lines, if produced, will meet on that side." Euclid, being

stated the equivalent of this in the form, "Two intersecting

and Lobachevski

The lack of correspondence between the historic

1

Gow, pp 272, 312-313.

2

See Heiberg, Euclidis Elementa, Leipzig, 1883. Also De Morgan's

article, Eucleides, in Smith's Dictionary of Greek and Roman Biography