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1


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Final Report
of the
National Committee of Fifteen
on
Geometry Syllabus.




CONTENTS.
Preliminary  Statement............................................  3
(A) Historical Introduction:
Bibliography................................................  5
F rance......................................................  6
Germany................................................... I3
Italy........................................................  9
England....................................................  20
United States.......................................   29
(B) Logical Considerations:
Axioms....................................................  32
D efinitions..................................................  35
Informal Proofs.......................................       38
Limits and Incommensurables............................... 40
Time and Place in the Curriculum.......................... 40
Purpose in the Study of Geometry......................... 42
Points on Solid Geometry.................................. 44
(C)  Special  Courses.............................................   46
(D) Exercises and Problems:
Distribution, Grading, and Nature.......................... 49
Sources of Problems........................................ 53
Problems Involving Loci.................................... 6I
Algebraic Methods in Geometry............................ 63
(E) Syllabus of Geometry:
Preface to Lists of Theorems.............................. 66
Theorems on Plane Geometry............................. 70
Theorems on Solid Geometry................................ 79
Conclusion.................................................  87




FINAL REPORT OF THE NATIONAL COMMITTEE
OF FIFTEEN     ON   GEOMETRY      SYLLABUS.
PRELIMINARY STATEMENT.
At the meeting of the National Education Association in
Cleveland in I908, the mathematics Round Table of the Secondary Department, numbering some two hundred members,
unanimously called for a national committee to study and report
upon the question of a syllabus for geometry. In December,
I908, the American Federation of Teachers of the Mathematical
and National Sciences at its meeting in Baltimore authorized the
appointment of a national committee of fifteen on geometry
syllabus. At the Denver meeting of the National Education
Association in I909, the secondary department authorized the
committee which had already been appointed by the American
Federation to proceed under the joint auspices of the two
national bodies.
The committee thus constituted was made up of eight representatives of secondary schools and seven representatives of universities, as follows: William Betz, East High School, Rochester,
N. Y.; Edward L. Brown, North High School, Denver, Colo.;
William Fuller, Mechanic Arts High School, Boston, Mass.;
Walter W. Hart,* University of Wisconsin; Frederick E.
Newton, Andover Academy, Andover, Mass.; Eugene R.
Smith, The Park School, Baltimore, Md.; Robert L. Short,
Technical High School, Cleveland, O.; Mabel Sykes, Bowen
High School, Chicago, Ill.; Charles L. Bouton, Harvard University; Florian Cajori, Colorado College; Herbert E. Hawkes,
Columbia  University; Earle  R. Hedrick, University   of
Missouri; Henry L. Rietz, University of Illinois; David Eugene
Smith, Teachers College, Columbia University; Herbert E.
Slaught, Chairman, University of Chicago.
* At that time head of the department of mathematics in the Shortridge High School, Indianapolis, Ind.
3




4


THE MATHEMATICS TEACHER.


After an extended preliminary correspondence, the investigations of the committee were conducted in three subdivisions of
five members each, the first dealing with "logical considerations," under the chairmanship of David Eugene Smith; the
second having charge of "exercises and problems," under the
chairmanship of Henry L. Rietz; and the third determining the
"lists of theorems," under the chairmanship of Earle R.
Hedrick.
These subcommittees carried on their work by correspondence
and in some cases by meetings of their members during a period
of a year and a half, the results being submitted from time to
time to the general committee for suggestions and criticisms.
A meeting was then held in Cleveland, Ohio, on November 24,
25, 26, I9IO, at which were present the three subchairmen, the
general chairman, and three other members. All the recommendations of the subcommittees were submitted to a searching
examination in the course of which they were either passed,
eliminated, amended, or reconstructed until finally substantial
agreement was reached on all points.
These conclusions were then submitted to the remaining members of the committee for their approval and the report as thus
evolved was put into 'type and a preliminary edition of two
hundred copies was placed in the hands of a carefully selected
list of critics for further suggestions. In this way the reportfinally reached the provisional form in which it was presented
at the San Francisco meeting of the National Education Association in I9II, where it was discussed at great length and
adopted with the understanding that the committee would again
work it over in the light of all criticisms and suggestions and
present it in final form at the next meeting. These instructions
were followed with great care, and when all modifications had
been made, upon which the members of the committee could
reach substantial agreement, the report was again printed and
distributed directly to two thousand members of mathematics
associations throughout the country and indirectly, through the
Bureau of Education at Washington, to some three thousand
people upon personal request. In this final form it was presented at the Chicago meeting of the National Education Asso



FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.


5


ciation in I912, where it was unanimously adopted, and an
appropriation was made for its further distribution.
In the preparation of this report the various syllabi published
in this country and in foreign countries have been consulted.
With one of these, the syllabus published by the Association of
Mathematics Teachers in New England, this report makes direct
comparison, indicating both omissions from, and additions to,
that document.   The historical introduction was prepared by
Florian Cajori.
SECTION A. HISTORICAL INTRODUCTION.
ATTEMPTS MADE DURING THE EIGHTEENTH AND NINETEENTH
CENTURIES TO REFORM THE TEACHING OF GEOMETRY.
BIBLIOGRAPHY.
The following six books bearing on the history of the teaching
of geometry have been found most useful in making this compilation:
I. V. BOBYNIN-" Elementare Geometrie," being Chapter XXII. in Cantor's "Vorlesungen fiber Geschichte der Mathematik," Vol. IV.,
Leipzig, I9O8, pp. 321-402. (Covers the second half of the eighteenth century.) Referred to as "Bobynin."
2. F. KLEIN-"Elementarmathematik vom    h6heren Standpunkte aus.
Theil II.; Geometrie." Leipzig, I909, pp. 433-515. Referred to as
"Klein."
3. J. PERRY-" Discussion on the Teaching of Mathematics." British
Association Meeting, I901. Referred to as "Perry."
4. H. SCHOTTEN-" Inhalt und Methode des planimetrischen Unterrichts." Leipzig, I890. Referred to as "Schotten."
5. M. SIMON —"Ueber die Entwicklung der Elementar-Geometrie im
XIX. Jahrhundert." Leipzig, I906. Referred to as "Simon."
6. A. W. STAMPER-"A History of the Teaching of Elementary Geometry," New York, I906. Referred to as "Stamper."
Other useful sources of information on the history of the
teaching of geometry are as follows:
i. Reports of the Association for the Improvement of Geometrical
Teaching (in England). The name of the association has been
changed to "The Mathematical Association," and its present organ
is the Mathematical Gazette.




6


THE MATHEMATICS TEACHER.


2. Zeitschrift fir mathematischen und naturwissenschaftlichen Unterricht, Leipzig and Berlin.  Formerly called "Hoffmann's Zeitschrift," now "Schotten's Zeitschrift."
3. L'Enseignement Mathermatique. Revue internationale. Paris.
4. La Revue de l'Enseignement des Sciences. Published monthly. Paris.
5. Nature (London). See Indexes for "Geometry."
6. D. E. SMITH-"The Teaching of Geometry." Boston, I9II.
7. G. LORIA-" Vergangene und kiinftige Lehrplane." Deutsch von H.
WIELEITNER, Leipzig, I906.
8. G. LORIA-" Della varia fortuna di Euclide." Rome, I893.
9. R. FRICKE —"Ueber Reorganisationsbestrebungen des mathematischen
Elementarunterrichts in England." Jahresbericht d. deutsch. Math.
Vereinigung, Vol. XIII., I904, p. 283, etc.
Io. KLEIN-SCHIMMACK-"Vortrage fiber den mathematischen Unterricht an den h6heren Schulen." Leipzig, I907.
II. Plan d' etudes et programmes d' enseignement dans les lycees et
colleges de garcons. Paris, I903.
12. A. W. Stamper's list of references at the end of his book.
13. The various histories of mathematics.
FRANCE.
France began to maintain a critical attitude toward Euclid as
a text-book in geometry for beginners as early as the time of
Petrus Ramus (I580). Ramus treated geometry as the art of
accurate measurement.   In the eighteenth century this spirit of
independence was intensified by the publication of Clairaut's
"Elements de Geometrie" (I74I), in which surveying and other
practical matters received marked attention.  In the latter half
of the eighteenth century Euclid ceased to be used as a textbook in France.
Williamson, in his edition of Euclid, 1781, criticises Clairaut
as follows: "Elements of geometry carefully weeded.of every
proposition tending to demonstrate another; all lying so handy
that you may pick and choose without ceremony. 'This is useful in fortification;' 'you cannot play at billiards without this.'
'You only look 'through a telescope like a Hottentot until this
proposition is read,' with many such powerful strokes of
rhetoric to the same purpose. And upon such terms, and with
such inducements, who would not be a mathematician? Who
would go to work with all that apparatus which I have described
as necessary for understanding Euclid, when he has only to take
a pleasant walk with Clairaut upon the flowery banks of some




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.


7


delightful river, and there see, with his own eyes, that he must
learn to draw a perpendicular before he can tell how broad it
is?"  About 1836 De Morgan remarks that these arraignments
are not "without their force, 'when directed against experimental geometry as an ultimate course of study, [but] lose their'
ironical character and become serious earnest, when applied to
the same as a preparatory method."     De Morgan strongly
favors a geometry like Clairaut's as a preparatory course.
The critical attitude of Ramus and Clairaut toward the " Elements " of Euclid brought Ito the mind of D'Alembert the questions: "What are the elements of a science? What should be
the contents of a book called elements? " D'Alembert gives his
answers in two articles, "Elements des sciences," and "Des
elements de geometrie" in the Encyclopedie methodique (about
I784).*
D'Alembert distinguishes between two kinds of elements of a
science:
(I) If all truths or theorems of a science which are the foundation for all others are brought together, so that these truths or
theorems potentially comprise the whole science, then these constitute, when properly co6rdinated, the elements of the science.
In geometry, such elements embrace not merely the principles
of mensuration and the properties of plane figures, but also the
application of algebra to geometry, and the differential and integral calculus in its application to curved lines.
(2) The elements of a science may be defined also as comprising those truths or theorems which treat the subject matter
in the simplest way, and which constitute, together with their
deductions, a detailed study lof the simplest parts of the science.
By the elements of geometry, elements of this kind are usually
meant; they include only the properties of plane figures and the
circle.
Dissatisfied with the elements of geometry known in his day,
D'Alembert sets up the following demands which such texts
should fulfill:
*Encyclopedie methodique, Mathematiques I., 617-625, III., I33-I36.
We have used the Italian translation of this dictionary, Padova, 8oo00,.and also a full abstract of these articles, given by Bobynin. See Bobynin,
p. 325, etc.




8


THE MATHEMATICS TEACHER.


(I) The text should develop the subject along the path pursued by the discoverers of the science, so as to show the truths
in their natural relations to each other.
(2) The usual division of the subject into longimetry, planimetry, and stereometry does not provide for the circle and sphere
and is therefore inadequate. The division into plane geometry
and solid geometry, D'Alembert does not consider at all. He
suggests the division into the geometry of the straight lines
(considered with respect to position and relative magnitude)
and circles, the geometry of surfaces and the geometry of solids.
The straight line and circle must be taken up together. The
circle renders immense service in considering the position of
lines. The measurement of angles by circular arcs and the principle of congruence constitute the basis of the first part of the
geometry of lines, upon which other theorems of this part rest.
The second part in the geometry of the straight line has as its
fundamental theorem the one on the section into proportional
parts of two sides of a triangle by a line parallel to the third
side. This involves incommensurables.
(3) Incommensurable relations must be treated by the apagogic method, according to which it is shown that one ratio
cannot be greater or smaller than a certain other ratio, hence it
must be equal to that other ratio. He uses this for the following reasons: Incommensurable magnitudes involve the idea of
the infinite and therefore, he claims, cannot be treated by any
direct method. Notwithstanding this difficulty presented by incommensurable lines, he maintains that they should be taken up
early in geometry, because of their importance. He states that
the whole theory of incommensurables demands only one
theorem, concerning the limits of quantities, viz.: " Magnitudes
which are the limits of one and the same magnitude, or magnitudes which have one and the same limit, are equal to each
other." In the geometry of the circle, of surfaces and solids,
he feels that the method of exhaustion or that of limits should
be used.
(4) A suitable text on the elements of geometry can be prepared only by a mathematician of the first rank. D'Alembert
complains that most elementary geometries are written by men
of little ability.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.


9


(5) To lay down definitions at the beginning without any
analysis of the subject is not only contrary to sound philosophy
but contrary to the natural march of thought.*  Axioms are
useless.
Ideas similar to those of D'Alembert are embodied in a text
on geometry by Louis Bertrand of Geneva,t who in Berlin had
been close to Euler. Bertrand's book antedated D'Alembert's
articles in the Encyclopedie Methodique. Like D'Alembert he
divides geometry into three parts: (I) Geometry of line and
circle, (2) Measurement of parts of a plane bounded by straight
lines and circles, (3) Measurement of curved surfaces and
solids. Bertrand ignored the classification of geometry into
plane and solid. His second theorem is: "When two planes
intersect, their common section is a right line." The straight
line and circle are taken up together at the beginning as D'Alembert would have it. The incommensurable case is treated by
the reductio ad absurdum method. In the latter part of the
geometry he uses also the method of exhaustion. Bertrand
reduces the number of theorems, in one instance, by replacing
theorems on the mensuration of prisms, pyramids, cylinders,
cones, and spheres with the corresponding problems.
Bertrand's work was published in two unwieldy volumes and
had little sale, yet exercised some influence, particularly upon
Lacroix, whose "Cours de Mathematiques," published at the
close of the eighteenth century, has been used until recently.
Lacroix divides his geometry into geometry of the plane and
geometry of space, and does not follow D'Alembert closely.
According to Lacroix there are only two kinds -of theorems that
should find a place in an elementary geometry: (I) Theorems
necessary for the comprehension of the line of argument, developed synthetically. (2) Theorems which grow out of the
practical operations in geometry (drawing and measuring). He
objects to placing all axioms at the beginning, believes in the
omission of the definition of an angle, favors "a straight line is
the shortest path between two points" as growing out of the
* See "Axiome " and " Courbe" in Encycl. Meth.
t Developpment nouveau de la partie elementaire des mathematiques,
Geneva, 1778.




IO


THE MATHEMATICS TEACHER.


child's experience, and uses the apagogic method for incommensurables.
Another author of note was Bezout, who followed D'Alembert's plan quite closely, but was criticized for his lack of rigor
and for his endeavor to lighten the work of the examiner as
well as of those being examined.*
-The most celebrated work on elementary geometry is that of
Legendre (I794). He came nearest to fulfilling D'Alembert's
requirement that the elements be written by a mathematician of
the first rank. He does not follow D'Alembert's plan for a book
on geometry, nor does he heed the philosophic demand that the
author should follow the path of the originators of the science.
Impressed by the lack of rigor in the works of his day, he aims
at greater rigor and approaches closer to Euclid than his predecessors had done. He does not divide geometry in the manner
of D'Alembert and Bertrand.' Like Euclid, Legendre begins
with definitions and axioms. The first four chapters are given
to plane geometry, the last four to solid. The first book treats
of the equality of angles and triangles, the second of the circle
and the measurement of angles, the third of proportional figures,
the fourth of regular polygons and the measurement of the
circle. Legendre uses in measurement the terms equal and
equivalent. He uses the reductio ad absurdum method for incommensurables and the method of exhaustion for curved lines.
What was it that made this book so successful? In the first
place must be mentioned his great clearness of exposition and
his attractive style. A great advance of Legendre over Euclid
was the fuller treatment of solid geometry.  He leans less
toward logic and more toward intuition than does Euclid. In
place of Euclid's famous fifth book on incommensurables,
Legendre borrows rational and irrational numbers from arithmetic, even though in arithmetics no scientific treatment of those
subjects was given in his day. A theorem true for rationals is
assumed to be true for irrationals. Thus, if A: B  C: D, then
AD    BC in all cases. Klein says that this is in accordance
with the practice of the best mathematicians of his day, that even
Lagrange works out the expansion of (x- +h)n   when n is
*Bobynin, p. 355.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  I I


rational and assumes the results thus obtained to be true for
irrational values of n. Legendre stands for a fusion of geometry, not only with arithmetic, but also with trigonometry. As
late as 1845 Legendre's geometry still contained trigonometry,
but as Klein remarks,* the trigonometry and the practical applications of geometry were gradually filtered out. Comparing
A. Blanchet's edition of 1876 with an edition of 1817, we find
also that the twelve "notes" on topics of elementary geometry,
covering 55 pages in the older edition, are omitted in the later
edition. The later edition has a somewhat fuller treatment of
solid geometry and a list of exercises in original proofs, loci, and
constructions. Other notable changes were made in the 1845
edition by J. B. Balleroy and A. L. Marchand. They state that
Legendre uses the reductio ad absurdum method to excess, a
method which "convinces but does not satisfy the mind."
Legendre's text is, however, left intact, alternative proofs being
given in notes at the end. These alternative proofs, as well as
the proofs given in the modified text of the I876 edition, are
rough applications of the theory of limits.
During the first half of the nineteenth century, and even later,
the works of Legendre, Lacroix, and Bezout were used extensively in France. 'In later editions less stress was laid upon
practical applications and numerical computation. Otherwise
few changes occurred. In general, school organization, based
on the regulations of the time of Napoleon I., was quite fixed in
France until I870. France has a rigid centralization of authority in education. If the " Conseil d' instruction superieure"
decides upon a change, the whole country adopts it at once. As
compared with the German, the French teacher has little individual freedom. France is a country with a "system of
revolutions from above."t Since 1870 the movement has been
toward greater individual freedom. The later tendencies in
geometry are imaged in the work of Rouche and de Comberousse, which contains a large amount of new material and meets
the demands of the one year course of the classe de mathematiques speciales during which as much as seventeen hours
* Klein, p. 470.
+ Klein, p. 457.




12


THE MATHEMATICS TEACHER.


per week are given to mathematics and a degree of specialization is allowed in preparation for university courses, as in no
other country. In I902 and I905 new official courses of study
were adopted in France in which greater stress is laid upon
graphic representation, the idea of a variable and a function,
and upon the practical applications of mathematics. The introduction of the derivative in algebra in the regular secondary
classes, the use of motion in geometry, and emphasis on
geometric drawing are other characteristic features of this reform. This new tendency is mirrored in the geometry of E.
Borel, a valuable book, in which the practical receives due
emphasis and in which intuition meets with fuller recognition.
With Borel the concept of motion is prominently used. There
is an introduction of eight pages on the use of the ruler, compasses, and protractor, and ten pages on the mensuration of
surfaces and solids, treated empirically. Applications are skillfully interwoven with theory throughout the book. He has
well-selected practical exercises involving symmetry, the nets
of regular polygons, the use of pulleys, and so on. Algebraic
geometry and the development of metric properties come last in
the book. He introduces the rudiments of trigonometry. The
usual division into plane geometry and solid geometry is not
rigidly maintained. A similar tendency is shown in the recent
geometries by C. Bourlet (I908), Fort and Dreyfus (I908), and
Niewenglowski (I9I0).
A parallel and somewhat different tendency in France is seen
in the geometry of Ch. Meray of Dijon, which was first brought
out in 1874 but has only in recent years received much attention.
Meray represents the severely logical mode of exposition;* he
uses in his proofs no fact of observation which has not been
previously set down in an axiom; he formulates a complete list
of axioms, but introduces each only when it is needed; nor does
he aim to limit their number to a minimum. Characteristic of
Meray is the complete fusion of plane and solid geometry, and
the use of motion, not only as a means of proof, but also to
define parallels. Recently there has been considerable discussion in France on the question whether in laying the foundations to geometry, motion should be used or not. The defenders
* Klein, p. 475.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  13


of a static theory of parallels claim that motion cannot be
visualized on the board, rendering intuition more difficult. The
defenders of the kinematic theory advocate the use of movable
figures.*  Prepared under the influence of Meray, so far as the
use of motion is concerned, are the geometries by Borel, Bourlet,
Fort and Dreyfus, and Niewenglowski, above mentioned.
Influenced by the Perry movement in England and America,
France is experimenting on the laboratory method of instruction.t A mathematical laboratory has been recently founded by
J. Tannery and E. Borel at the Ecole Normale superieure in
Paris in order to give prospective instructors an insight into the
possibilities of this method of teaching.
GERMANY.
Klein$ expresses surprise that, during the Renaissance, Euclid
should have come to be looked upon as a text suitable for the
first introduction in geometry. Perhaps the reason for this attitude toward Euclid lies in the fact that geometry was first
taken up in the universities by students of maturer years. As
geometry came gradually to be taught to younger and younger
pupils, Euclid was still retained. Thus the misconception arose
that Euclid was a suitable geometrical text for young boys.
While D'Alembert formulated his ideas on elementary geometry in France, A. G. Kiastner evolved in Germany a type of
his own, in his work, "Anfangsgriinde der Arithmetik, Geometrie, Trigonometrie und Perspectiv," G6ttingen, I758. /Kastner
begins with definitions and axioms in Euclidean style, develops
the geometry of the plane (69 pages) and ends this part with
practical applications (47 pages). The second part of the
geometry begins with the geometry of space (60 pages), continues with 31 pages given to plane trigonometry and its applications to the solution of triangles, and with 9 pages of practical geometry. Then follow spherical trigonometry and 24
pages on perspective. The method of exhaustion is used. It
* Schotten, Zeitschrift, Vol. 40, I909, p. 445; La Revue de l'Enseignement des Sciences, many articles in Vols. I-3.
t Schotten, Zeitschrift, Vol. 40, 1909, pp. 444-5; L'Enseignement Mathemnatique, II, p. 206.
t Klein, p. 434, 435.




IA


THE MATHEMATICS TEACHER.


was the opinion of Kastner that "the newer works on geometry
lose the more in clearness and thoroughness, the farther they
depart from  Euclid." He complains that modern authors,
particularly the French, have departed from the ancient rigor,
" to make the study of mathematics easier for people whose main
occupation is not study, namely for soldiers."
Not without interest is W. J. G. Karsten's "Lehrbegriff der
gesamten Mathematik," in eight volumes, I767-77, the first two
volumes of which are given to arithmetic and geometry.
Karsten begins with arithmetic, then proceeds to plane geometry,
closing with simple arithmetical applications. He proceeds
thereupon to solid geometry, returns to arithmetic, and gives the
rudiments of algebra with logarithms, followed by trigonometry
and its applications to plane geometry. Finally are given the
rudiments of spherical trigonometry and a fuller treatment of
solids. Nowhere are heavy demands made upon the pupil.
That this exposition was intended for students of university
grade, rather than those in the preparatory school, testifies to the
low state of mathematical instruction in German universities of
the eighteenth century.  Close relation between arithmetic,
geometry, and trigonometry is also maintained in the works of
J. G. Biisch (I776) and G. S. Kliigel (I798), the aim being to
make the subject easy of comprehension.
In the nineteenth century, until near its end, advanced mathematicians in Germany took little or no part in the improvement
of the teaching of elementary mathematics. In geometry,
Euclid's text was not usually taught, but the dogmatic method
of Euclid was in vogue during the first half. About the middle
of the century Euclid's order of the theorems came to be criticized as chaotic. It is interesting to see the Germans attack
Euclid's order as arbitrary and the English defend it as the only
order worthy of serious consideration.  The grouping of
theorems according to subjects came to be discussed in Germany.* The advocacy of object teaching by Pestalozzi, the
championing of Pestalozzianism by Herbart, the attacks upon
mathematical reasoning and particularly upon Euclid that were
made by Schopenhauert conspired to influence the teaching of
geometry.
* Schotten, p. II.
t Klein, p. 503.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.   I 5


About I860 the genetic method (called "heuristic" when the
inventional side was emphasized) came to be discussed. This
makes a plea of being a natural method, since it incites self-activity in the pupil. With the genesis of a theorem the pupil sees
intuitively its inner relation to other theorems; he not only sees
whence he came but also whither he is going; the reader of
Euclid is blindfolded, so to speak, and then somehow transported to the next station. It is difficult to prepare text-books
for the genetic method. The teacher by careful questioning one
moment leads the student, the next moment follows him, and no
one can foresee the exact path which this mode of advance will
mark out. It is not strange, therefore, if many teachers proceeded heuristically while the texts retained mostly the dogmatic
form.*  Moreover, experience made it plain to teachers that the
dogmatic statement of theorems has a high mnemotechnic
value.f  While the genetic method in its pure form has not
succeeded in establishing itself, it has exerted a strong influence
by shifting the emphasis from the memorizing of proofs to the
cultivation of originality and logical reasoning.
Another movement that sprang from the teachings of Pestalozzi and Herbart was the adoption of preliminary courses on
observational geometry and drawing, about I870. Such courses
had been recommended long before this time. This movement
was stronger in Germany than in England and France. In their
propaedeutic courses the geometry of solids was to receive consideration and a taste of the genetic method was recommended.
The pupils acquired dexterity in the use of ruler and compasses.
Propaedeutic courses have maintained their place to the present
time.
Herbart made strong endeavors to remove the superstition
that had arisen in early days when Euclid was placed in the
hands of young and immature students, to the effect that mathematics could be learned only by a few pupils endowed with
special gifts. According to his view the fault lies as a rule in
the abstract character of the early instruction; the introduction
of propaedeutic courses and the greater emphasis upon "Anschauung" at all stages had shown that most students can
* Schotten, p. 96.
t Schotten, p. 13.




THE MATHEMATICS TEACHER.


master mathematics. Whether "amathematicians" do exist in
rare instances, is a question which Klein refers to experimental
psychologists for reply.*
A third movement, agitated in Germany, was in favor of the
introduction into elementary instruction of the concepts of the
modern projective geometry. It originated about I870.t The
criticism was made that Steiner, M6bius, and von Staudt had.
been so busy with their researches as to make no attempt to
reform elementary instruction, and that text-book writers had"
ignored the researches of these great men. The leaders in this
attempt to incorporate modern methods were Schlegel and
Fiedler. A concomitant of this programme was the breaking
down of the division of geometry into plane and solid, and the
effort by the use of models, etc., to make geometry more concrete. To effect this reform, a number of texts by Schlegel,
Miiller, Kruse, Becker, Worpitzky, Henrici, and Treutlein
sprang into existence.t Aside from the production of interesting text-books this agitation has had little success. The books
in question were seldom used.~ Can it be that D'Alembert's
dogma is, after all, based upon truth-the dogma that the historical order of development of geometry is the pedagogical
order; that is, the easiest approach to the science for the young
mind? Are the concepts of projective geometry more difficult
to grasp than those of the older geometry, or did the texts just
named overtax the pupils, and perhaps in other ways violate
the demands of sound pedagogy?
Most interesting are the statistics gathered in Prussia in I880
which showed the following distribution of geometrical texts:
Kambly was used in 217 institutions; Koppe in 54; Mehler in
44; Reidt in 29, while 55 texts were used in one institution each.
Kambly's "clever but unscientific book" was first issued in
Breslau in I850 and a few years ago reached the Ioist edition
in the revision by Roeder. Koppe was looked upon as an inferior work, yet it enjoyed great popularity. On the other
hand, books like those of H. Miiller and even Henrici and
* Klein, p. 499.
t Schotten, p. I8.
$ Schotten, p. I9.
~ Schotten, p. 20.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.   17
Treutlein seldom passed beyond the second edition. This most
astonishing success of works considered as scientifically inferior requires explanation. Schlegel says* "that the quality
of the books most widely adopted allows one to draw an inference respecting the scientific level of the instruction generally
reached in that subject."  But this remark considers merely one
phase of this question. May not the mass of -teachers have had
a feeling or insight concerning text-books which involved questions of intuition or other psychologic matters that the writers
of the more scientific books overlooked?  Simont points out
that until recently the German teacher, unlike the French, enjoyed complete freedom in teaching, and that small texts, like
Kambly, allow his individuality much wider play.
Kambly's "Elementar-Mathematik" was made up of four
parts: first, arithmetic and algebra; second, planimetry; third,
plane and spherical trigonometry; fourth, stereometry. Of interest here is the interpolation of trigonometry.  We have
before us Kambly's Planimetrie, 43d edition, Breslau, 1876.
Among the points of popularity we mention the following:
I. The book contains only as much matter as a class can conveniently finish in one year. Skipping parts of a book, says
Kambly, has a bad effect upon both pupils and parents.
2. The diction is clear and simple. Mathematical symbols
are used freely. The setting of the 'type is such as to enable the
eye more quickly to see the relations set forth.
3. The arrangement of the book is such as to allow the
teacher much freedom. He may, for instance, omit incommensurables altogether, or else substitute for certain proofs in
the regular text others given in the footnotes where rough
proofs are found for incommensurable cases.
4. Easy arithmetical applications, original theorems, and original constructions are given at the end of the book, so that some,
or all, may be conveniently taken or omitted, according to the
preference of the teacher.
Koppe's "Planimetrie" made somewhat greater demands
upon the powers of the pupil than did Kambly, but incommensurables were treated only in footnotes or in remarks fol* Schotten, p. 21.
t Simon, p. 25.




I 8


THE MATHEMATICS TEACHER.


lowing the proofs of theroems.   In Liibsen's "ElementarGeometrie," I have not been able to find a reference to incommensurables. It differs from Kambly and Koppe in having
better figures and in having them on the page where they are
needed, instead of at the end of the book on separate sheets
that unfold. A  clever feature in Liibsen are the practical
applications introduced from the very beginning. How to run
a straight line over undulating country by the use of poles, is
explained in several diagrams on the first pages. Other figures
show how to determine the distance between points on opposite
banks of a river.
Since about I890 the activity of Felix Klein of G6ttingen, in
mathematical reform, has been very great. For the first time
since the death of Kastner, is the influence of university professors upon the teaching of elementary mathematics in Germany beginning to be strongly felt. Among the defects of
geometrical instruction, he points out the insufficient fusion of
the various branches of elementary mathematics.*  Thus, too
little attention is given to drawing of solids and to projection,
to the idea of motion in a figure to replace Euclidean rigidity,
to the fusion of arithmetic and geometry, to the introduction of
the coordinate representation of analytics. On the other hand,
the construction of triangles from given data is over emphasized,t as is also the study of the curious points and lines
in the geometry of the triangle. This last criticism applies even
more strongly to English text-books.
Klein points out that modern demands in geometric teaching,
first, emphasize the psychologic point of view,$ which considers
not only the subject matter, but also the pupil, and insists upon
a very concrete presentation in the first stages of instruction,
followed by a gradual introduction of the logical element;
second, call for a better selection of the material from the viewpoint of instruction as a whole; third, insist on a closer alignment with practical applications; fouth, encourage the fusion
of plane and solid geometry, and of arithmetic and geometry.~
* Klein, p. 439.
t Klein, p. 442.: Klein, p. 435.
~ Klein, p. 437.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.


I9


The "Lehrbuch der Mathematik nach modernen Grundsatzen"
by Behrendsen and G6tting, a secondary school text published in
I909, carries out Klein's ideas on the teaching of algebra and
geometry.
A piece of research of vital importance in the advanced study
of geometry is the " Foundations of Geometry," brought out in
1899 by Professor Hilbert of G6ttingen.* Though widely read
by mathematicians, it has exerted no direct influence upon elementary teaching in Germany. It has been felt that this mode
of treatment is not suitable for pupils first entering upon demonstrative geometry.
ITALY.
Since the unification of Italy, great mathematical activity has
existed in that country. Before that event, very different practices in geometrical teaching existed in different parts of the
country.t In 1868 Cremona and Battaglini were members of
a government commission to inquire into the state of geometrical teaching. They found it unsatisfactory, and the number of bad text-books so great, and so much on the increase,
that they recommended for classical schools the adoption of
Euclid, an edition of which was brought out by Betti and
Brioschi. Later other works of scientific merit replaced Euclid.
Cremona's great emphasis upon projective geometry reached
from the universities down into secondary schools. A typical
work is that of A. Sannia and E. d'Ovidio, I869, which uses
the theory of limits and retains the division of geometry into
plane and solid. It stands closer to Euclid than to Legendre.
The blending-of plane and solid geometry, which received great
emphasis in Italy, is typified in the "Elementi di Geometria"
of R. de Paolis, I884.
A very remarkable school came into being in Italy, the purpose of which is to render geometry still more rigorous than in
the Euclidean text. Starting with a single basic concept, the
point, all other concepts are to be logically developed. This
movement is typified in the works of G. Veronese.+  Of ele* D. Hilbert, " Grundlagen der Geometrie," in Festschrift zur Feier
der Enthiillung des Gauss-Weber-Denkmals in Gottingen, Leipzig, 1899.
t Simon, p. 43.
t Klein, p. 482.




20


THE MATHEMATICS TEACHER.


mentary works he has prepared " Nozioni Elementari di Geometria Intuitiva," I902, and "Elementi di Geometria," I904,
the first of these being a propoedeutic work. Demonstrative
geometry is taken up in Italy with older pupils than in Germany
and the United States; hence works of greater rigor can be
used. Veronese endeavors to state all the necessary postulates
of geometry, no matter how obvious, as for instance, "There
exist different points," to make it plain that we do not consider
a geometry in which only one point exists.* As regards the
selection of material, Veronese confines himself mainly to that
of Euclid, thus receding from the tendency of the School of
Cremona. He avoids all fusion with arithmetic. Somewhat
similar in character is the "Elementi di Geometria" of F.
Enriques and U. Amaldi, I905.
The effort at rigor, due to Veronese, has been intensified in
the great school of Peano, which endeavors to eliminate all intuition. It seems that this school has influenced even elementary instruction and the teaching in technical schools.t  This
recent Italian emphasis upon extreme rigor has led to deplorable results with the less gifted pupils, and a reaction appears
to be setting in. Under the leadership of Loria and Vailati a
movement is on foot favoring greater emphasis upon intuition,
the introduction of some modern geometrical notions, the fusion
of geometry with arithmetic, and the concession to the demands
for practical applications made by this age of industrial development. In fact, Italy is entering upon a reform much like that
of Germany and France.t
ENGLAND.
Roger Bacon says that toward the close of the thirteenth
century the definitions and a few of the theorems in geometry
were studied by some pupils at Oxford.~ About I570 Sir
Henry Savile began to lecture at Oxford on Greek geometry,
and in 1619 Briggs at Cambridge on Euclid. In I665, Isaac
* Klein, p. 483.
t Klein, p. 486.
t For additional details see W. Lietzmann's article in Schotten's Zeitschrift, Vol. 39, pp. 177-191; Vol. 40, pp. 227-228.
~ Ball, " Mathematics at Cambridge," 1889, p. 3.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  21


Barrow at Cambridge prepared a complete edition of Euclid,
which was the standard for fifty years. Gow says that "the
seventy years or so, from I660 to 1730, when Wallis and Halley
were professors at Oxford, Barrow and Newton at Cambridge,
were the period during which the study of Greek geometry was
at its height in England."* In 1703, William Whiston became
the successor of Newton at Cambridge.- He brought out an
edition of Tacquet's Euclid. Robert Simson's edition of Euclid
first appeared in I756. Simson was professor of mathematics
at the University of Glasgow. In the universities of Great
Britain, Euclid met with no competition. Ward's "Young
Mathematician's Guide," I707, may have been used to some
extent, but probably more for its arithmetic and algebra than
for its geometry. Practical men, holding positions as excise
officers, had to be familiar with practical geometry. For them
practical treatises existed, some of which gave explanations of
the slide rule. A departure from Euclidean rigor might be
expected in the education of men for the army or navy. We
have seen that Kastner criticized the French for making mathematics easy for men interested in war. England has had since
1722 an academy at Portsmouth where men spent one or two
years studying navigation, drawing, etc. England has had also,
since I74I, a military academy at Woolwich, where sons of
noblemen and military officers were taught fortification, gunnery and mathematics. Among the mathematical professors at
Woolwich, during the eighteenth century, were Thomas Simpson, John Bonnycastle and Charles Hutton, all three authors of
text-books, including geometries. Hutton's works went through
several editions in the first half of the nineteenth century.
From this it is evident that Euclid did not hold universal sway
in England. Yet the forces opposing him were utterly unable
to dislodge him.
In 1822, Sir David Brewster brought out an English translation of Legendre's geometry. Did teachers rally in favor of the
introduction of this text?  We shal see that DeMorgan suggested the use of some parts of it on solid geometry; DeMorgan
deplored that solid geometry was seldom or never taught before
trigonometry. But otherwise we are not able to find any
serious reference to this translation of Legendre.
* Gow, " History of Greek Geometry," Cambridge, I884, p. 208.




22


THE MATHEMATICS TEACHER.


During the second half of the eighteenth century England
had come to be the only country where Euclid was practically
the only geometrical text used. During the eighteenth century
the average age of freshmen in the English universities was
gradually increasing, and perhaps at this time, Euclid passed
from the universities to the lower schools. There is no explicit
proof, however, that in the great "public schools" Euclid was
studied before the nineteenth century.*
Very recentlyt some interesting information has been published about one of the "public schools"-Christ Hospitalwhich paid more than usual attention to mathematics in the
courses for boys preparing to enter the royal navy. It seems
that as early as I680 such boys were required to study the
earliest parts of the first book of Euclid, the Ioth, I th, and I2th
propositions of the sixth book, and to learn arithmetic.
Perhaps this represented all the theoretical mathematics taught,
for Sir Isaac Newton, whose advice about changes in the course
was sought, notes the following omissions: There was no "symbolic arithmetic," no "taking of heights and distances and
measuring of planes and solids," no "spherical trigonometry,"
nothing of "Mercator's chart." In other "public schools"
probably no courses in geometry were given during the eighteenth century. Says Stamper: "It was not until about the
middle of the nineteenth century that the study of Euclid became common in the secondary schools of England."
It would be instructive to secure more information explaining
how it was possible for Euclid to maintain its supremacy as a
text, when geometry was being transferred from the universities to the schools. What were the experiences of teachers in
secondary schools with the Euclidean text? The desirability of
modifying Euclid must have arisen early, for in I795 John
Playfair brought out a revised Euclid containing the first six
books and adding the computation of 7r and a book on solid
geometry drawn from modern sources. Playfair endeavored to
give the geometry a form which would render it more useful.
Euclid's fifth book, which had never been used successfully
* Stamper, p. 88.
t "A School Course in Mathematics in the XVII Century," by W. W.
R. Ball, in the Mathematical Gazette, Vol. V., 1910, Part I., pp. 202-205.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  23


with beginners in geometry, as far as we can ascertain, was
modified by Playfair by replacing Euclid's prolix explanations
by the more concise language of algebra. But Playfair did not
try to change the nature of the reasoning. Had there been a
strong movement against Euclid in England at this time, Playfair would probably have joined it. In his review of Leslie's
"Geometry" in the Edinburgh Review, Vol. 20, 1812, p. 79, he
says: "A question has been sometimes agitated whether it is
most advantageous, for the study of geometry, to possess a
number of elementary treatises, or to have one standard work,
like that of Euclid... the same lessons are not suited to every
intellect, and on these accounts it may be of advantage that
different elementary texts should exist. We are very much
inclined to the latter opinion."
William George Spencer's unique booklet on "Inventional
Geometry" was brought out about 1830 or 1835, but "received
but little notice" at that time. A noteworthy device for aiding
the young mind through sensuous stimulus was the use of
colored diagrams, suggested by Oliver Byrne, in his edition of
Euclid, London, 1847. The failure of this book is doubtless
due to the want of moderation in the use of colors.
The ablest writer on the teaching of elementary geometry
during the first half of the nineteenth century in England was
Augustus DieMorgan. His articles published in the Quarterly
Journal of Education, in 1831, 1832, and 1833, display a pedagogical insight which would have prevented many calamities in
English teaching, had his views been more promptly and widely
accepted.  Elsewhere we quoted DeMorgan's remarks on
Williamson's.riticism of Clairaut's geometry, which showed
that DeMorgan firmly believed in a preliminary course in geometry, as an introduction to a logical course like that of Euclid.
It will appear that England was the last country actually to
introduce propaedeutic courses in elementary instruction.
DeMorgan did not hesitate to recommend radical changes in
Euclid. Here is what he said in I83I, in an article in the
Quarterly Journal of Education, entitled "On Mathematical
Instruction":
"With regard to the fifth book of the 'Elements,' we recommend the teacher to substitute for it by the common arithmetical




24


THE MATHEMATICS TEACHER.


notions of proportion. Admitting that this is not so exact as
the method of Euclid, still, a less rigorous but intelligible process
is better than a perfect method which cannot be understood by
the great majority of learners. The sixth book would thus
become perfectly intelligible."
Two years later, in an article in the same journal "On the
Methods of Teaching the Elements of Geometry," DeMorgan
dares to suggest that certain parts of Legendre might be profitably substituted for parts of Euclid. "The eleventh book of
Euclid may, in our opinion, be abandoned with advantage in
favour of more modern works on solid geometry, particularly
that of Legndr, which the English reader will find in Sir
David Brewster's Translation." In the same article DeMorgan
gives utterance to a difficulty experienced by young students,
which 1has been referred to by many writers in different countries, the reductio ad absurdum. DeMorgan says: "The most
serious embarrassment in the purely reasoning part is the reductio ad absurdumr, or indirect demonstration. This form of
argument is generally the last to be clearly understood, though
it occurs almost on the threshold of the 'Elements.' We may
find the key to the difficulty in the confined ideas which prevail
on the modes of speech there employed." As regards the difficult fifth book, DeMorgan said, in I833, "We would say to all,
teach the fifth book, if you can; but we would have all remember that there is an if." In another place he adds: "We
strongly suspect that Euclid, as studied, does as much harm as
good." To the credit of teachers be it said, that the fifth book
was quite generally omitted. But DeMorgan's activity in this
line did not end here. In I836 he published "The Connexion
of Number and Magnitude; An attempt to explain the fifth
book of Euclid." For fifty years this tract was not duly appreciated; later it began to wield a wide influence; it is on this
tract that the substitute for the fifth book given in the Syllabus
of the Association for the Improvement of Geometrical Teaching is modeled; it is on this tract that the revised fifth book in
the more recent editions of Euclid by Nixon and by Hall and
Stevens is based.
The need of modifying the text of Euclid is brought out by
DeMorgan in the "Companion to the British Almanac" of




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  25


I849, page 20, as follows: "If the study of Euclid has been
almost abandoned on the continent, and has declined in England, it is because his more ardent admirers have insisted on
regarding the accidents of his position as laws of the science."
How little influence DeMorgan's views wielded in England
before about 1870 as regards the revision of Euclid's fifth book
and the study of solid geometry, appears from the fact that the
most popular edition of Euclid for many years was the one
brought out in 1862 by Todhunter. This author reproduces
Simson's text, though he greatly assists the pupil in overcoming
the difficulties by breaking up the demonstrations into their constituent parts. In an Appendix are given notes, supplementary
propositions and original exercises. Todhunter was quite out
of sympathy with the purposes of the Association for the Improvement of Geometrical Teaching.*
Opponents of Euclid existed in England at all times. Thus
in I860 W. D. Cooley brought out a rival text. Eleven years
later he expressed himself regarding this venture as follows:t
"In I860 there was published for me, by Messrs. Williams
and Norgate, a little volume entitled, 'The Elements of Geometry Simplified and Explained,' adapted to the system of empirical proof, and of exhibiting the truth of theorems by means
of figures cut in paper. It contains in 35 theorems the quintessence of Euclid's first six books, together with a supplement not
in Euclid. There was no gap in the sequence or chain of
reasoning, yet the 32nd and 47th propositions of Euclid were,
respectively, the 3d and I7th of my series. This book proved
a failure, for which several reasons might be given, but it will
be sufficient here to state but one, namely, that it came forth
ten years before its time."
The reformers found a champion in Sylvester, who in I869,
before Section A of the British Association, exclaimed: "I
should rejoice to see mathematics taught -with that life and
animation which the presence and example of her young and
buoyant sister (natural science) could not fail to impart, short
roads preferred to long ones, Euclid honorably shelved, or
buried 'deeper than e'er plummet sounded' out of the school* See "Conflict of Studies," by Todhunter, London, 1873.
t Nature, Vol. 4, 1871, p. 486




26


THE MATHEMATICS TEACHER.


boy's reach...." The reform forces finally organized themselves, in 1871, into the "Association for the Improvement of
Geometrical Teaching" (A. I. G. T.).
It is a curious circumstance that England's great mathematician, Arthur Cayley, opposed this reform movement. His
admiration for Euclid was so ardent that he even expressed a
preference for the original treatise without Simson's additions.
In the opinion of Langley, Cayley "overshot the mark and his
opposition told in favor of the Association."*
The second "Report" of the A. I. G. T. recommended practical exercises in geometrical construction, easy originals and
numerical examples. Two years were given to the preparation
of the Geometrical Syllabus on proportion. A double syllabus
was prepared: A Syllabus on Geometric Constructions, and a
Syllabus on Plane Geometry. Most of DeMorgan's suggestionst on the revision of the fifth book of Euclid were adopted.
This Society, after long labors, finally issued a substitute text,
"The Elements of Plane Geometry."   This was not used at
home, but was used with success in the British colonies. Klein
expresses himself, as follows, in regard to it: "This is essentially merely a smoothed down and polished presentation of the
first six books of Euclid's elements; thus the rough places at the
beginning of the first book... are removed by a consistent use
of the concept of motion, but in general the sequence and the
contents of Euclid are adhered to, in deference to the examinations. It is therefore only a tame reform, that is here attempted; nevertheless, it has met with sharp opposition by the
adherents of the old English system. As proof of this, I refer
to an amusingly written book of Dodgson, 'Euclid and His
Modern Rivals.'"   Here Euclid comes out victorious, and all
reformers, particularly Legendre and members of the A. I. G. T.,
are put to the rout. J. MD.Wilson's "Elementary Geometry,"
Ist edition, I869, came in for a large share of the criticism. At
Oxford, where Dodgson had given instruction in geometry for
many years, this same Wilson had, at one time, read a critical
paper before the Mathematical Society, on "Euclid as a TextBook of Elementary Geometry." Wilson was a prime mover
in the organization of the A. I. G. T.
* Fifth Report of the A. I. G. T., p. 21.
t" Companion to the British Almanac," 1849, pp. 5-20.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  27


Since about I870, many editions of Euclid have been printed
containing revisions with the object of better adapting Euclid
to school use. They exhibit all possible gradations of departure from  the original text. There appeared sequels to
Euclid like that of F. Casey. Professor Klein expresses himself in regard to these as follows: "The necessity has been felt
to consider modern research, going beyond Euclid; this has
been done by pressing it by force into the rigid Euclidean form,
whereby a good part of the modern spirit is, of course, lost."*
During thirty years the A. I. G. T. appeared to have accomplished comparatively little. It had secured the concession that
proofs different from Euclid's shall be accepted in examinations and had brought about a sentiment favoring some modification and enrichment of the Euclidean text. In reality, it
had accomplished much more, for it had prepared the way for
the great agitation of I9OI, known as the Perry Movement,
which called for a complete divorce from Euclid. The discussion of the teaching of mathematics at the Glasgow meeting of
the British Association marks an epoch. The following are
suggestions and criticisms that were contained in Perry's
Syllabus.t
I. Experimental geometry and practical mensuration to precede demonstrative geometry. Use of squared paper. Rough
guessing at lengths and weights to be encouraged.
2. Some deductive reasoning to accompany experimental
geometry.
3. More emphasis on solid geometry; this subject has been
postponed too long.
4. Adoption of co-ordinate representation in space.
5. The introduction of trigonometric functions in the study
of geometry.
6. Emphasis upon the utilitarian parts of the subject.
7. Examinations conducted by any other examiner than the
pupil's teacher are imperfect examinations.
In criticism of previous practices, Perry held that a boy
should be educated through the experiences he already pos* Klein, p. 447.
" Discussion on the Teaching of Mathematics," edited by John Perry,
I90I, p. 97.




28


THE MATHEMATICS TEACHER.


sesses, and should be allowed to assume the truth of many
propositions. He held that the teacher must recognize that
boys take unkindly to abstract reasoning. He criticized Oxford because, for the pass degree there, two books of Euclid
must be memorized, even including the lettering of figures, no
original exercises being required. In the discussion that followed, all favored the preliminary experimental course and
some advocated a second experimental course to accompany
Euclid. Hudson and Forsyth still believed in maintaining the
Euclidean sequence of theorems. Minchin declared Euclid's
order bad. S. P. Thompson and MacMahon favored the retention of Euclid. Miall did not see why we should have a recognized geometry any more than one arithmetic, or one trigonometry. Minchin, Magnus, Pressland, Workman, and Lamb declared themselves against Euclid as a text-book..
The immediate result of Perry's address of I900, at Glasgow,
was the appointment of two committees, one of the British Association and the other of the Mathematical Association. The
former committee confined its work to the more general aspects
of geometrical teaching. The latter, which was composed
mainly of schoolmasters, formulated a set of detailed recommendations, which were published in the Mathematical Gazette
of May, I902. They include an experimental introductory
course, requiring the use of instruments, practical measurement
and numerical work. In the formal study of geometry is
recommended the retention of Euclid as a framework, the admission of hypothetical constructions, definitions not to be
taught en bloc, the omission of incommensurables in the ordinary school course, and the use of algebra in the treatment of
areas.
The Perry laboratory method has led to the preparation of
some severely practical works, but as Lodge says, Perry "overemphasized fact divorced from principles." A middle ground
has met with greater favor. The plans recommended by the
Mathematical Association have been embodied very successfully
by Godfrey and Siddons in a text-book entitled, "Elementary
Geometry, Practical and Theoretical" (Cambridge University
Press, I904). The recommendations of the Mathematical Association have met with favor among teachers, and the general




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  29
effect has been beneficial. A circular issued in I908-I909, by
the Board of Education, on "The Teaching of Geometry and
Graphic Algebra," showed the wide departure made since the
beginning of the twentieth century. We quote two sentences:
"Axioms and postulates should not be learnt or even mentioned."  "It should be frankly recognized that unless the
power of doing riders has been developed, the study of the
subject is a failure."*
The greatest obstacle to reform in England has been the
system of examinations. After thirty years of failures the
Mathematical Association, at last, has been remarkably successful in persuading examining bodies to give up their insistence
upon Euclid, and now Euclid's proofs and arrangement are no
longer required by the universities. "Any proof of a proposition will be accepted which appears to the examiners to form a
part of a logical order of treatment."
THE UNITED STATES OF AMERICA.
During the seventeenth century, arithmetic and geometry received some attention in the last year of the college course at
Harvard College. In 1726 Alsted's "Geometry" is mentioned
as a text-book studied by Harvard seniors, but as soon as geometry came to receive serious attention in American colleges,
Euclid became the text used. The first mention of Euclid that
we have seen at Yale is in I733; at Harvard, in I737. In the
latter part of the eighteenth century, geometry was taught to
lower classmen. According to a member of the Harvard class
of I798, "the sophomore year gave us Euclid to, measure our
strength." In I801 Professor Webber said, "A tutor teaches
in Harvard College Playfair's 'Elements of Geometry.'"
In I813 the "Analytical Society" was formed at Cambridge
in England, which aimed to encourage in Britain the rigorous
study of French higher mathematics. The influence of this
movement reached the United States.   In about ten years
American teachers began to adopt French texts.  Collateral
events at West Point had the same tendency. There elementary
mathematics was taught from I808 to I8IO by F. R. Hassler,
who was a graduate of the University of Berne in Switzerland.
* Nature, Vol. 8o, 1909, May 27, p. 374.




30


THE MATHEMATICS TEACHER.


In 1817 Crozet, -of the Polytechnic School in Paris, introduced
descriptive geometry into West Point.
In I819, John Farrar, of Harvard, brought out a translation
of Legendre's Geometry, which, with translations made by him
of other French and Swiss texts on mathematics, were at once
widely adopted in the leading American colleges. American
teachers were willing to turn to the French, not only for works
on the calculus and celestial mechanics, but also for books on
elementary mathematics. So it came about that Euclid was
replaced by Legendre. In 1828 Charles Davies, professor at
West Point, brought out an edition of Brewster's translation of
Legendre's Geometry. Davies did not enunciate propositions
with reference to and by the aid of the particular diagram used
for the demonstration, and to that extent returned to the method
of Euclid. Davies' edition became widely popular under the
name of "Davies-Legendre," and was much used in the United
States as late as the '70's.
One of the earliest American geometries worthy of note was
that of Benjamin Peirce. The Harvard catalogue of 1838 announces that Freshman take Peirce's Geometry.     Peirce
favored the use of infinitesimals and also the use of the term
direction, a concept probably first used in this country by a
Harvard teacher named Hayward in his geometry of I829.
Peirce's text did not become widely popular, for, like his other
elementary books, it was too condensed for immature students.
In 1843 or 1844, Harvard first made geometry a requirement
for admission to College.
In 1851, Professor Elias Loomis, of Yale, issued a geometry
which was revised in 1871. Loomis came under French influences as a student in Paris. In the second edition of his text he
says: "The present volume follows substantially the order of
Blanchet's Legendre, while the form of demonstrations is
modeled after the more logical method of Euclid." It has been
said of American writers, that while they have given up Euclid,
they have modified Legendre's geometry so as to make it resemble Euclid as much as possible. This applies to Loomis
with greater force perhaps than to any other author.
In 1871 Professor Olney, of the University of Michigan,
published a " Geometry" under two main heads:




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  3 I


I. Special or Elementary Geometry, comprising (I) Empirical Geometry, (2) Demonstrative Geometry, (3) Original
Exercises in the Application of Algebra to Geometry, (4) Trigonometry.
II. General Geometry (Plane Loci).
Olney was a self-educated man. He was a great teacher and
had original ideas about teaching. It is said that he was prevented by his publishers from departing very far from the traditional classification. His ideas were novel and forecasted in
many ways the present tendencies in mathematical teaching.
His geometry shows that he attempted to correlate the various
mathematical topics and to introduce applications to everyday
affairs. Olney's books were used quite extensively in the
Middle West, but acquired no firm foothold in the East.
Just before the death of William Chauvenet, in 1870, appeared his "Geometry," the only elementary book he wrote.
Closely following French models, exhibiting a wonderful ease
and grace of style, Chauvenet produced a remarkable book,
which was used in many of the best schools. He included as
a part of the work, an introduction to modern geometry.
Perhaps no work on geometry ever published in the United
States has 'been so highly respected as this.
In 1878 appeared the "Geometry" of G. A. Wentworth,
which is still in use. We omit all discussion of it, as also of
later books which have been published in this country.
The researches on non-Euclidean geometry, begun in the
eighteenth century in Italy and Germany, and brought to
fruition in the early part of the nineteenth century, did not produce appreciable effect upon the teaching of elementary geometry until the last quarter of the nineteenth century. It was
in 1867 and 1868 that Baltzer, Battaglini, Grunert, and Hofiel
brought Bolyai and Lobatchevsky to the attention of the mathematical public at large.
The new ideas have not affected the teaching of elementary
geometry except in some of the definitions and postulates. They
have assisted in the rejection of the definitions, "parallel lines
are lines everywhere equally distant," and "parallel lines are
straight lines which have the same direction." They have
shown the futility of "proving" the parallel-postulate and have




32


THE MATHEMATICS TEACHER.


led to the use of the word "axiom," not as a "self-evident
truth," but as a synonym for "postulate."
In conclusion, we note that, with the beginning of the
twentieth century, England began once more to influence the
teaching of geometry in the United States, through the so-called
" Perry movement," and that Germany, which at no time during the nineteenth century affected geometrical teaching in
America, makes itself felt at the present time through the pupils
of Klein and Hilbert and through the international movement
towards reform in the teaching of mathematics, headed by
Klein.
The Committee recommends the foregoing historical sketch
to the careful consideration of teachers of geometry. Special
attention is called to the age-long contest between the extreme
formalists and the extreme utilitarians. The committee stands
for neither extreme position. It recommends a reasonable attention to exercises in concrete setting, such, for instance, as
simple problems involving the trigonometric ratios in connection
with similar triangles, or such applications as those shown on
pages 98-102 of this report. But, in so doing it does not recommend diminishing attention to the logical side of the subject, but
rather a quickening of the logical sense through a more rational
distribution of emphasis which will make for economy of both
time and mental energy in mastering the standard theorems and
leave opportunity for a broader view of the subject in its concrete relations. See Sections D and E.
SECTION B. LOGICAL CONSIDERATIONS.
AXIOMS.
(a) Nomenclature.-The best historical usage distinguishes
between Axioms (Euclid's "Common Notions") and Postulates (Euclid's "aitemata" or "requests") by including in the
former certain general statements assumed for all mathematics,
and in' the latter certain specifically geometric concessions.
These names and this distinctibn are now in general use and
there seems to be no good reason for attempting to change them.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  33


However, teachers who may wish to use the single term
"assumptions" to cover both, or to use the term " axiom" to
mean any proposition whose truth is postulated, thus making
axiom and postulate synonymous, should be free to do so.
(b) General Nature.-It is evident that strict mathematical
science would lead us to seek and to recommend an " irreducible
minimum " of assumptions, while educational science leads us to
see that such a list would be unintelligible to pupils and therefore unusable in the schools. Since we cannot recommend the
adoption of a set of assumptions along the Hilbert line, we
therefore lay down the general line of axioms and postulates
needed in geometry, without insisting upon an exact list or upon
any particular phraseology.
(c) General List of Axioms.As to the nature of the quanltities, positive quantities are to be
understood. When the negative quantity enters into elementary
geometry it is in the discussion of propositions and not in cases
in which the axioms are directly employed. For example, it
is desirable not to confuse beginners in geometry by the question of dividing unequals by negative numbers.
Operations upon equal quantities.-It should be stated, preferably in a series of axioms, that if equals are operated upon
by equals in the same way, the results are equal; i. e., if a — b
and x =y, then a+- x   bl+y, a-x ~-b -y (where a > x),
ax = by, etc.
Operations upon unequal quantities.-It should be stated that
if unequals are operated on by equals in the same way, the results are unequal in the same order; i. e., if a > b and x -y,
then a +x> b +y, etc. These various cases enter into elementary geometry, and this assumption should be stated in such
a manner that the student can easily refer to it in his work.
There is also the assumption that if unequals are added to
unequals in the same order the sums are unequal in the same
order, and that if unequals are subtracted from equals the remainders are unequal in the reverse order, these being the only
ones relating to inequalities that are needed in elementary
geometry.
As to substitutions.-In geometry it is continually necessary
to make use of the assumption that a quantity may be sub



34


THE MATHEMATICS TEACHER.


stituted for its equal in an equation or in an inequality. Often
this assumes the common form that "quantities that are equal
to the same quantity are equal to each other." The committee
recommends this axiom.
Inequality among three quantities.-It is necessary to say in
geometry that if a> b and b > c then a > c, and an axiom
to this effect is necessary.
The whole and its parts.-Although the definition of " whole"
might be given in such a manner as to render unnecessary the
usual axiom, it seems advisable to make the statement in the
ordinary form.
It is to be understood that in applying these axioms to geometric magnitudes, the letters used refer to the numerical measures of such magnitudes, and as such belong to arithmetic and
algebra, thus giving the theoretical basis for the correlation of
geometry with these subjects.
Summary.-Axioms covering the above points are of advantage in the practical teaching of geometry, but this committee
has no recommendation to make as to order or phraseology.
They may be summarized as follows:
If a-b and x =y, then
(I) a+x     b+y.              (3) ax=by.
(2) a-x-=b-y (a > x).         (4) a/x=b/y.
(5) anb==    and a/a=z -/b, where n is a positive integer.
(6) If a>b and x=-y, then a+x>b+y, ax>by,
a/x > b/y, and if a > x, then a-x > b-y.
(7) If a > b and c > d, then a + c > b + d, and if x=-y,
then x-a < y-b.
(8) If a=x, and b-=x, then a-=b.
(9) If x=a, we may substitute a for x in an equation or
in an inequality.
(Io) If a > b and if b > c, then a> c.
(II) The whole is greater than any of its parts, and is
equal to the sum of all its parts.
(d) General List of Postulates.(i) One straight line and only one can be drawn through
two given points.
Corollary I. Two points determine a straight line.
Corollary 2. Two straight lines can intersect in only one
point.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.


35


(2) A straight line-segment may be produced to any required
length.
This includes one postulate and one problem of Euclid, and
so manifestly depends upon the simplest uses of straight edge
and compasses as to be a proper geometric assumption.
(3) A straight line is the shortest line between two points.
(4) A circle may be described with any given point as a center
and any given line-segment as a radius.
(5) Any figure may be moved from one place to another,
without altering its size or shape.
(6) All straight angles are equal.
This and the following corollaries may be included among the
theorems for informal proof under Section E.
Corollary I. All right angles are equal.
Corollary 2. From a point in a line only one perpendicular
can be drawn to the line.
Corollary 3. Equal angles have equal complements, equal
supplements, and equal conjugates.
Corollary 4. The greater of two angles has the less complement, the less supplement, and the less conjugate.
The above axioms and postulates may be recommended for
use as soon as the formal proof of propositions is begun, the
postulate of parallels being introduced when needed, as follows:
Postulate of Parallels. Through a given point one line and
only one can be drawn parallel to a given line.
The question of limits is considered later. It it not deemed
desirable to postulate explicitly the existence of such concepts
as point, line, and angle, nor to assume that a line drawn through
a point in a triangle must cut the perimeter twice, nor to add
a postulate of continuity. It is well, however, for teachers to
mention that such assumptions are always tacitly made.
In any case the committee feels that a certain amount of care
should be taken in fixing the location of points and lines and
proving that lines intersect, when the accuracy of the proof in
question might be affected by ignoring such details.
DEFINITIONS.
(a) New Terms.(I) General principle.-It is unwise for individual teachers
or writers to introduce terms beyond those actually in common




36


THE MATHEMATICS TEACHER.


use in geometry, or to change the accepted meaning of common
terms, unless there seems to be a very definite advantage in the
new term and an unquestionable sanction in the mathematical
world. In particular, the substitution of a new term for an old
one, to denote the same concept, is undesirable.
(2) Type of terms that may safely be added to those of the
older elementary geometry: congruent, because this is so widely
used both here and abroad, and because it avoids the loose use
of equal and the long forms of identically equal, and equal in
all their parts.
(3) Types of terms that may safely be dropped: scholium,
because this has been so generally abandoned, and because it is
unnecessary; mixed line, an antiquated term of no value in elementary geometry. Other such terms are trapezium and rhomboid.
(4) Type of terms that seem of too doubtful advantage to be
recommended definitely by this committee, teachers being left
free to use them if they desire, the terms thus being given an
opportunity to make their way if they possess real merit: ray, a
term that has abundant sanction in higher geometry, but may
be dispensed with in elementary work. Other such terms are
mid-join (for median), cuboid (for rectangular parallelopiped),
n-gon (for polygon of n sides), and sect (for segment of a
straight line).
(5) Types of terms that are used with a different meaning
in higher geometry, and that may properly be used in elementary geometry with the more recent signification: circle, as meaning the line, which is, indeed, the primitive Greek meaning; circumference, as meaning the length of the circle-these usages
requiring a redefining of segment of circle, semicircle, area of
circle, and other obvious terms.
The committee recognizes also the tendency to unify the usage
of such terms as polygon and sphere in elementary and higher
geometry. This tendency should be encouraged. In any case,
it is essential that the pupil should understand clearly what the
terms mean in the statements and proofs. of the propositions.
(b) Symbols.(i) General principle.-No symbols should be recommended
beyond such as are already in wide use in elementary geometry




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN. 37
37


and any that are unnecessary or are not generally accepted
should be abandoned. The elaboration of personal symbolism,
sometimes to the point of eccentricity, is such as to be cumbersome in the mathematics of the present.
(2) Recommendations.-The committee feels that the common symbols of algebra, most of which are known to the pupil
beginning geometry, and such obvious symbols as those for.
perpendicular, triangle, circle, square, and parallel, are all that
are needed in a course in elementary geometry, and that it is unnecessary to specify these symbols in detail. It appears that
there is no generally received symbol for congruence, the symbols, -, and    all being' in use, and it seems best to recognize this fact, leaving teachers at present to decide the question
for themselves. In due time a general consensus of opinion
may lead to some definite usage, and it is the feeling of the committee that the second symbol given is a desirable one.
(c) Distribution of Definitions.-The committee recommends
that new terms be taught when the time arrives for using them.
This allows a teacher to use a book in 'which the definitions are
massed or one in which they are scattered, but it encourages
teaching them on the latter plan. It is recognized that the
massed plan has the advantage of a dictionary arrangement,
and this is a plan that 'a text-book writer might reasonably
adopt, but it is not a plan to be followed in the actual teaching
of the terms.
(d) The Defined and the Undefined.-The attention of teachers is called to the fact, now coming to be well recognized, that
certain terms in geometry must be looked upon as undefined.
Certain concepts are so elementary that no simpler terms
exist by which to define them, although they can easily be explained. For example, point, line, surface, space, angle, straight
line, curve.  The committee recommends that teachers give
more attention to instilling a clear concept of such terms and
none to exact definition.
On the other hand the committee recommends the careful
definition of readily definable terms, where these definitions are
parts of subsequent proofs, such definitions to be memorized
exactly or in their essentials. For example, right angle, square,
isosceles triangle, parallelogram.




38


THE MATHEMATICS TEACHER.


There is a further class of easily defined terms, where the
definition is not made the basis of a proof, and it seems obvious
to the committee that the memorizing of the exact wording
of such definitions is not a wise expenditure of time. Such
terms are hexagon, heptagon, re-entrant angle, concave polygon,
etc.
(e) The Form of Definition.-A definition may begin with the
term defined, as in a dictionary; or it may close with the word
defined; or it may at times contain the word in the midst of
the sentence. The committee feels that it is of no moment
which of these forms is taken, or that the definition be embodied
in a single sentence. A definition that is to be memorized as the
basis of a proof should be as nearly scientific as the powers of
a beginner in geometry -will justify, containing only terms that
are simpler than the term defined, not being tautological, and
being reversible-but further than this it seems unwise to attempt. to specify the form of a definition.
INFORMAL PROOFS.
(a) Justification.-It is not pretended that elementary geometry is a perfect piece of logic. In general, the modern departures from Euclid have sacrificed logic for other ends, and
even Euclid's "Elements" was not without numerous logical
imperfections. That is to say, it has always been considered
justifiable to sacrifice logic to a greater or less degree. The
principle is that a logical sequence should be maintained, and
formal proofs of propositions necessary to the sequence should
be required, so far as this is consonant with the educational
principle of adapting the matter to the mind of the learner.
Now in many cases it happens that an informal and confessedly
incomplete proof is more convincing to a beginner than a formal
and complete one, and is less discouraging because it postpones
the minor and seemingly unimportant steps to a time when their
importance may be appreciated and the proofs understood.
(b) Types.-To be specific, the following are types of propositions that are better passed over by the beginner without
formal statement, being introduced at the proper points in the
development, or with informal proof, than proved in the euclidean fashion:




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  39


If one straight line meets another the sum of the two adjacent
angles is a straight angle, and conversely (and related propositions);
All straight angles are equal (a proper postulate with related
corollaries);
Two straight lines can intersect in only one point;
A straight line can have but one point of bisection (and the
related case for angles);
The bisectors of vertical angles lie in one straight line;
Polygons similar to the same polygon are similar to each
other;
If one angle is greater than another, its complement is less
than the complement of the other (and related propositions);
A straight line can cut a circle (circumference) in two points
only;
Circles of equal radii are equal (and related statements);
All radii of the same circle are equal (and similarly for diameters);
A circle can have but one center;
And propositions relating to the conditions under which two
circles (circumferences) intersect.
It should be understood that these propositions are merely
types, and that others of the same type may be treated in the
same way, as specified in Section E of this report.
(c) Experience of Other Countries.-It is the experience of
all countries where Euclid is not taught that good results follow
from the use of a reasonable number of such informal proofs.
The German and Austrian text-books are especially given to
such procedure, and the results seem to have been favorable
rather than otherwise. The number of propositions formally
proved in a German text-book is notably less, for example, than
in a corresponding French text-book.
(d) Dangers.-It is evident, however, that we may easily go
to a dangerous and ridiculous extreme in this matter. With
all of the experiments at improving Euclid the world has really
accomplished very little except as to the phraseology of proposiltions and proofs; the standard propositions remain, and if
geometry has any justification, apart from its kindergarten aspect (which requires but a short time), most of these proposi



4o


THE MATHEMATICS TEACHER.


tions will continue to be proved, and should continue to be
proved. These propositions, whether in the Euclid or Legendre
arrangement, number in the neighborhood of I60 for plane
geometry. Of this number upwards of one hundred must receive formal proof in any well-regulated course in geometry.
TREATMENT OF LIMITS AND INCOMMENSURABLES.
(a) Present Status.-It is generally agreed that the present
treatment of this subject is open Ito two objections, (I) it is not
sufficiently understood by the student to make it worth the
while, and (2) it is not scientifically sound.
(b) Remedies Proposed.-Corresponding to the two defects
mentioned two remedies have been proposed, (I) to make it
less formal and technical, so that it shall be better understood,
and (2) to abandon the incommensurable case altogether in
secondary education.
(c) The Position of This Committee.-This committee recommends that in'elementary geometry the nature of incommensurables and limits be explained, but that the subject no longer
be required for entrance to college or be included in official
examinations. It recommends that the schools treat the subject
as fully beyond this point as circumstances seem to demand,
and to this end reference is made to the syllabus given in
Section E.
The prime object is to relieve the schools of the necessity of
teaching the subject, while leaving them free to do so if they
wish.
TIME AND PLACE IN THE CURRICULUM.
(a) Conventionally.-At present, in America, plane geometry
is generally taught in the tenth school year (not counting the
kindergarten).
In the East it is completed in the eleventh school year, and
in the West solid geometry is completed in the eleventh or
twelfth year. In spite of all the discussion about constructive
geometry (intuitive, metrical, etc.) in the first eight grades,
carried on in the past half century, no generally accepted plan
has been developed to replace the old custom of teaching the
most necessary facts of mensuration in connection with arithmetic. We have, therefore, at this time, algebra in the ninth




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  41


school year, plane geometry in the tenth, and algebra and geometry in the eleventh and sometimes in the twelfth.
(b) Changes Suggested.-Certain changes in this conventional plan have been suggested.
(I) To provide for preliminary (inductive, constructive, observational) work in geometry in the elementary grades. This
topic is discussed in Section C of this report.
(2) To precede the work in plane geometry by some definite
work in geometric drawing. Attention may be called to the
fact that the recent great advance in art education has ha'd one
disadvantage from the standpoint of geometry, in that geometric
drawing has been abandoned, and that therefore some little
work in handling compasses and ruler must now form part of
the first steps in this subject.
(3) To unite geometry and algebra, or geometry and trigonometry. This committee does not feel that the experiments
along this line, which have been made in only a few schools,
have been sufficient to determine whether or not geometry
should run parallel with algebra in the ninth, tenth and eleventh
school years.
(c) Position of This Committee.-This committee recommends that plane geometry be assigned not less than one year
nor more than one and one half years in the curriculum, being
preceded by at least one year of algebra except where the individual teacher desires to carry it along with algebra.
*It should be distinctly understood that owing to the condition
of unrest in the entire field of secondary education it is at
present impossible to give any final advice along any of these
lines of change. It is probable that many of the readjustments
now under general discussion will influence every high school
curriculum in the course of time. It is also possible that some
of the proposed changes will be adapted by the different types
of secondary schools to their own needs, and that they will
receive greatly varying emphasis in different localities. A certain amount of experimentation will undoubtedly be necessary
to test the feasibility of some of the proposed plans. Great care
should be taken to make all such experimentation with due
regard for all that was good in the past, so that the new
curricula may be the result of evolution, and not of revolution.




42


THE MATHEMATICS TEACHER.


The most noteworthy tendency in secondary education is the
desire for more organic teaching and hence the desire for more
time. This tendency finds its most significant expression in the
movement toward a six-year curriculum. It is undoubtedly
true that in a six-year curriculum many of the problems of
correlation would be brought nearer to a solution, that many
difficulties arising from the present tandem system would disappear, and that mathematics would be given a place in the
curriculum more nearly commensurate with its importance.
For a brief account of the six-year curriculum the reader is
referred to the book of Hanus entitled "A Modern School,"
published by the Macmillan Company; and also to the Proceedings of the National Education Association for I908.
PURPOSE IN THE STUDY OF GEOMETRY.
(a) Historical Review.-Geometry was originally, as its name
indicates, purely a practical subject. This phase of its history
remains in the work in mensuration in arithmetic to-day. It
then became a philosophical subject, connecting with mysticism
in the Pythagorean school, being put upon 'a more solid scientific
basis by the Platonists, and being crystallized by Euclid about
300 B. C. Since that time the formal side has dominated. But
this formal side has been attacked time after time, by the
astrologers and mystics, by the cathedral builders of the Middle
Ages, strongly by the French writers of the seventeenth and
eighteenth centuries, recently by an extreme school in England,
and at present in a less formidable fashion in our own country.
The results of these attacks in so far as they have meant the
abandoning of formal proofs have been futile.
(b) The Practical Side.-In the high school geometry has
long been taught because of its mind-training value only. This
exclusive attention to the disciplinary side may be fascinating
to mature minds, but in the case of young pupils it may lead to
a dull formalism which is unfortunate. On the other hand
those who are advocating only a nominal amount of formal
proof, devoting their time chiefly to industrial applications, are
even more at fault. The committee feels that a judicious fusion
of theoretical and applied work, a fusion dictated by common
sense and free from radicalism in either direction, is necessary.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  43


As to the nature of the applications, the committee feels that
there are several types o'f genuine problems, but that many of
the so-called real applications either are too technical to be
within the grasp of the young beginner, or represent methods
of procedure that would not be followed in real life. Moreover, it should be remembered that the very limited time devoted
to plane geometry (usally a single year) renders it impracticable to introduce many of the applications that might be desirable if the time were not so restricted.
(c) The Formal Side.-No reference to the applications of
geometry is to be construed to mean that the committee feels
that the formal side should suffer, or that geometry is wanting
in a distinct disciplinary value. A formal treatment of geometry, to about the traditional extent, is necessary purely as a
prerequisite to the study of more advanced mathematics, and
still more because such treatment has a genuine culture value,
for example, in assisting to form correct habits in the use of
English.
Certain writers on education have claimed that geometry has
no distinctive disciplinary value, or that the formal side is so
intangible that algebra and geometry should be fused into a
single subject (not merely-taught parallel to each other), which
subject should occupy a single year and be purely utilitarian.
These writers fail to recognize the fundamental significance of
mathematics in either its intellectual or its material bearing.
(d) Claims for Geometry.-Among the claims in behalf of
geometry the committee would emphasize the following:
Geometry is taught because of the pleasure it gives when
properly presented to the average mind.
Geometry is taught because of the profit it gives when
properly presented. For example:
(I) It is an exercise in logic, and in types of logic not generally met in other subjects of the school course, and yet types
which occur in geometry in unusually simple setting and which
are easily carried over into the actual affairs of life. Closely
connected with the logical element is the training in accurate
and precise thought and expression and the mental experience
and contact with exact truth.
This logic may be no more practical than literature or art




44


THE MATHEMATICS TEACHER.


or any other great branch of learning, but its general effect on
the human mind has been doubted by such a small number of
scholars as to render it worthy of the highest confidence.
(2) The study of geometry leads also to an appreciation of
-the dependence of one geometric magnitude upon another, in
other words to the tangible concept of functionality.
(3) The study of geometry cultivates space intuition and an
appreciation of and control over forms existing in the material
world, which can be secured from no other topic in the high
school curriculum.
(4) The value of the applications of geometry to mensuration.and the satisfaction derived by the pupil in verifying the formulas of mensuration already met by him in arithmetic are well
recognized by all teachers.
If we had to justify the position of any other subject in the
curriculum, history, rhetoric, geography, biology, etc., it is
doubtful whether equally specific and cogent reasons could be
found. If we were to dismiss geometry with a few practical
lessons, much more should we be compelled to. dismiss most
other subjects in the curriculum with the same treatment.
HISTORICAL NOTES.
Of the stimulating effect of occasional bits of historical information given by the teacher or the text-book there can be
no question. There is plenty of material to be found in the
well-known elementary histories of the subject.  The discoverers of particular propositions are known in a few cases,
and the general story of the subject, told informally as the pupil
proceeds in his study, adds a human interest that is valuable.
Portraits of famous mathematicians may be recommended for
-the schoolroom.
POINTS RELATING TO SOLID GEOMETRY.
(a) Axioms and Postulates.-The list of axioms already
given need not be increased, but the following postulates may be
added:
(I) One plane and only one can be passed through two intersecting straight lines.
Corollary. A plane is determined by three points not in the




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  45


same straight line, by a straight line and a point not in it, or by
two parallel lines.
This postulate, which is the analogue of the first postulate in
plane geometry, may also be given as a theorem for informal
proof.
(2) Two intersecting planes have at least two points in
common.
(3) A sphere may be described with any given point as center
and any given line-segment as radius.
It is tacitly assumed that the figures described in the course
in solid geometry exist and can be made the subject of investigation; e. g., the prism, pyramid, cylinder, cone, etc.
It may also be assumed, tacitly or explicitly, that the various
closed solids have definite areas and volumes; e. g., that a sphere
has a definite volume which is less than that of any circumscribed convex polyhedron and greater than that of any inscribed convex polyhedron.
(b) Definitions.-Latitude is left to the teacher in regard to
the use of such terms as prismatic space, cylindrical space,
nappes of a cone, and some of the names suggested for a
rectangular parallelopiped, which are convenient but not necessary in an elementary course.
After the analogy of the circle defined as a line, it is proper
that the sphere be defined as a surface but the more common
definition may be retained if desired.
(c) Purpose.-In solid geometry the utilitarian features play
an increasingly important part. The mensuration involved in
plane geometry is so simple as to be fairly well understood as
presented in arithmetic. Solid geometry, however, offers a
rather extended field for practical mensuration in connection
with algebraic formulas. The subject is therefore particularly
valuable for high school classes. A further application is found
in the power afforded to visualize solid forms from flat drawings, a power that is essential to the artisan and valuable to
everyone. The committee therefore summarizes the purposes
of solid geometry as follows:
(I) To emphasize and continue the values of plane geometry,
mentioned above;
(2) To present a reasonable range of applications to the field
of mensuration;




46


THE MATHEMATICS TEACHER.


(3) To cultivate the power of visualizing solid forms from
flat drawings, without entering the technical domain of descriptive geometry.
SECTION C. SPECIAL COURSES.
(a) Courses for Different Classes of Students.One of the topics which this committee undertook to consider
was that of different courses for various classes of students
in the high schools.
After investigation, it is the belief of the committee that there
should be no attempt to outline such courses. The syllabus as
recommended in Section E may be altered in special cases by
the omission of the theorems printed in small type and by increased emphasis upon theorems which admit of direct practical
applications.
The preceding recommendation, together with the possible
omission of solid geometry, would reduce the course to less than
half the traditional length. It seems probable that no greater
reduction would be desirable even for students in purely commercial courses, or indeed in any course in which formal geometry is a required subject.
(b) Preliminary Courses for Graded Schools.A portion of the report of this committee was to deal with
preliminary courses to be undertaken in graded schools.
Recommendations.-It is of the utmost importance that some
work in geometry be done in the graded schools. For this there
are at least two very strong reasons. In the first place, geometric forms certainly enter into the life of every child in the
grades. The subject matter of geometry is therefore particularly suitable for instruction in such schools.
Moreover, the motive for such teaching is direct. The ability
to control geometric forms is unquestionably a real need in
the life of every individual even as early as the graded school.
For those who cannot proceed further this need is pressing;
the direct motive involved compares very favorably with any
other direct motive for work in the grades. For those who
are going on to the high school, the development of the appreciation of geometric forms is almost an absolute prerequisite
for any future work in geometry.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.   47
Informal Work.-It is quite obvious that no work of formal,
logical character should be undertaken in the graded schools.
The earliest work in geometry will doubtless be so informal
that it will not constitute a separate course. Instruction in
drawing, in pattern making, and in elementary manual training
furnishes a basis for considerable geometric work even in the
first grades of the primary school.
Such work as this should be encouraged, though no special
outline of it can be given on account of its dependence upon
other courses. The constructions for erecting perpendiculars,
bisectors of angles, etc., can and should be given in connection
with such manual training work as making boxes, patterns, etc.,
though no technical nomenclature need be used. In such work
paper folding and the use of simple instruments should be encouraged, including the compasses, the ruler, and in later years
the protractor and squared paper.
Mensuration.-In connection with arithmetic much geometric
work may be taken up which is consistent with the child's
real interests and life. Measurement may be introduced very
early and the mensuration of simple forms such as the square,
rectangle, and triangle need not be long delayed. After this,
other geometrical forms and solids may be introduced under the
head of mensuration even earlier than is now customary. In
properly conducted schools, the students will become familiar
at the same time with such figures as the circle, cube, sphere,
etc., in manual training and in other elementary courses, such as
nature study, geography, etc.
In the later grades practically all of the simple geometric
forms will find their place in arithmetic under the head of mensuration, in drawing, and in manual training.
Work in the Higher Grades.-A special course in geometry
in the graded school is desirable, if at all, only in the last grade
or the last two grades. In such a course no work of demonstrative character should be undertaken, though work may be
done to convince the student of the truth of certain facts; for
example, by paper folding or cutting a variety of propositions
may be made evident, such as the sum of the angles of a triangle is I80~, etc.
The theorem just named is typical of the theorems which the




48


THE MATHEMATICS TEACHER.


student should know as facts before he leaves the graded school.
Many others of the theorems printed in black faced type in the
syllabus submitted herewith may be taught in this course without formal proof.
Theorems as Facts.-Emphasis should be laid upon the facts
with- which the student is already familiar through the work
described above. The course should be regarded partially as a
-classification and a systematization of the knowledge previously
acquired. Thus, simple geometrical forms should be brought
up in the connection in which they have arisen in the student's
past experience. In taking up constructions, explicit mention
should be made of the previous work in which a given construction occurred, and further practical instances of the use of such
constructions should be given.
Drawing to Scale.-Emphasis should also be laid on other
work of a concrete nature which involves direct use of geometric facts. Thus the propositions concerning the similarity
of triangles should be introduced by means of the drawing of
figures to scale. Attention should be called to the case in which
figures have been drawn to scale in the past. The usefulness
and the necessity of the operation should be emphasized, and
such applications as the drawing of house plans, the copying
of patterns on a smaller scale, etc., should be given. The use
of cross section paper for this purpose may be encouraged.
Finally after the notions involved are very clear indeed, and
after actual measurements have been made and reduced to scale
the precise facts regarding similar triangles may be given. This
should follow and not precede the work described above. The
applications to elementary surveying should, if possible, be
made in actual field work.
Models and Patterns.-The situation just described for similar triangles should be carried out so far as possible in other
instances. Thus the important theorems on the measurement
of angles can be illustrated in many ways. The Pythagorean
theorem, without formal proof, can be illustrated and made real
to every student by reference to the pattern forms in which it
occurs, the calculation of distances, and other real applications.
Finally the mensuration formulas can all be given. In the
latter, concrete illustrations should abound and verification by




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  49


means of models and measurements upon them    should be
encouraged.
Forms of Solid Geometry.-Contrary to the traditional procedure, the forms of solid geometry should be emphasized even
more than those of plane geometry, for they are more real and
more capable of concrete illustration.
Not only should formal demonstration be avoided but also
long lists of definitions which tend to confuse rather than enlighten. Definitions should be stated formally only after the
concept is clearly formed in the student's mind. No axiom
should be stated as such at any point, though frequent assumptions should be made without an attempt at proof. In all such
cases care should be taken that 'the statements made seem reasonable to the student and no forward step should be taken
until he is absolutely convinced of the truth of the statement.
Justification.-That such work is of vital value to the student
can scarcely be doubted; that it 'is absolutely legitimate will
probably be admitted by all interested in primary education. Its
value, its real direct motives, its contact with life, the legitimacy
of its subject matter exceed incomparably those of the traditional course in advanced arithmetic. At least, the course in
arithmetic may be vitalized by a liberal infusion of such geometric work.
If such a course is not given in the grades-perhaps even
though it is-a course of similar character but very much
shorter may be given in the high school before formal work in
demonstrative geometry is attempted. In any event it is desirable that the course in formai geometry should not proceed
in its traditional groove until the teacher is assured that the
ideas mentioned above are thoroughly familiar to the student.
SECTION D. EXERCISES AND PROBLEMS.
DISTRIBUTION, GRADING, AND NATURE OF EXERCISES.
(a) Increasing Number of Exercises.-There has been a
growing tendency in the last two decades to increase abnormally
the number of exercises to be considered by each pupil under
the following heads: (I) long lists of additional theorems
(beyond the full set usually given in the texts), (2) long lists
of problems of construction having at best remote connection




50


THE MATHEMATICS TEACHER.


with any uses of geometry within reach of the ordinary high
school pupil, (3) long lists of numerical exercises given in the
abstract, that is, unrelated to any concrete situation familiar to
the pupil or arousing his interest.
To give a single illustration of each:
(i) The squares of two chords drawn from the same point in a circle
have the same ratio as the projections of the chords on the diameter
drawn from the same point.
(2) To construct a triangle having given the perimeter, one angle
and the altitude from the vertex of the given angle.
(3) Through a point P in the side AB of a triangle ABC, a line is
drawn parallel to BC so as to divide the triangle into two equivalent
parts. Find the value of AP in terms of AB.
(b) The Distribution of Exercises.-It is recommended that
there should be treated in connection with each theorem such
immediate concrete questions and applications as are available,
and especially early in the course should such theorems be given
as easily lend themselves to this class of exercises.
For example, in a treatment in which the theorems on congruence of triangles are placed early, there is the opportunity
to bring in at once the simplest schemes for indirect measurement of heights and distances. Then later as similarity of triangles is taken up, there is the chance to recur to the same
problems and let the pupil see how the principle adds power and
facility in making indirect measurements. There is thus a progressive development in the facility for solving concrete problems along with the theory.
This principle can be carried out in many different lines. For
example, in connection with triangles, circles, and squares, there
are many applications immediately available and easily found in
tile patterns, window tracery, grill work, steel ceiling patterns,
etc. These afford fine exercises in construction early in the
course, and are equally available later in the computation and
comparison of areas. When such exercises are given they
should be distributed as far as possible in connection with the
theorems used in the construction and comparison of the figures
involved.
However, only the simplest uses of the theorems can be shown
in the immediate connection, both because of the space occupied




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  5 I


by them and the danger of interrupting the continuity of the
theorems by too many exercises thrown in between them, and
also because most of these applications make use of various different theorems, and hence must come after certain groups of
theorems, thus making necessary occasional lists of problems
and applications scattered through the various books, as well
as sets of review exercises at the end of each book.
The whole question of distribution is thus to be determined
by the relation of the problems and applications to the single
theorems or groups of theorems to which they belong. The
important question of emphasis in Section E of this report is
best brought out by the grouping of many exercises around the
basal theorems.
On the basis of distribution we have all extremes in the
various texts, including: (I) The purely logical presentation,
that is, the continuous chain of theorems with practically no
applications in concrete setting in connection with them and
almost none at the end of the books; (2) the same as the foregoing, except that the long sets of exercises are placed at the
end of each book, where they loom up before the pupil as great
tasks to be ground through, if, indeed, they are not omitted
altogether; (3) the psychological presentation in which the
more difficult exercises are either postponed to a later part of
the course or are omitted altogether, and the easier ones are
brought into more immediate connection with the theorems to
which they are related.
The time and space made available by the third method of
presentation provides an opportunity for the pupil to gain some
acquaintance with the uses of the theorems as he proceeds and
to become genuinely interested in the development of the subject. The committee strongly recommends this latter method
of presentation. 'In expressing its disapproval of method (2),
it is not to be understood that the committee objects to any
text-book because it offers a large number of exercises, placed
at the end of each book, from which the teacher is to make a
selection. The objection to (2) should be clear from reading
(3), which the committee approves.
(c) The Grading of Exercises.-Too much cannot be said in
favor of a large number of simple cases rather than too many




52


THE MATHEMATICS TEACHER.


difficult questions, especially early in the course, but also even
throughout the secondary course in geometry.
The average high school pupil is not likely to become adept at
proving difficult and abstruse theorems independently or in
solving complicated problems. On the other hand, the rank
and file are bound to become discouraged and hopelessly lost in
the so-called "originals," unless the grading is carefully done,
and steps of difficulty are kept down to a very reasonable lower
limit.
The ideal treatment would seem to be: (I) To make a proposition appeal to the pupil as reasonable by simple illustrations,
after which should follow the deductive proof; (2) to apply the
theorem to more difficult situations, involving problems which
the pupil regards as interesting and worth while. It is recognized that this ideal cannot be attained with reference to all the
theorems of geometry but it is believed that it can be attained
in very many cases; and, wherever this is possible, great interest
and incentive are given to the pupil.
As a matter of fact, familiarity with the elementary truths
pertaining to angles, parallelograms, and circles, when consistently tried out and seasoned by applications to numerous comparatively simple and interesting geometric forms suggested by
figures which abound in concrete setting on every hand within
reach of all, is usually of more value to the average pupil (and
even to the better pupils) than is the study of a larger number
of abstract theorems or problems through which they are often
forced. Nevertheless, for the benefit of the brighter pupils, it
is desirable that a few comparatively difficult problems be given,
especially at the ends of the various books, or in a supplementary list.
(d) The Nature of Exercises.-This topic has been referred
to under (a), (b), and (c). It is recognized that a fair proportion of the traditional exercises given in abstract setting should
find a place in a course in geometry, but the committee believes
that, in accordance with the common practice of the past twenty
years, this class of exercises has been magnified and extended,
especially with reference to the more difficult exercises, beyond
the interest and appreciation of the average pupil.
The committee therefore recommends that a judicious selec



FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  53


tion of a reasonable number of abstract originals be made in
order to leave time for an equally reasonable number of problems, particularly those with local coloring, stated in concrete
setting.
Since ample lists of abstract originals are within easy reach
of all teachers of geometry, it seems unnecessary to supply
illustrations of such exercises. But as the teacher must generally depend upon his own initiative to supply problems in
concrete setting, it seems desirable to indicate a few sources
from which such problems may be obtained. The committee
believes that a reasonable number of problems of this character
creates an interest in the minds of the pupils that reacts strongly
in augmenting his understanding and appreciation of the logical
side of the subject. But it is not to be understood that the
committee regards these problems as practical in the narrow
sense of the word.
SOURCES OF PROBLEMS.
(a) Architecture, Decoration, and Design.-Industrial design
and architectural ornament ard replete with details that may be
used as a source of supply for geometry problems. These
problems are of three kinds: (I) The problems involved in the
construction of the figures themselves; (2) the demonstrations
necessary to establish numerous relations which are visible to
the mathematician and which must occasionally be assumed by
designers; (3) problems in computation.
Among the industrial products that involve geometric ornament are tile and mosaic floors, parquetry, linoleum, oilcloth,
steel ceilings, ornamental iron, leaded glass, cut glass, and the
like. Figures for problems from these sources may be made
from the cuts in trade catalogues.
Problems based on architectural ornament are largely from
details of Gothic tracery and can be obtained only by a study
of the buildings themselves or of the photographs of them that
may be seen in architectural libraries. Gothic tracery is found
in windows, in ornamental iron, in carved stone and wood on
the outside and inside of buildings, on furniture, choir screens,
rafters, and the like, that abound in medieval cathedrals and
churches and in their modern imitations.




54


THE MATHEMATICS TEACHER.


These problems have distinct advantages. In many cases
their comprehension'and solution require no technical knowledge beyond the elementary mathematics needed. These designs abound, are largely within the reach of pupils, and their
use in the class room brings before pupils as nothing else can
the beauty and widespread application of geometric forms. In
them may be found applications of many topics of elementary
mathematics and from them may be obtained numerous exercises of all grades, from the simplest to the most complex. By
their use it is possible, therefore, to introduce anywhere in the
work problems that are within the reach of the average pupil
and appeal to him with a minimum of experiment, explanation,
discussion, or previous special preparation.
(b) Problems of Indirect Measurement.-It should not be
considered that the types of applications under (a) are relatively of greater importance than numerous others. Any application that adds interest to the study of rigorous geometry is
of value. Of special interest are all simple means of effecting
indirect measurements of distances, such, for instance, as the
numerous applications of the congruence theorems and the
theorems on similarity of triangles. Here the teacher will find
much assistance in Principal Stark's "Measuring Instruments
of Long Ago." Again in considering the isosceles triangle, the
universal leveling instrument (aside from the spirit level) offers
a number of applications. The form is that of an isosceles
triangle bisected by a line from the vertex.
Many simple and interesting problems in indirect measurement are made available by the introduction of the trigonometric
ratios, sine, cosine, and tangent. This can be done as soon as
the theorems on similar triangles are known, and 'the computation by measurement of a two place table of natural functions
at intervals of 5~ affords, of itself, a fine drill in the application
of these theorems, at the same time providing material for solving concrete problems of great interest to young pupils.
It may not be possible to find time for this in connection with
the usual course in plane geometry. Schools that devote most
of their time and effort to preparing pupils to pass entrance
examinations for college would certainly find it difficult to




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.


55


meet any added requirement. In view, however, of the omissions suggested by this committee and the readjustment of emphasis on basal theorems, time may be found, as the experience
of an increasing number of teachers has shown. Where this
can be done it constitutes an important step in the closer correlation of the subjects in elementary mathematics. In any case
only the natural functions should be used, and the applications
should -be limited to those involving right triangles.
(c) Other Sources.-Problems may also be obtained from
physics, mechanics, and other sciences, from engineers' and
builders' manuals and works on carpentry and masonry, as, for
instance, problems derived from various common forms of
trusses and the construction of arches. But there is a danger
in connection with problems from these sources, that aside from
the geometry involved, they may contain technical terms and
mechanical features unknown to the average pupil and not easily
understood without more explanation and consequent distraction from the geometry itself than is warranted in the ordinary course. Problems of this class should be carefully tried
out and sifted before being adopted for use.
(d) Illustrative Problems.-There are given below a few
typical problems which are suggested for the purpose of making
clear what the committee has in mind. Through simple problems of these types and many others which might be suggested
much interest can be imparted to the study of demonstrative
geometry, even though the problems be not practical in the strict
sense of the word. The danger of using too many problems in
any narrow field is, however, apparent.
(I) The theorem regarding the angle sum in a triangle has a large
number of applications. For example, to measure PC, stand at some
convenient point A and sight along APC and (by
c^a6     the help of an equilateral triangle cut from pasteboard) along AB. Then walk along AB until a
- AH^ N point B is reached from which BC makes with
BA an angle of the equilateral triangle (60~).
A Z-   ------ ^oLL   Then AC==AB, and since AP can be measured
we can find PC. This is an example of a problem that adds interest to
the work without being itself a practical application that would be used
by a surveyor.




THE MATHEMATICS TEACHER.


(2) A problem of the same nature is the following: To measure AC,
first measure the angle CAX, either in degrees with
a protractor or by sighting across a piece of paper
and marking it down. Then walk along XA pro[ x   ~    ~ ^duced until a point B is reached, from which BC
makes with BA an angle equal to half of angle
CAX. Then it is easily shown that AB-A=C.
(3) The sailor makes use of this principle when he " doubles the angle
on the bow" to find his distance from a
\ L              lighthouse or promontory. If he is sailing on the course ABC and he notes a
lighthouse L when he is at A, and takes
the angle A, and if he notices when the
C         B         A   angle that the lighthouse makes with his
course is just twice the angle noted at A, then BL  AB. He has AB
from his log, so he knows the distance BL.
(4) To measure the line XY, when the observer is at A, we may
measure any line AB along the stream.
xf^            o       Then the observer may take a carpenter's
square, or even a large book, and walk.13p^^ \s_ p -    along AB until a point P is reached from
which X and B can be seen along two
sides of the square. Similarly the point
rPZ       xt'      Q may be fixed. Then by walking along
YM to a point Y' that is exactly in line with M and Y and also with
P and X, the point Y' is fixed. Similarly X' is fixed. Then X'Y' = XY.
(5) A field containing 9 acres is represented by a triangular plan
whose sides are 12 in., I7 in., and 25 in. On what scale is the plan drawn?
Conant.
(6) Assuming the earth to be a sphere of which the radius is 3,960
miles, find the length of one degree of longitude at 60~ north latitude,
and compare its length with that of one degree of longitude at the
equator.


K




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  57


(7) ABCD is a square. Equal distances AE, BF, CG and DH are
measured off on the sides AB, BC, CD and DA respectively. If the lines
AF, BG, CH and DE are drawn intersecting at Y, Z, W and X, prove
that XYZW    is a square. If A4B=a and 4E is A of AB, prove that
AF==a/3VIO, AX=-a/IoVo, XY=a/5VIo, FY=a/3oVIo; and that
the area of XYZW   is 2a2/5.
This figure is the basis of an Arabic design used for parquet floors.
The solution involves both algebraic and geometric work in concrete
setting.
'0
E   /       F
A     x      D           B
(8) ABC is an equilateral arch, and CD its altitude. A is the center
of the arc BC and B the center of the arc AC. The equilateral arches
AED and DFB are erected on AD and BD respectively. D is the center
of arc AE and FB and A and B are centers of arcs ED and DF, each
drawn with 2AB as radius. What is the locus of centers of circles
tangent to CA and CB? To ED and DF? To AC and DF? To CB and
ED?   Construct a circle tangent to the arcs AC, CB, ED and FD.
This figure is the basis of a common Gothic window design. The
solution involves the intersection of loci.
(9) CD is the perpendicular bisector of
AB.   Equal distances AX   and BY are
Ejc     ^Ts c -measured off on AD and BD respectively.
EF is perpendicular to CD at C. Circles
fB  \ \,.4 z   \ fare drawn with X and Y as centers and
iD - -- ------     AX and BY as radii. Construct a circle
\    |\              tangent to EF at C and to circle X.
X\~ t\ IProve that this circle is also tangent to
the circle Y.
If CD is less than ~AB the part of
'~              this figure between lines CD and AB is
one form of a three centered arch.




58


THE MATHEMATICS TEACHER.


(IO) In the drawing above, which is the basis of the mosaic floor
design to the right, the circle with center N is inscribed in one of the
squares whose side is SH. The arcs ORQ and OKL are drawn with
the vertices of the square as centers and half the side as radius. The
semicircles LMN and NPQ are drawn on LN and NQ as diameters.
Find the areas of the various figures bounded by circular arcs within
this square. Note the symmetry of the whole figure within the square
ABCD.
C
H          A          D s 
(II) ABC is an equilateral triangle. A and B are centers of arcs BC
and AC respectively. CD is the altitude of triangle ABC. Arcs DF and
DE, constructed with radii equal to AB, are tangent to CD at D and
intersect AC and CB respectively at E and F. Construct a circle tangent to the arcs DE, DF, AC and BC.
Suppose the problem solved. Let O be the center of the circle. Connect O with B, the center of arc AC, and with H, the center of arc DE.
From triangles ODB and OHD the following equation is derived:
(s-r)-2 - (S/2)-2= (s + r)2  S2,


where s is the length of AB and r is. radius of the required circle.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  59


This figure is the basis of a church. window design. Many problems
of this type may be easily obtained.
(12) A quarter mile running track has two parallel sides and semicircular ends. Each straight away section is equal in length to one of
the ends. If the track measures exactly one-fourth of a mile at the curb,
or inner edge, how much distance does a runner lose in running two feet
from the curb? Six feet? What is the area of the track if it is I5 feet
wide? What is the area of the enclosed field? What are the dimensions of a rectangular field sufficiently large to contain such a track?
What will it cost at $2.00 per cubic yard to cover such a track with
cinders to a depth of 2 inches? Pettee.
(e) References to Sources of Problems.-In connection with
the recent search for real applied problems in elementary mathematics numerous bibliographies have been compiled to which
reference is here made, as well as to a few other books, aside
from current texts, which may be helpful to teachers. Concrete problems should be selected carefully and used wisely.
Those which may appear to one class of pupils as real applied
problems may seem highly. abstract to another class. Probably
few problems in the following lists would appear real to all
pupils and yet all are likely to find increased interest in any
problem which has a concrete origin.
Printed bibliographies.I. A list of 38 titles of books and 2I titles of trade journals, School Science and Mathematics, Vol. IX., No. 8, I9o9, pages 788-798.
2. A more extended list of books on the whole range of applied problems, School Science and Mathematics, Vol. VIII., No. 8, November,
I908, pages 641-644.
From this list Saxelby, Godfrey and Siddons, and Perry may be
mentioned especially.
3. A comprehensive list of books and journals relating to the uses of
geometry in architecture, decoration, and design in a forthcoming
volume entitled "A Source Book of Problems for Geometry," by
Mabel Sykes. Allyn and Bacon, Boston, 1912.
4. A vast bibliography of suggestive titles, with a classification and discussion of some phases of industrial problems by a committee of the
National Education Association on "The Place of Industries in
Public Education." Proceedings of the Association, I910, pages
652-788.
Collections of problems.I. Real Problems in Geometry, Teachers College Record, March, I909.
A classification and discussion of types of applied problems, by
James F. Millis.




6o


THE MATHEMATICS TEACHER.


2. Real Applied Problems in Algebra and Geometry, School Science and
Mathematics. A collection begun in I909 by a committee of the
Central Association of Science and Mathematics Teachers. The
work is still in progress. The problems collected up to November,
1909, have been classified and published in pamphlet form.
Other selected titles.I. "Lessons in Experimental Geometry," Hall and Stevens. The Macmillan Company, New York, 1905.
2. "Numerical Problems in Geometry," J. G. Estill. Longmans, Green
and Company, New York, 1908.
3. "Mensuration," G. B. Halsted. Ginn and Company, Boston, I9I0.
4. " Elementary Mensuration," F. H. Stevens. The Macmillan Company,
New York, 1908.
5. "A Notebook of Experimental Mathematics," Godfrey and Bell.
London, Edward Arnold, I905.
6. "Elements of Mechanics," M. -Merriman. John Wiley and Sons,
New York, I905.
7. "Shop Problems in Mathematics," Breckenridge, Mersereau and
Moore. Ginn and Company, New York, I9Io.
8. "Pocket Companion containing Tables," etc. Carnegie Steel Company, Pittsburgh, Pa., 1903.
9. "Leitfaden der Geometrie," Jahne and Barbisch. Vienna, 1907.
io. "Raumlehre fur Mittelschulen," Martin and Schmidt. Berlin, 1898.
II. "Geometrie fur die Zwecke des practischen Lebens," G. Ehrig.
Leipzig, 1906.
12. "Mathematische Aufgaben," Schulze and Pahl. Leipzig, 1908.
I3. "Cours abregee de Geometrie," Bourlet and Baudoin. Paris, 1907.
I4. "Cahiers d' execution de dessins geometriques," M. P. Baudoin. Paris.
I5. "Geometria Intuitiva," P. Pasquali. Milan.
I6. "Regole di Geometria Pratica," F. Aandreotti. Florence, 1897.
I7. "The Power of Form    Applied to Geometrical Tracery," R. W.
Billings. London, I851.
I8. "Gothic Architecture in England," Francis Bond. London, B. T.
Botsford, I905.
I9. "Les elements de l'art Arabe," Jules Bourgoin. Paris, 1879.
20. "Pattern Design," Lewis F. Day. London, B. T. Botsford; New
York, Scribner's Sons, 1903.
21. "Geometrische Ornamentik," L. Diefenbach. Berlin, Max Spielmeyer.
22. "Romano-British Mosaic Pavements," Thomas Morgan. London,
I886.
23. "Decorated Windows, A Series of Illustrations," Edmund Sharpe.
London, 1849.
24. "Specimens of Tile Pavements," Henry Shaw. London, I858.
25. "Specimens of Geometrical Mosaics of the Middle Ages." Sir
Matthew Wyatt. London, 1848.
26. "The Teaching of Geometry," David Eugene Smith. Boston, Ginn
& Co., 1911.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  61


PROBLEMS INVOLVING LOCi.
(a) Phraseology.-While the committee does not wish to prescribe the exact phraseology of any definition, it would recommend greater care in the formation of the definitions underlying the subject of loci. It is suggested that any definition
used should be substantially equivalent to the following:
The locus of a point (or the locus of points) satisfying given conditions is a configuration such that:
(I) All points lying on the configuration satisfy the conditions;
(2) All points satisfying the conditions lie on the configuration.
It would seem desirable to make all proofs on loci conform
to this definition. It is of course understood that the teacher
will lead the pupil up to such a definition through varied forms
of concrete description, such as "path of a point in motion," etc.
(b) Motion in Geometry.-It seems well to give some consideration to the place of motion in a well-rounded course in
elementary geometry, and to bear in mind that this course is all
the geometry to be studied by the majority of high school pupils.
It has recently been urged by prominent European mathematicians that motion should be given a more prominent place at
this stage. We may well recall that the space concepts dealt
with in our usual courses in geometry are almost entirely to be
described as static. There is in theorems and problems on loci
a dynamic element that is of importance. The pupil is pretty
familiar with motion as a concrete experience, and it seems of
first-class importance to idealize some such concrete experiences, until they possess the precision of geometry.
For example, in a given plane we may consider in a way well
described as a static configuration the perpendicular bisector of
a line-segment joining two points; but when we consider this
line as generated by a point moving in the plane in such a way
that it is always equidistant from the two given points, we add
a dynamic element.
As to phraseology, the expression "locus of points" suggests
a static configuration, while the expression "locus of a point"
emphasizes the dynamic element, and is equivalent in thought
to the "path of a point moving with certain prescribed conditions." Both of these phases should have a place in the treatment of loci problems, and thus both forms of expression should




62


THE MATHEMATICS TEACHER.


be used, the one or the other being more suggestive in different
cases. It is even desirable to use different forms in describing
a given case to make clear the idea and to cultivate facility
in expression.
(c) Concrete Nature of Loci.-Contrary to the usual conception, the locus idea is one that may very easily be made concrete
and brought down to the comprehension of young pupils. For
example, the opening of a book or of a door suggests a variety
of loci. The same may be said of many concrete illustrations
easily accessible to the pupil.
In this way, loci problems may and should be introduced at
certain stages of the subject. For example, in Book I: The
locus of a point equidistant from two fixed points, equidistant
from two intersecting lines, or from two parallel lines, or at a
given distance from a fixed line. In the book on circles, the
locus of all points equidistant from a fixed point, the locus of
the centers of circles of fixed radius and tangent to a given line,
the locus of the centers of all circles tangent to two parallel
lines or two intersecting lines, and the locus of the vertices of
all triangles having a common base and equal vertex angles.
In solid geometry, the locus of points equidistant from a given
point, from two given points, from a given plane, from two
intersecting planes, from two parallel planes, etc.
(d) Loci in Problems of Construction.-Important features
of the construction problems in geometry are dependent upon
loci considerations which should be emphasized in this connection. For example:
(I) To find the locus in the plane of all points equidistant
from three given points, it is necessary to determine the intersection of two loci both of which are straight lines.
(2) To find the locus in the plane of all points equidistant
from a fixed point and at a given distance from a given line, it
is necessary to find the intersection of two loci one of which is
a straight line and the other a circle.
The discussion of the various possibilities in connection with
such problems is one of the most valuable exercises for the pupil.
For example, as to whether there are one, two, or no points fulfilling the conditions in the second example above. While it
may be possible to solve and discuss such problems without




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  63


using the term locus at all, yet this leads to roundabout and
awkward explanations while the language of loci is elegant and
concise.
Moreover, facility in the use of this language is not only
desirable from the standpoint of the high school pupil but is of
the utmost importance for those who may continue the study
of geometry in college.
(e) To summarize, the locus idea is deserving of a careful
and systematic treatment for the following reasons:
(I) It introduces a dynamic element through the consideration of the idea of motion.
(2) It presents an elegant language for the statement of those
propositions on which nearly all of our problems of construction
are based.
(3) It aids greatly in the cultivation of space intuition and in
emphasizing the important concept of functionality.
(f) Additional Illustrations Appropriate for Use.i. Find the locus of all points at a fixed distance from the sides of a
triangle, always measuring from the nearest point of a side.
2. Find the locus of points such that the sum of the squares of the distances from two lines intersecting at right angles is Ioo.
3. Find the locus of the vertices of a regular polygon of a given number of sides that can be circumscribed about a given circle.
4. Find the locus of the midpoints of the sides of regular polygons
of a given number of sides that can be inscribed in a given circle.
5. Find the locus of all points from which a given line-segment subtends a given angle.
6. Find the locus of a point the sum of the squares of whose distances
from two given points is constant.
7. Find the locus of a point the difference of the squares of whose
distances from two given points is constant.
8. Find the locus of all lines drawn through a given point, parallel to
a given plane.
9. Find the locus of a point in space equidistant from three given
points not in a straight line.
ALGEBRAIC METHODS IN GEOMETRY.
The committee feels that the use of algebraic forms of expression and solution in the geometry courses may well be
extended, with advantage to both algebra and geometry, and
that this may be done without in any way encroaching upon




64


THE MATHEMATICS TEACHER.


the field of analytic geometry, which belongs to a later stage of
development.
(a) The Notation Should be More Algebraic.-While it is not
feasible or desirable to lay down hard and fast rules to standardize the notation of geometry, an examination of current texts
makes it evident that some notations in common use are unnecessarily awkward when compared with the notations used in
elementary algebra. The notation of geometry is, in general,
improved by much use of lower-case letters to represent numerical values, leaving capitals to represent points. This notation
is here called algebraic because the student will recognize the
relations of equality and inequality much more readily in the
familiar notation of algebra than if these relations are presented
in a notation not used in algebra.
(b) Algebraic Statement of Propositions.-Many of the
theorems of geometry may be stated to advantage in algebraic
form, thus giving definiteness and perspicuity and especially
emphasizing the notion of functionality. This mode of expression can be made of much value to the student if he is required
to translate into English all the symbols involved.
The following are illustrations of the algebraic statement of
propositions:
(I) In any triangle, a= bh/2, where a is the area, b is the base and
h is the altitude.
(2) In a right triangle, c2 = a'+ b2 where c is the hypothenuse and
a and b are the sides including the right angle.
(3) In any triangle, c2= a2+ + b -+ 2ap, where a, b, c are sides of the
triangle and p is the projection of b on a.
(4) For any secant and tangent drawn from a point to a circle, we have
t2 = sx, where t is the length of the tangent, s is the length of the secant
and x is the length of the external part.
It is not intended to convey the impression that the usual
statement of propositions should be replaced by the algebraic
statements but rather that the student should be required to
translate the one form of statement into the other. The algebraic statements are often superior to the usual statements in
point of brevity and conciseness. Moreover, the algebraic statement prepares for the idea of functionality which is too little
understood by persons who are not trained in mathematics beyond the high school course. That is to say, some appreciation




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  65


of the influence of changing one part of a configuration on other
parts of the configuration can often be gained readily from the
algebraic statement.
(c) Geometrical Construction of Formulas.-Some propositions can be proved simply and elegantly by methods involving
algebra. It is somewhat usual in text-books on geometry to
give a proof of the geometrical statement of such an algebraic
formula as (a + b)2= a2 +b2b-+2ab, where a and b are the
numerical measures of the line segments, but to neglect the geometrical construction of the formula. The latter seems to be
the point of greatest importance. It is not additional evidence
of the validity of the theorem that is sought. That is established in algebra. What is of first-rate importance is to give a
geometrical picture of the formula, thus showing a certain geometrical interpretation and to have the student put the result
into geometrical phraseology when a and b are line-segments.
The construction of line-segments a + b, a - b, and of areas
ab, (a + b)2, (a-b)2, where a and b are line-segments, should
come early in the course. Later, when the requisite theorems
are being developed, the further elementary expressions
a ab, Vlab, Va2 + b2, Va2-b2, aV k,
ka, k' c
where a, b, and c are line-segments and k is a positive integer,
should be constructed.
This interdependence of algebra and geometry is a matter of
no small importance both historically and for subsequent mathematical work. It should be brought out by suitable exercises
that the use of algebra often enables one to establish relations
from which a geometrical construction can be made readily or
to show the nature of a difficulty involved.
For example, to inscribe a square in a semicircle:
If x represents the side of the square and r the radius of the
circle, we have at once from a right triangle that r2 = x2 -+ n2/4
and hence x=    2/,~5g r, which can be constructed from exercises given above.
(d) Geometric Exercises for Algebraic Solution.-Some
exercises for algebraic solution, such as are found in many recent texts, should find a place in any course in geometry. For




66


THE MATHEMATICS TEACHER.


example, the following is a suitable exercise after the proposition stating that a=  bh, where a is the area, b and h are sides
of a rectangle:
The area of a rectangle is 480 square inches. Each side of the rectangle is increased I inch and, by this change, the area is increased 45
square inches. Find the sides of the rectangle.
Similarly, after the proposition pertaining to secants and tangents to a circle, the following is suitable:
A secant line which passes through the center of a circle of radius IO
is intersected by a tangent of length 15. Find the length of the external
part of the secant.
Such exercises do much to unify geometry and algebra, and
may well replace some of the usual exercises.
Finally, after the theorem on the volume of a frustum of a
pyramid, a problem like the following has value as an algebraic
exercise, although it is in no sense a real applied problem.
A pier is built of solid concrete construction, in the form of a frustum
of a pyramid with square bases. The altitude is twice an edge of the
lower base and the area of the lower base is four times that of the upper
base. Find the dimensions of each base if the pier contains 600 cubic
feet of solid concrete.
SECTION E. SYLLABUS OF GEOMETRY.
PREFACE TO LISTS OF THEOREMS.
(I) Lists not Exhaustive.-The lists of theorems which follow
are not to be taken as exhaustive, and it is distinctly understood
that theorems may be added at the discretion of the teacher.
For example, the theorem on the existence of regular polyhedra
may find a place in certain courses. Some theorems are omitted
only with the understanding that they may be inserted as exercises for the student; some such possible exercises are:
In any triangle, the product of any two sides is equal to the product
of the segments of the third side formed by the bisector of the opposite
angle, plus the square of the bisector.
The medians of a triangle meet in one point which divides each median
in the ratio I:2.
To divide a given straight line-segment in extreme and mean ratio.
To find the area of a triangle in terms of its sides.
To construct a square having a given ratio to a given square.
The surface of a sphere is equivalent to the area of four great circles.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  67


(2) Logical Order.-Although there is some indication of a
possible logical order in the lists, there is no intention of specifying any definite order. It would be impossible to carry out
as a whole precisely the order stated below.
In several connections the words "corollary to " or " synonymous to " may seem to imply an order. These phrases are used
only to indicate the reason for putting the theorem quoted in the
group in which it appears. Thus in Group II., on page 71, it
would not be clear in every case that each theorem is a corollary
of plane geometry without such a suggestion of possible derivation.
It should be noticed that some logical arrangements would
necessitate the insertion of the theorems omitted in this list.
Such an insertion is entirely in the spirit of this report, as is
also any conceivable change in the order, except where specified
explicitly in the report.
(3) Subsidiary Theorems.-A number of theorems omitted
in the lists below may well be given as ordinary statements in
the course of the text as corollaries, or as remarks, without the
emphasis which attaches to formal theorems. Among such general statements which should by all means be made at the proper
points are the following:
No triangle can have more than one right angle or more than one
obtuse angle.
The third angle of a triangle can be found if two are known.
An equilateral triangle is equiangular.
The square on a side of a right triangle adjacent to the right angle is
equal to the square on the hypotenuse minus the square on the other side.
Through three points not in a straight line not more than one plane
can be passed.
The areas of two spheres are to each other as the squares of. their
radii; their volumes as the cubes of their radii (like statements for other
solids).
The number of such statements is exceedingly large and all
of them could not be given in any syllabus. A large majority
are at the present time stated, if at all, in the course of the
reading matter, or in exercises, and not as explicit theorems.
It is understood, and indeed expected, that these statements, together with many which are omitted from the lists of theorems
below, should be treated in this manner.




68


THE MATHEMATICS TEACHER.


(4) Informal Proofs.-The theorems given below under the
heading: "Theorems for informal proofs," should be stated at
the proper points in the text and in theorem form, or as postulates. Their proofs, however, can well be omitted where this
omission is suggested, or be made exceedingly informal by the
insertion of a single phrase which will give the proper suggestion for the proof. Many other theorems which are equally
obvious are not stated because they occur more naturally as
corollaries or as exercises. (See the preceding paragraph.)
Regarding the method of proof in general, while the demonstrations should remain as logical as they are at present, it is
suggested that the formalities of logic, as such, be frequently
dispensed with to a very considerable extent and that the propositions be frequently stated and proved in language resembling
that to be found in any other mathematical text-book. This is,
indeed, the style of many classical treatises, such as Legendre's
or Euclid's. It is certainly satisfactory and there is no reason
why the proof should not remain quite as logical when the older
style is followed.
The symbolic form of demonstration which appears in many
texts should be regarded simply as a shorthand expression of a
complete proof in ordinary English phraseology. The latter
should be given by the student in all cases. The ability to pass
from the symbolic form to ordinary English, that is, to translate
the shorthand into the language of everyday life, should be
constantly tested by the teacher, for the same reason that the
formulas of algebra derive their real meaning and power from
the thought content which the student can attach to them.
(5) Arrangement for Emlphasis.-The main list of theorems
is divided into several heads, each group being introduced by a
theorem of suitable importance upon which the rest of the
theorems in that group depend more or less closely. This
arrangement has been selected in order to emphasize the importance of a few major propositions, namely, those which
carry a maximum of applications and from which the rest can
be derived, thus serving as a nucleus for the whole of geometry.
This effort to gain emphasis has been carried out still further
by printing the theorems in different grades of type so that
those of fundamental importance and of basal character are




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  69


printed in black face type; those of considerable importance
which are secondary only to the preceding ones are printed in
italics. A number of other theorems are printed in Roman
type, while the least important are printed in small type. The
latter (small type theorems) may be omitted without serious
danger, or they may be used as corollaries or exercises instead
of receiving the emphasis which attaches to a theorem; in fact,
probably no injury would result from a similar treatment of
many of the theorems stated in Roman type.
The distinction in emphasis is desirable not only for guidance
in omitting theorems in courses which are necessarily abbreviated, but it is also of the highest importance in courses in
which all of the theorems are given. An orderly classification
of theorems in the student's mind, a notion of the'dependence
of the minor theorems on the more basal ones and an appreciation of their relative importance is of the utmost direct value
to the student and furnishes him with the only possible means
of permanently retaining geometrical knowledge in usable form.
The direct value mentioned arises both from the power acquired and also from the essential grasp of the subject, which
is the purpose of education. It is a fundamental characteristic
of the mind from which there is no escape that any clear impression of a vast field must have exactly such distinctions in
emphasis as are outlined here for geometry. These statements
and this arrangement are intended to be of assistance to the
teacher in guiding him as to the emphasis to be laid upon
theorems during the course and especially at the completion of
a given book or chapter.
(6) Trigonometric Ratios.-Attention is called to the paragraphs under XIII., 2-4, on the computation of two-place tables
of sines, cosines, and tangents from actual measurements, provided the pressure of time due to examining bodies is not too
great. This work can be done with about the same amount of
effort that is expended by the student on the ordinary geometrical theorems of the same class. Its importance is due to the
fact that such a small table will really present the fundamental
ideas of trigonometry and will enable the student to solve right
triangles in the trigonometric sense.




70


THE MATHEMATICS TEACHER.


(7) Abbreviations.-In a large number of instances theorems
are stated in condensed or abbreviated form and the statement
of a number of theorems is often combined into one. This is
done only for the purpose of reducing the length of this report.
It is to be understood that such abbreviated statements are made
only for the teacher and should not be presented to the student
in this form. In particular, it is probably preferable to use
words instead of letters in statements for high school pupils of
such theorems as those in II., I-4.
The numbers which follow each of the theorems are references to a syllabus prepared by a committee of the Association
of Mathematics Teachers of New England, I906.
(8) Omissions.-Since the quantities which appear in geometry are often treated by means of their numerical measures,
the introduction of any useful algebraic facts is completely
justifiable; in particular, the algebraic theory of proportion may
well be employed; but such algebraic material is not included in
this syllabus since the syllabus deals primarily with geometric
topics.
The following list shows the omissions from the New England Syllabus:
Plane geometry omissions: C2, GI2 (2nd part), G20, JI3, L2,
N6 (2nd part), NI3, PI, P2 (see note to XV., I), P6, PI2, T2,
T4, T5.
Solid geometry omissions: E7, E8, E9b, FI2, FI3, FI4, H2
(but see IV., 7), K2, K3 (see note to I., 2) M6, M7, M8, M9,
Q3, Q4, Q8b, Q9 (see VI., note), RI, R2 (see note to VI., 13),
R9, Rio, RI3, RI4 (see note to VI., 22), S3 (see preface,
3), S7.
THEOREMS OF PLANE GEOMETRY.
I. Theorems for Informal Proof.
(The following theorems may be stated as assumptions, or
may be given such informal proof as the circumstances may
demand.)
I. All straight angles are equal.* [*]
*Reference numbers are to the New England Syllabus. Where an
asterisk [*] replaces the reference number, the theorem is not contained
in that syllabus.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  71


2. All right angles are equal. [*]
3. The sum of two adjacent angles whose exterior sides lie in the same
straight line equals a straight angle. [Ji.]
4. If the sum of two adjacent angles equals a straight angle
their exterior sides form a straight line. [J2.]
5. Only one perpendicular can be erected from a given point
in a given line. [G3.]
6. The length of a circle (circumference) lies between the lengths
of perimeters of the inscribed and circumscribed convex polygons.
[PI3.]
(It is recommended that this statement be used as a definition to be
inserted at context.)
7. The area of a circle lies between the areas of inscribed and circumscribed convex polygons. [Pi4.]
(It is recommended that this statement be used as a definition to be
inserted at context.)
8. Two lines parallel to the same line are parallel to each
other. [*]
9. Vertical angles are equal. [J3.]
(Very informal proof sufficient.)
O.. Complements of equal angles are equal.     [*]
II. Supplements of equal angles are equal. [*]
12. The bisectors of vertical angles lie in a straight line.
[j4]
I3. Any side of a triangle is less than the sum of the other
two and greater than their difference. [*]
14. A diameter bisects a circle. [A5.]
I5. A straight line intersects a circle at most in two points. [G6.]
II. Congruence of Triangles.
I. Any two triangles* ABC and A'B'C' are congruent if:
* In this syllabus the angles of a triangle ABC are denoted by the
capital letters A, B, and C; the sides are denoted by small letters a, b,
and c, where a is the side opposite the angle A, etc.
(I)     a=a'       b=b'        C =C'           [AI.]
(2)     a    a'    B    B'     C =C'           [A2.]
(3)     a =a'      b=b         c=c'            [A3.]
(4)     a =a'       c-=c'      C= C'-9o~       [A4.]
(State these in detail and in English. See preface, article 7.)




72


THE MATHEMATICS TEACHER.


2. A triangle is determined when the following are given:
(I) a, b, C; (2) a, B, C; (3) a, b, c; (4) a, c, C=9o~. [*]
(Synonymous to I.)
3. Construction of triangles from given parts; measurement
of unknown parts by ruler and protractor. Given: (I) a, b, C;
(2) a, B, C; (3) a, b. c; (4) a, c, C, possibly two solutions. [*]
(This is the fundamental, elementary idea of trigonometry.)
4. In any two triangles if a= a' and b =b', either of the inequalities
c > c' or C > C' is a consequence of the other. [03, 04.]
III. Congruent Right Triangles.
i. Two right triangles are congruent if, aside from the right
angles, any two parts, not both angles, in the one are equal to
corresponding parts of the other. [A4.]
(Very important subcase of II, I.)
2. If two oblique lines c and c' be drawn from a point in a
perpendicular p to a line AA', cutting off
'B/  tdistances d and d', then any one of the
~/  \c      equalities, c=-c', d —d', A=A' B==B',
p          is a consequence of any other.   [G5.]
3. A diameter perpendicular to a chord
J  a-'  d A      bisects the chord, the subtended angle at
the center, and the subtended arc; conversely, a diameter which
bisects a chord is perpendicular to it.  [G5b, G8.]
(Corollary to 2. See also IV, 3.)
4. If two oblique lines c and c' be drawn from a point in a perpendicular p to a line AA', cutting off unequal distances d and d', then
either of the inequalities c > c', d> d', is a consequence of the other.
[05, 06.]
(In particular, c is greater than p.)
5. If, in a triangle ABC, a= b, the perpendicular from C on
c divides the triangle into two congruent triangles.  [*]
6. In a triangle ABC, either of the equations a= b, A =B,
is a consequence of the other. [GI, G2.]
7. In a triangle ABC, either of the inequalities a > b, A >B, is a consequence of the other. [Oi, 02.]




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.


73


IV. Subtended Arcs, Angles and Chords.
i. In the same circle, or in equal circles,
any one of the equations d =d', k          k',
c =\ c', 0 -  0', is a consequence of any other
o0         one of them.   [A6, 7, 8, 9, G9.]
2. Any one of the inequalities (see figure),
<d<d', 0 >               ', c> c', k > k',
is a consequence of any other one of them. [07, 8.]
3. In any circle an angle at the center is measured by its
intercepted arc.  [J8.]
(Only the commensurable case.)
4. If a circle is divided into equal arcs, the chords of these
arcs form a regular polygon. [GI2, last part.]
5. To construct an angle equal to a given angle. [JI4.]
(Regular polygons may be constructed approximately by means of a
protractor. In the same way other approximate constructions may be
introduced which depend upon the protractor.)
V. Perpendicular Bisectors.
I. The perpendicular bisector of a line-segment is the locus
of points equidistant from the ends of the segment.     [SI.]
2. To draw the perpendicular bisector of a given line-segment.
[GI4.]
3. To erect a perpendicular at a given point in a line. [*]
(Corollary to 2.)
4. To drop a perpendicular from     a given point to a given
line. [D5.]
(Corollary to 2.)
5. To bisect a given arc or angle. [Gi5, 15.]
(See III, 3.)
6. To inscribe a square in a circle. [Gi8.]
7. One and only one circle can be circumscribed about any
triangle. [GI3.]
8. Three points determine a circle. Two circles can intersect, at most,
in two points; this will happen when the distance between their centers
is less than the sum of the radii and greater than the difference of the
radii. [G7.]
(Corollary to 7.)




74


THE MATHEMATICS TEACHER.


9. Given an arc of a circle, to find its center. [*]
(Corollary to 7.)
10. A circle may be circumscribed about any regular polygon. [GI3,
third part.]
II. The perpendicular bisectors of the sides of a triangle meet in a
point. [T3.]
VI. Bisectors of Angles.
I. The bisector of any angle is the locus of points equidistant from the sides of the angle. [S2.]
2. A  circle can be inscribed in any triangle.    [GI3, second
part.]
(Construction to be given.)
3. A circle can be inscribed in any regular polygon. [GI3, last part.]
4. Of the inscribed and circumscribed regular polygons of n and 2n
sides for a given circle, to draw the remaining three polygons when one
is given. [GI7.]
5. The bisectors of the angles of any triangle meet in a point. [Ti.]
(Corollary to 2.)
VII. Parallels.
I. When two lines are cut by a transversal the alternate
interior angles are equal if, and only if, those two lines are
parallel. [Half of Di, 2.]
When two lines are cut by a transversal, the alternate interior angles
are unequal if, and only if, the lines are not parallel.
(Synonymous to I.)
2. When two lines are cut by a transversal the corresponding angles
are equal, and the two interior angles on the same side of the transversal are supplementary if, and only if, the two lines are parallel.
[Half of Di, 2.]
(Corollary to I.)
3. Two lines perpendicular to the same line are parallel.
[D4, G4.]
(Only one perpendicular can be let fall from a point without a line to
that line. Synonymous to 3.)
4. A line perpendicular to one of two parallels is perpendicular to the other also.  [D3.]
(Corollary to I.)
5. If two angles have their sides respectively parallel or respectively perpendicular to each other, they are either equal or
supplementary.    [J7.]




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  75


6. Through a given point to draw a straight line parallel to
a given straight line. [D6.]
7. A parallelogram is divided into two congruent triangles by
either diagonal. [*]
8. In any parallelogram the opposite sides are equal, the opposite angles are equal, the diagonals bisect each other.  [D7.]
(Corollary to 7.)
9. In any convex quadrilateral, if the opposite sides are
equal, or if the opposite angles are equal, or if one pair of
opposite sides are equal and parallel, or if the diagonals bisect
each other, the figure is a parallelogram. [D8.]
VIII. Angles of a Triangle.
I. In any triangle the sum of the angles is two right angles.
[J5(b).]
2. In any triangle, any exterior angle is equal to the sum of
the two opposite interior angles. [J5(a).]
(Synonymous to I.)
3. The sum of the interior angles of any polygon of n sides is 2(n -2)
right angles. [J6.]
4. To inscribe a regular hexagon in a circle.  [GI9.]
To construct an angle of 60~. (Synonymous to 4.)
IX. Inscribed Angles.
i. An angle inscribed in a circle is measured by half of its
intercepted arc. [J9.]
2. Angles inscribed in the same segment are equal to each
other. [*]
3. An angle inscribed in a semicircle is a right angle. [*]
4. The two arcs intercepted by parallel secants are equal.
[GII.]
5. The angle between a tangent and a chord is measured by
half the intercepted arc. [Jio.]
6. The angle between any two lines is measured by half the
sum, or half the difference, of the two arcs which they intercept
on any circle, according as their point of intersection lies inside
of, or outside of, the circle. [JII, 12.]
7. The tangent to a circle at a given point is perpendicular
to the radius at that point.  [LI, 3.]




76


THE MATHEMATICS TEACHER.


8. For a given chord, to construct a segment of a circle in which a
given angle can be inscribed. [JI5.]
9. To draw a tangent to a given circle through a given point. [L4.]
IO. The tangents to a circle from an external point are equal.
[Gio.]
(Corollary to 7.)
X. Segments Made by Parallels.
I. If a series of parallel lines cut off equal segments on one
transversal, they cut off equal segments on any other transversal. [Dg.]
2. The segments cut off on two transversals by a series of
parallels are proportional. [See NIo.]
(Only the commensurable case.)
3. A line divides two sides of a triangle proportionally, the
segments of the two sides being taken in the same order, if and
only if it is parallel to the third side.  [NI, 2.]
(Only the commensurable case.)
4. To divide a line-segment into n equal parts or into parts
proportional to any given segments. [N9, io.]
5. To find a fourth proportional 'to three given line-segments.
[Nil.]
XI. Similar Triangles.
I. Two triangles ABC and A'B'C' are similar if
(I)    A=A'         B =B'       C    C'     [N3.]
or (2)     a =ka'      b    kb'    C =C'       [N4.]
or (3)     a   ka'     b    kb'    c ==kc'     [N5.]
where k is a constant factor of proportionality.
(See preface, article 7.)
2. Given a fixed point P and a circle C, the product of the
two distances measured along any straight line through P, from
P to the points of intersection with C, is constant.  This product
is also equal to the square of the tangent from P to C if P is an
external point.  [NI8.]
3. The bisector of any angle of a triangle divides the opposite side into segments proportional to the adjacent sides.
[Half of N6.]




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  77


4. To construct a triangle similar to a given triangle. [*]
(Drawing triangles to scale; measurements of remaining parts to
scale. Basal in trigonometry.)
XII. Similar Figures.
I. Polygons are similar if and only if they can be decomposed
into triangles which are similar and similarly placed.  [N7, 8.]
2. Regular polygons of the same number of sides are similar.
[N14.]
3. The perimeters of similar polygons are proportional to
any two corresponding lines of the polygons. [NI5.]
4. The circumferences of any two circles are proportional to
their diameters, thus c = 2rr,, where 7r is constant.  [PI5.]
(7r = 3.14... to be computed later.)
5. To construct a polygon similar to a given polygon. [*]
(Drawings to scale; maps, house plans; readings from drawings;
plotting of measurements. Essential in surveying.)
XIII. Similar Right Triangles.
(The committee feels that numbers 2, 3, 4 following should have a
place where time for their discussion can be secured, which will doubtless be the case except under pressure from examining bodies.)
i. Any two right triangles are similar if an acute angle of
the one is equal to an acute angle of the
B   other, or if any two sides of one are proportional to the       corresponding   sides of the
other.  [*]
A;  b  C^    2. For a given acute angle A, the sides of
any right triangle ABC (Cg=o~) form fixed
ratios, called the sine (a/c), the cosine (b/c), the tangent
(a/b). [*]
3. Computation of a two-place table of sines, cosines, tangents from actual measurements. [*]
(Probably a two-place table for every 5~ or IO~; to be done by students, preferably 6n squared paper.)
4. Solution of right triangles with given parts by use of the preceding
table of ratios. [*]
(Height and distance exercises.)




78


THE MATHEMATICS TEACHER.


XIV. Right Triangles.
I. In any right triangle ABC the perpendicular let fall from
the right angle upon the hypotenuse divides the triangle into
two similar right triangles, each similar to the original
triangle. [*]
C          2. The length of the perpendicular p is the,//4\Q       mean proportional between the segments m
>^ |L a\ 8and n of the hypotenuse; i. e., p2=mn.
[P8.]
3. Either side, a or b, is the mean proportional between the
whole hypotenuse c and the adjacent segment m or n; that is,
a2 = cm; b2    cn.  [Pg.]
4. To find a mean proportional between two given line-segments. [NI2.]
5. The sum of the squares of the two sides of a right triangle
is equal to the square of the hypotenuse; a2 + b2 =c2.  [Pio.]
(It should be noticed that the proposition can be proved either algebraically or geometrically.)
6. In any triangle ABC, if B is less than go~, then b2= a + c2  2cm;
if B is greater than 90~, then b2 = a2- 2 + c 2cm, where m is the projection of a on c. [PI7.]
(See figure under 2.)
7. Given the radius of a circle and the perimeter of an inscribed regular
polygon of n sides, to find the perimeter of the regular inscribed polygon
of n sides and the perimeter of the regular circumscribed polygon
of 2n sides. [GI7. See also X4.]
8. To calculate 7r approximately. [*]
XV. Areas.
I. The area of a rectangle is the product of its base and its
altitude; i. e., a: bh.  [PI, 2, 3.]
(This formula may be taken as the definition of area.)
2. Parallelograms, or triangles, of equal bases and altitudes
are equivalent. [Ci.]
3. The area of a parallelogram is the product of its base and
its altitude; i. e., a  bh.  [P4.]
4. The area of a triangle is one half the product of its base
and its altitude; i. e., a  I/bh.  [PS.]




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  79


5. The area of a trapezoid is one half the product of its altitude and the sum of its bases; i. e., a=  (bl + b2)h.  [P7.]
6. The areas of similar triangles or polygons are proportional
to the squares of corresponding lines. [Ni6, 17.]
7. The area of a regular polygon is one half the product of
its perimeter and its apothem.  [PII.]
8. The area of any circle is one half the product of its circumference and its radius; i. e., a  rrr2.  [PI6.]
9. The areas of two circles are proportional to the squares of their
radii. [*]
(May be treated as suggested in preface, article 3.)
Io. To construct a square equivalent to the sum of two given squares.
[*]
(Pythagorean proposition.)
II. To construct a square equivalent to a given rectangle. [C3.]
(Mean proportional. See X., 6.)
THEOREMS OF SOLID GEOMETRY.
In this part the same general principles apply as were stated
in the preface above.
Throughout, but particularly in divisons I and I below, very
great emphasis should be laid upon the student's real grasp of
the conceptions, of the space figures, and of the significance of
the theorems. While the theorems in division I will be seen to
need little or no suggestion of proof, it is a mistake to suppose
that they can be hastened over; on the contrary, even in these,
the teacher should spare no pains to make sure that the student's
mental picture is quite vivid, resorting to formal proof when
necessary. To this end, illustrations, figures, models, various
forms of presentation, and all such aids are legitimate throughout the course in solid geometry.
I. Theorems for Informal Proof.
I. If two planes-cut each other, their intersection is a straight
line.  [S4.]
2. Two dihedral angles have the same ratio as their plane
angles.  [K2, 3, 4.]
(Equivalent to K3.)
3. Every section of a cone made by a plane passing through
its vertex is a triangle.  [M4.]




80


THE MATHEMATICS TEACHER.


4. Every section of a cylinder made by a plane passing
through an element is a parallelogram.   [M2.]
5. The area of a sphere lies between the areas of circumscribed and
inscribed convex polyhedrons. [*]
(It is recommended that this statement be used as a definition to be
inserted at context.)
6. The volume of a sphere lies between the volumes of circumscribed
and inscribed convex polyhedrons. [*]
(It is recommended that this statement be used as a definition to be
inserted at context.)
7. The projection of a straight line upon a plane is a straight line.
[S8.]
II. Corollaries from Plane Geometry.
(The ability to make the transfer from plane geometry to
solid geometry, and vice versa, in forming conceptions and in
logical deductions is of the utmost importance.  The following
theorems are easily reducible to plane geometry in at most two
or three planes. The intention is that careful proofs be given,
but the student should see that these theorems result immediately from known theorems of plane geometry.)
I. The intersections of two parallel planes with any third
plane are parallel. [Fi.]
2. A plane containing one and only one of two parallel lines
is parallel to the other.  [F7.]
3. If a straight line is parallel to a plane, the intersection of
the plane with any plane drawn through the line is parallel to
the line. [*]
4. Through a given point only one plane can be passed
parallel to two straight lines not in the same plane. [Fio.]
(Derived from 2.)
5. Through a given straight line only one plane can be passed
parallel to any other given straight line in space, not parallel to
the first.  [FII.]
(Derived from 2.)
6. Through a given point only one plane can be drawn parallel to a given plane. [F9.]
(Synonymous to 4.)




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  8I


7. If a perpendicular PO be let fall from a point P to a plane
L, any one of the equalities
a=a', c==c', B    B', A= A'
is a consequence of any other of them; and
-   any one of the inequalities
yhc     K/            a > a', c > c', B < B', A > A'
is a consequence of any other one of them.
[Oio, H7. See also S8.]
8. The perpendicular PO is shorter than any oblique line.
[09.]
9. Two straight lines are parallel to each other if and only if
they are both perpendicular to some one plane. [F2, 3.]
IO. If two straight lines are parallel to a third, they are
parallel to each other. [F4.]
(Derived from 9.)
II. Two planes are parallel to each other if and only if they
are both perpendicular to some one straight line. [F5, 6.]
(Derived from 9.)
12. The locus of points equidistant from the extremities of a
straight line is a plane perpendicular to that line at its middle
point. [S5.]
13. If two straight lines are cut by three parallel planes, their
corresponding segments are proportional. [See MI.]
I4. The locus of points equidistant from  two intersecting
planes is the figure formed by the bisecting planes of their
dihedral angles. [S6.]
III. Planes and Lines.
i. If a straight line is perpendicular to each of two other
straight lines at their point of intersection, it is perpendicular
to every line in their plane through the foot of the perpendicular. [Ei.]
2. Every perpendicular that can be drawn to a straight line
at a given point lies in a plane perpendicular to the line at the
given point. [E2.]
(Corollary to I.)




82


THE MATHEMATICS TEACHER.


3. Through any point only one plane can be drawn perpendicular to a given line. [E5.]
(Corollary to I, and II., II.)
4. Through a given point only one perpendicular can be drawn
to a given plane. [E6.]
(Corollary to I.)
5. If two angles have their sides respectively parallel and
lying in the same direction, they are equal, and their planes are
parallel. [KI, F8.]
6. If a line meets its projection on a plane, any line of the
plane perpendicular to one of them at their intersection is perpendicular to the other also. [*]
7. Between any two straight lines not in the same plane, one and only
one common perpendicular can be drawn, and this common perpendicular is the shortest line that can be drawn between the two lines. [EI2.]
8. Two planes are perpendicular to each other if and only if a
line perpendicular to one of them at a point in their intersection
lies in the other.  [E3, 4.]
9. If a straight line is perpendicular to a plane, every plane passed
through the line is perpendicular to the first plane. [Ega.]
(Corollary to 8.)
Io. If two intersecting planes are each perpendicular to a third plane,
their intersection is also perpendicular to that plane. [Eio.]
(Corollary to 8.)
11. Through a given straight line oblique to a plane, one and
only one plane can be passed perpendicular to the given plane.
[EII.]
12. The acute angle which a straight line makes with its own
projection on a plane is the least angle which' it makes with any
line of the plane. [OI3.]
13. Two right prisms are congruent if they have congruent
bases and equal altitudes.  [Bi.]
I4. If parallel planes cut all the lateral edges of a pyramid, or
a prism, the sections are similar polygons; in a prism, the sections are congruent; in a pyramid, their areas are proportional
to the squares of their distances from the vertex.  [MI.]
(See II., I3.)
I5. Every section of a circular cone made by a plane parallel to its
base is a circle, the center of which is the intersection of the plane with
the axis. [M5.]
I6. Parallel sections of a cylindrical surface are congruent. [M3.]




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.  83


IV. Spheres.
I. Every section of a sphere made by a plane is a circle.
[Mio.]
(Several corollaries may be added.)
2. The intersection of two spheres is a circle whose axis is the line of
centers. [H4.]
3. The shortest path on a sphere between any two points on
it is the minor arc of the great circle which joins them.  [OI4.]
4. A plane is tangent to a sphere if and only if it is perpendicular to a radius at its extremity. [MII, I2, I3.]
5. A straight line tangent to a circle of a sphere lies in a plane tangent
to the sphere at the point of contact. [*]
6. The distances of all points of a circle on a sphere from its
poles are equal. [Hi.]
7. A point on the surface of a sphere, which is at the distance
of a quadrant from each of two other points, not the extremities
of a diameter, is the pole of the great circle passing through
these points. [H3.]
8. A sphere can be inscribed in or circumscribed about any given tetrahedron. [H5.]
9. A spherical angle is measured by the arc of a great circle
described from   its vertex as a pole and included between its
sides (produced if necessary).    [K5.]
V. Spherical Triangles and Polygons.
(Every theorem stated here may also be stated as a theorem on polyhedral angles.)
I. Each side of a spherical triangle is less than the sum of
the other two sides.  [OII (b).   See also (a).]
2. The sum of the sides of a spherical polygon is less than 360~.
[O12 (b). See also (a).]
3. The sum of the angles of a spherical triangle is greater than I80~
and less than 540~. [K8.]
4. If A'B'C' is the polar triangle of ABC, then, reciprocally,
ABC is the polar of A'B'C'.    [K6.]
5. In two polar triangles each angle of the one is the supplement of the opposite side in the other.   [K7.]
6. Vertical spherical triangles are symmetrical and equivalent. [C8,
H6.]




84


THE MATHEMATICS TEACHER.


7. Two triangles* on the same sphere are either congruent
* The same notation is used as in the plane triangles.
or symmetrical if
a = a'    b = b'     c == c'   [B2 (b).   See also (a).]
or a =a'      b = b'    C=      C'  [B3 (b).  See also (a).]
or a=a'       B -B'     C    C'    [B4 (b).   See also (a).]
or A =- A'    B    B'    C-      C'  [B5 (b).  See also (a).]
8. Either of the equations a -b, A = B is a consequence of the
other. [*]
VI. Mensuration.
(The relation between the areas and volumes of similar solids may be
treated as corollaries in individual cases. See preface, article 3. It is
understood that certain statements concerning limits may be assumed
either explicitly or implicitly; these are not stated as theorems. See
Q3, 4, 9, R9, Io.)
I. An oblique prism is equivalent to a right prism whose
base is a right section of the oblique prism and whose altitude
is a lateral edge of the oblique prism. [C4.]
2. A plane passed through two diagonally opposite edges of a
parallelepiped divides it into two equivalent triangular prisms.
[C6.]
3. The lateral area of a prism is the product of a lateral edge
and the perimeter of a right section. [QI.]
(Corollary of Plane Geometry.)
4. The lateral area of a regular pyramid is one half the
product of the slant height and the perimeter of the base.
[Q2.]
(Corollary of Plane Geometry.)
5. The lateral area of a right circular cylinder is the product
of the altitude and the circumference of the base; i. e., s= 2nrrh.
[Q5.]
6. The lateral area of a right circular cone is one half the
product of the slant height and the circumference of the base;
i. e., s    7rrl.  [Q6.]
7. The lateral area of a frustum of a regular pyramid is one half the
product of the slant height and the sum of the perimeters of the bases.
[Q7.]
8. The lateral area of a frustum of a right circular cone is one half
the product of the slant height and the sum of the circumferences of the
bases. [Q8 (a).]




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.        85
9. The area of a zone is the product of its altitude and the
circumference of a great circle; i. e., s= 27rrh.  [Qio.]
(Lemma for Io below.)
o0. The area of a sphere is the product of its diameter and
the circumference of a great circle; i. e., s = 4rr2.  [QII.]
ii. The area of a lune is to the surface of a sphere as the angle of the
lune is to 360~. [QI2.]
I2. The area of a spherical triangle is to the area of the sphere as its
spherical excess is to 720~. [QI3.]
I3. The volume of a rectangular parallelepiped is the product
of its three dimensions. [Ri, 2, 3.]
(This may be taken as a definition.)
I4. The volume of any parallelepiped is the product of its
base and its altitude. [C5, R4.]
I5. The volume of any prism is the product of its base and
its altitude. [Rs, 6.]
16. The volume of any pyramid is one third the product of
its base and its altitude. [C7, R7, 8.]
I7. The volume of a circular cylinder is the product of its
base and its altitude; i. e., v-v  rr2h.  [RII.]
18. The volume of a circular cone is one third the product
of its base and its altitude; i. e., v- 3krr2h.  [RI2.]
19. The volume of a spherical sector is one third the product
of the radius and the zone which is its base; i. e., v = 37rr2h.
[Ri5.]
20. The volume of a sphere is one third the product of its
radius and its area; i. e., v    4/3rr3.  [RI6.]
(The wording suggests a proof, but that proof is by no means prescribed. The wording is convenient, the proof may even preferably
follow 2I below.)
The following two theorems, while not thought by the committee to be indispensable, offer both for student and teacher an
outlook for that larger view of geometry and of mathematics as
a whole which is very desirable. They forecast important principles in future mathematical courses; they are capable of the
most practical direct applications; they offer a possibility of
organizing and retaining the important mensuration formulae
given above.




86


THE MATHEMATICS TEACHER.


21. If two solids contained between the same parallel planes
are such that their sections by a plane parallel to those planes
are equal in area, the two solids have the same volume.  [*]
("Cavalieri's Theorem." Formal proof should not be given.)
22. The volume of any sphere, cone, cylinder, pyramid, or prism, or of
any frustum of one of these solids intercepted by two parallel planes, is
given by the formula v= -h(t+ 4m + b), where t is the area of the
upper base, b that of the lower base, m that of a base midway between
the two, and where h is the perpendicular distance between the two
parallel planes. [See RI3, 14.]
(This formula also.applies to any so-called prismatoid; it is conveniently useful in practical affairs. It should not be proved for the general
case, but each separate solid mentioned above, numbers I to 20, can be
shown to conform to this rule, by a direct check.)
MAXIMUM OR MINIMUM LISTS.
The committee feels that it would be impossible to set up a
minimum list, or a maximum list, of theorems for college entrance examinations or for other broad purposes, which would
meet with any widespread approval; and that the influence of
such a list, if it were generally accepted, would be pernicious in
leading to a total disregard of many minor facts of geometry
which deserve at least passing notice.
Recognizing, however, a practical demand for some criterion
on the part of the college examiner, as well as on the part of the
teacher, the committee makes the following recommendations
for the guidance of examiners and teachers:
(a) All theorems in black face type in this syllabus should be
thoroughly known. The student should be able to state each
of these upon a clear suggestion of its topic; and to demonstrate
each of them without hesitation in accordance with some definite
logical order. He should know why they are important: in particular, he should be ready to mention other theorems in geometry closely connected with them.and he should know any important concrete applications of these theorems in ordinary life.
(b) All theorems in italics should be known to, the student
substantially in the form here given, but some latitude may be
allowed in combining parts of these with parts of other
theorems. The student should be able to prove each of these
theorems on fairly short notice and he should have a reasonable
idea of the importance of each of them.




FINAL REPORT OF NATIONAL COMMITTEE OF FIFTEEN.   87
(c) The theorems printed in ordinary Roman type should be
familiar to the student when they are stated by the examiner,
and the student should be able to make a proof for any one of
them if allowed a reasonable interval for thought.
(d) The theorems printed in small type, and indeed many
other facts of geometry not given in the syllabus, may be used
by the examiner with the understanding that they are to be
regarded in examinations as of the nature of exercises rather
than as theorems with which the student is supposed already to
have considerable familiarity.
(e) However, the committee would suggest (I) that examination questions involving the trigonometric ratios, XIII., 2,
3, 4, p. 77, be accompanied by alternative questions on other
topics, since some schools may not find time for these applications; and (2) that no questions be given by examiners involving proofs of theorems which may properly be taken as assumptions or as definitions, such, for instance, as I., 6, 7, page 70;
XV., I, page 78; VI., 13, 2I, pages 84, 85.
CONCLUSION.
It should be said that the members of the committee are not
entirely agreed as to certain minor details of this report. For
example, some would place among the exercises certain propositions now in small type; others would prefer to put some
theorems in black faced type which are now in italics; others
would prefer three types of propositions instead of four; and
some would modify certain postulates and would consider as
postulates or as propositions to be demonstrated certain
theorems included in the list of those requiring only informal
proof. The committee does not regard these minor matters of
any great consequence, and therefore wishes to be considered as
approving the spirit and general tenor of the report, rather than
as giving individual sanction to all such details.
HERBERT E. SLAUGHT, Chairman,
The University of Chicago, Chicago, Ill.
WILLIAM BETZ,
East High School, Rochester, N. Y.
EDWARD L. BROWN,
North High School, Denver.




88


THE MATHEMATICS TEACHER.


CHARLES L. BOUTON,
Harvard University, Cambridge, Mass.
FLORIAN CAJORI,
Colorado College, Colorada Springs, Colo
WILLIAM FULLER,
Mechanic Arts High School, Boston, Mass.
WALTER W. HART,
University of Wisconsin, Madison, Wis.
HERBERT E. HAWKES,
Columbia University, New York City.
EARLE R. HEDRICK,
University of Missouri, Columbia, Mo.
FREDERICK E. NEWTON,
Phillips Academy, Andover, Mass.
HENRY L. RIETZ,
University of Illinois, Urbana, Ill.
ROBERT L. SHORT,
Technical High School, Cleveland, Ohio.
DAVID EUGENE SMITH,
Teachers College, Columbia University, New York City.
EUGENE R. SMITH,
The Park School, Baltimore, Md.
MABEL SYKES,
Bowen High School, Chicago, Ill.
Additional copies of this report may be secured gratis upon
application to the COMMISSIONER OF EDUCATION, DEPARTMENT
OF THE INTERIOR, WASHINGTON, D. C.