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SiMON FRASER UNIVERSITY
0
?
MEMORANDUM
SENATE
?
From
Subject
_CURRICULUM AND CALENDAR CHANGES
?
-
Date—
DECEMBER 16, 1971
TICS
MOTION: ?
"That Senate approve, as set forth in
S.72-15
1.
New Course Proposal - Mathematics
305-4.
2.
New Course Proposal - Mathematics 361-3 -
with
discontinuance of Mathematics 261-3.
3.
Changes in Prerequisites for Mathematics Courses.
4.
Change in Calendar Description for Mathematics
152-3."
0
 
SiMON FRASER UNIVERSITY
?
T 7, 7 ,
Xf
0-'
MEMORANDUM
To ?
SENATE
?
From
SENATE COMMITTEE ON UNDERGRADUATE _STUDIES
Subject CURRICULUMANDCALENDARCHANGES
-_J
Date DECEMBER 16._1971
DEPARTMENT OF MkTHEMATICS
The Senate Committee on Undergraduate Studies
approved the submission of the Department of
Mathematics, as set forth in SCUS 71-29, and
recommends approval to Senate.
0
 
• ?
..
?
•
SiMON FRASER UNIVERSITY
MEMORANDUM
To........
... ......
.
.
I.Evans,,..S.ecre.taxy ..................................................... .
From ..... ....... .S...
Aronoff..................................................................
...... S.Qna.t. ... Commi.t.tee...on ... Un.dergraduate. .S.tudie
?
/........Dean. .of...Scien.ce
........................................................
Subject ...... ..
Agenda Ite.ms,....Department ... of ... Mathemat:
Cs
Date................
December...i,....1.971...................................................
The attached proposals have been approved by the Faculty
of Science Undergraduate Curriculum Committee and the
Executive Committee, acting on behalf of the Faculty.
They are now being forwarded for the approval of the Senate
Committee on Undergraduate Studies and Senate.
Included:
1.
New Course Proposal, Math 305-4 (see memo of
?
I
October 22, 1971)
2.
Changes in Prerequisites for Mathematics
Courses (see memo of October 27, 1971)
?
.
3..
New Course Proposal, Math 361-3 (see memo of
October 27, .1971)
4.
Change in Calendar Description of Math 152-3
(see memo of November 1, 1971)
la
Enclosure
cc: J. Chase, Chairman of SC1JS
R. Lardner, Mathematics
 
.
• ?
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a
•
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•. ? S ?
FACULTY
OF
SCIENCE
NEW
COURSE PROPOSAL
• ?
I
CALENDAR INFORMATION ?
•
•
bepartment:
?
Mathematics ?
Course Number:
?
305-4 ?
Title: ?
Statistical
•Analysis of Sample Surveys
•
Sub-title or Description-:
An
introduction to the major sample survey designs and their statistical
•
analysis. ?
Considerations of cost and the use of prior information will
•
be included. ?
S
•
Credit
Hours: ?
4 ? Vector
Description: ?
4-1-0
•
Pre-requisite(s) Two courses in probability or statistics, one of which -
may be taken concurrently. ?
e.g. ?
Math 101-3 plus
302-3 (102-3);
Math
101-3 plus Psych
210-3;
Econ
235-3 plus 236-3;
Math
371-3 plus 489-4.
II
E ?
LMEfl'
AND
SCHEDULING ?
•
.Estated Enrolment: ?
20 per offering
Semester Offered (e.g. Yearly, every Spring; twice yearly, Fall and
Spring):
Yearly, Fall
?
'S
?
:
•
When course will first be offered:
?
Fal). 1972
?
• •
-
•..
?
•
?
III
JUSTIFICATION ?
•. ?
• ?
• . ?
•
?
•
?
•
A. ?
What
is
the detailed description of the course including differentiation
•
•
• ?
• from lower level courses, from similar courses in the same department
• ?
0
, ?
and
from courses in other departments in the University?
(See attached)
?
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.
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••• ?
• ?
• ?
• ?
• ?
•
?
•
B.
?
What
is
th ?
range of topics that may be dealt with
in
the course?
Completely randomized surveys, stratified sampling, cluster sampling,
•
• ?
double sampling, ratio estimation.
?
S
?
• ?
•
 
I
•
.
.• ?
C.
How
does this course fit the goals of the department?
•
It will
provide Mathematics students with an introduction
to an area in which statistical theory is heavily used and
will provide a service to students in Economics and Social
Sciences.
D..
How does this course affect degree requirements?
•
Itis not required for any degree. ?
••
1 ? • ?
S
E.
What are the calendar changes necessary to reflect the addition of
• this course?
? ..
?
.
?
;
:: ?
..
New entry. ? . • ?
.
?
•• ?
•
F.
What
?
, if any, is being dropped from the calendar if this
course.is approved?
None. ?
• ? • ?
;•
?
• ?
•.
?
• ?
IT
C.
What is
the nature of student demand for this course?
• ?
. Students in Economics have requested ?
•. • -
•
such a course. ?
Mathematics students interested in
statistics will also take the course.
?
•
?
• •
?
•
H.
• Other reasons for introducing the course.
• ?
•
•
?
.
?
• It
is
in an area in which students have a reasonable chance of
employment with a Bachelor's degree.
?
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•
•
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•.
•
•
?
•• ?
••
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••
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..... ?
•
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7.
..•
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•..• ?
• ?
.,..
IV
?
BUDGETARY AND SPACE FACTORS
?
.
?
.•..
S.
? •
.5
S
.,
•• ?
S•.
•
• ?
. ?
••
•
. ?
A. Which
faculty will be available to teach this -course?
•
•
Dr
?
D.
Mallory, Dr. R. Rennie, Dr. C. Villegas
• ?
.
?
S
?
•
?
•
?
•
?
.
?
.5 ? . ?
•
?
• ?
•
?
•
?
•
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r.
 
I
III. (A) The course will cover methods of choosing sample units, probability
distribution of estimators, major designs and their statistical
analysis.
The course is not similar to any lower level course or to any other
Mathematics course. A course on survey design (Psych 315-3) is
listed by the Psychology Department; it is designed for
Psychologists (Psych 201-3 is a prerequisite) and is not offered
on a regular basis.
o
 
MAThEMATICS 305-4
S
i. STATISTICAL CONCEPTS
Types of Variables
Review of Probability, Distributions, Parameters, Estimators, Confidence
Intervals. Central Limit Theorem
2.
BASIC SAMPLE SURVEY CONCEPTS
Reasons for Sample Surveying
Sample Survey Terms (population, frame, etc.)
.
,.,
Types of Sample Survey Sampling (Simple, Random,stratified, Multi-stage, etc.)
3.
SIMPLE
RANDOM SAMPLING
Use of Random Number Tables and Generators -
Estimators, Variance of Estimators, confidence Intervals
Choice of Sample Size
4.
STRATIFIED RANDOM SAMPLING
Purpose, Choice of Strata
S ?
Estimators, Variance of Estimators, Confidence Intervals
?
Methods of Allocating samples among strata
Choice of Sample Size
Comparison with Simple Random Sampling (Accuracy, Cost, etc.)
Post Sample Stratification
S. CLUSTER SAMPLING
Purpose, Sample Units, Examples
Estimators, Variance of Estimators, Confidence Intervals
Choice of Sample Size
Comparison with Simple Random Sampling (cost, etc.)
6. MULTI-STAGE SAMPLING
Purpose, Choice of Sample Units, Examples
Two Stage Sampling
Estimators, etc.
Choice of' Sample Size
Extension to three or more stages
5 ?
Comparison with single stage methods
. . . 2
 
-2-
. 7. ?
RATIO
-
AND
flEGRESSION
ESTIMATION
Revicw of Covariance,
Correlation, Regression
Purpose and examples
Ratio Estimates, Variance
of Estimates, etc.
Regression Estimates,
Variance of Estimate
Choice of Sample Size
Comparison with other methods
8. ?
SYSTEMATIC SAMPLING
Purpose and examples
Estimators, Accuracy
of Estimators
Choice of Sample Size
Problems of Inference
9.
?
APPLICATIONS
Census Surveys
Market Surveys
Opinion Polls
Wildlife Surveys
LI
0
 
tk..(-
U
SiMON FRASER UNIVERSITY
MEMORANDUM
Dr. S. Aronoff, Chairman
?
.
From
?
Dr. R. W. Lardner,
To
d ?
idüfe...
?
At1n ?
head.....................................
Faculty of Science
?
Mathematics Department
. ?
PROPOSAL
Subject...............
MathematiCs
305-4
Date
...............
............................................
ctober22
?
1
97
.
1
...........................................
I have attached copies of a new course proposal for a
course entitled "Statistical Analysis of Sample Surveys'.
?
The
proposal is the result of requests from the Department of Economics
and Commerce and from individual students. We expect to receive
a memo from the Department of Economics and Commerce in support of
this proposal in the near future.
?
We will forward a copy of this
memo to you when it is received.
1^1
?
S
?
i
'S
0
 
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•
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. ?
.
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•
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..
?
•. ?
.
It
SiMON FRASER UNIVERSITY.
MEMORAtDU1A
To
.................
................
D•
?
119'?T..................................................................
From
...................HO1n.es
Mathematics
Economics and Commerce
Subject
........................ .samp1.
course
?
.Dat
e
...........OctPbe.?
5.,
....
]9Jl
As you know, I think that a course in sampling methods would be
very useful for students of economics and commerce. I have had
feedback from a graduate of this Univerity who is employed by the
Census Bureau and his major complaint is that he ws inadequately
prepared in this area through his course work at Simon Fraser. More
and more, research in economics and commerce involves sample survey
work, and at present our course offerings are inadequate in this
area. The text you mentioned by Mendenhall, Ott and Scheaffer which
would have our present Economics
236
(proposed Economics
33)
as its
prerequisite izculd, I thick, suit our prcgrammc very well.
I
,.lh
 
SIMON FRASER UNIVERSITY
DEPARTMENT OF
?
KE?I ?
BURNABY 2, BRITISH COLUMBIA
ECONOMICS AND COMMERCE
?
Tdephone 291-3111 Area code 601
December
3, 1971
D. Sharma
Undergraduate Program Chairman
Mathematics Department
Simon Fraser University
Dear Professor Sharma:
I understand that the mathematics department is considering
offering a course in the statistical analysis of sample surveys.
It is
my belief that such a course would be of great value to economics
and commerce students who are quantitatively oriented, or who are
interested in employment in government (or other) agencies using
. ?
sample surveys.
We would certainly recommend such a course to any of our
students in either of the above two categories should the mathematics
department offer it as a regular or as a selected topics course.
I feel equally certain that such a course would fit similar interests
of students in other departments.
Yours truly,
• ?
.
• ?
P. E. Ke nedy
• ?
Undergraduate Program Chairman
lh
N
 
?
S ?
-
.
?
I ?
....
FACULTY OF
SCIENCE ?
S
NEW COURSE PROPOSAL
?
•
?
I ?
CALENDAR INFORMATION
Department:
Mathematics
?
?
Course Number:
361_3
Title:
Mechanics of
Deformable Media-
Sub-title or Description:
Analysis of deformation and stress and an introduction to constitutive
• ?
equations for different materials.
• ?
Credit Hours: ?
3 ?
Vector
Description: (3-1-0),
Pre-requisite(s):
> (See attached)
?
II
?
ENROLMENT AND SCHEDULING
(
Estimated Enrolment:
?
15 per offering
Semester Offered (e.g. Yearly, every Spring; twice yearly, Fall and
Spring):
?
•
?
Yearly, every Spring
When course will first be offered:
Spring 1973
?
III ?
JUSTIFICATION ?
S
A.
What is the detailed description of the course
including
differentiation
from lower level courses, from similar courses in the same department
and from courses in other departments in the University?
No sim.lar course is offered either in this Department or other
Departments in the University. There is no overlap with other courses.
.1 ?
• ? . ?
S
B.
What is the range of topics that may be dealt with in the course?
?
( -
?
?'lechar.ics of deformation, materials and their mechanical behaviour,
stress analysis, kinematics of dcfrmation, infinitesimal deformations. r.
 
0•
... ?
. .
•
? / ?
. ?
.
. ?
.
? .. ?
..••
• ?
1•
• ?
• ?
C. HOW
•
?
does this course fit the goals
. ?
of the department?
•
? t.
This
course is essential to our undc;'graduate program in
Applied Mathematics. The level of the material formerly
covered in 261-3 will be increased and the orientation
• ?
of
the courses will be changed from-an engineering emphasis
(Continued on the attached page)
D.
How does this course affect degree requirements?
This course has no effect on degree requirements.
E.
What are the calendar changes necessary to reflect the addition of
this course?
New entry. ?
. • -
F.
Whatcourse, if any, is being dropped from the calendar if this
course is approved?
0 ?
• . •..
?
•0
Mathematics 261-3.
? . ?
•
G.
What is the nature of student demand for this course?
It provides necessary background for undergraduate applied
mathematics students.
-
H.
Other reasons for introducing the course. ?
*
It is
?
prerequisite for most of the upper level applied
• ?
• ?
mathematics courses.
? • •
IV ?
BUDGETARY'AND SPACE FACTORS
?
0
A. Which faculty will be available to teach this course?
• ?
All
applied mathematics faculty: Drs. Graham, Lardner, Pechlaner,
S ?
Shadman, Sharma and Shoemaker.
? • •
• ? . ?
•
?
• ? • ? •
?
0 ?
•
?
r.• ? -
 
• ?
Page
,• ?
.
?
•
1 ?
••
?
..•
.1/.
B.
What are the special space and/or equipment requirements for
V ?
this course?
No special requirements.
C. Any other budgetary implications of mounting this course:
Since this course will replace Mathematics 261-3 and will be
offered less frequently than that course was, this teaching
commitment can be met from existing resources.
(
.
f ?
.•
 
/ ?
S
I;
Of
• ?
1. ?
pre-requisite(s): Mathematics 161-3 or quiva1ent* and
Mathematics 25-4. *Students who hav not had Math 161-3
but who have taken Physics 201-2, may be allowed to take
Math 361-3 (provided they have completed Math 253-4) with
the permission of the Department.
III.
C. to a 'strength' materials emphasis. This will allow us
• ?
to
assume greater mathematical maturity on the part of
students taking the course which will therefore allow
the course to be more mathematical in nature.
I.
 
.
?
3THU.MT1CS 361-3
1. 1th;:tticai I'z•1iinar$.es; ?
Suition convention. Vector trcr:natjon.
Gradient, Curl and Divergence. C:rtcsicn
tenscr.'3 of
?
--A two.
?
(2
2.
Forcc; in Space: Moment of a force about a point, about an
axis., vector !.onent of a
ccile,
C!)OSi
tion of couples in space, reduction of a
force systcm to a
givcn centre, resultant
of a force system in space, conditions of
equi libri. ua.
(CHAPTER 7, Targ's book).
3. Curvilinear notion of a Particle
(Ci!APTER ).O, Targ's book).
4.
mechanics of Defo
y tion:
?
Forces in a deforrabie body. 1policaLicn
of equi].ihritrn conditions to dor:mble
body. Concept of a Continunn. Fornulatiori
of probleirs in.Cor.tinutr Mechanics.
?
( ;eek)
5.
JI als and their
. .echarL1ca1 3ehavicur: ?
!icrcsca.c noinit of view.
Cij'itallin so).ics, ic3:phous so.icis,
Iii
t
i
polvoers, fluid:. 01--servedncJ
Behaviour of clids nacroscopic ncint of
view. Uniaial lo-
:,
ding of a bar. Strecs
strain curve. Linear elastic, n3nl!n2ar
elastic, plastic behaviours in uniaxial
loading. Time de p
endent and rate inritht
behaviours. Stress
concentration,
ter: erati.r.e
effects, failure end fracture, atigu, i'uc:-
hug. Ic3ealisd odls of
p
erfe:t fluid
s
,
perfectly .elastic, perfectl
y nlatic, ricid
perfectly plact.tc, linear elastic - perfeciv
• ?
plastic beiaviours. Care and cation in
employing idealised models to real nysica'
•
?
problems in rvchanics. ?
(15 wce;
G. St:rcs J\nalvs
j c: ?
Surface tractions and body
forces. The stress
vector. The ddcr.cc' of the Strcs Vector
on Oricnt:aUon. The Stre:s Tensor. The
rqatios of •otion. Sy:-..etry
of st; tcn:or.
Proper
ties of stress tensor, pr
1
?
sc -;s,
nx%imxrP and iinimu: stss , strc:s condit
o:
• ?
at the boundar y
. Plane state of stxcss. (' ;:)
7. JJner:.t1: of Defcrrnaticn:.
?
Field quantities. The material timc 'rivtive.
\1ocj Lv
p
d acceleration of a
q
c'ncri. a pr
tide.
Laji.an.izin .mi
?
cri .n vari:blcs.
and Lu)t:r.t .n c scription of ocrntie:, ?
•'-•
 
4
p.3. corari;on of a
Of
?
tlie ?
c'Lfor;.1(1
bc_sc'.'
y
jth ?
t.h;:t• of
?
th
b.'
OY 1:fl
?
o,
Stin
V'^nso:. ?
:tensiom,
?
ang].e chucs,
vo) ?
:vjs,
?
z- and
?
locd.l.. rotations.
Pr5.rci"ai strain
s
.
?
cc'idl Dofor
?
t.3c; -
such
?
tS
rig.ci
c:ortcm,
deford:ion, unifon.i trns1at0n, p'.rc
•
rotation. Dispiacem
.
nt ccnditior. ?
at
-
th ?
};rry. ?
(4 ?
k)
. ?
infini.tccinal Doritons:
7tions of .infinitosinal theory.
Linear strain tnor.
Ixtcn5i.ons,
Angle
changes,
vohi ?
changes, local
rot
atiori:; in tcrris of linear strain
ten
so
r.
?
Co!tt.nui.ty
of deforEtion
(copathili cy coniit:ons) .
?
oluticn
•
for ?
placnts with given strains.
Strain rates. Deformation of flo,.
?
(1 week)
TEXTS:-
' -
1cth:2nics of SOU
IL
and
Flui(tf -by RobrtR. Long, Prc::tic.-1iail Inc.
"Theoretical Mechanics, a
Short Course" by S. Targ, Gordon
?
Breach, Science
Publishers.
 
e
.4
SiMON FRASER UNIVERSITY
MEMORANDUM
0
To.........
.
Pr.,... S....
Aronoff .......................................................................
.
From
.......... Dr...
R.. W.... Lardner
Acting Head
...............
D.e.an ... of - science
.....................................................................
.......... Mathematics ... Department
Subject
...................................................................................................................... I
?
Date
..........
October. .21,...1971 ............................................ ..............
As requested in your memo of October 13, 1971, the following is a
list of the proposed modifications to the prerequisite listings for
mathematics courses offered in the Spring Semester 72.
Economics 235-3 should be added as an alternate prerequisite for this
course, and this should be noted as follows:
1.
Mathematics 302-3, ELEMENTARY STATISTICAL METHODS:
?
?
Prerequisite: Mathematics 101-3 or 371-3 or Economics 235-3.
2.
Mathematics 152-3, CALCULUS II:
Mathematics 150-3 should be added as a conditional alternate .prerequisite
for this course and this should be noted as follows:
.
Prerequisite: Mathematics 150-3 (with a grade of A or B), or
Mathematics 151-3.
3.
Mathematics 241-2, PURE MATHEMATICS III:
The present prerequisite listing should, be deleted and replaced by the
following statement:
Prerequisite: Mathematics 152-3.
4.
Mathematics 470-4, VARIATIONAL CALCULUS:
Mathematics 161-3 should be substituted for Mathematics 261-3 as
one of the prerequisites for this course, and this should be noted
as follows:
Prerequisites: Mathematics 161-3 or Physics 203-2. Mathematics
?
11-4
or Physics 382-4 should precede or be taken concurrently.
S
?
f ?
I
?
j7
 
p
Mathematics Department
October 27, 1971
CHANCES IN PREREQUISITES FOR UNDERGRADUATE MATHEMATICS COURSES
1.
MATHEMATICS 241-3, PURE MATHEMATICS III:
?
Delete: ?
'Mathematics 141-2 or 142-2 or 151-3.'
?
Add: ?
'Mathematics 152-3.'
2.
MATHEMATICS 352-2, INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS:
?
Delete: ?
'Mathematics 152-3.'
?
Add:
?
'Mathematics 253-4 should precede or be taken concurrently.'
3.
MATHEMATICS 443-3, COMBINATORIAL THEORY:
?
Delete: ?
'Mathematics 432-4 should precede or be taken concurrently.'
?
Add: ?
'15 semester hours of Mathematics recommended.'
4.
MATHEMATICS 467-3, VIBRATIONS:
?
Delete: ?
'Mathematics 251-3 (or 214-3) and 261-3 and 352-2.'
?
Add: ?
'Mathematics 161-3 and 352-2, and 361-3 is recommended strongly.'
5.
MATHEMATICS 468-4, CONTINUUM MECHANICS:
?
Delete:
?
'Mathematics 251-3 and 261-3 and 352-2 (Mathematics 214-3
may be used in place of 251-3 and 352-2).
?
Add: ?
'Mathematics 311-4 and 361-3.'
6.
MATHEMATICS 469-4, FLUID DYNAMICS:
?
Delete:
?
'Mathematics 261-3. Mathematics 411-4 or Physics 382-4
should precede or be taken concurrently.'
?
Add:
?
'Mathematics 361-3. Mathematics 311-4 or Physi'382-4
should precede or be taken concurrently.'
.
 
Page two: Changes In Prerequisites for Undergraduate Mathematics Courses:
7.
MATHEMATICS 471-4, SPECIAL RELATIVITY:
?
Delete:
?
'Mathematics 261-3 or Physics 203-2. Mathematics 411-4 or
Physics 382-4 must precede or be taken concurrently.'
?
Add: ?
'Mathematics 311-4 or Physics 382-4.'
8.
MATHEMATICS 486-4, PROBABILITY:
?
Delete: ?
'Mathematics 251-3 and 371-3 and 241-2 or permission of the
instructor.'
?
Add:
?
'Mathematics 251-4 and 371-3.1
.
F
c._.
 
SiMON FRASER UNIVERSITY
MEMORANDUM
00 ?
Dr.
?
S.
?
Aronof......Chafrin.........................................
.
. ...
From ..... . ...... Dr... .R.W..
Lardner........................................................
Undergraduate Studies Committee
?
Acting head
fty ... of
?
.
.............................................................
ence
...................
.Mathematics..Depar.tme.nt......................................
Y.
SubIect..,.t.S1QNS...TOTHE MA.U.EMA.ICS ............................ ?
.
.ate...............O,c.tober...2.7.,...i.9.71......................................................
UNDERGRADUATE CALENDAR SUBMISSION
I am now forwarding to you the remainder of the proposed changes and
additions to the Mathematics undergraduate Calendar submission for next
year. I have attached a new course proposal form for a course entitled,
Mathematics 361-3, Mechanics of Deformable Media, a memo from Dr. Holmes
of the Department of Economics and Commerce in support of the new course
proposal, Mathematics 305-4 - Statistical Analysis of Sample Surveys, and
a list of changes in prerequisites for mathematics ccurses. The recommen-
dations for changes In prerequisites have been made on the basis of the
changes in syllabus for the calculus courses and the changes in requirements
for mathematics majors and honors students.
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SiMON FRASER UNIVERSITY
MEMORANDUM
Dr. S. Aronoff, Chairman
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From. ?
Dr. R. W. Lardne
Undergraduate Studies Committee
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Acting Head
Faculty of Science
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..çpçpment
Subject
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MATHEMATICS
UNDERGRADUATECALENDAR ?
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Date
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9ymbqrLJ7
SUBMISSION - Mathematics 152-3
The recently approved changes in the syllabus for Mathematics 253-4
(Calculus III) make it necessary to change
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the calendar description for
Mathematics 152-3 (Calculus II). Further, in recent attempts to clarify
the prerequisite listings for the latest edition of the Course Guide,
we have been informed by the Registrar's Office the Senate approval is
required for the requested change in the prerequisite listing for Mathem-
atics 152-3. ?
Even though the present calendar states that Mathematics
151-3 (Calculus I) may be either a prerequisite or a co-requisite for
Math 152-3, it has not been possible for students to take these courses
concurrently since their lecture times have been scheduled to clash.
The requested changes are as follows:
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1. DELETE:
"152-3 Calculus II
Conic sections. Definite and indefinite integrals, techniques of
integration. Improper integrals. Taylor's formula. Sequences
and series. Conditional, absolute and uniform convergence.
Differentiation and integration of series. Introduction to
vectors. ?
(3-1-0)
Prerequisite: Mathematics 151-3 must precede or be taken concurrently."
2. ADD:
11
152-3 Calculus II
Conic sections. Definite and indefinite integrals. Techniques and
applications of integration. Polar co-ordinates.
(3-1-0)
Prerequisites: Mathematics 150-3 (with a grade of A or B), or
Mathematics 151-3."
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R. W. Lardner
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