Google
This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project
to make the world's books discoverable online.
It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject
to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books
are our gateways to the past, representing a wealth of history, culture and knowledge that's often difficult to discover.
Marks, notations and other maiginalia present in the original volume will appear in this file - a reminder of this book's long journey from the
publisher to a library and finally to you.
Usage guidelines
Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the
public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing tliis resource, we liave taken steps to
prevent abuse by commercial parties, including placing technical restrictions on automated querying.
We also ask that you:
+ Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for
personal, non-commercial purposes.
+ Refrain fivm automated querying Do not send automated queries of any sort to Google's system: If you are conducting research on machine
translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the
use of public domain materials for these purposes and may be able to help.
+ Maintain attributionTht GoogXt "watermark" you see on each file is essential for in forming people about this project and helping them find
additional materials through Google Book Search. Please do not remove it.
+ Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just
because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other
countries. Whether a book is still in copyright varies from country to country, and we can't offer guidance on whether any specific use of
any specific book is allowed. Please do not assume that a book's appearance in Google Book Search means it can be used in any manner
anywhere in the world. Copyright infringement liabili^ can be quite severe.
About Google Book Search
Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers
discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web
at |http: //books .google .com/I
£-«!^5^f, ;;?
Satbattr Sniieraits
LIBRARY OF THE
SCHOOL OF ENGINEERING
TRANSFERRED
HARVARD COLLEGE
LIBRARY ,—
^
1£
^ r r •
" •*• 'J "SJ .
^ Y^ """ ^--^.^^ '^-y-^u J.^.a^.A^.. . C^
-vctti^ /v^. J5^ .tA^'.,y^^r 0^ ^'
N * ^-V' '' -. -V'. ^
SYLLABUS OF MATHEMATICS
A SYMPOSIUM COMPILED BY THE (COM-
MITTEE ON THE TEACHING OF MATHE-
MATICS TO STUDENTS OF ENGINEERING
Accepted by the
Society for the Promotion op Engineering Education
AT THE Nineteenth Annual Meeting held at
Pittsburgh, Pa., June, 1911
Office of the Secretary
ITHACA, N. Y.
1912
^ 1/ MIVAI* CVtL£«l LIMAIT
SCHOOL ^; rv.M ,-r- ...-^
T^
"46
^
Press or
tHi New Era Printin« compani'
LANCASTER. PA.
TABLE OF CONTENTS.
Page.
Bepobt of the Committee on the Teaching of Mathematics to
Students of Enoineebing 1
Syllabus of Elementaby Algebba 5
Chap. I. Transformation of Algebraic Expressions 6
Chap. II. Solution of Equations 13
Chap. III. Miseellaneous Topics 16
Syllabus of Elementaby Geometby and Mensubation 19
Syllabus of Plane TBigonometby 27
Chap. I. Sine, Cosine and Tangent of Acute Angles .... 28
Chap. II. The Trigonometric Functions of any Angle .... 33
Chap. III. General Properties of the Trigonometric Functions 43
Syllabus of Analytic Geometby 47
Chap. I. Bectangular Coordinates . 4S
Chap. II. The Straight Line 60
Chap. ni. The Circle 62
Chap. IV. The Parabola 63
Chap. V. The Ellipse 67
Chap. VI. The Hyperbola 61
Chap. VII. Transformation of Coordinates 66
Chap. VIII. General Equations of the Second Degree in x
and y 66
Chap. IX. Systems of Conies 70
Chap. X. Polar Coordinates 71
Chap. XI. Coordinates in Space 72
Syllabus of Differential and Integral Calculus 75
Chap. I. Functions and Their Graphical Eepresentation . . . 77
Chap. II. Differentiation. Bate of Change of a Function 82
Chap. HI. Integration as Anti-Differentiation. Simple Dif-
ferential Equations 96
Chap. IV. Integration as the Limit of a Sum. Definite
Integrals 102
Chap. V. Applications to Algebra. Expansion in Series;
Indeterminate Forms 109
Chap. VT. Applications to Geometry and Mechanics 114
Discussion at the Pittsburgh Meeting 119
Syllabus on Complex Quantities 134
• • •
lU
REPORT OF THE COMMITTEE ON THE TEACH-
ING OF MATHEMATICS TO STUDENTS
OF ENGINEERING.
To the Society for the Promotion of Engineering Education:
The committee was appointed at a joint meeting of mathe-
maticians and engineers held in Chicago, December 30-31,
1907, under the auspices of the Chicago Section of the Ameri-
can Mathematical Society, and Sections A and D of the
American Association for the Advancement of Science,* and
on the suggestion of officers of the Society for the Promotion
of Engineering Education who were there present, the com-
mittee was instructed to report to this Society.
The membership of the committee is as follows :
Alger, Philip R., professor of mathematics, U. S. Navy,
Annapolis, Md.
Campbell, Donald F., professor of mathematics, Armour
Institute of Technology, Chicago, 111.
Engleb, Edmund A., president of the Worcester Polytechnic
Institute, Worcester, Mass.
Haskins, Charles N., assistant professor of mathematics, Dart-
mouth College, Hanover, N. H.
Howe, Charles S., president. Case School of Applied Science,
Cleveland, Ohio.
KuiCHLiNG, Emil, consulting civil engineer. New York City.
Magbuder, William T., professor of mechanical engineering,
Ohio State University, Columbus, Ohio.
Modjeski, Ralph, civil engineer, Chicago, 111.
Osgood, William F., professor of mathematics. Harvard Uni-
versity, Cambridge, Mass.
Slichteb, Charles S., consulting engineer of the U. S. Recla-
mation Service, professor of applied mathematics, Univer-
sity of Wisconsin, Madison, Wis.
* For an aeeount of the Chicago meeting, see Bdenoe for 1908 ( Jolj
12, 24, and 31; August 7 and 28; and September 4).
1
2 COMMITTEE ON TEACHING MATHEMATICS.
Steinmetz, Charles P., consulting engineer of the General
Electric Company, professor of electrical engineering,
Union University, Schenectady, N. Y.
Swain, George F., consulting engineer, professor of civil
engineering. Harvard University, Cambridge, Mass.
Townsend, Edgar J., dean of the College of Science and pro-
fessor of mathematics. University of Illinois, Urbana, lU.
Tubneaurb, Frederick E., dean of the College of Mechanics
and Engineering, University of Wisconsin, Madison, Wis.
Waldo, Clarence A., head professor of mathematics, Washing-
ton University, St. Louis, Mo.
WiLUAMS, Gardner S., consulting engineer, professor of civU,
hydraulic and sanitary engineering, University of Michigan,
Ann Arbor, Mich.
Woodward, Calvin M., dean of the School of Engineering and
Architecture and professor of mathematics and applied
mechanics, Washington University, St. Louis, Mo.
Woodward, Robert S., president of the Carnegie Listitutionof
Washington, Washington, D. C.
ZiWET, Alexander, professor of mathematics. University of
Michigan, Ann Arbor, Mich.
Huntington, Edward V., chairman, assistant professor of
mathematics, Harvard University, Cambridge, Mass.
After deliberation, the committee decided that it could best
carry out the purpose for which it was appointed by preparing
a synopsis of those fundamental principles and methods of
mathematics which, in the opinion of the committee, should
constitute the minimum mathematical equipment of the stu-
dent of engineering, and the chairman was instructed to sub-
mit for discussion a preliminary draft of such a synopsis. Li
order to secure as wide a discussion as possible, the committee
voted to accept the invitation of the Society for the Promotion
of Engineering Education to publish this preliminary draft in
the Bulletin in advance of its presentation to the Society,
and to solicit criticisms and suggestions from all members of
the Society, whether members of the committee or not.
It should be distinctly understood that this advance publica-
COMMITTBE ON TEACHING MATHEMATICS. o
Hon is merely a tentative draft, prepared by the chairman, and
still under discussion by the members of the committee; the
committee as a whole is not to he held responsible for any part
of the synopsis in its present form.
The synopsis, as now planned, will consist of six parts:
1. A Syllabus of Elementary Algebra ;
2. A Syllabus of Plane Trigonometry ;
3. A Syllabus of Differential and Integral Calculus;
4. Numerical Computation and the Solution of Equations;
5. Geometry (Elementary and Analytical) ;
6. A Syllabus of Elementary Dynamics ;
preceded by the following introductory note, explaining its
scope and purpose :
* * It is hoped that this report may be serviceable in two ways :
first, to the teacher, as an indication of where the emphasis
should be laid ; and secondly, to the student, as a syllabus of
facts and methods which are to be his working tools. It does
not include data for which the student would properly refer
to an engineers' hand-book; it includes rather just those
things for which he ought never to be obliged to refer to any
book — ^the things which he should have constantly at his
fingers' ends.
''The teacher of mathematics should see to it that at least
these facts are perfectly familiar to all his students, so that
the professor of engineering may presuppose, with confidence,
at least this much mathematical knowledge on the part of his
students. On the other hand, if the professor of engineering
needs to use, at any point, more advanced mathematical meth-
ods than those here mentioned, he should be careful to explain
them to his class.
**The committee has not found it possible to propose a de-
tailed course of study. The order in which these topics
should be taken up must be left largely to the discretion of
the individual teacher. The committee is firmly of the
opinion, however, that whatever order is adopted, the principal
part of the course should be problems worked by the students,
and that all these problems should be solved on the basis of a
4 COMMITTEE ON TEACHING MATHEMATICS.
small number of fundamental principles and methods, such
as are here suggested.
''The defects in the mathematical training of the student of
engineering appear to be largely in knowledge and grasp of
fundamental principles, and the constant effort of the teacher
should be to ground the student thoroughly in these funda-
mentals, which are too often lost sight of in a mass of details.
**A pressing need at the present time is a series of synoptical
text-books, which shall present, (1) the fundamental prin-
ciples of the science in compact form, and (2) a classified and
graded collection of problems (which would naturally be sub-
ject to continual change and expansion) . It is the hope of the
committee that this report, which is confined to the first part of
the desired text-book, will stimulate throughout the country
practical contributions toward the second."
In the early part of its investigation the committee collected
a large amount of information in regard to the present status
of mathematical instruction for engineering students. Since
that time, however, a much more inclusive inquiry has been
undertaken by the International Commission on the Teaching
of Mathematics, of which the American Commissioners are
Professors D. E. Smith, J. W. A. Young and W. F. Osgood.
In order to avoid unnecessary duplication, this committee
voted to turn over all the results of its own inquiry in this field
to the larger commission, to be worked up in accordance with
the general scheme adopted by that commission, and to be
incorporated in their report. This material is therefore not
included in the present report.
Respectfully submitted,
Edward V. Huntington,
Chairman,
June, 1911.
A SYLLABUS OF THE FORMAL PART OF
ELEMENTARY ALGEBRA.
This syllabus is intended to include those facts and methods of ele-
mentary algebra which a student who has completed a course in that
subject should be expected to "know by heart" — that is, those funda-
mental principles which he ought to have made so completely a perma-
nent part of his mental equipment that he will never need to "look them
up in a book. ' '
It is not intended as a program of study for beginners, and no at-
tempt has been made to arrange the topics in the order in which they
should be taught. In reviewing the subject, however, either at the end
of the course in algebra, or at the beginning of any later course^ such a
syllabus will be found serviceable to both teacher and student; and in
the hands of a skillful teacher, and supplemented by an adequate collec-
tion of problems, it might well be made the basis of a course of study
conducted by the "syllabus method."
One of the chief defects in the present-day teaching of algebra is the
multiplicity of detached rules with which the student 's mind is burdened;*
and every successful attempt to knit together a number of these detached
rules into a single general principle (provided this principle is simple
and easily applied) should conduce to economy of mental effort, and di-
minish the liability to error.
TABiiE OF Contents.
CHAPTEB I. TBANSrOBMATION OF ALGEBRAIC EXPRESSIONS.
General laws of addition and multiplication.
Type-forms of multiplication (Factoring).
Fractions.
Negatives.
Badicals and Imaginaries.
Exponents and Logarithms.
Chapter II. Solution of Equations.
Legitimate operations on equations.
To solve a single equation.
Quadratic equations.
Exponential equations.
To solve a set of simultaneous equations.
Chapter III. Miscellaneous Topics.
Batio and proportion.
Variation.
Inequalities.
Arithmetical, geometric, and harmonic progressions.
* For example, in a recent prominent text-book there are no less tluoi
fifty italicized rules in the part of the book preceding quadratic
equations I
CHAPTER I.
Transformation of Algebraic Expressions.
1. The ordinary operations of transforming and simplify-
ing algebraic expressions should be so familiar to the student
that he performs them almost instinctively ; at the same time
he should be able, whenever called upon, to justify each step of
his work by reference to some one or more of a small number
of well established principles.
For example, if the student is asked hy what authority he replaces
? . ^ by T» or Va* + ^* by a + h (to mention only two of the common-
est blunders), he will be forced to recognize either that he is making use
of methods that he has never proved, and that are in fact erroneoiis,
or else (which is more likely) that he is working altogether in the dark,
without any conscious reason for the steps he has taken.
The following list of such principles, while making no pre-
tense at logical completeness, will be sufficient for all practical
purposes.
2. Gteneral laws of addition and multiplication.
a-}-J = 6-}-a. ab=sba. ( Commutative laws. )
(a-f 6) -^6 = 0+ (6 + c). {db)c = a{hc).
(Associative laws.)
a{i -\- c) =db + ac. (Distributive law.)
a-fO=a. aXl = a. aXO=0.
These laws hold when a, h, e are any of the quantities that occur in
ordinary algebra, whether ''real'' or "complex."* The student should
he constantly encouraged to test general algehraie statements hy substi-
tuting concrete numerical values.
3. Type-forms of multiplication (Factoring).
The following type-forms of multiplication are the ones that are most
important to remember:
* This syllabus is confined chiefly to the algebra of real quantities; the
algebra of complex quantities will be treated only incidentally.
6
AliQBBSA. I
o« — 6*=(a— 6)(a + 6),
a» — 6*=(o— 6)(o» + a6 + 6*),
and 80 on; the general case is best remembered in the form
1— «"=(!— «)(l + a; + a5» + a5»H h*""^)-
Note also that in the algebra of real quantities, a" + 6» is
divisible by a + 6 when and only when n is odd. Thus :
a» + 6»=(a + 6)(a* — a6 + 6*).
Further : {x + a){x + h) =«* +{a + h)x + ab,
and the '^ binomial theorem":
(a— 6)* = a* — 2a6 + 6^ (a— 6)" = a» — 3a'6 + 3a6« — 6»;
a^ + na-6 + ^-^^a-6» + r^n ^ l)in ^ 2) ^^_,^, ^ ^ ^ ^
where kl = ''k faetorial'' = l X 2 X 3 X ••• X &.
4. Fractions.
a
Def. If bx=^a, then and only then we write x= T(or a/6,
ora-~6).
Here a is called the numerator and "b the denominator of the fraction.
A fraction with a zero denominator, as a/0, does not represent anj
definite quantity. For, if a is not zero, there is no quantity x such that
X ^ = a; and if a = 0, then every quantity x will have this property.
Hence, the denominator of a fraction must always he different from aero.
From the definition, a/1 = a ; also
a
- « 1, - = 0, (a + 0).*
* The symbol >f means "not equal to.^'
8 AIXJEBBA.
To add two fractions with common denominator :
a b a + b
- + - = —1—.
c e c
To multiply two fractions :
a X ax
b y'^ by'
To divide by a fraction, ** invert the divisor and multiply" :
a X a y ay
b ' y^b a;"" bx'
The value of a fraction is not changed if we multiply (or
divide) both the numerator and the denominator by any
quantity not zero:
This is the most important principle concerning fractions.
For example, to reduee two fractions to a common denominator, we
have merely to multiply numerator and denominator of each fraction by
a suitable factor.
Again, to simplify a complex fraction, we multiply the whole numera-
tor and the whole denominator by any quantity which will *' absorb" all
the «ibsidiary denominators. Thus, by multiplying by xye, we have
- 4--
X y ayz -\- bzz /
c -\- d (c-|-c?)ay '
z.
at once, by a single mental process. (The common practice of reducing
the numerator and denominator separately, and then inverting the denom-
inator and multiplying, is tedious and dumEfy.)
Def . If bx = 1, then x = 1/6, which is called the reciprocal
of 6. To divide by 6 (6 4= 0) is the same as to multiply by the
reciprocal of 6.
5. Negatives.
Def. If a + a;=0, then and only then we write a;= — a.
In particular, — ( — a) =a.
If a is not zero, — o is always opposite to a ; that is, if a is
positive, — o is negative, and if a is negative, — a is positive.
ALQEBBA. 9
Thus, if a= — 3, which in a negatire qnantitj, then ^a=3, which
la positive.
The notation | a |, which is coming into use more and more
widely, means the absolute value of a, that is, the numerical
value of a regardless of sign; thus, 1 5 | = 5, | — 5 1 = 5.
The laws of operation tvith the minus sign are best remem-
bered by regarding — a as the product of a and — ^1 :
(— l)Xa=— a,
whence, in particular (putting a= — 1),
(-1)X(-1)=1.
When this is done, the customary formulas:
(— o)(— 6)=a6, (— o)(5)= — aft, ^ = -^^ = — ^, ^ = ^,
become immediate consequences of the general laws of multiplication
and division, and therefore need not be separately memorized; and the
same is true of the formula
a — (x + y — 0)=za — x — y + z,
which, when remembered in the following form, becomes an immediate
application of the distributive law: ''a minus sign in front of a paren-
thesis must be 'distributed' through every term within, if the parenthe-
ses are to be taken away."
By knitting together in this way the rules for negatives with the
general rules of operation, the total number of processes to be remem-
bered iind applied, and hence the liability to error, is materially reduced.
Def . If a + a? = 6, then and only then we write x = b — a.
It is easily shown that 6 — a = 6 + ( — <*) ; that is, subtracting
any quantity a is the same as adding the opposite of a.
6. Radicals.
Def. If a is positive, and n is any positive integer, there
will always be one positive value of x such that x^ = a. This
value X is denoted by y/a, and is called the (principal) nth
root of a.
It should be noticed that while there are (for example) two square
roots of 9, namely 3 and — 3, it is only the positive one of these two
values that is denoted by V^; that is, the mark V^means 3 and not — 3.
10 ALGEBBA.
The radical sign, except in the case of square roots, and
sometimes in the case of cube roots, should always be replaced
by fractional exponents (see below) when it is desired to com-
pute with these quantities; this done, no special rules for the
manipulation of radicals need then be remembered beyond the
general laws of exponents.
Square roots. If a and 6 are positive.
y/a^b = ay/b, and y/ayb^^y/ab.
Note also the process called '' rationalizing the denominator
(or numerator) of a fraction"; for example,
c c Va — V6 cNa^-ylb)
— rr = s=s ' X — i= ^ — ^ *
Vl— 03 Vl— « Vl— 03 1—05
X
Vl + aj Vl + aj Vl — 05 Vl — aj^'
Def . If a is negative, and n is odd, there will always be one
negative value of x such that x^ = a; this value is denoted by
^"a, and is called the (principal) nth root of a.
Thus {/^^^S = — 2.
7. Imaginaries.
If a is negative, and n is even, then there is no positive or negative
nth root of a. Hence, such quantities do not occur in the algebra of
positive and negative quantities. They occur only in the more general
algebra of complex qitantities; in this algebra every quantity a (except
zero) has n distinct nth roots, the notation ^a being applied, as occasion
requires, to any one of these n values. The detailed study of this general
algebra is probably too difficult for a first course; for such applications
as occur in elementary work, the following working rules are sufficient :
1) In manipulating a complex quantity of the form V — 6>
where b is positive, write V — ^ == V — 1 y/b^is/b, and treat
♦ like any other letter ; then simplify the result by the relation
i^=—l.
2) Every complex quantity can be written in the form
a-^ib, where a and b are **real'' (that is, positive, negative,
or zero) ; and if a + t6 = a' + i6', then a=a' and 6 = 6'.
ALQEBBA. 11
In electrical engineering the letter i ia used to denote current, and
V — 1 i« denoted by j.
8. Exponents.
The subject of negative and fractional exponents is a part of algebra
in which the preparation of the student is apt to be especially unsatis-
factory.
Definition of negative and fractional exponents. If a is positive, and
p and q are any positive integers, then
a«-=l, «^=^> a^i9={/^ a^l9=:{/^.
9. Laws of operation with exponents.
If a and b are positive, then :
a»*« = a^a"^, a~* = ( a» ) •, ( a6 ) » = a»6«
All these laws hold for any values of m and n; the three fundamental
ones can readily be recalled to mind through simple special cases, such as
aV, (a^)«, and (ab)«.
The three other laws commonly mentioned, namely
a«-*»=:a»/a», a»/»= '^o"*, (a/5)«=:a»/5»,
are immediate corollaries of those just mentioned.
If a is negative, and m not an integer, a^ will, in general, be a complex
quantity. In such cases, let a' = — a, so that a' is positive, and write
a«= ( — l)**a'*», where ( — 1)*» must then be handled according to the
rules of operation in the algebra of complex quantities.
10. Logarithms.
The subject of logarithms should be taught in logical connection
with the subject of exponents. The common practice of separating these
subjects, and treating logarithms as a part of trigonometry, is unfortu-
nate. Numerous applications of logarithms can be found that have
nothing to do with trigonometry; moreover, the training in the use of
logarithms which a student gets in trigonometry is usually quite inade-
quate as a preparation for the applications of logarithms in any of his
later work outside of surveying.
Def. The logarithm of a (positive) number, to any (posi-
tive) base, is the exponent of the power to which the base must
be raised to produce that number.
12 ALGEBBA..
Thus, the notation
X = logtN
means
Note that negative numbera in general have no logarithms in the
algebra of real quantities.
From the laws of exponents we have, whatever the base
may be :
log {ab) =log a + log 6, log l^j =log a — log 6,
log (a») =n log a, log ^~a= -log a,
Tit
log 1 = 0, log ( base) = 1.
Only two bases are in common use. For purposes of
numerical computation, the base chosen is 10, and in this
system log(10*)=n.
(See chapter on numerical computation.) In higher mathe-
matics, the base e = 2.718 • • • is used, for the reason that the
use of this base simplifies certain formulas in the calculus ; in
this system log (e") =n.
Change of base. To find logeN when log^^^N is known, let
x=logeN, that is, e*=N. Then take the logarithm of
both sides of this equation to base 10, and solve for x.
The resulting formula, loge^= (logieN')/(logi«6), is so easily obtained
in this way that it is not worth while to remember it separately. The
approximate values
logio« = .4343, and log«^'= (2.3026) logi^y,
however, are useful to remember.
CHAPTER 11.
Solution op Equations.
11. Legitimate operations on equations. If a given equa-
tion is true, it will still be true if we
(a) add any quantity we please to both sides;
(6) subtract any quantity we please from both sides;
(c) multiply both sides by any quantity we please ;
(d) divide both sides by any quantity we please except zeroj
(e) raise both sides to any positive integral power;
(/) •extract any positive integral root of both sides, except
that if an even root is extracted, the double sign + must be
used;
{g) •take the logarithm of both sides (provided both sides
Prepositive).
In regard to (d), we must never divide both sides by an unknown
quantity without first excluding the possibility that that quantity is zero.
In (/), the restriction stated means, for example, that from ^'=B
we can infer merely that ^ = ± V^; that is, that either A = y/^, or
-4 = — ^/B; but we cannot tell which.
12. To solve a single equation in x, means to find all the
values of x that satisfy the equation, or to show that none
such exist.
Any value of x that satisfies the equation is called a root of
the equation.
In testing a root, the only safe method is to substitute the given
value in each side of the equation separately, and see whether the re-
sults, when reduced, are equal. Thus, we should find that x= — 2 is a
root of the equation a? = 2 — V12 — 2a?, and that a; = 4 is not a root.
In this connection it should be noticed that if we square both sides
of a given equation, the new equation will, in general, have more roots
than the given equation. Thus (to use the same example), by squaring
X — 2 = — V12 — 2a; we have a^ — 2a? — 8 = 0. This equation has of
course the root — 2, since a? = — 2 satisfies the original equation from
* In the algebra of complex quantities (/) and (g) are not applicable.
2 13
14 ALGEBRA.
which this was derived; but it has also the root 4, which was not a root
of the original equation.
The formal process usually called ''solving the equation'*
means merely transforming the equation, by a judicious choice
of the legitimate operations, into a form in which the solutions
are obvious.
If this is not possible, we must have recourse to the method
of trial and error which, while often laborious, is always
applicable in numerical cases. (See chapter on numerical
computation.)
If an equation is given in the factored form:
(x — a) (a? — /3) (a? — 7) . . . = 0,
then the roots are obviously a; = a, a; = /3, a; = 7, . . . . Thus, the roots
of x(x + 2) = are and — 2.
13. Quadratic equations. To solve the quadratic equation
ax^-{-bX'{'C = 0,
we may divide through by a:
, b c
a a
and then ** complete the square'':
whence,
-6dbV6'-4ac
^— 2^ '
or, we may use the general result just obtained as a formula.
The quantity which must be added to both sides in '' completing the
square ' ' is obvious by analogy with a^ + ^^na; + ^'» so that this method
requires less effort of the memory than the method of solution by formula.
The ' ' method of factoring ' ' is very convenient in certain special cases,
when the factors can be obtained by inspection.
The method still sometimes used, of first multiplying through by 4a to
avoid fractions, is apt to lead to confusion, and should be discouraged.
From the formula it is evident that the sum of the roots is
ALGEBRA. 15
aji + x,= — 6/<*> *^d the product of the roots is x^X2=c/a;
also, if the coefficients, a, b, c, are real, the roots will be real-
and-distinct, real-and-coincident, or imaginary, according as
6* — 4ac is positive, zero, or negative.
14. Exponential equations. To solve an equation of the
form 0^^=^, when a and b are positive, take the logarithm of
both sides: x log a=log b; and then solve for x.
15. To solve a set of simultaneous equation in z, y, z • • •
means to find all the sets of values of. x,y, z, • • •, that satisfy
all the equations at once, or show that none such exist.
Two fiimnltaneoas equations of the first degree, as ax + by^^e and
Ax + By=s C, can always be solved in a couple of lines, if the work is
arranged as follows:
7«— 6y = l
14x — 10y = 3
5
3
2
1
(— 35-f 42)x = — 6-f 9
(12 — 10)y = — 2 + 3
whence the values of x and y are obvious, provided aB — bA is not aero.
(If aB — hA^z 0, there is either no pair of values x, y that satisfies both
the equations, or else there are an infinite number of pairs of values that
do so ; in this latter ease, the equations are not independent, that is, either
of them can be derived from the other.)
The theory of simultaneous equations, and sometimes the numerical
computation, is facilitated by the use of determinants.
In general, n independent equations will suffice to deter-
mine n unknown quantities.
CHAPTER III.
Miscellaneous Topics.
16. Ratio and Proportion. The ''ratio of a to &'' means
simply the fraction a/6/ and a ** proportion'' is simply an
equation between two ratios.
The notation a'.hllold should be replaced bj the equation afb^zo/d)
and all special terminology, such as "taking a proportion by alterna-
tion/' ''by composition," etc., should be dropped in favor of the
ordinary language of the equation.
17. Variation. The statement **y varies as a?," or ''y varies
directly as «," or **y is proportional to a?," means y = kz,
where k is some constant. Similarly, ''y varies inversely as
a?," means y = k/x; **y varies inversely as the square of x,''
means y = k/x^. The constant k can always be determined if
we know any pair of values of x and y that belong together.
The statement "y varies as u and v," means y varies as the product of
« and i;, that is, y = Jcuv,
18. Inequalities. The notions of ' ' greater and less ' ' are thoroughly
familiar when we are dealing only with positive quantities, but the ex-
tension of these terma to the algebra of all real quantities (positive,
negative, and zero) is apt to cause some confusion.
(a) All real quantities (positive, negative, and zero) may
be represented by the points of a directed line (running, say,
from left to right) :
0— — o 0—0 >
_3 _2 —1 +1 +2 +3
and the notation a<b (read: ^*a algebraically less than &")
means simply that a precedes &, or a lies on the left of b, along
this line.
Similarly, a> & (read: "a algebraically greater than h") means that
a eome$ after h, or lies on the right of h, along the line. (The idea that
a negative quantity is a magnitude whose siee is in some way ''less than
nothing'' should be carefuUy avoided.)
16
ALGEBRA. 17
Obviously, if a and b are any real quantities, one and only
one of the three relations: a = h, a < 6, and a>b, will hold
between them; further, if a < 6 and 6 < c, then a<,c.
(b) Complex quantities require for their representation the points of
a plane instead of the points of a line, and the symbols < and > are not
used in eonnection with these quantities.
^ Legitimate operations on ineqiiaUties. If a given inequality
is true, it will still be true if we
(a) add any quantity we please to both sides;
(b) subtract any quantity we please from both sides;
(c) multiply both sides by any positive quantity;
(d) divide both sides by any positive quantity;
(e) raise both sides to any positive power (integral or
fractional), provided both sides are positive.
(/) take the logarithm of both sides, provided both sides
are positive.
If we multiply or divide both sides by any negative number,
we must reverse the sense of the inequality.
The neglect of the rules for handling inequalities is the source of many
common errors.
19. Arithmetical Progression.
In an arithmetical progression :
a, a + d, a + 2(2, a + 3(2, •••,
each term is obtained from the preceding by adding a con-
stant quantity.
The nth term is obviously Z = a+ (n — l)d.
a + l
The sum of n terms is 8= ^ n.
This formula is most easily remembered in the form:
8=s (average of the first and last terms) X (number of terms).
The arithmetic mean between a and &isA=^(a + &).
20. Oeometric Progression.
In a geometric progression :
a, ar, ar^, ar*^ •••,
18 ALGEBRA.
each term is obtained from the preceding by multiplying by a
constant quantity.
The nth term is obviously I = ar^K
a(l — r»)
The sum of n terms is 8= — = .
1 — r
This formula is best remembered in eonneetion with tbe mle for
factoring: •
1— f*=(l— r)(l+r + r«4-r»H h'^M-
The geometric mean between a and b is 0=^'\/ab.
The geometric mean is also called the mean proportional.
Infinite geometric progression. If | r | < 1, the sum of n
terms approaches the limit
a
1 — r
as n increases indefinitely (since, in the expression for 8, if
I r I < 1, r» approaches zero).
21. Harmonic Progression.
A harmonic progression is a series of terms whose recip-
rocals are in arithmetical progression. (The harmonic pro-
gression is not of great importcmce.)
The harmonic mean between a and 6 is H=^ -,
a+
A SYLLABUS OF ELEMENTARY GEOMETRY AND
MENSURATION.
This syllabus is intended to include those facts and methods of ele-
mentary geometry which a student should have so firmly fixed in his
memory that he will never think of looking them up in a book.
1. Bight Triangles.
In a right triangle, the square on the hypotenuse is equal
to the sum of the squares on the other two sides (Pythagoras,
580-501 B.C.) ; and the sum of the acute angles is 90**.
Examples of right triangles with integral sides: 3, 4, 5; 5, 12, 13.
Two right triangles are congruent when they agree with
respect to (a) any side and an acute angle; or (6) any two^ '
sides.
In the "45** triangle'' and the ** 30-60** triangle,'' the ratios
of the sides are as indicated in the figure.
2. Oblique Triangles.
In any plane triangle, the sum of the angles is 180**. Hence,
an exterior angle of a triangle equals the
sum of the opposite interior angles.
Of two unequal sides in a triangle, the greater is opposite
the greater angle. t
A plane triangle is, in general, wholly determined when any
three of its parts (not all angles) are given.
19
20 ELEMENTABY GEOMETBY AND MENSUBATION.
There are four cases :
(a) two angles (provided their sum is less than
180®) and one side;
(&) two sides and the included angle;
(c) the three sides (provided the
largest is less than the sum of the other
two);
(d) two sides and the angle opposite one of them (the ''ambiguous
ease/' in which we maj have two solutionS| or one, or none).
Hence the usual rules for testing the equality of two plane
triangles.
The center of gravity of a plane triangle is the
intersection of the three medians, and is two
thirds of the way from any vertex to the middle
point of the opposite side.
3. Angles in a Circle.
An angle inscribed in a semicircle is a right angle.
^^
An angle subtended by an arc of a circle at any point of the
circumference is equal to half the angle subtended by the same
arc at the center.
4. Similar Figures. Proportion.
If any two lines are cut by a set of parallels,
the corresponding segments are proportional.
(Hence the usual rule for dividing a given line
into any number of equal parts.)
In all problems in proportion, the notation aihiicid, and all special
terminology, such as "taking a proportion by alternation," "by com-
ELBMBNTABY GBOMBTBY AND MBNSUBATION.
21
position/' etc.| should be abandoned in faror of the ordinary language
of the equation. For example, if a/5 = o/d, then, bj adding 1 to both
sides, (a + b)/'b^=(o + d)/d; and bj subtracting 1 from both sides,
(a — 5)/5=(o--d)/d; etc.
If two plane triangles are similar, their corresponding sides
are proportional.
In a right triangle, the perpendicular
from the vertex of the right angle to the
h3rpotenuse is a mean proportional be-
tween the segments of the hypotenuse:
p« = tnn.
Any two similar fig-
ures, in the plane or in
space, can be- placed in
** perspective,'* that is,
so that lines joining
corresponding points of the two figures will pass through a
common point. In other words, of two similar figures, one is
merely an enlargement of the other.
In two similar figures, if i is the factor of proportionality,
any length in one =& X (the corresponding length in the
other) ; any area in one == ft* X (the corresponding area in
the other) ; any volume in one = ft* X (the corresponding vol-
ume in the other).
5. Lines and Planes.
If a line is perpendicular to a plane,
every plane containing that line is perpen-
dicular to the plane.
22
ELBMBNTABY QEOMETBT AND MSNSUBATION.
A dihedral angle is measured by a plane angle
formed by two lines, one in each face, perpen-
dicular to the edge.
6. Plane Areas.
Area of parallelogram
= base X altitude.
Area of triangle
= i base X altitude.
Area of trapezoid
= i sum of II sides X alt.
= mid-section X altitude.
7. The Circle. (ir = 3.1416 • • • = 22/7, approximately.)
Circumference of circle = 2?rr.
(Proved by regarding the circle as the
limit of an inscribed or circumscribed
polygon; proof rather long.)
Area of circle = irr^.
(Proof by regarding drcle as limit of sum of triangles radiating out
from the center, the altitude of each triangle being the radius of the
circle; hence, area of circle =} circumference X radius.)
Area of sector angle of sector
area of circle four right angles
; hence,
Area of sector = ^r**, where is the angle in radians.
For area of segment, subtract triangle from sector.
blbmbhtab; gboubtbt Aim mensusahon. •s3
S. The Ojlinder.
Volume of any cylinder (or
priam)^ base X altitude.
Area of curved surface of any
right cylinder (or right prism) =
perimeter of base X altitude.
(Proof b7 regarding the area as the limit of a anm of reetanglM
trhooe eommoa altitttde ia the altitude of the cylinder; or, b^ slitting
the ejlinder along an "element" and rolling the Barf ace ont into a
rectangle.)
9. The Oone.
Volume of any cone (or pyra-
mid) ^ 1/3 base X altitude.
(Proof bj diiwecting a triangular
prism; or, more simplj, hy the in-
tegral calenlas.)
Area of curved surface of a right circular cone (or a regular
pyramid) = 1/2 perimeter of base X slant height.
(Proof bj regarding the area as the limit of a sum of triangles whose
eommon altitude ia the slant height of the eone.)
Area of frustum of a right circular cone (or of a regular
pyramid)
=1/2 sum of perimeters of bases X slant he^ht.
= perimeter of mid-section X slaut height.
(Proof bj regarding the area as tile limit of the trapesoids whose
eommon attitude is the slant bri^t of the f rtistnm.)
24
ELEMENTABY GEOMETBY AND MENSUBATION.
10. The Sphere.
Area of a zone = circumference of great circle X altitude
of zone.
In other words, the area of the sphere cut out bj two parallel planes
is equal to the area of the portion of the circumscribing cylinder inter-
cepted between the same pair of parallel planes. (Proof by regarding
the zone as the. limit of a sum of conical frustums.) Hence»
Area of sphere = 47rr*
= area of four great circles of the sphere.
In other words, the area of the sphere is equal to the area of tJie
cwrved surface of the drcwnMcrihing cylinder.
Volume of sphere ^ f ^r*.
(Proof bj regarding sphere as limit of a sum of pyramids radi-
ating out from the center, the altitude of each pyramid being the radius
of the sphere; hence, volume of sphere =} area of sphere X radius.)
Area of a lune _ angle of lune
area of sphere "" four right angles*
Area of spherical triangle is proportional to its
spherical excess (that is, the excess of the sum of its
angles over 180**).
(Proof by considering three lunes which have the given triangle in
common.)
XLXMENTABT QEOMETBT AND MEKStnUTIOK. 25
TA« fotlenoing further theoremi, the proof of lehieh involvet the inte-
gral edlcvlua, are mentioned here alio, beeaute they are eaty io remember
and are often terviceable tn elementary work.
11. Oavalieri'a Theorem (1598-1647).
Sappoae two solids have their bases in the same pUtne, and
let the sections made in each solid by any plane parallel to the
base be called "corresponding sections." If then the corre-
sponding sections of the two solids are always equal, the toI-
nmes of the solids will be equal.
(Proof bj ragardiDg each of the solidf aa the limit of a pile of thin
■labs.)
12. Theorenu of Guldin (1577-1643), or
of Pappus (about 290 A.D.).
1. Suppose a plane figure revolves about
an axis in its plane but not cutting it.
Then the volume of the solid thus generated
is equal to the area of the given figure
times the length of the path traced by its
center of gravity.
2. Suppose a plane curve revolves
about on axis ia its plane but not cutting
it. Then the area of the surface thus
generated is equal to the length of the
given curve times the length of the path
traced by its center of gravity.
26
ELEMENTABY GEOMETBY AND MENSUBATION.
13. The Prismoidal Fonniila.
The prismoidal formula holds for any solid lying between two parallel
planes and such that the area of a section at distance x from the base is
expressible as a, polynomial of the second (or third) degree in x.
It A, B = areas of the bases, M = area of a plane section
midway between the bases, and k = altitude, then
Volume of prismoid = - (A + B + 4M).
b
A SYLLABUS OF PLANE TRIGONOMETRY.
Table or Ck)NTENTS.
Chapter I. Sine, Ck)siNEy and Tangent or Acute Angles.
Definitions of sine, cosine, and tangent of an acute angle as ratios
between the sides of a right triangle:
sin a? = opp/hyp; cos a? = adj/hyp; tan fl; = opp/adj.
To trace the changes in these functions, as the angle changes from
0® to 90® (circle of reference).
Use of tables. Exact values of functions of 30", 45", and 60".
To find remaining functions of an angle when one function is given
(draw right triangle). To construct an angle from its tangent.
Fundamental relations: sin' x + cot? x = l, tan a; = 8in 2;/cos a;, etc.
Solution of right triangles.
Problems in orthogonal projection.
Problems in composition and resolution of forces, etc.
Chapter n. The Tbigonomstuo Functions or ant Angle.
Angles in generaL Congruent, complementary, and supplementary
angles.
Units of angular measurement: degree, grade, radian.
Definitions of sine, cosine, and tangent of any angle.
To trace the changes in these functions, as the angle changes from
0" to 360" (circle of reference).
Definitions of cotangent, secant, and cosecant:
cot fl; = l/tan x, sec fl7 = l/cos x, cse fl; = l/sin x.
Definitions of versed sine and coversed sine:
vers 0? =5 1 — cos x, covers x = l — sin x.
Use of the tables: reduction to first quadrant.
Solution of oblique triangles.
Law of sines: a/& = sin A/an B.
Law of cosines: a* = &* + c* — 2l>e cos A,
Chapter III. General Properties or the Trigonomstrio Funotions*
Belations between the functions of a single angle.
Functions of ( — x). Functions of (aj±n90"), etc.
Functions of tJie sum and difference of two angles:
sin (oj + y) = sin x cos y + cos a? sin y,
cos (oj + y ) = cos a; cos y — sin a? sin y.
Functions of twice an angle, ^nd of half an angle.
The inverse functions, sin-*a?, cos""a?, tan"*a:, etc.
Solution of trigonometric equations.
27
^i
CHAPTER I.
SnfE, CosmB, and Tangent op Acute Angi-es.
1. Definition of sine, cosine, and tangent of an acute angle
X. — In any right triangle, if a; is one of the acute angles, the
Bine, cosine and tangent of x are defined as ratios between the
sides of the triangle, as follows:
side opp. side adj.
BmiE=-r T^ COSiB^-v: r^
hypot hypot.
side opp.
™"*"8ide adj. Pio- i.
These ratios are pore numbers, depending only on the size of
the angle.
2. To trace the changes in these nom-
bers when the angle changes from 0° to
90°, draw the figure so that the denomi-
nator of the ratio is kept constant, say
equal to 1 inch, and trace the changes in
the numerator. Thus, from Fig. 2, when
X goes from 0* to 90°, sin x goes from
to 1, and cos x goes from 1 to ; from
Fig 3, when x goes from 0° to 90°, tan x
goes from to infinity.
3. Tables. — The ratios thus defined
are called ' ' trigonometric functions' '
of the angle, and their valaes have
been tabulated, to 4, 5, or 6 places of
decimals, in the "tables of trigonometric
functions." Before using the printed
tables, the student should make his own
table, for a few angles, by graphical con-
j,jg 3 struction, with a protractor, to two
places of decimals.*
It is dear from tlie figure that the values of eoa x from 0* to 90° are
[1 X in rererw order; note how this fact !■
the tablet.
28
FiQ. 8.
th« nme aa the values of
made nn of to aaT« space
TBIGOKOMBTBY.
29
Fio. 4.
4. The functions of 30°, 45% and 60° can be found exactly,
without the use of the table. Thus, in the
triangles which occur in Fig. 4, it is readily
proved by the Pythagorean theorem that
if the hypotenuse is 1 inch, the shortest
side is ^ in., the longest side is ^V^ ^'f
and the middle-sized side j^y/H in. Hence
any function of 30°, 45°, or 60° can be read
off the figure by inspection. For example,
sin 30° = i, tan 45° = 1, tan 60° = V3 ; etc.
5. It is frequently required to find the remaining functions
of an angle when any one function is given. To
do this, draw a right triangle, mark one of the
angles, and mark two sides to correspond to
the given function. Then compute the remain-
ing side by the Pythagorean theorem, and read
off any desired function from the completed
figure. For example,
Given, tan a; = J. From the figure, sin x = 2/Vi3; etc.
Given, sin a; = a. From the figure, tan x = a/y/1 — a* ; etc.
To construct an angle when any one of its functions is
given, first find the tangent of the angle; when the tangent is
known, the construction of the angle is obvious.
6. The notation sin^ x, etc., is used as an abbreviation for
(sin a?)*; etc.
The following fundamental relations are easily proved and
remembered from the figure : for any angle x,
TiQ. 7.
sin* a; -f cos* a; = 1, tanx
3
Fio. 8.
sin a; sin (90° — a;) = cos a;,
cos a;' cos (90° — a;) = sin x.
30 TBIGONOMETBY.
7. The student should be thoroughly drilled in the defini-
*. tions of the sine, cosine and tangent, in right
/^l/ triangles in all possible positions in the plane
V^ ^V-» regardless of lettering. Thus, the mental proc-
ess should be as follows : pointing at the figure,
^'the tangent of ilm angle is ilm side, divided
by this side"; etc.
®' • The following forms of the original equations
are especially useful, and should be emphasized :
side opp. = hypot. X sine ; side adi. = hypot. X cosine.
Solution of Bight Triangles.
8. We recall that in any right triangle, the sum of the
squares on the two legs is equal to the square
on the hypotenuse, and the sum of the acute
angles is 90^. Hence, when either acute angle
is known, the other may be found; and the
sine of either acute angle is the cosine of the
other:
c^=a^'\' b^^ sin A = cos B.
9. By the aid of a table of sines, cosines and tangents, when
any two parts of a right triangle, besides the right angle, are
given, the remaining parts may be found (except in the case
where the given parts are the two acute angles, in which case
the triangle is not determined) .
For, we have merely to remember the definitions of the func-
tions, selecting the equations so that only one unknown ap-
pears in each equation ; then solve for the unknown quantity,
and compute by the aid of the tables. The results should be
checked by substituting in some relation not used in the direct
computation.*
* This computation, like xnanj other numerical computations, can often
be shortened bj the use of the slide rule, or by the use of logarithms;
in fact, tables are provided which give the logarithms of the trigono-
metric functions directly in terms of the angles; but the student should
thoroughly understand the use of the fiinctions themselves before he
begins to use the logarithmic tables.
TBJGONOMETBY. 31
10. Numerous problems inyolvii^ right triangles : isosceles
triangles, polygons, oblique triangles solved hy means of right
triangles, heigbts and distances, surveying problems, etc.
Obthogonal Phojbction. Cohponxnts op Forces, etc.
11. The projection of a length AB on any line is the given
length times the cosine of
the angle between the lines.
(Proof from the definition of
cosine.)
The projection of a plane
area upon any fixed plane is
the given area times the cosine
of the angle between the
planes. Proof by the theorem
oflimito.) ^'■"-
12. The component of a force along any fixed axis is ihe
y * magnitude of the force times the cosine of the
£>• — ^ angle between the force and the axis.
y. Since we usually require the eomponents
p^~™ along two rectangular axes, it is important to
leii r^™™^''^' *li** "^08 C®^" — a:)=sina;. The
mental process should be as follows ;
In Fig. 12, the component of F along the jz-axis is F times
the cosine of 9; the component of F along the x-axis is F
times the cosine of the other angle, which is F times the sine
of 0; that is, Fg=F&y&9; F,=Faae. Similarly, in Pig.
13, F» = FeoB^; Ff=: — Fsin^ (minus, because it pulls
backward along that line).
The components of velocities, accelerations, or any other
vector quantities are to be handled in the same way.
13. Every problem should be accompanied by a sketch or
diagram, to show that the student understands the meaning
of each step of his work. And in many cases, an accurate
graphical solution on a drawing board may be used as a valu-
able check on the correctness of the nmnerical computation.
32 TBIGOKOMBTBY.
14. Note. That portion of trigonometry which has been
outlined up to this point is so elementary in character, and so
readily understood and appreciated by the student, thai it
ma/y well be introduced much earlier in the course than is
usually done — perhaps even as early as the elementary course
in plane geometry.
CHAPTER II.
The Trioonometric Functions of Any Angle.
16. Angles in general. — ^An angle, as the term is used in ap-
plied mathematics, is the amount of rotation of a moving
radius OP about a fixed point 0, measured from a fixed line
Fio. 14.
OX. Here OX is called the initial line and OP the terminal
line of the angle. Counterclockwise rotation is positive, and
angles are added and subtracted as algebraic quantities. The
quadrants are numbered as in the figure; an '^ angle in quad-
rant 77" for example, means an angle whose terminal line lies
in quadrant 77.
16. Congruent angles are angles differing by any multiple
of 360^
17. Complementary angles are angles whose sum is 90°;
supplementary angles are angles whose sum is 180°.
18. Units of angular measurement are: the degree, sub-
divided into minutes and seconds, or decimally; the grader
33
34 TBIGONOMETBY.
subdivided decimally; and the radian, subdivided decimally*
1 degree = 1^ = l/90th of a
right angle;
1 grade = 1/lOOth of a right
angle (used in France) ;
1 radian = angle subtended by
Fio. 15. an arc equal to the radius.
Since ratio of semi-circumference to radius ^ir (where
ir= 3.1416 ••• =3% approximately), we have
TT radians = 180®, and hence 1 radian = about 57.3*".
19. The radian is especially important in problems concern-
ing the motion of a particle in a circular path. Thus, if
r ft. ^radius of the circle,
s ft. ==^ length of arc traversed, and
tf radians = angle swept over by the moving radius, then
s = r0.
This important equation is not true unless the angle is meas-
ured in radians. Again, if
V ft. per sec. = linear velocity of the particle in its path, and
CD radians per sec. = its angular velocity, then
v = r<i>.
Further, if the angular velocity = a> radians per sec. = N
rev. per min., then the relation between the numbers a> and
N is given by
0)
30'
In all higher mathematics, when a letter is used for an
angle, without designating the unit, it is understood that the
letter means the number of radians in the angle.
TRIQONOMETBY.
35
20. Definition of sine, cosine, and tangent of any angle, —
Let X be any angle, swept over by a moving radius revolving
from OX to OL, and suppose for convenience of language that
OX extends horizontally to the right. Assume, for the moment
that OX and OL are not perpendicular. Prom any point P
of the moving radius drop a perpendicular on the initial
line (or the initial line produced), thus forming a right tri-
Pio. 16.
#
angle, called the triangle of reference for the given angle x.
In this triangle, the perpendicular MP is called the side oppo-
site 0, and is positive if it runs up, negative if it runs down ;
the base OM is called the side adjacent to 0, and is positive if
it runs to the right, negative if it runs to the left, and the
radius OP is called the hypotenuse of the triangle and may
always be taken as positive. The sine, cosine and tangent of
the angle x are then defined as follows :
side opp.
sm X = -r T^
hypot.
cosa; =
side adj.
hypot.
tana; =
_ side opp. _ sin x
"" side adj. "" cos a?
These ratios are positive or negative numbers, depending
only on the position of the terminal side of the angle x, and
36 TRIGONOMETBY,
are called trigonometric functions of x. The functions of any
angle congruent to x are the same as the functions of x^ so
that we need consider only the angles in **the first revolu-
tion," that is, angles between 0** and 360**.
21. To trace the changes in each function as the angle
changes from 0** to 360 "*, draw the figure so that the denomi-
nator of the ratio is kept constant, say equal to 1 inch, and
trace the changes in the numerator (Fig. 17 for the sine and
cosine ; Fig. 18 for the tangent) . Obviously, the sine will be
positive for angles in the upper quadrants; the cosine will
be positive for angles in the right hand quadrants; and the
tangent will be positive in quadrants I and III.
The definitions of the functions of 0% 90**, 180% and 270%
which were not included above, can now be readily obtained
by noting what becomes of the function of a variable angle
X when x approaches one of these values as a limit.
In using the ** circle of reference" be careful to have every
angle start from the initial line that extends horizontally to
the right.
Other Trigonometric Functions.
22. Definition of other trigonometric functions. — ^Besides
the sine, cosine, and tangent, other functions in common use
are the cotangent, the secant, and the cosecant, which are
most conveniently defined thus :
1 1 1
cot a; = 7 , secaj = , cscaj==-; — .
tan X cos x^ sm x
Less important, but often convenient, are the versed sine and
the coversed sine:
vers a? = 1 — cos x, covers a; = 1 — sin x.
23. It is worth remembering that the sine and cosine are
always less than (or equal to) 1, in absolute value; their
reciprocals, the secant and cosecant, are always greater than
(or equal to) 1, in absolute value; the tangent and cotangent
may have any value, positive or negative; while the versed
sine and coversed sine are always positive, ranging from to 2.
IBIOONOMETBY.
37
TlQ. 17.
FlO. 18.
38
TBIGONOMETBY.
Fig. 19.
24. Use of the tables: reduction to the first quadrant. — The
tables in comiuon use give the values of the functions only
for angles between 0° and 90**, that is, only for angles in the
first quadrant. To find the functions of an angle x in one
of the other quadrants, find first the "reduced angle*' in
quadrant I (that is, a; — 90% or «— 180% or a; — 270**), and
then proceed as in the following examples:*
(a) To find coso;, when a; is in quadrant 77. Draw any
angle in quadrant 77 to represent the angle x (avoiding,
however, lines near the middle of the
quadrant) and draw the ** reduced
angle" x — 90** in quadrant 7. Then,
pointing at the figure, cos x is this line
(VW) [divided by the radius], which
is the same in length as this line (^)
[divided by the radius], which is the
sine oi X — 90**; but the first line is
negative ; hence
cosa; = — sin {x — 90°),
where sin (a; — 90**), of course, can be found in the table.
(6) To find tanx^ when x is in quadrant 77. Pointing at
the figure, tan x is this line ( < ) divided
by this line (|||), which is the same as
this Kne (VVV) divided by this Une (M)i
which is the cotangent of {x — 90**);
but the signs are unlike; hence
tan a; = — cot {x — 90**),
where cot {x — 90**) can be found from
the table.
Similarly for any other case.
25. The converse problem of finding the angle correspond-
ing to any given function is complicated by the fact that there
will be (in general) two angles between 0** and 360*^ corre-
sponding to any given function. The most satisfactory way
* The given angle is supposed to be already reduced to an angle be-
tween 0** and 360".
Fig. 20.
TBIQONOMETEY.
39
to find these two angles, in any numerical case, is to draw
the figure, and proceed as in the examples below, in which rco
in each case represents an angle in the first quadrant which
can be found in the table.
Fig. 21.
Given sin a? =^0.5;
x = Xq or 180° — Xq.
Fig. 23.
Given cos a; =0.8;
a;=a;o or 360° — ajo-
Fig. 25.
Given tan a? =0.8;
x=Xq or 180° + Xq.
Fig. 22.
Given sin re = — 0.5;
a; = 180°+a?o or 360'
X,
Given cos re = — 0.5;
aj=180° — iTo or 180°+a;<^
Fig. 26.
Given tan 3?=: — 0.8;
a;=180°— rco or 360°
— Xa.
40 TBIGONOMBTBY.
These results are not forxnulae to be memorized; it is much
safer, and more intelligent, to draw the appropriate figui^e,
or to visualize it in the mind, for each case as it arises. The
student should be thoroughly drilled in numerical cases,
especially for angles in the second quadrant.
Notice that an angle is completely determined when we
know the value of any one of its functions, and the sign of
any other function (not the reciprocal of the first).
It we restrict ourselves to angles between 0° and 180®,
as in the case of angles in a triangle, then an angle is wholly
determined by either its cosine or its tangent ; but there will
be two angles, x and 180° — x, corresponding to a given sine.
26. The functions of certain angles in the later quadrants,
corresponding to 30®, 45®, and 60® in quadrant I, may be
found exactly, without the use of the tables, by inspection
of the figure (see § 4).
For example, cos 120®= — i.
27. If it is required to find the remaining functions of an
angle when one function is given, draw a right triangle and
proceed as in §5, considering only the absolute values of
the quantities, without regard to sign; then adjust the sign
of the answer according to the quadrant in which the
angle lies. Or, the angle may be drawn at once in the proper
quadrant.
Solution op Obliqub Triangles.
28. In any plane triangle the following theoriems are easily
proved from a figure :
(1) The ''Law of Sines.*' — ^Any side is to any other side as
the sine of the angle opposite the first side is to the sine of the
angle opposite the other side ; in the usual notation :
•
a __ sin A
6"" 5^5'
with two analogous formulae obtained by '^ advancing the
letters. ' '
TBIGONOMETEY. 41
(2) The ''Law of Cosines.'* — ^The square of any side is
equal to the sum of the squares of the other two sides, minus
twice their product times the cosine of the included' angle:
a* = 5*-j-c* — 25c cos ii,
with two analogous formulae obtained by ''advancing the
letters. ' '
These two laws, with the fact that the sum of the angles is
180°, sufiSce to ''solve" any plane triangle, and are important
in many theoretical considerations.
The following formulae which are especially* adapted to
logarithmic computation, give the tangents of the half-angles
in terms of the sides, and are included here for reference :
, A
tan-rr
2 « — a'
where
and
tan- =
2 « — c
a+b + c
«= 7i
= ^-^ — ?:1j^ L}d — ?2 as radius of inscribed circle.
From these formulae we have at once.
Area = r5= V^(^ — ^) (^ — &)(^ — c)-
29. The only case which is likely to give any difiSculty, is
the "ambiguous case" in which the given parts are two sides
and the angle opposite one of them. Here we must remem-
ber, at a certain point in the work, that when the sine of an
angle is given, there will be, in general, two angles corre-
sponding to that sine, one the supplement of the other; so
that from that point on, the problem breaks up into two
separate problems. But if the sine of an angle is 1, then
the only value for the angle is 90° ; and if the sine is greater
than 1, there is no corresponding angle, and the problem is
impossible. It is advisable to construct a fairly accurate
figure.
44 TBIGONOMBTBY.
This method requires the memoriziiig of no rules or for-
mulae, besides the definitions of the functions; a yery little
practice will develop all the speed and accuracy that can be
desired, and the method is one which is readily recalled to
mind after long disuse. The special case of complementary
angles, however, is worth remembering as a separate formula :
Any function of (90® — 35)= the co-named function of x.
Formulas fob the Sum op Two Angles, Etc.
34. In simplifying trigonometric expressions which occur
in calculus, mechanics, etc., the following formula are so fre-
quently required that they should be thoroughly memorized.
The ability to recognize those relations readily, regardless of
the special lettering employed, is a necessary condition for
rapid progress in almost any branch of analysis, but it is
highly undesirable to extend the list beyond the limits here
given.
The fundamental formulae from which all others are derived
are these two, the proof of which is obtained from a figure :
(1) sin (a? + 3/) = sin a? cosy + cos a; sin y,
(2) cos (a? + y) =cosa?cosy — sino^sinj/.
^ - . « -
These and the following formulae should be memorized in
words, not in letters : thus, ''the sine of the sum of two angles
is the sine of the first times the cosine of the second, plus the
cosine of the first times the sine of the second," etc.
Dividing (1) by (2) and then dividing numerator and de-
nomerator by the product of the cosines, we have
xoN ^ • . \ tan a? -f tan y
(3) tan (a? -f y) = = j. j. -
Changing the sign of y in these three formulae, and remem-
bering the relations for negative angles, we have the corre-
sponding formulae for sin (a; — y), cos (a; — y), tan (a; — i/),
which will be exactly the same as (1), (2), and (3) with all
the connecting signs reversed:
TBIGONOMETBY. 45
(4) sin {x — y) =smrDcosy — cosrDsiny,
(5) cos (a; — y } = cos a; cos y -{- sin a; sin y,
^ ^ w«iva^ y; i + tan* tany
Putting x=y in (1), (2), and (3) we have at once
(7) sin 2a; = 2 sin a; cos rD, c^ ^ >.
(8) cos2x = cos*a; — sin'a?
2tanx
(9) tan 2z =
1 — tan* x'
Solving (8) first for sin a; and then for cos re, and putting
2a;=y, or x=y/2, we find
i'-i/^N • y "^_i_ /IJ— COS y ^
(10) gin| = ±^-^-^,*
J
i'-i-iN ^y _i_ /l + cosy
(11) co^^^ztyj ^-J.
whence,
i'-iftN ^ y _i_ /I — cos y
(12) tan^ = ±^/:i—; ^.
^ "^ 2 A/l + cosy
This last formula may be transf ormed, by rationalizing
numerator or denominator, into
^ « l-cosy _ riny_
2 sin y 1 + cos y
Other formulas, useful for special purposes, should not be
memorized, but should be derived as needed.
36. In proving the identity of two trigonometric expres-
sions, it is best to reduce each expression separately to its
simplest form.
* The plus sign ia to be used when sin }y is positive, the miniu sign
when sin iy is negative. Similarly in the next two formulas.
4
*\
46 TBIGONOMBTBY.
The fallacy of supposing that because a true relation can be
deduced from a given equation, the given equation is there-
fore necessarily true, should be carefully explained.
For ezample, from the false equation 3 = — 3 we can obtain the tme
equation 9=: 9 by squaring both sides; or, from the false equation
30® = 150® we can obtain the true equation % =: % by taking the sine
of both sides; but in each of these cases tiie step taken is not reversible.
36. The following device for transforming an expression
of the form a cos a; -|- ^ sin a; is often useful :
a cos X + 6 dn a; = Va' + ^ [ ^(/^ y^ co8» + ^J_^^ ^^ dn x^
= J.cos (a; — -B),
where A = i/(a* + 6') and tan 5 « -.
37. The inverse functions.
The angle between — 90® and +90® whose sine is a; is de-
noted by sin-^ x.^
The angle between 0® and 180® whose cosine is a; is denoted
by cos"^ X.
The angle between — 90® and +90® whose tangent is x
is denoted by tan-^ x.
In simplifying expressions involving these ''inverse func-
tions/' it is well to take a single letter to stand for each in-
verse function ; as, y = sin-^ x, whence, by definition, sin y=x ;
etc.
38. Solution of trigonometric equations. Many trigonomet-
ric equations can be solved only by the ** method of trial and
error" (see chapter on numerical computation). In other
cases, however, it is possible, by the use of the formulas given
above, to transform the given equation into a form involving
only a single function of a single angle ; if this equation can
be solved for the function in question, then the required value
(or values) of the angle can be found from the tables or.it
can be shown that no solution exists.
*The STmbol sin*^ x (or arc sin a;) is often defined as simply ''the
angle whose sine is a;"; but since there are many such angles, it is neces-
sary to specify T^ch one is to be taken as "the" angle, if the li^Tmbol
is to have any definite meaning.
A SYLLABUS OF ANALYTIC GEOMETRY.
This syllabus is intended to include those facts and methods of ana-
lytic geometry which a student who has completed an elementary course
in that subject should have so firmly fixed in his memory that he will
never think of looking them up in a book.
A course of study in analytic geometry should consist chiefly of
problems solved by the students, each problem being solved on the basis
of a small number of fundamental formulas such as are here mentioned.
This syllabus is confined mainly to the conic sections; but a satis-
factory course in analytic geometry should include also the study of
many other curves, both in rectangular and in polar coordinates. The
syUabus takes up only those properties of curves which can be readily
investigated without the aid of the calculus; but the present tendency
to introduce the elements of the calculus before any elaborate study
of geometry is attempted is to be much encouraged.
Table of Contents.
Chapter I. Bectanoulab Co-obdinates.
Chapter n. The Straight Line: Equations of the Form
Chapter HE. The Circle: Equations of the Form
«* + y' + i>« + ^y + y=o.
Chapter IV. The Parabola: y*=:2px.
Chapter V. The Ellipse: 6V + ay = a*6*.
Chapter VI. The Hyperbola: b*a^ — aV=:a*b\
Chapter YII. Transformation of Co-ordinates.
Chapter VIII. General Equation of the Second Degree in
X and y.
Chapter IX. Systems of Conics.
Chapter X. Polar Co-ordinates.
Chapter XI. Co-ordinates in Space.
47
CHAPTER I.
BECTANQULAB COORDINATES.
1. In many geometrical problems it is convenient to describe
the position of a point in a plane by giving its distances from two
fixed (perpendicular) lines in the plane.*
For example, on a map, the distance of a point to the east or
west from a fixed meridian is called the longitude of the point, and
its distance north or south from the equator is called its latitude.
So in general, in any plane, the distance of a point to the right
or left from a fixed vertical axis is called the abscissa, x, of the point,
and its distance up or down from a fixed horizontal axis is called
its ordinate, y. The x and y together are called the coordinates of
the point.
*» The value of a;(=Oilf) will be positive to the ^
right, negative to the left; the value of y (=zMP)
will be positive upward, negative downward. The ^^
point for which x = oci and y = yi is denoted by -
Pi,OT(xi,yi).
M
2. To express the distance between two points in terms of their
coordinates : — -^
D=y/(x, — xir+ (2/a — 2/1)'.
3. To find the coordinates of the point half way between two
given points:
X = i(xi + X2),
y = Kyi + y^)'
* We restrict ourselves here to rectanguLar axes; oblique axes are, however,
oocajsionally useful.
48
ANALYTIC GEOMETRY.
49
4. To find the coordinates of a point P on the line through two
fixed points, and such that its distance from the first point is n
times the distance between the two points y
(PJ> = nP^,):
» = 0^1 + n(x2 — Xi),
y=yi+ n{y^ — yO.
Here n may be any real nimiber (positive, negative, or zero).
5. To find the slope of a line through tivo given points:
m = tan^ = ^^=^\
X2 Xi
The angle ^ is called the inclination
of the line; tan <f> is the slope.
6. If two lines are parallel, their slopes are equal : mj = ^a.
If two lines are perpendicular,
the product of their slopes is minus one:
mints = —1 .
7. To express (he areas of triangles and polygons in terms of the
coordinates of the vertices, consider the trapezoids formed by the
ordinates drawn to the vertices.
8. In any problem involving an imknown point, remember that
two conditions are necessary to determine the coordinates of the point
(simultaneous equations in two unknown quantities).
CHAPTER II.
THE STRAIGHT LINE: EQUATIONS OF THE POBM AOJ + -By -j- C = 0.
9* We have seen that if two conditions are imposed on x and y,
the position of the point (x, y) is wholly determined. If only one
condition is imposed, the position of the point is only partially
restricted. (Examples : a; = 5, x* + ^ = 25, etc.)
The collection of all points which satisfy a given condition im-
posed on X and y is called the Iocils of that condition; and the con-
dition itself, expressed in algebraic language, is called the eqiuttion
of the lociis. Thus, the equation of a straight line is the algebraic
expression of the condition which x and y must satisfy in order that
the point (a;, y) shall lie on the line; in other words, the eqvxUion of
a line is an equation which is true when the coordinates of any point
on the line are substituted for x and y, and false when the coordinates
of any point off the line are substituted for x and y; and so in general
for the equation of any locus.
10. To find the coordinates of the points of intersection of
two loci whose equations are given, we have simply to find the
pairs of values of x and y (if any) which satisfy both the equations
at once (simultaneous equations in x and y).
11. To find the equation of a line (not perpendicular to either
axis), when its slope, m, and the coordinates of
one of its points {xi, yi), are given:
y — yi = M^ — a?i)-
The equation of a line perpendicular to
the X-axis (or the y-axis) is, by inspection,
a; = a (or 2/ = 6). «
The equation of any straight line b of the form Aa; + By + C = 0, and
the locus of every equation of the form Ax + By + C — ba straight line.
Hence, to plot the locus of such an equation, it b sufficient to find any two of
its points.
12. To find the slope of a line whose equation is given (the line
being not perpendicular to an axis), write the equation in the
form j/=( )a;+( ); then the coefficient of x will be the slope.
50
AKALYTIO GEOMETRY.
51
13* To find the equation of a line parallel or perpendicular to a
given line and through a given point, remember that mi = m^ for
parallel lines, and mirn^ = — 1 for perpendicular lines (see § 6).
Special method: if the given line is Ax + By + C ^=s 0^ then the parallel
\b Ax + By ^=^k and the perpendicular is Bx — Ay s= K, where k and K are to
be determined.
14. To find the angle between two lines
whose slopes are given:
tan ^ = ,"^~"" .*
1 + mjmi
15. To find the distance between a given point (xq, yo), and a
given line:
(a) When the inclination of the line,
<f>, and the coordinates of one of its
points, (xi, ^i), are given, we have
from the figure :
QPo=(Xii —xi) sin <l>—(yo—yi) cos 4>,
(6) When the equation of the line is
given in the form Ax + J5y + C = 0, use the following formula if
Axq -^Byo + C
\/A^ + B'
Here the vertical bars mean " the absolute value of."
D =
* Proof: By trigonometry, tan (0, — ^) = tan»» — tan^i^
1+ tan ^ tan ^i
t Proof: Show that the foot of the perpendicular from Po to the line
Ax +By + C ^0 has the coordinates a^i s a:^ — hA, ytsssyo — hB, where
h^ziAxo + Byo+Q/iA^ + B*).
CHAPTER III.
THE CIBCLB I EQUATIONS OF THE POBM 7^ + y^ -{-Dx -|- JS?y + F = 0.
16. The equation of a circle is the algebraic expression of the
condition which x and y must satisfy in order that the point {x,y)
shall lie on the circle (see § 9 and § 10).
V
17. To find the equation of a circle when its radius, r^ and the
coordinates (a, ^) of its centre are given:
When the centre is at the origin (0, 0); this
equation becomes
a:* -j- y* = r*.
18. The equation of any circle is of the form x^+y^+Dx+Ey+F=sO.
Conversely, every equation of the form j^ + y' + Dx + Ey + F =0 can be re-
D El
duced to the form (x + -^y + (y + -j)' « -^D' + -B» — 4F), and therefore rep-
resents a circle with centre at ( — D/2, — E/2), or a single point, or no locus, ao*
cording as />" + ^ — 4F is positive, zero, or negative. When we say, in brief,
that the locus of any equation of the form a^ + 1/* + Dx + Ey +F =sOiatk
"circle," we must understand that the "circle" may be "real," "null," or
"imaginary."
19. To find the centre and radius of a circle whose equaJtion is
given, do not use a formula, but *' complete the squares" of the terms
in X and y in each case, and compare with the standard equation in
the manner just indicated.
20. In problems concerning tangents to a circle, use the fact that
the tangent is perpendicular to the radius drawn to the point of con-
tact.
52
CHAPTER IV.
THE pababola: ^ = 2px.
21, Definition: The locus of a point which moves so that
its distance from a fixed point
its distance from a fixed line
= 1
is cdled a parabola.
The fixed point is called the focua and the fi^ed line the directrix.
The line perpendicular to the directrix
through the focus is called the principal
axis. There is evidently only one point
of the principal axis which is also a point
of the curve, namely the point half way
between the focus and the directrix; this
point is called the vertex.
22. If we take t^e vertex as the ori-
gin and the principal axis as the axis of x^
the equation of the parabola is
y^ = 2px,
where p = the distance between focus and
directrix.*
23. The form of the curve is therefore that shown in the
figure, t By definition PF = PH for every point P on the curve.
The breadth of the curve at the focus is called the lattLS rectum,
and is equal to 2p.
M
u
-^
/.
r
J>
(.
i
i«
IT
\
e
k
s.
* proof: If («, y) is any point on the curve, then
Many British authors write the equation in the form ^ ss 4ax, to avoid
fractions. Other writers use ^ = ^px for the same purpose; this latter form,
however, is unfortunate, since 2p is a fairly well-established notation for the latus
rectum in each of the conies.
t Thus when x is 0, y is 0. When x increases, y increases, plus and minus;
the curve is symmetrical with respect to the x-axis. When x is negative, y is
imaginary. Whenx » p/2, y ss ± p; whenx a8 2p, y = rt 2p,
53
54
AITALYTIC GEOMETBY.
24. To find the eqtuUum of a tangent to the parabola y^ = 2px,
use one of the following formulas :
(a) When the point of contact, (x^, t/J , is given :*
ViV = P(x + Xi) ;
When the slope, m, is given :t
(6)
y = mx +
2m'
A line perpendicular to a tangent at the point of contact is
called a normal.
If the tangent and normal at
any point P meet the principal
axis at T and N, the projections
of PT and PN on the principal axis
are called the subtangent and sub-
normal, respectively. The subtan-
gent is bisected by the vertex. The
subnormal is constant, equal to the
semi-latus rectum, p.
25. The locus of the middle points of a
set of parallel chords in the parabola is a
straight line parallel to the principal axis; such
a line is called a diam^er. In the parabola
y^ = 2px, if the slope of the parallel chords is m,
then the equation of the diameter is t/ = p/m,X
* Proof: Let Pa = (xi + h, yi + A;) be a second point on the curve, near Pi;
then the dope of the tangent at Pi will be the limit of k/h as P9 approaches Pi
along the curve, namely m = p/pi. Then use S 11. The slope of the curve maj
also be found by the general method of the differential calculus.
t Proof: Determine p so that the line y=imx + p shaU have only one
point in common with the curve. [Bemember that a quadratic equation
Ax' + Bx + C = will have equal roots if B*— 4i4C=0.]
t Proof: Let (Xq, y^) be any point of the required locus; find the points of
intersection of the curve and a line through {Xq, y^ with slope m; then
express the condition that (xo, y^ shall be the middle point between these two
points. [Remember that the sum of the roots of a quadratic equation
ila:« + Bx 4. C = is — B/A.]
V
AKALYTIC GEOMETBY.
55
26a. Among the many properties of the parabola which should be worked
out as problems, the following may be mentioned as espeeiallj important, and
easj to remember:
1. The normal at any point P biseets the angle formed
by the line from P to the f oens and the line through P
parallel to the principal axis (parabolic mirror).
2. If Pt, Pa, . . . are any points on a parabola,
the distances of these points from the principal
axis are proportional to the squares of their dis-
tances from the tangent at the vertex.
3. If the tangents at P and Q inter-
sect at T, and if Jf is the middle point of
the chord FQ, then the line through T
and Jlf is a diameter, and the segment
TM is bisected by its point of intersec-
tion with the curve.
4. The locus of the foot of the perpendicular from the
focus on a moving tangent is the tangent at the vertex.
66
ANALYnO GEOMETBY.
5. The loeos of tbe point of intersection of perpen-
dicular tangents is the directrix.
Vote. The usual methods for constructing a parabola should also be given.
CHAPTER V.
THE ellipse: 6V -f- «y = «V*
26. Definition: The locus of a point which moves so that
its distance from a fixed point _
its distance from a fixed line '~
(where e is a constant less than 1), is called an ellipse.
The fixed point is called the focus, the fixed line the directrix,
and the constant, e, the eccerUricUy. The line perpendicular to the
directrix through the focus is called the principal axis. There are
evidently two points of the principal axis which are also points of
the curve; these two points are called the vertices, and the point half
way between them is called the centre.
J)l VF 'C V
I-- ^ — 4 — -dT— 4
27. If we let 2a =: the distance
between the vertices, then .•*
the distance between the centre and either vertex is a;
the distance between the centre and the focus is ae;
the distance between the centre and the directrix is a/e,
28. If we take the centre as the origin and the principal axis
as the axis of x, the eqiuition of the ellipse is
a^^ V ^'
where h is an abbreviation for a ^\ — e^.f Note that b < a.
♦ Proof: Since the vertices, V and F', are points of the curve, VF/VD = t
and V'F/V'D = e; that is,
a^CF _ a + CF _
^g-33^-« and ^^cj)-e,
whence CF as ae and CD = a/e,
t Proof: If (x, y) is any point on the curve, then
V(x + oe)' + (y -^O? _ ^
, a
57
58
ANALYTIC GEOMETBY.
29, The form of the curve is therefore that shown in the
figure.* The right triangle enables us to find any one of the three
quantities a, b, and e, when the other
two are given.
The symmetry of the equation
shows that the curve might equally
well have been obtained, with the same
eccentricity, e, from a second focus and
directrix, shown on the right.
The breadth of the curve at either
focus is called the latiis rectum, and is
equal to 2a(l — ^), or 2b^/a.
30. Let P be any point of the ellipse, F and F' the foci, and PH
and PJ?' the perpendiculars from P to the directrices; then
(a) PF/PH = e and PF'/PW = e,
by definition of the curve. Further-
more ."t
V>)
PF + PF = 2a.
In fact, the ellipse is often defined as the locus of a point which
moves so that the sum of its distances from two fixed points is constant,
31. If a circle be described upon the major axis of an ellipse
as diameter, each ordinate in the ellipse is to
the corresponding ordinate in the circle as b is
to a,X In fact, the ellipse is often defined as
the locus of points dividing the ordinates of a circle
in a constant ratio.
From this property it follows that the area of
an ellipse is irab.
* Thus, when t/ = 0, a; = ± a; when a? a= 0, y = ± 6. The curve is sym-
metrical with respect to both axes. In first quadrant, as x increases, y decreases
(slowly when x is small, and rapidly when x approaches a).
t Proof: PF = e{PH) and PF' = e{PW), so that PF + PF' = e{HW)
= e(2a/e) = 2a.
JProof : In the ellipse, y = - ^c? — a;*; in the circle, y as ^a' — a*.
ANALYTIC GEOMETBY.
59
32. To find the eqtuUion of a tangent to the ellipse ::5 + tj = If
a'
use one of the following formulas f**
(a) When the point of contact, (xi, 2/1), is
given:
XiX
o» -r- 63 -A.
(6) When the slope, m, is given:
y = mx dc ^a^m^ + 6^.
33. The locus of the middle points of a set of parallel chords
in the eUipse is a straight line through the centre; such a line is
X^ t/2
called a diameter. In the ellipse -^-^-j^^l,
if the slope of the parallel chords is m, then
the slope of the diameter is
a^m'
Any two lines through the centre, such
that the product of their slopes is — b^/a^, are
called a pair of conjugate diameters, because
each bisects all chords parallel to the other.
34. The circle described in § 31 is called the auxiliary circle.
If P is any point on the ellipse, and Q the
corresponding point on the auxiliary circle
(see figure), then the angle 4> which CQ makes
with the axis is called the eccentric angle of
the point P. From the figure, and § 31,
x^ a cos<f> and y ^ b sin^^
where x, y are the coordinates of P.
The eccentric angles of the ends of two conjugate diameters
differ by 90^
* Proof as in the case of the parabola.
60
ANALYTIC GEOMETBY.
34ft. Among the many properties of the ellipse that should be worked
out aa problems, the following are eapeciallj easy to remember:
1. The normal at any point P bisects the angle
formed bj the lines joining P with the foci
2. The locus of the foot of the perpendicular from
the focus. on a moving tangent is the circle on the major
axis as diameter.
3. The locus of the point of intersecti on of pe r-
pendicular tangents is a circle with radius Va' + &'•
4. The area of a parallelogram bounded hy tan-
gents parallel to conjugate diameters is constant.
Note, The usual methods for constructing an ellipse should also be given.
\
CHAPTER VI.
THE HYPERBOLA I 6V — oV ^ aV.
85« Definition: The locus of a point which moves so that
its difliance from a fixed point ^
its distance from a fixed Une
(where 6 is a constant greater than 1), is called a hyperbola.
The fixed point is called the focus, the fixed line the directrix^
and the constant, e, the eccentricity. The line perpendicular to the
directrix through the focus is called the principal axis. There are
evidently two points of the principal axis which are also points of
the curve; these two points are called the vertices, and the point
half way between them is called the centre.
V" € J>
— *—• Ck- 4,
V fl
36. If we let 2a = the distance
between the vertices, then.** K-— fl'
the distance between the centre and either vertex is a;
the distance between the cevire and the focus is ae;
the distance between the centre and the directrix is a/e.
37. If we take the centre as the origin and the principal asds
as the axis of x, the equaiion of the hyperbola is
where b is an abbreviation for oye* — l.f Note that 6 may be
greater or less than a, or equal to a.
* Proof: Since the vertices, V and V, are points of the curve, VF/VD = s
and V'F/V^D = e; that b,
CF^a . a + CF
jzjz =s e ana — _. ^-. ^ e,
a — CD a -T CD '
whence CF ss ae and CD ss a/e.
t Proof: If ix, y) is any point on the curve, then
V(x~oe)» + (y — 0)» _,
a
X
15 61
62
ANALYTIC GEOMETBY.
€? 6» "" ^•
38. The form of the curve is therefore that shown in the figure.*
The two lines through the centre
with slopes zfc 6/a are called the
CLsymptotes of the hyperbola; the
two branches of the curve ap-
proach these lines more and more
nearly as they recede from the
centre, t The right triangle enables
us to find any one of the three
quantities, a, b, and e, when the
other two are given.
The sjrmmetry of the equation
shows that the curve might equally
well have been obtained, with the same eccentricity, e, from a
second focus and directrix, shown on the left.
The breadth of the curve at either focus is called the lotus rectum,
and is equal to 2a(e* — 1), or 2VJa,
30. Let P be any point of the hyperbola, F and F' the foci, and
PH and PH' the perpendiculars
from P to the directrices; then
(a) PF/PH = e and PF'/PHf= e,
by the definition of the curve.
Furthermore : J
(6) \PF —PF^l =z2a.
In fact, the hyperbola is often defined as the locus of a point
which moves so that the difference of its distances from two fixed points
is constant.
* Thus, when y = 0, x = ± a; when x = 0, or a: < o, y is imaginary; when x
increases beyond a, y increases, plus and minus (most rapidly when x is little
greater than a). The curve is symmetrical with respect to both axes.
V h I c? , & .
t For, the slope - = - a/1 — -^ approaches - as x mcreases; more-
over, if tA is the ordinate of any point on the curve, and 1/2 the ordinate of the
corresponding point on the asymptote, then the difference 7/2 — yi approaches
zero; for y^? — t/i' = 6', and therefore t/2 — yi = b^/(y2 + yi).
t Proof: PF = e{PH) and PF = e{PH% so that IPF--PF'\^ e{HH^
= e {2a/ e) = 2a.
ANALYnO GEOMETBY.
63
40, The product of the distances from any point of a hjrperbola
to the asymptotes is constant. Hence, the hyperbola is often
defined as the locus of a point which
moves 80 that (he jyroduct of its distances
from two fixed lines is constant. (The
distances here may be the perpendicular
distances; or, the distance to each line
may be measured parallel to the other.)
4i. An important special case is that of the "rectangular'!
hyperbola, whose asymptotes are perpendicular
(o ^ 6) ; the eqiuition of the rectangular hyper-
bola referred to its asymptotes as axes is (by § 40)
a'
42. To find the eqiuition of a tangent to the h3rperbola
-5 — |i = 1; ^^'se one of the following
formulas:*
(o) When the point of contact, (x^^, y^),
is given :
^_M— 1.
(6) When the slope, m, is given:
y^mx± ^c?ni? — 6*.
43. The locus of the middle points of a set of parallel chords
in the hyperbola is a straight line through the centre; such a line is
called a diameter. In the hyperbola
"2 — ?2 ^ ^'^^ *^® slope of the parallel
chords is w, the slope of the diameter
a^m
Any two lines through the centre,
such that the product of their slopes is b^/a^; are called a pair of
conjugate diameters, because each bisects all chords parallel to the
other.
* Proof as in the case of the parabola.
64
ANALYTIC GBOMBTEY.
43a. Among the properties of the hyperbola, the following are eai^ to
remember:
1. If a line cuts the hyperbola and its aifjrmptotes,
the parts of the line intercepted between the curve and
the ai^ymptotes are equal. In particular, the portion
of any tangent intercepted between the asymptotes
is bisected by the point of tangency.
2. The area of the triangle bounded by any tan-
gent and the ai^ymptotes is constant.
Note, The usual methods of constructing a hyperbola— especially the
rectangular hyperbola — should be given.
CHAPTER VIL
tt
TRANSFORMATION OF COORDINATES.*
44. The equation of a curve can often be simplified by a
change of axes,'' either changing to a new origin (xq, yo)>
or turning the axes through an angle Oy or both.
If {Xy y), (x', 2/'), {x^', y"), are the coordinates of the same
point P, referred to three different sets of axes, as in the
figures, then
t1
V'
X
X — ^ Xq "J" X
*▼*
% *
%
%
%
%
%
•
•
%
\
%
Q. \B
z
"^^
X
a; = a;'' COS tf — y" sin tf t
2/ = a?"sintf + 2/"costf
Suppose now that the point P is allowed to move under cer-
tain conditions given by an equation in x and y. The same
condition can be expressed in terms of x' and y' or in terms
of x" and y" by substituting in the given equation the values
of X and y just found. This process is called a transforma-
tion of coordinates, from the axes x, y to the axes x', y', or to
the axes x^^ y" ; and the new equation can often be made sim-
pler than the given equation by a suitable choice of x^^ and
yo> or 6.
* 8ee also the chapter on polar co-ordinates.
t These last formulas are most easily remembered as foUows:
X ^ easterly displacement of F,
= (easterly component of x^) -J- (easterly component of y")
^Qcf toBB — y^ sin ^,
y = northerly displacement of P
^ (northerly component of s^) 4- (northerly component of y^)
=a;^ sin ^ -f- y'^ cos 0.
65
CHAPTER VIII.
GENERAL EQUATION OF THE SECOND DEGREE IN X AND ]/.
45. The general equation of the second degree in x and y
is of the form
Ax^'\-Bxy-\-Cy^ + Dx + Ey-\-F = 0,
By a suitable transformation of coordinates this equation can
always be brought into one or other of the following forms :
A V + CY + -P' = 0, C V + ^'''a? = 0, C V + -P" = 0,
and hence can be shown to represent a conic section, using
this term in a general sense to include (1) an ellipse, which
may be real, null, or imaginary; (2) a hyperbola, or a pair
of intersecting lines; (3) a parabola, or a pair of parallel lines
(distinct, coincident, or imaginary).*
The student should be able to plot readily the locus of an
equation of the second degree in any of the simple cases men-
tioned below — ^these being the cases which occur most often
in practice.
46. To plot Ax^ + Ci/^ + JP = 0, where A and C have the
same sign. Find the intercepts on the axes (by putting
x = and y = 0) ; if both are real, we have an ellipse in which
a = the larger of the two intercepts, and 6 = the smaller; or
if A = C, the ellipse becomes a circle. If both intercepts are
zero, or imaginary, the locus is a single point, or imaginary.
•Proof: If B^ — 44 C is not zero, transform to parallel axes with
origin at {Xq, yo)f and choose Xq and y^ so that the terms of the fLrst
degree in the new equation shall vanish; then turn the axes through nn
angle 0, and choose 6 so that the term in xy shall vanish. If B* — 4AC
= 0, turn the axes through an angle d, and choose $ so that the term in
xy shall vanish; then transform to a new origin (Xq, y^), and choose z^
and i/o so that the constant term and one of the terms of the first degree,
or so that both the terms of the first degree shall vanish. For special
methods of abbreviating the computation in numerical eases see § 55,
note.
66
ANALYTIC GBOMBTEY. 67
47. To plot Ax^ + Cy* + -P = 0, where 4 and C have oppo-
site signs. Unless i^ = 0, one of the intercepts will be real
and the other imaginary, and the curve will be a hyperbola
whose principal axis is the axis on which the intercepts are
real. To find the slopes of the asymptotes, divide by a?* and
find the limit of y/x as x increases indefijiitely . If i^ = 0, the
locus is a pair of intersecting lines.
48. To plot Ci/2 + Da; + JP = 0. Write this as
This is a parabola with vertex at a?o= — -^/A mm! running
out along the positive or negative axis of x. Plotting one or
two points will fix the direction, and comparison with the
equation i/^ = 2pa? will give the semi-latus rectum, p.
49. To plot Ax^ + Cy* + Dx + J5y + -P = 0. Write this in
the form
and ''complete the squares"; then reduce to the form
Ax'^ + Cy'^ + JP= by an obvious change of origin.
50. To plot Cy* + Dx + -By + P' = 0. Complete the square
of the terms in y and reduce to the form Cy* + -P^ + -^=0
by an obvious change of origin.
51. To plot Bxy + JP = 0. This is a rectangular hyberbola
referred to its asymptotes (see §41). The equation
Bxy-\-'Dx-\-Ey'\-F^=-^ can be reduced to this form by
moving the origin to a;o= — -B/B, 3/0= — D/B.
52. If the equation to be plotted does not come under any
of the forms just considered, a fair idea of the position of the
curve may be found by the following very elementary method.
Solving the equation for y in terms of Xy we have, if C is not
zero, '
^ 2C-^2C^^'
where X is an expression involving x alone. Finding the
68 ANALYTIC GEOMETBY.
values of — (Bx + E)/2C, and adding and subtracting the
values of yJX/iCy for various values of a?, we can find as
many points (a;, y) on the curve as we please. Or, again,
solving for x in terms of y, we have, if A is not zero,
where 7 is an expression involving y alone. From this equa-
tion we can find values of x corresponding to as many values
of y as we please.
This method is very easy to remember, but does not give
jreadily the exact dimensions of the curve.
63. The center of the curve will be the point of intersection
of the two lines
2iia? + B2/ + I> = 0, ^
Bj? + 2Cy+J5 = 0,
except when B^ — dLlC = 0, in which case these lines will be
parallel, and the curve has no center.
54. The slopes of the lines joining the origin with the infi-
nitely distant points of the curve (if any) are given by writ-
ing tiie terms of the second degree equal to zero :
Ax"" + Bxy + Cy^ = 0,
dividing through by x^ (or y^), and solving for y/x (or x/y).
65. If a more detailed discussion of the curve is required,
it is best to follow the special methods of reduction given in
the text-books (compare foot-note in §45).t
66. The student should be familiar with the geometric
proof that all the ^^ conic sections " can be obtained as plane
* The student of the calculus will recognize these equations as
ay/aa; = and aF/ay = o,
wheie
is the equation of the curve.
t The resulting formula are given here for reference, although the
problem is not one of common occurrence.
Bequired, to plot the equation Aa^ -f- Bxy -f- Cy" -J- Pa? -J- Ey -f- F = 0.
ANALYTIC GEOMETBT. 69
sections of a right circular cone. It is a profitable exercise
to construct a cone, given the vertex and a hyperbolic section.
It should also be made thoroughly clear why an elliptic sec-
tion is a symmetrical figure instead of egg-shaped.
Case I. Central canie. If B* — 4^C is not zero, transform to tbe
eenter as a new origin:
then turn the axes through a positive acute angle B given by
tan 20 = B/{A — C).
The transformed equation wiU be
where F' = Dxo/2 + Epo/^ + F, while A' and C are found by solving
the equations A' + C'=A + C, ^' — C' = ± V(4 — C;)» + jB», where
the sign before the radical is to be + or — aeording as B is positive
or negative. The reduced equation can be plotted as in §§46, 47.
Case H. Pardbolio type. If B' — 4^C^0, the equation may be
written in the form {ax + ep)* + Dx + By + F^sO, where a = VA
while c=VCorc = — VC according as B is positive or negative. The
locus will be of the parabolic type. Take as a new axis of x* the line
ax + oy + m = 0,
where m=(aD + eE)/2(A + C), and choose the positive direction
along this line so that it shall make a (positive or negative) acute angle
with the axis of x. This line will be the principal axis of the curve.
Two subcases may now occur.
(a) If a/o is not equal to D/E, take as axis of y* the line
ex — ay + n =0,
where n=(A + C) (m" — F)/(aE — oD), This line will be the tan-
gent at the vertex, and the transformed equation will be
y^' = 2pa/, .
where 2p= (cD — aE)/V{A + cy. The locus is a true parabola.
Of) It a/e=sD/E, the equation referred to the axis of x' will be
which represents a pair of distinct, coincident, or imaginary parallel
lines.
CHAPTER IX.
SYSTEMS OF GONICS.
57. If TJ and V are expressions of the second degree in x
and y, the equations 27=0 and 7=0 will represent conies;
then (a) the equation 17 -{- %7 = 0, where k is any constant,
will represent another conic passing through all the points
of intersection of the first two, and having no other points in
common with either of them; and (&) the equation 277^0
will represent a curve made up of the two conies 27 = and
7=0 taken together. Corresponding theorems hold good if TJ
and 7 are any expressions in x and y (not necessarily of the
second degree).
58. To find the equation of a conic through five points, let
w = and t; ^ be the equations of the lines PiP, and PJP^^
and let w' ^ and v' = be the equations of the lines P^Pz
and P2P4. Then uv + ku'v' = 0, where k is any constant,
will be the equation of a conic through these four points. It
remains to determine k so that this conic shall pass through P^.
59. The equation
where & is an arbitrary constant, represents a family of con-
focal ellipses and hyperbolas, which intersect at right angles.
70
CHAPTER X
POLAB COORDINATES.
This chapter, placed here for convenience of reference, may
well be introduced, in teaching, much earlier in the course.
60. It is often convenient to represent the position of a
point P by giving the angle, ^, which the line through and
P makes with the ^r-axis, and the distance, r, from to P along
this line. The angle ^ is called the vectorial angle, or simply
the angle, of the point P, and is measured from the positive
direction of the axis of x to the positive direction of the line
through and P, The distance r=OP is called the radius
vector of the point P, and is positive or negative according as
it runs forward or backward along the line through and P.
It is customary to take r positive, and let 4> range from 0°
to 360^.
61. From the figure,
a; = r cos ^, y = r sin <f>,
a?* + 2/^ = r^, y/x = tan <l>.
By the aid of these relations, we can trans-
form any equation from rectangular to
polar coordinates, and vice versa.
62. The polar equation of a conic, referred to the focus as
origin, and the principal axis as axis of x
(see figure) is
P
r =
1 — e cos '
where p is the semi-latus rectum, and e the
eccentricity.
63. Plotting curves in polar coordinates is an excellent
exercise in reviewing the trigonometric functions. The work
should be so arranged that no critical value of the function
occurs between two successive assigned values of 6.
71
CHAPTER XI.
Coordinates in Space.
64. Four methods are in use for representing numerically
the position of a point in space. If Ox, Oy, Oz are three mu-
tually perpendicular axes, the position of any point P may
be determined by :
(1) Rectangular coordinates, ^, y, z;
(2) Polar coordinates in space, r, a, p^ y, where the angles
a, p, y are subject to the restriction cos* a -j- cos* p + cos* y = 1 ;
(3) Spherical coordinates, r, <^, tf, where <^=the latitude of
P, and its longitude :
(4) Cylindrical coordinates p, 0, z.
The relations between the various sets of coordinates are as
follows :
a? = r cos <^ cos ^ p = r cos ^,
y = r cos ^ sin p* = a;* + 2/^>
21 = r sin ^ ic* + 2/* + ^^ = ^«
As there is no well-established uniformity in the use of the
letters in spherical coordinates, or in the choice of the positive
directions along the axes, it is important, in reading any
72
ANALYTIC GEOMETBY«
73
author^ to note, on a fi^re, the exact meanings of the letters
he employs.
65. Distance between two points, in terms of their coordi-
nates:
P1P2 = V{x, — x,y + {y, — t/,)» + («, — z,y.
66. Angle ^ between two lines whose direction cosines are
given:
cos iff = I1I2 4" ntjin^ -(- Witij,
where l^ = cos oi, m^ = cos ft, n^ = cos yi, etc.*
67. Equation of a plane:
Ix + my + nz = p,
where p = perpendicular distance from
the origin, and I, m, n = the direction
cosines of the normal to the plane, t
Every equation of the form
j^x -{- By -{- Cz '\- B =^Q represents a
plane ; for, it can be thrown into the form
lx-{'my'\-m=p by dividing through by V-^* + -8* + C^*-
* Proof: let (1) and (2) be lines through the origin, parallel to the
given lines; on these lines take points Pi, P, at a distance r from the
origin; then
PiP,« = f»4-f» — 2fr cos ^=(Wi — rZa)»+ {m^-^ rm;)* •\' (nH — m,)*.
t Proof: The foot of the perpendicular is A'=: (pZ, pm, pn) ; take N'
=3 (2pl, 2pm, 2pn) and express the condition that the point {x, y, 0)
shall be equidistant from and N\
74 ANALYTIC GEOMBTBY.
68. Equation of sphere with center at the origin :
69. Equation of ellipsoid, with center at the origin :
«*/«* + y V6* + «Vc' = 1.
70. Any equation in x, y, z will represent a surface (real
or imaginary), the form of which can be investigated by the
method of plane sections. Thus, putting x = x^, the equation
becomes an equation in y and 2, which represents a curve in
the plane x = Xj^; similarly for y= y^ and z = Zi.
71. Any equation of the second degree iax,y,z represents
a (real or imaginary) surface of the second degree, or coni-
coid. The types of real conicoids are as follows:
(1) Ellipsoid, with semi-axes a, b, c. Special case: ellip-
soid of revolution, generated by rotating an ellipse about its
major axis (prolate spheroid) or about its minor axis (oblate
spheroid).
(2) Hyperboloid of two sheets. Special case: generated
by rotating a hyperbola about its principal axis.
(3) Hyperboloid of one sheet, or ruled hyperboloid. Spe-
cial case : generated by rotating a hyperbola about its conju-
gate axis. Two sets of straight lines can be drawn on this
surface.
(4) Elliptic paraboloid. Special case : generated by rotat-
ing a parabola about its principal axis.
(5) Hyperbolic paraboloid, or ruled paraboloid. A saddle-
shaped figure, on which two sets of straight lines can be
drawn.
(6) Cone, generated by a straight line always passing
through a fixed point called the vertex, and always touching a
fixed conic, called the directrix. If the directrix is a circle,
the cone is a circular cone (right or oblique). If the vertex
recedes to infinity, t>ie cone becomes a cylinder. On any cone
a single set of straight lines can be drawn.
The student should become familiar with at least the shapes
of these surfaces, through diagrams or models.
Any plane section of any surface of the second degree is a
conic.
A SYLLABUS OF DIFFERENTIAL AND
INTEGRAL CALCULUS.
This syllabus is intended to include those facts and methods of the
calculus which every student who has completed an elementary course
in the subject should have so firmly fixed in his memory that he will
never think of looking them up in a book. The topics here mentioned
are therefore not by any means the only topics that should be included
in a course of study, nor does the arrangement of these topics, as
classified in the following table of contents, necessarily indicate the
order in which they should be presented to a beginner.
Table of Contents,
part i. runonons of a single vabiablb.
Chapter I. Functions and theib Graphical Beprxsentation.
Function and argument. Tables. Graphs.
The elementary mathematical functions.
Continuity.
To find a mathematical function to represent an empirically given
curve.
Chapter II. Differentiation. Bate of Change of a Function.
A. Definitions and notation. Bate of change of a function, or slope
of the curve. Derivatives. Increments and differentials. Higher
derivatives.
B. To find the derivative when the function is given.
Bules for differentiating the elementary functions. Differentiation
of implicit functions, and of functions expressed in terms of a param-
eter.
C. To find the derivative when the function itself is not given; set-
ting up a differential equation.
Useful theorems on infinitesimals.
D. Applications of differentiation in studying the properties of a
given function. Slope; concavity; points of inflexion. Maxima and
minima. Multiple roots. Small errors.
Chapter III. Integration as Anti-Dipperentiation. Simple Dif-
ferential Equations.
Definition of an integral. Constant of integration.
Formal work in integration. Use of tables of integrals. Method of
substitution, and method of integration by parts.
Simple differential equations.
75
76 CALCULUS.
CHAPTEB rV. iNTEOSiLTION AS THE LiMIT Or ▲ STTIC. DXTlKm
INTEOBALS.
Definition of the definite integral of f(x)dx from x:=aio x^h.
Fundamental theorem on the evaluation of a definite integraL
Duhamel's theorem*
Approximate methods of integration: squared paper; Simpson's
rule; the planimeter; expansion in series.
Definite integral as a function of its upper limit.
Chaptsb Y. Applications to Algebka: Expansion in Sebibs; In-
DXm&BMINATB FOSMS.
Taylor's theorem with remainder. Maclaurin's theorem. Im-
portant series.
Theorem on indeterminate forms.
Chapter YI. Applications to Geometry and Mechanics.
Tangent and normal. Subtangent and subnormal.
Differential of arc (in rectangular and polar coordinates).
Badius of curvature.
Velocity and acceleration in a plane curve.
PART n. FUlfCTIONS OP TWO OR MORE VARIABLES.*
* In preparation.
CHAPTER I.
Functions and Theib Qbaphical Bepbesentation.
1. Function and argument. — ^In many problems in prac-
tical life we have to deal with the relation between two variable
quantities, one of which depends on the other for its value.
For example, the temperature of a fever patient depends on the time;
the velocity acquired by a falling body depends on the distance faUen;
the weight of an iron ball depends on its diameter, etc.
In general, if any quantity y depends on another quantity
X, then y is called a function of x, written, for brevity,
y=f{x)f and the independent variable x is called the argu-
ment of the function. More precisely stated, the notation
y=f(x) means that to every value of the argument x (within
the range considered), there corresponds some definite value
of the function, t/; the value of y, or f{x), corresponding to
any particular value re = a is denoted by /(a).
If several values of y correspond to each value. of x, we have what
is called a "multiple valued function of x," which is really a eoUection
of several distinct functions. For example, if y'=:x, then y = ± V^y
which is a double valued function of x.
Any mathematical expression involving a variable a; is a
function of x ; but there are many important functional rela«
tions which cannot be expressed in any simple mathematical
form.
2. A function is said to be tabulated when values of the
argument (as many as we please, preferably at regular inter-
vals) are set down in one column, and the corresponding
values of the function are set down in another column, op-
posite the first. For example, in a table of sines, the angle
is the argument, and the «ine of the angle is the function.
3. A function may also be exhibited graphically, as follows :
Lay off the values of the argument as abscissas along a (hori-
zontal) axis, Ox, and at each point of the axis erect an ordi-
16 77
78 CALCULUS.
nate, y, whose length shall indicate the value of the function
at that point; a curve drawn through the tops of these ordi-
nates is called the curve, or the graph, of the function. It
should be clearly understood, however, that it is the height of
the ordinate up to the curve, rather than the curve itself, that
represents the function.
In plotting the curve for any function, it is important to
indicate on each axis the scale which is used on that axis, and
the name of the unit. For example, if we plot distance as
a function of the time, the units on the y-axis may represent
feet, and those on the a;>axis, seconds.*
The obvious method of obtaining the graph of the snm or differenee of
two functions directly from the graphs of those functions should be
noted.
4. The elementary mathematical functioBS. — In many im-
portant cases, the relation between the function and the argu-
ment can be expressed by a simple mathematical formula.
For example, if « = the distance fallen from rest in the time t,
then s = igt^. In such cases, the value of the function for
any given value of the argument can be found by simple sub-
stitution in the formula.
The most important elementary mathematical functions are
the following:
Algebraic functions: ex, c/x; x^, x*; y/x {x positive).
Here V^ = ^® positive value of y for which y^ = x.
Trigonometric functions: sin x, cos x, tan x (xin radians).
Exponential function: e'' (e = 2.718 . . .).
Logarithmic function: loge x (x positive).
The student should be thoroughly familiar with the curves
of each of these functions, so as to be able to sketch them, or
visualize them, at any moment; many of the essential prop-
erties of the functions can be obtained by inspection of the
curve.
* It is not necessary that the lengths representing the imits of x and y
shall be equal; scales should be so chosen that the completed graph is of
convenient size to fit the paper. In applications to geometry, however
(see Chapter YI), the scales must be equal.
CALCULUS. 79
He should also be familiar with the formulas necessary for
handling expressions involving these functions. The better
drilled the student is in this formal algebraic work, the more
rapid progress can he make in the really vital parts of the sub-
ject. (See chapters of this report on algebra and trigo-
nometry.)
5. Next in importance are the following: the hyperbolic
functions, which are coming more and more into use :
sinh x= (e* — 6"*)/2, cosh x= (e* + ^"*)/2,
tanh x= (e" — e"*)/(e* + e"*) ;
the inverse trigonometric functions:
sin-^a;=the angle between — 7r/2 and +7r/2 radians
(inclusive) whose sine is a;;*
cos-^ir=the angle between and v
(inclusive) whose cosine is x\
tan"^rc = the angle between — 7r/2 and +9r/2
(inclusive) whose tangent is x ;
and the inverse hyperbolic functions:
sinh"^ x = the value of y for which sinh y = re ;
cosh-^rc = the positive value of y for which cosh y = x;
tanh"^ic = the value of y for which tanh y=x.
It should be noticed that the curves for the inverse functions
can be obtained from the curves for the direct functions by
rotating the plane through 180° about the line bisecting the
first quadrant.
Formulas for the hyperbolic functions resemble those
for the trigonometric functions, but the differences are so
* The symbol sin'^ x is often defined as simplj ' * the angle whose sine
is X "; but sinee there are manj such angles, it is necessary to specify
which one is to be taken as ** the " angle, if the symbol is to have any
definite meaning. Thus, if sin a; = ^, a; may equal ir/6, or 5ir/6>, etc.;
but only one of these values, namely ir/6, is properly denoted by the
symbol sin"* i. Similarly for cos"* x and tan"* x; and also for cosh"* x,
which is like Vx in this respect. The conventions adopted to avoid am-
biguity may be readily recalled from the figure, if we note that in each
case the complete curve consists of two or more ''branches," and that
that one is taken as the "principal branch" which passes through the
origin, or which lies nearest the origin on the positive side of the rr-azi?.
80 CALCULUS.
confusing that it is better not to try to memorize any formulas
for the hyperbolic functions, but to look them up whenever
they are needed. (The list in B. 0. Peirce's Table of Integrals,
for example, is entirely adequate.)
6. Continuity. A function y=f(x) is said to be conUnn-
ous at a given point a; = a, if a small change in x produces
only a small change in y; or more precisely, if f{x) always ap-
proaches /(a) as a limit when x approaches a in any manner.
A function may be c^t^continuous at a given point in three
ways: (1) it may become infinite at that point, as y = \/x at
a; = 0; or (2) it may make a finite jump, as t/ = tan-^ (!/«) at
aj = 0;* or (3) the limit Jjfix) may fail to exist because of the
oscillation of the function in the neighborhood of x = a, as
t/=sin 1/x at ir = 0. In each of these cases, the function is,
properly speaking, not defined at the point in question.
A good example of a discontinuous function is the velocity of a
shadow cast bj a moving object on a zig-zag fence.
In what follows, we shall confine our attention to functions
that are continuous, or that have only isolated points of dis-
continuity.
7. To find a mathematical function to represent an em-
pirically given curve. — In many cases the form of the func-
tion is given only empirically; that is, the values of the func-
tion for certain special values of the argument are given by
experiment, and the intermediate values are not accurately
known (for example, the temperature of a fever patient, taken
every hour). In such cases, the methods of the calculus are
not of much assistance, unless some simple mathematical law
can be found which represents the function sufficiently accu-
* This function approaches 7r/2 when x approaches from above, and
— ir/2 when x approaches from below.
CALCULUS. 81
rately.* This problem of finding a mathematical function
whose graph shall pass through a series of empirically given
points is a very important one, which is much neglected in the
current text-books. The complete discussion of the problem
involves, it is true, the theory of least squares, which would
undoubtedly be out of place in a first course in the calculus ;
but an elementary treatment of the problem in simple cases
would be very desirablcf
The curves which are most likely to be worth trying, in any
given case, are these :
y = a'{'bx (straight line) ;
t/ = a + 6a5 + ex* (parabola) ;
y = a + c/(rc + 6) (hyperbola);
i/ = a sin (bx + c) (sine curve) ; and
y=ax^.
In testing this last curve, put y' = log y, re' = log x, and
a'=log a, and see whether y' and x' satisfy the straight line
relation y' = a^ -{-mx^; the use of ** logarithmic squared
paper " greatly facilitates the process.
The student should be familiar with all the possible forms
of these curves, for various values of the constants a, b, c,
and m.
*If no sunple law can be found to represent the entire curve, it is
sometimes possible to break np the curve into parts, and find a separate
law for each part.
t Numerous examples may be foimd in John Perry's ''Practical
Mathematics," and in F. M. Sazelby's "Practical Mathematics"
(Longmans, 1905).
CHAPTER 11.
DiFFEBENTIATION. BatE OF ChANOE OF A FUNCTION.
For the sake of eleameas; this chapter is divided into four parts,
A, B, C, D.
A. DEPINmONS AND NOTATION.
*
8. Bate of change of function; slope of curve. — Given a
function, y = f{x)f one of the most important questions we
can ask about it is, what is the rate of change of the function
at a given instant?
For example, the distance of a railroad train from the starting point
is a function of the time elapsed, and we maj ask, what is the rate of
change of this distance? The answer is, so-and-so many miles per hour.
Again, the volume of a metal sphere is a function of the temperature,
and we may ask, what is the rate of change of this volume? The answer
is, so-and-so many cubic inches per degree.
If the graph of the function is a straight line, then clearly
the rate of change of the function will be constant; for, at
any instant, (change in t/) /(change in x) =the slope of the
line.
If the scales along x and y are the same^ the slope of the line = tan ^,
where is the angle which the line makes with the x axis. If the
scales are not the same, the slope of the line may still be interpreted as
the ratio of the ' ' side opposite ' ' to the ' ' side adjacent ' ' in the triangle
of reference for <f>, provided each side is measured in the proper units.
For example, in the figure, slope = 7/3.
82
CALCX7LUS. 83
If the graph is not a straight line, the meaning of ''rate
of change" at a given instant must be made more precise, as
follows: Consider a particular value, x=Xq; give x an arbi-
trary change, Aa;, and compute the corresponding change in
y, namely, At/=/(iCo + ^^) — /(^o)- Then the ratio Ay/Aa?
may be called the average rate of change of the function dur-
ing the interval from x=XQto x = Xq-\' Ax, (Qeometrically,
Ly/Lx is the slope of the secant PQ in the figure.) Now let Ax
approach zero, so that the interval in question closes down
about the point x = Xq. Then the ratio Ay /Ax unll in general
approach a definite limit, and this limit is called the actual
rate of change at the point x=Xq. (Geometrically, the limit
of Ay /Ax is the slope of the tangent at P.*)
9. Derivatives. The rate of change of a function y=f{x)
at any point, or the slope of the curve at that point, is called
the derivative of the function at that point, and is denoted by
/'(a?), or Dory, or y'.
The. notation y is also used, but only when the independent
variable is the time.
This definition of the derivative of a function as the limit
of Ay /Ax is the fundamental concept of the differential cal-
culus. It is desirable that the meaning of the definition be
made perfectly clear, by numerous and varied illustrations,
before any formal work in differentiation is taken up.
10. Increments and Differentials. — The value Ax is called
the increment given to x, and Ay the corresponding increment
* The sense in w^ch the tangent line is the * * limit ' ' of the secant
lines should be made thoroughly clear. First, the tangent is a fixed
line; secondly, the secant is a variable line, depending on the value
given to Air (that is, for every value of ^, except the value 0, there
?? a corresponding position of the secant) ; thirdly, the angle between the
tangent and the secant can be made to become and remain as small as we
please by taking Ax sufficiently small. The tangent line itself does not
in general belong to the series of secant lines; it is not in any sense the
"last one" of the secants; it is a separate line, which bears a special
relation to the series of secant lines, as described. The student may
readily convince himself that the tangent is the only line through P
that has the property just stated.
84
CALCULUS.
produced in y. The value that Ay would have if the curve
coincided with its tangent (see figure) is called the differential
of y and is denoted by dy.
In case of the independent variable a?, the differential of x
is, by definition, the same as the increment: dx=Ax.
The use of differentials gives us a new notation for the de-
rivative,
_ dy
ra^) =
dx'
Both these notations are in common use.
Notice that Ay and dy are both variables which approach
zero when we make Ax approach zero; dy/dx is a constant,
equal to tan <^; Ay /Ax is a variable, approaching tan ^ as a
limit. Hence we may write :
lim^ = r(x) = D.y==g = tan*,
and
^x^O
Ax
dy=f(x)dx.
These relations between increments, differentials, and deriva-
tives should be thoroughly mastered; they are readily recalled
by the figure. Note especially that Ax and dx are quantities
measured in the same unit as x ; and Ay and dy in the same
unit as y; while the derivative, dy/dx, that is, the slope, is
(in general) measured in a compound unit (like miles per
hour) .
i
CALCULUS. 85
If the lengths representing the units of x and y are not equal, the
slope of the curve, or tan if>, must be understood in the generalized sense
explained above.
The process of finding the derivative, or the equivalent
process of finding the differential of the function in terms of
the differential of the argument, is called differentiation.
11. Higher derivatives. Since the slope of the curve varies^
in general, from point to point, the derivative, f'{x), is itself
a function of x (often called the derived function) ; the de-
rivative of f'{x) is called the second derivative of the given
function, and is denoted by /"(a?), or I>«^y, or y" (or by y in
case the independent variable is the time) ; and so on for the
higher derivatives.
It is also easy to define second, third, . • . differentials, but
they are not of great importance. One matter of notation,
however, should be carefully noticed, namely that d^y/dx^ is
commonly used to denote f (x), that is ^ ^^ — - , and not,
{dxy
As an example where the distinction is important, consider
xz=$ — sin tf and y = 1 — cos 0,
where is the independent variable.
86 CALCULUS.
B. To Find the Debivativb when the Function is Given.
12. Formal work in differentiation. The student should
be thoroughly familiar with the results of differentiating all
the elementary functions. A list of the formulas which should
be memorized is given below; any other formulas should be
worked out as needed, or looked up in a book.
To establish these formulas, first prove the following im-
portant limits :
-. sin Jw ^ J T 1 — cos Jm ^
lim — -. — = 1, and lim = 0,
provided u is in radians ; and
lim^l + ^y=« = 2.718---;
and hence prove the formulas for differentiating the sine and
the logarithm.
The proofs of the other formulas present no difficulty.
* These lunits having been established, it can then be shown that
^ — -. Bin(w + Aii) — sinii ir
lim — ^ — . ^ = 73;;«» u, if v is measnred in degrees,
Au-U) ^« loO
= 008 u, if u is measnred in radians ;
lim '°«<" + f>-^°g" = (0.4343--)J,iftheb«aeialO.
= -, if the base is « = 2.718 • •.
u
The reason for choosing the radian as the unit angle, and e as the base
of the "natural" system of logarithms is the simplification in the
formulas for the derivatives of the sine and the logarithm which results
from this choice.
CALCULUS. 87
Rules for Differentiating the Eusmentart Functions
OF A Single Variable.*
{The first four of these rules are the fundamental ones, from
which all the others can he derived,)
The differential of a constant is zero: —
dk^Q.
The differential of the logarithm to the base e of any function
is one over that function, times the differential of the function : —
d(logea;) = — dx (e =2.718. . .).
The differed|ial of the siNk of any function (in radians) is
the cosine of that function, times the differential of the function:
<i(sin x) = cos x dx.
The differential of the sum [or difference] of two functions
is the differential of the first plus [or minus] the differential of
the second: —
d(u ± y)=du ± dv.
The differential of a constant times any function is the con-
stant times the differential of the function: —
d(kx) = k dpc.-\
The differential of a function to any constant power is the
exponent of the power, times the function to the power one less,
times the differential of the function: —
d(x^) = nx^'^d^.
Useful special cases of this rule are: —
'^^ = 2^'^' ^(i) h^'
The differential of e with a variable exponent is e with the
same exponent, times the differential of the exponent: —
d(e^)=e^dx (e =2.718. . .).
•All these niles remain valid when the word ''derivative" is pnt in
place of ** differential," and the symbol *'D" in place of *'d."
t To prove this and the next ^Ye niles, let y = the function, and take
the logarithm of both sides before differentiating.
88 CALCULUS,
THb differential of the product of two functions is the first
times the differential of the second, plus the second times the
differential of the first: —
d(uv) = udv + V du.
The differential of the quotient of two functions is the denomi-
nator times the differential of the nimierator, minus the numer-
ator times the differential of the denominator^ all divided by
the denojninator squared: —
The differential of the cosine of any function is minus the
sine of that function, times the differential of the function: —
(f (cos 0?) = — sin X dx,*
The differential of the tangent of any function is the secant-
square of that function, times the differential of the function: —
d(teLn x) = sec'ic dx.^
The differentials of the inverse sine, the inverse cosine, and
the inverse tangent, of any function, are given by the following
formulas, which the student may put into words for himself: —
d(sin-* x) = - dxy X (— !«• ^ sin-*a^ ^ Iv)
l/l — x^
d(cos-* a;) = =. dxj (0 ^ cos-^a: ^ v)
d(tan-' x) = --^ dx. (— !» ^ tan-^a: ^ lie)
[To find the differential of u to the trth power, where u and v
are any functions, let
and take the logarithm to base e of both sides before differ-
entiating. — Similarly, to find the differential of the logarithm of
u to any base Vy let
y ~ logt)^> whence "ifl =u;
then differentiate both sides.]
♦Proof: cosa; = siii (iir — x), f Proof: tan a? = fdn ir/cos a?.
J Proof: Let y = sin-*a?, that is,"* sin y = a;; then differentiate both
sides. — Similarly for the next two formulas.
CALCULUS. 89
The rules on these two pages suffice for the differentiation of
any elementary function; they should he carefully memorized.
The differentials of the hyperbolic functions are given by
the following formulas, which are also worth remembering:
d sinh a;=cosh xdx; d cosh x=sinh xdx;
d tanh a; = sech2 xdx;
hence,
d sinh""^ 3?=-' . r, d eosh~^ a;=s — =r , dtanh-^a;™
13. Differentiation of implicit functions, and of functions
expressed in terms of a parameter.
(a) Suppose we have an equation connecting x and y, but
not giving y explicitly as a function of a;; as, for example,
9a;* + 42/* = 36. In finding dy/dx in cases of this kind, in-
stead of first solving the equation for y in terms of x, and then
differentiating, it is usually better to differentiate the equation
through as it stands (remembering that both x and y are
variables) ; thus, in the present example we have
18xdx-{-8ydy = 0, whence, dy/dx = — 9x/4y.
This result can then, if desired, be expressed wholly in terms
of X, by aid of the original equation.
(b) Again, suppose y is given as a functioii of u and v,
where u and v are both functions of x; as, for example,
y=zu^^v sin u. Differentiating both sides by the regular
rules, we have dy = 2udu-]-v cob u du + mn u dv, whence,
collecting the iSrma in du and dvy and dividing by dx,
dv r^ , ^ du , , , ^dv
^=(2« + i»co8«)^ + (8m«)^.
This result shows how the rate of change of y depends on the
rates of change of u and v, which are supposed to be known.
(c) Finally, both x and y may be given as functions of a
third variable, t; as, x=F{t)y y=f{t). To every value of
this auxiliary variable, or ** parameter,*' t, there corresponds
a pair of values of x and y^ so that here again y is indirectly
determined as a function of x. Of course if we can eliminate t
90 CALCULUS.
we shall have a single equation connecting x and y ; but it is
often more convenient to keep the equations in the parameter
form. Thus, to find dy/dx, we have merely to diflferentiate
both of the given equations: dx==F'(t)dt, dy=f(t)dt; and
then divide the second result by the first: dy/dx=f(t)/F'(t),
C. To Find the Derivative when the Function Itself is
NOT Given ; Setting up a Differential Equation.
14. In many cases it is required to find the rate of change
of a function when the function itself is not directly given;
in fact it is often easier to find the derivative of a function
than it is to find the function itself.
For example, a hemispherical bowl of radius r, full of
water, is being emptied through a hole in the bottom ; find the
rate of change of the volume of water drawn off, regarded as a
function of the distance, y, between the level of the water and
the center of the bowl. To compute this value directly from
the definition, we notice first that the increment AV produced
in y by an increment Ay given to y will have a value between
ir(r* — 2/^) Ay and ir[r^ — (y + ^y)^]^y; dividing either of
these values by Ay, and taking the limit of the ratio AT/Ay,
we find at once dV/dy='n' (r* — y^), which gives the re-
quired value of dV/dy for any value of y from y ==0 to y=r.
This process of finding the derivative directly from first
principles, as the limit of the ratio of the increments, when
the function itself is not given, is called ** setting up a dif-
ferential equation," since the result of the process is an
equation between the differentials of the function and of the
argument.*
Every problem of this kind is a problem in finding the
limit of the ratio of two variable quantities, each of which is
approaching zero; and in this connection the following theo-
rems on infinitesimals are extremely useful, if not indis-
pensable.
* The problem of finding the relation between the quantities them-
selves* when the relation between their differentials is known will be dis-
cussed in the next chapter.
CALCULUS. 91
15. Theorems on infinitesimals.
Def . Any variable quantity that approaches as a limit is
called an infinitesimal. For example, A^;^ ^y, dx, dy, are
infinitesimals.
The erroneous notion that an infinitesimal is a constant quantity which
is '' smaller than any other quantity, however small, and yet not zero''
should be carefully avoided.
Notation. The notation lim x=a, or x-^a (read: '*a; ap-
proaches a as a limit ")» means that a; = a-j-c> where c is a
variable approaching zero. Thus a statement expressed in
terms of **lim" or ** — ^ " can always be translated into an
equation, which can then be handled by the ordinary rules of
algebra. The symbol — » is preferable to = and seems likely to
replace it.
Def. If a and p are infinitesimals, and lim (a/p) = 0, then
a is said to be an infinitesimal of higher order than p.
For example, if Au ^ e . At;, where e itself approaches 0, then Au is
of higher order than Ai;. Again, 1 — cos Atf is of higher order than A^.
If the diflference between two infinitesimals is of higher
order than either, then their ratio approaches 1 as a limit; and
conversely, if the ratio of two infinitesimals approaches 1,
then their diflference is of higher order than either. Two
infinitesimals having this relation may be called ** similar "
or ' * equivalent ' ' infinitesimals.
Important examples are the following: a convex arc of a
curve, and the chord of that arc, are ** similar " infinitesimals.
Again, sin Ax and tan Ax are both ** similar " to Ax, provided
Ax is in radians.
FmsT Replacement Theorem for Infinitesimals. In
finding the limit of the ratio of two infinitesimals, either of
them may he replaced by a ^^similar" infinitesimM, without
affecting the value of the limit.
As explained above, two infinitesimals are '' similar " : (1)
if the difference between them is of higher order than either;
or (2) if the limit of their ratio is 1. (Sometimes the first test
is more convenient, sometimes the second.)
92 GALGX7LUS.
This theorem frequently enables us to replace a complicated
infinitesimal, like ir(r -]- Hr)*^, by a simpler one, as irf^Ax;
hut it justifies this replacement only in the case expressljfi
stated in the hypothesis of the theorem, namely the case in
which we are finding the limit of a ratio.* (The fallacy that
^'infinitesimals of higher order can always be neglected"
should be carefully guarded against.)
* A second replacement theorem for infiniteBimalB will be given in the
chapter on Definite Integrals.
OALOTJL.TJS. 93
D. Applications op Difpebentiation in STUDTiNa the
Properties op a Qeten Function.
16. That a knowledge of differentmtion is of fundamental
importance in studying the variation of a given function is
evident from the following theorems.
Let the given function be y=f(x).
I. The value of the derivative at any point shows the slope
of the curve at that point.
Hence, if the derivative is positive at any point, the curve
is rising at that point (as we move in the positive direction
along the axis) ; that is, the function is increasing. And if
the derivative is negative at any point, the curve is falling at
that point ; that is, the function is decreasing.
II. If the second derivative is positive at any point, the
slope is increasing at that point, and hence the curve is con-
cave upward; and if the second derivative is negative at any
point, the slope is decreasing at that point, and hence the
curve is concave downward.
17
94
CALCULUS.
A point where the concavity changes sign is called a point
of inflexion; at every such point, the second derivative is zero.*
17. Maxima and minima. — The application to problems in
maxima and minima is inunediate. In seeking the largest or
smallest value of a given function in a given interval, we need
consider only (1) the points where the slope is zero; (2) the
points where the slope is infinite (or otherwise discontinuous) ;
and (3) the end-points of the interval; for among these points
the desired point will certainly be found. In most practical
cases it will be a point where the slope is zero.
The conditions of the problem will usually show clearly
which of these points, if any, is a maximum (or a minimum).
18. Multiple roots. — The roots, or the zeros, of a function,
are the values of the argument for which the function becomes
zero. An inspection of the figure will show that any value of
X for which f{x) and f{x) are both zero simultaneously, will
count as at least a double root.
* But the second derivative may be zero at points which are not
points of inflexion; for example, y=fl^ at a; = 0.
CALCULUS. 95
19. Small errors. — The following theorem is very useful in
discussing the effect, on a computed value, of small errors in
the data :
III. If dx is small, dy and Hy are nearly equal.
That is, the difference between dy and ^y can be made as small as we
please, in comparison with dx, by making dx sufficiently small (except
at points where dy/dx does not have a finite value).
Thus, if we wish to find approximately the error At/ pro-
duced by a small error in x, it will usually be sufficiently
accurate to compute, instead of At/, the simpler value, dy.
In problems concerning the relative error, dy/y, or dx/x,
it is often convenient to take the logarithm of both sides of
the given equation y==if{x) before differentiating.
This class of problems is of great practical value.
CHAPTBB ni.
INTEGRATION AS THE INVEBSE OF DIFFERENTIATION. SIMPIiB
DIFFERENTIAL EQUATIONS.
20. In many problems in pure and applied mathematics, we
have given the derivative [or differential] of a function, and
are required to find the function itself.
Suppose f(x) [or f{x)dx] is the given derivative [or differ-
ential] ; it is required to find a function F{x) which, when dif-
ferentiated, will give f(x) [or f(x)dx]. Clearly, if one such
function F(x) has been found, then any function of the form
F(x)']-C, where C is any constant, will have the same
property.
Definition. — Any function F{x) whose differential is
f{x)dx is denoted by
jf(x)dx,
read: an integral of f{x)dx. The process of finding an inte-
gral of a function is called integration, or the inverse of
differentiation.
If J (x) is any particular integral of f(x)dx, then every
integral of f(x)dx can be expressed in the form F(x) -{-C,
where C is a constant, called the constant of integration.
It can be shown that every continuous function has an inte-
gral; but this integral may not (in general, will not) be ex-
pressible in terms of the elementary functions.*
Most of the functions that occur in practice can, however,
be integrated in terms of elementary functions, by the aid of
a table of integrals, such as B. 0. Peirce's well-known table
of integrals. The entries in such a table can be verified by
differentiation.
21. Formal work in integration. — The time devoted to the
formal work of integration should not be longer than is neo-
* In such cases, an approximate expression for the integral may be
obtained by infinite series.
96
CALCULUS. 97
essary to give the student a reasonable degree of expertness in
the use of the tables.
The following integration formulas should be memorized;
they are derived immediately from the corresponding formulas
for differentiation.
I cudx = c j udx ; I (u + v + — )dx = 1 vdx + 1 vdx + • • • ;
af'dx = (provided n ^= — 1) ;
(in words: an integral of any function raised to a constant
power, ^ — 1, times the differential of that function, is equal
to the function raised to a power one greater, divided by the
new exponent) ;
i Bin xdx^ --COB x; | cosa;(2a;=s8ina;; j sec*a;da:= tan«;
The constant of integration must be supplied in each case.
A large number of integrals can be brought under the form
fx^dx by a simple transformation. For example,
/cos* xdx = /cos^ XQOBxdx=f(l — sin* x) cos x dx
= /cos xdx — /sin*xcosa?dte=/cosxda; — /(sin a?)*d(sin x)
= mnx — (sina?)V3.
Similarly for any odd power of the sine or cosine.
The following integrals are also important, though it is not
worth while to memorize them when a table is at hand:
I 8in*«da;=:^(a?— sinrccosa;); j cos*a;da;=B^(a; + 8ina;cosa;);
/^ 1 i. /'»' . ^\ n 1 + sin a; r dx x
=»log.tan (- + o)=il^g.^i ' — ; I - — =log.tan ^\
jsinh xdx=^ cosh x) | cosh xdx=i sinh x; I BecVxdx=i tanho;.
98 CALCULUS.
22. Among the other formulas of integration, the following
are perhaps the ones that occur most often in practice; they
are inserted here for reference, and especially to illustrate the
usefulness of the hyperbolic functions.
/dx 1 __^x
a^ + a^~^ a a*
-r^ = ^ log. ^^^ = - tanh-^ -,
or — sir 2a 'a — X a a
/dx 1_ re — a 1 ^. .x
XT — a Jia 'rc + a a a
dx . _i 3? ,x
= Bin^-, or =— cos*
dx
f
f r^-^— , = lQg«(^ + >^^ + ^')> ^r =8inh-^^,
•/ ^JQr + or **
/-F== = log. (^ + ^^ — «')> or = cosh-' -,
fV** + aM« = |[a? V** + a'+ aMog. (x + V?+a')],
or = ^ a? V?+a'+ a* sinh"^ - ,
j Va^ — a'd« = ^[^ V^— a'— a' log, (a; + V?--fl?)] ,
or = pr he V«*— a*— a* cosh~^ - .
23. Methods of Integration. Among the methods by which
a given integral may be reduced to a form in the tables (or
an integral in the table to one of the fundamental forms) , the
most important are (1) the method of substitution and (2)
the method of integration by parts.
In the method of substitution, the given integral, ff(x)dx,
is expressed wholly in terms of some new variable y (and dy),
in the hope that the new integral may be easier to handle than
the old one. The substitutions which are most likely to be
useful are the following:
CALCULUS. 99
(a) y = any part of the given expression whose differential
occurs as a factor; y = aj*; yz=z\/x\ y=sina;; y=cosa;;
t/ = tan(a?/2).
(6) X = a sin y, o r = a tan y^ or = a sec y, in expressions
involving ya^ — x^ or Va* + x^, or V^---^ respectively.
But much can be done without formal substitution of a new
letter, if one remembers that the **a?*' in the formulas of
integration may stand for any function.
The method of integration by parts is an application of the
formula
Ciidv = uv —Cvdu.
Take as dv a part of the given expression which can be
readily integrated; on applying the formula, the new integral
may be simpler than the old one.
The student should be practiced in both of these methods.
24. Simple differential equations. In a large number of
problems in pure and applied mathematics, it is possible to
write down an expression involving the rate of change of a
desired function more readily than to write down the expres-
sion for the function itself. (Compare Chap. 11, B.) In
other words, it is often easier to write down a relation between
the differentials of two variables than to write down the rela-
tion between the variables themselves. Such a relation con-
necting the differentials of two or more quantities, is called a
differential equation^ and any function which satisfies the
equation, when substituted therein, is called a solution of the
equation.
Every such problem, then, breaks up into two parts: (1)
setting up the differential equation; (2) solving that equation.
The first part of the problem has already been treated in
Chap. II, B. This^ part of the problem is too apt to be neg-
lected in elementary courses; there is scarcely anything that
develops real appreciation of the power of the calculus more
effectively than practice in setting up for one's self the differ-
ential equations for various physical phenomena.
100 CALCULUS.
As to the second part of the problem, namely, the solu-
tion of the differential equation, the general plan is to reduce
the given equation, by more or less ingenious devices, to the
form dy=f{x)dx, or y=ff{x)dx, and then to complete the
solution, if possible, by the aid of a table of integrals. In a
technical sense, the differential equation is said to be ''solved''
when it is thus reduced to a simple ''quadrature" ; that is, to a
single integration.
The solution of a differential equation of the nth order, that
is, an equation involving the nth derivative, will contain n
arbitrary constants ; to determine these constants, n conditions
connecting re, y, y' . . ., y^*^ must be known (the " initial "
or " auxiliary " conditions of the problem).
25. The general discussion of differential equations is too
large and too difScult a topic to find a place in a first course
in the calculus, but two, at least, of the simpler equations are
so important that their solution should be given, as an exercise
in integration.
These equations are the following:
(1) ^ + n«y = 0, where 3^ = 1.
The solution is
2/ = Ci sin (n^ + Cj) or, y = C^smnt + C^coBnt,
where the C's are arbitrary constants.
The solution is
y = C?! sinh {nt + C,), or, y = C^e** + C^e-^*,
where the C's are arbitrary constants.
The method of obtaining these results, rather than the re-
sults themselves, should be remembered: namely, multiply
through by dy, noting that dy/dt = y', and integrate each term,
getting iy*^ + ^n^y^ = C ; then replace y' by dy/dt, "separate
CALCULUS. 101
the variables, ' ' and integrate again. By a similar method, any
equation of the form dy'/dt + f{y) = can be solved, if we
can integrate f{y)dy,
26. Another very important differential equation is the
equation for ' * damped vibration ' ' :
The solution is given here for reference :
Case 1. If a*— 6* > 0, let m = Vo*— 6*; then
y=(7i6-»*sin(m* + C,),
or y = [C, sin (mt) + C^cos (m*) ]«"**.
Case 2. If a*— 6*=0,
Case 3. If a» — 6* < 0, let n=y/b^ — a^; then
y = Ci6-»*sinh(ni + CJ,
or y = C,6-<**«>* + C^er^^^K
CHAPTEB IV.
mTEORATION AS TEE LIMIT OF A. SDH, DEFDOTB INTEaiUI^.
27. The limit of a Bum. Many problems in pure and ap-
plied mathematics csn be brought under the following general
form:
€Hven, a continuous function, y = f(x), from x^a to
x=b. Divide tM interval from z=:ato x = b into n equal
parts, of length Ax=(b — a)/n* Let Xy,Xt,Xi, . . . Xm be
values of x, one in each interval; take the value of the func-
tion at each of these points, and multiply by Ax; then form
the sum:
f(x^)^-\-fix^)Ax-^---+fiXn)^.
Required, the Umit of this sum, as n increases indefinitely,
and Ax j^ 0.
This problem may be interpreted geometrically as the prob-
lem of finding the area under the curve y = f{x), between the
ordinates x='a and x^b; each term of the sum represents
the area of a rectangle whose base is Ax and whose altitude is
the height of the curve at one of the points selected. It is
easily seen that the difference between the sum of the rec-
tangles and the area of the carve is less than a rectangle
* It is not ceeessaiy that the parts be equal, provided the largest of
them approaebea xero when n is made to increaee indefinitely.
102
CALCULUS. 103
whose base is Ax and whose altitude is constant. This dif-
ference approaches zero as Ax = 0; therefore the sum of the
rectangles approaches the area of the curve as a limit.
In this way, or by an analytic proof, it is shown that the
limit of the sum in question always exists. The problem then
is, to find the value of this limit.
The value of the limit can always be obtained by the fol-
lowing fundamental theorem, whenever an integral of the
given function f{x) can be found.
Fundamental Theorem of Summation. If x^, Xj, • • • Xn
are values of x ranging from x = a to x = h, as in the state-
ment of the gerleral problem above, then
lim Uix,)^ +Kx,)^ + .. . +/(OAa:] = F{h) - F{a),
where
Fix) = fKx)dx
is any function whose derivative is the given function /(re).
The proof of this remarkable theorem is best given by show-
ing that the right hand side of the equation, as well as the
left, is equal to the area under the curve from x=ato x^=b;
to do this, consider the area from x=a to a variable point
x=x, and find the rate of change of this area regarded as a
function of x; hence find the area itself as a function of x,
determine the constant of integration in the usual way, and
then put x = b in the result.
Depinition. The limit of a sum of the kind described above
is called the definite integral of f{x)dx from x = a to x = b,
and is denoted by
^ S/(^.)Ax, or r^'f{ix)dx>
The function obtained by the inverse of differentiation is
called, for distinction, an indefinite integral. By the funda-
mental theorem just stated, the definite integral is equal to the
difference between two values of the indefinite integral :
jrjV(^)<i»^= [//(^)*«^]^^^- [X^^'^)*'La-
104 CALCULUS.
The double nae of the term "integration" — ^meaning in one ease anti-
differentiation, and in the other case finding the limit of a sum — and the
fundamental theorem connecting these two distinct concepts, should be
made thoroughly dear.*
The concept of the definite integral is the most useful con-
cept in the application of the calculus, and the study of
problems which can be formulated as definite integrals may
weU occupy one third of the time of a first course.
For example, problems in areas, volumes, surfaces, length of arc,
center of gravity, moments of inertia, center of fluid pressure, etc.
Many of these problems require two applications of the fundamewtaX
theorem.
28. Properties of definite integrals. From the definition
of the definite integral we have at once :
£ Kx)dx ^ ^ j^ f(,x)dx ',
r fix)dx + r f{x)dx « r f(,x)dx ;
and, by the aid of a figure, the Mean Value theorem:
J F{x)f{x)dx « F{X)Cs(x)dx,
where X is some (unknown) value of x between a and &, and
F{x) and f{x) are any continuous functions, provided f{x)
does not change sign from rc==a to ic = &.
We have also the following important theorem on change of
variable:
In evaluating the integral
r
/(^)^,!
if re is a function of a new variable ^, we may replace f{x)dx
by its value in terms of t and dt^ and replace x=a and re = 6
* The use of the term in the sense of summation was historically the
earlier, and the symbol f is the old English "long s," the first letter
of "sum."
CALCULUS. 105
by the corresponding values t = a and ^=i8, without altering
the value of the integral, provided that thoroughout the inter-
val considered there is one and only one value of x for every
value of t, and one and only one value of t for every value of x,
29. All problems leading to a definite integral are prob-
lems in finding the limit of a sum, each term of which is
approaching zero, while the number of terms is increasing
indefinitely. Whenever a function /(re) can be found, such
that all terms of the sum are obtained by substituting suc-
cessively «!, rCjj 6tc-> i^ the expression f(x)dx, then the formu-
lation of the problem as a definite integral is immediately
obvious. The separate terms of the sum, of which f{xjc)dx is
a type, are called elements.
Thus, in finding the area under a curve, an obvious element
of area is the rectangle ydx; if the curve revolves about the
^-axis, the element of volume of the solid thus generated is
the cylinder iry^dx. Here y must be expressed as a function
of X before the integration can be completed. Again, in polar
coordinates, the element of area is the sector, ir^dO, where r
must be a known function of 0.
In many cases, however, the proper function is not so
immediately obvious. In such cases, the following theorem is
of great service :
Second replacement theorem for infinitesmals (Theo-
rem OF Duhamel) . In finding the limit of a sum of positive
terms, each of which approaches zero while the number of
terms increases indefinitely, any term may be replaced by a
^* similar '' term tvithout affecting the value of the limit. Two
variables a and p are called " similar " if
(1) lim^«l, or 1/(2) lim?— ^=0.
For example, let us find the weight of a rod whose density,
w, and cross-section. A, are both functions of x. The **true
element" of weight, AW, corresponding to a given length Ax,
will certainly lie between the values w'A'Ax and w^A^^Ax, where
106 CALOULTJS.
u/, A' are the smallest values, and w'\ A!' the largest values
of w and A within the interval from ^ = ^ to ^=^4'^^»
but either of these extreme values may be replaced by the
simpler value wALx^ where w^A are the values of w and A
at the beginning of the interval, for,
,. v/A'^x .. yxf'A^^x ^
Jim T-r = hm 7-r as 1.
Hence, AW itself, which lies between these extremes, can be
replaced by moAlx^ which is therefore the required ''differ-
ential element" of weight.* The total weight of the rod,
from ^=a to a; = &, is then equal to the definite integral
toAdx;
_ a
where w and A must of course be expressed as functions of x
before the integration can be completed.
In justifying replacements of this kind by Duhamel's
theorem, sometimes the first test is more convenient, some-
times the second. When once the common replacements have
been justified, the use of the theorem in practice rapidly
becomes almost intuitive.
30. Approximate methods of integration. — ^If the function
f{x) is given only empirically, the theorem on evaluating the
definite integral by purely mathematical means cannot be ap-
plied. In such cases, an approximate value of the definite
integral ^
fix)dx
£
may be found by plotting the curve y=f{x) on squared
paper, and estimating the area by counting squares (and frac-
tions of squares) .
Another method of approximation is by Simpson* s Rule:
'When X is the independent variable, it is immaterial whether w«
write Arr or dr.
Divide die ares into n panels, wliere n is even, and Dumber
the ordinateB from 1 to n + l; then, if Ad; is the width of
each panel,
Area = ^Ax (first ordinate + l^t ordinate
-]- twice the sum of the other odd ordinates
4- four times the sum of the even ordinates) .
The instrument known as a planimeter provides a meehan-
ieal means of integration, used especially in measuring the
areas of indicator cards.
Another and very important method of approximation is
by the use of series; see the next chapter.
31. Definite Integral as a function of its upper limit. — If
X is a variable, the definite integral
£^K')<h:
represents the area under the curve y=fix) from z = a to
the variable ordinate x=X, and is therefore a function of X,
say ^(X), By applying the definition of derivative to this
fonetion, it is easy to see from the figure that <^'(X) =f{X) :
Thus ^(X) is one of the indefinite integrals of /(X).
108 CALCULUS.
Any indefinite integral which cannot be expressed in terms
of known functions can always be written as a definite
integral regarded as a function of its upper limit, and its
value, for any given value of the argument, can then be found
by one of the methods of approximate integration.
The elliptic integrals, the most important of which are
,,, == and I Vl-(ik*)Bin'tfdtf,
are handled in this way, by the method of expansion in series.
The student should be made familiar with the construction
and use of tables of the elliptic integrals.
In such tables, lb* is usually expressed in the f onn sin' a, which empha-
sizes the fact that Jb*^!.
CHAPTER V.
APPLICATIONS TO ALGEBRA : EXPANSION IN SERIES ; INDETER-
MINATE FORMS.
Note. — ^This chapter may be taken, if preferred, immediately
after the chapter on differentiation. It is in reality an exten-
sion of the "formal work" of that chapter, since it deals with
changes in the form of algebraic expressions.
32. Taylor's theorem. — It is often desirable to obtain an
approximate expression for a given function, in the neighbor-
hood of a given point x = a, in the form of a series arranged
according to ascending powers of x — a, with constant coefift-
cients. For values of x near to a, the higher powers of x — a
will then become negligible.
The most convenient theorem for this purpose is the fol-
lowing:
Taylor's Theorem. If f{x) is continuous, and has deriva-
tives through the (n-\-l)st, in the neighborhood of a given
point x=ia, then, for any value of x in this neighborhood,
where X is some unknown quantity between a and x. The last
term,
^ (n+l)!^"' ""^ '
is the error committed if we stop the series with the term in
(re — a)", and the formula is useful only when this error be-
comes smaller and smaller as we increase the number of
terms.
18 109
110 0ALGULU8.
This fonn for the '' remainder" B is easily remembered since it
differs from the general term of the series onlj hy the fact that the
derivative in the coefficient of the power of (x — a) is taken for x=zX
instead of for x^a,^ (There are also other forms of the remainder
which are sometimes useful.)
33. The special case where a = is called Blaclaiiriii's
Theorem:
where X is some unknown qoantily between and x.
34. Another special case, obtained by patting n = 0, gives
nx)-na)=nX)(x-a),
where again X is some unknown quantity between a and x.
This theorem is called the Law of the Mean, and is of great
importance in the theoretical development of the subject.
36. If the error-term in Taylor's Theorem approaches zero
as n increases, the formula becomes a convergent infinite series,
called the Taylor's series for the given function, about the
given point x=a.
The series with which the student should be especially fa-
miliar are the following :
* The simplest proof of this theorem is by means of integration. For
example, for the case n = 2, we have
£f"{t)dl=f"{X){x-a),
where X is some (unknown) constant between a and x (as is evident from
a figure) ; but also
£f"\t)dl=f"{x)-f"(a),
by the ftmdamental theorem; so that
/'(«) -r(«) =/"'(X)(«-o).
Integrating this equation twice between the limits a; = a and x^x,
remembering that /"(a) and f"'{X) are constants, we have at once:
/"(«) -/"(«) -r{a){x—a) =f"'{X)H''-ay,
f(x)—f(a)-r(,a)(x—a)—f"{a)Ux-ay=f"'(X)iH<» — oy-
OALCX7LU8. Ill
Binomial series :
/-• , N-i -• , . m(m — 1) , . m(m— l)(m — 2) , ,
(l + a;)-=l + ma;+ ^^ ^ x" + -^ ^ ^ar» + ---,
Sine series :
provided |a; | < 1.
a^ «* «'
sinajsssa: — ^ + ^ — =-. + ••• (a;in radians).
Cosine series :
/p« a:* a^
cos « = 1 — ;rT + T-i — TTi + • • • (« ill radians).
2 14 16 1
Ik
Exponential series :
x" a^ X*
Next in importance are the series for log (1 + ^)9 tan*^ x, sinh x,
and cosh x.
From these series we have the following important approxi-
mations, when X is small :
Binx=s X — • • , cos x=s I — • • • , etc.
An important special case of the binomial series is the
geometric series :
= l + a; + rc' + a^+«", provided | re | < 1.
1 — a;
36. The student should also understand the comparison test,
and the test-ratio test, for the convergence of an infinite series,
and the following theorem on aUemaiing series : If the terms
of a series are alternately positive and negative, each being
numerically less than or equal to the preceding, and if the nth
term approaches zero as n increases, then the series is conver-
gent, and the error made by breaking off the series at any
given term does not exceed numerically the value of the last
term retained.
112 OALCULUB.
Further, a power series can be differentiated or integrated
term by term, within the interval of convergence.
87. Indeterminate forms. — ^The evaluation of indeterminate
forms can often be facilitated hy the use of the following
theorem, in which f{x) and F{x) are functions which possess
derivatives at a given point x=a.
Theorem of indeterminate forms. If f(x) and F{x) both
approach zero, or both become infinite, when x approaches a,
then
^mi-'isimi
The second limit may often be easier to evaluate than the first.
The student should thoroughly understand the meaning of
indeterminate forms, for which the common symbols^, l",etc.,
are merely a suggestive short-hand notation.
Thus, ^'0/0" means that we are asked to find the limit of
afunctiony=/(a?)/J?(a;),when /(re) and F{x) both approach
zero. Now the change in f{x) alone would tend to decrease
y numerically, while the change in F(x) alone would tend to
increase y ; hence we cannot tell, without further investigation,
what the combined effect of both changes, taking place simul-
taneously, will be.
Again, the symbol 1*" means that we are asked to find the
limit of a function y =/(a;)^<*>, when f{x) approaches 1 and
F(x) becomes infinite. Now the change in f(x) alone would
tend to make y approach 1, while the change in F(x) alone
would tend to make y recede from 1 ; hence we cannot tell, with-
out further investigation, what the combined effect will be.
The student should thoroughly master in this way the
meaning of all the seven types of indeterminate forms, namely,
5, -^, O.oo,0«, 1-, 00*, 00-00.
The cases involving exponents are best treated by first find-
ing the limit of the logarithm of y, from which the limit of y
CALCULUS. 113
can then be obtained. The form 0- oo, or y=f{x) •-?(«), can
f(x) Fix)
be written as y= ^ \l,/ > , or y =^ ^ , .\ \ , which then comes
under one of the first two forms. The last form, oo — oo, is
usually best handled by the method of series.
Before applying the theorem of indeterminate forms, one
should, of course, try first to find the required limit by a
simple algebraic transformation, if possible.
CHAPTER VI.
APPLICATIONS TO GEOMETRY AND MECHANICS.
In all applications to geometry, in which a curve is repre-
sented by an equation connecting x and y, the scales on the x
and y axes must he equal (compare §3, footnote).
38. Tangent and normal. — The equation of the tangent at
any point (and hence the equation of the normal) can be
written down at once when we know the slope and the coordi-
nates of the point of contact.
Again, to find the subtangent or subnormal at any point,
we have simply to find the ordinate and the slope at that point,
and then solve a right triangle.
39. Differential of arc. If 5= length of arc of the curve
y=/(rr), measured from some fixed point A of the curve,
then s, like y^ is a function of x, and we may ask what is the
x^-t-Ax
rate of change of s with respect to x, that is, what is the value
of ds/dx. Now ds/dx=]im (A^/Ao?), and in finding this
limit we may replace the arc As by its chord, V(^^)*+(^y)*;
hence ds/dx = lim VI + (Ay/Ax) « = y l -f (dy/dx) *, or
ds=^^/{dxy + {dyy,
114
CALCULUS. 115
as indicated in the figure. This formula, and the correspond-
ing relations
dx = ds Qoa <!>, dy = ds sin ^,
are important, and are readily recalled to mind by the figure.
In the case of a circle of radius r, if (20= the angle at the
center, subtended by the arc ds, then
ds = rd0,
provided the angle is measured in radians.
40. Again, in case of a curve whose equation is given in
polar coordinates, r =/(*), we see at once from the figure, by
the aid of the replacement theorem, that
rde
ds « V(dr)* + (rdey and tan^=-^,
where ^ is the angle which the tangent makes with the radius
vector produced.
41. Radius of Curvature. — Consider the normal to a given
curve at a given point, P, and also the normal at a neighbor-
ing point, Q. These two normals will intersect at some point
C on the concave side of the curve ; and as Q approaches P,
along the curve, this point C will (in general) approach a
definite position C as a limit. The circle described with a
center at this point C and radius equal to CP wiU fit the
given curve more closely, in the neighborhood of the point P,
than does any other circle. This circle is called the osculating
circle, or the circle of curvature, at the point P; its center C
is called the center of curvature, and its radius CP is called
the radius of curvature, at the point P.
116 OAIiOTTLUS.
The radius of curvature may thus be taken as a measure of
the flatness or sharpness of the curve ; the smaller the radius
of curvature, the sharper the curve.
The length of the radius of curvature, B, at any point P is
most readily found as follows : In the triangle PC'Q, we have
C'P/PQ = sin <?/sin A<^, where A<^ is the angle between the
normals (or between the tangents) at P and Q. Therefore
B=lim CT=lim (chord P<?/sin A<^) sin Q; or, replacing
the chord by the arc A^, and sin A<^ by A<^, and noticing that
Q is approaching 90"*, so that lim sin Q = l, we have
iJ = lim (A5/A<^), or,
^ d$
This important formula is readily recalled to mind from the
figure, if one thinks of the arc A5 as approximately a circular
arc.
To express B in terms of x and y, we have only to remember that
ds = V(da?)*4- (^y)'=Vl -f^y^dx, and tan <f> = dy/dx = y', whence
dip = y^dx/ (1 + y^) ; then
jj_a±/y
OAIiOTTLUS. 117
Def . The curvature of a curve at a point is defined as the
rate at which the angle <l> is changing with respect to the
length of arc s ; that is,
d6 1
curvature = V- = ^ .
a$ R
If the slope of the curve is small, the curvature is approxi-
mately equal to y^\
Def. The locus of the center of curvature is called the
evolute of the curve.
The normals to the given curve are tangent to the evolute,
and the given curve may be traced by unwinding a string from
the evolute.
42. Velocity and acceleration. — Consider a particle moving
along a straight line. Its distance from the origin is a func-
tion of the time :
x = F{t).
The velocity of the particle is the rate of change of its
distance:
v = dx/dt = F'{t)=x\
The velocity will be positive or negative, according as the
particle is moving forward or backward along the line.
The acceleration of the particle is the rate of change of its
velocity:
A = dv/dt = F"{t)=x".
The acceleration will be positive or negative according as
the velocity is increasing or decreasing (algebraically).
If a particle is moving along a plane curve, we must
consider the components of its motion along two fixed axes.
The components of acceleration along the x- and y-axea
are a;" and y" ; the components of acceleration along the
tangent and normal are dv/dt and v^/B, respectively, where
v = y/x'^ -{-y'^=i}ie path velocity^ and jB = the radius of
curvature.
It should be carefullj noticed that dv/dt is not the whole acceleration,
but only that component of the acceleration which lies along the tangent.
118 OAIiOTTLUS.
The importance of this application in problems in mechanics
is obvious.
Note. — As explained in the preface of this report, these
pages are intended merely to give a r6siim£ of the working
principles of the calculus with which the student should be
perfectly familiar after having taken a course in this subject.
The main part of the work of such a course should be prob-
lems done by the students — each problem being solved on the
basis of the small number of fundamental theorems here
mentioned.
discussion on mathematics bepobt. 119
Discussion.
Professor Chas. 0. Gimther: It seems to me that in this
report some mention should be made of imaginary and com-
plex quantities. A little knowledge of these quantities can,
for instance, be utilized to good advantage by applying it to
that part of the calculus known as integration. In fact, in-
tegration can be simplified to the extent of eliminating the
usual ** reduction formulae " and rendering the use of tables
of integrals unnecessary.
As found in text-books in general, there are three cases for
which the expression
dy = co^e ma^edS (1)
can be easily integrated. Two of these cases include frac-
tional values for h and ft. All other cases in which h and h
are integers can either directly, or by means of a single
imaginary trigonometric substitution (tan tf=i sin a, in which
a is an imaginary quantity), be reduceid to one or more of the
three cases just referred to.
The general binomial differential expression
di/ = a;«»(a + 6a;»)»/«da? (2)
is only another form of (1) since y/a + hx'^ can always be
represented by one of the three sides of a right triangle and
therefore expressed as a trigonometric function of one of the
acute angles of the triangle.
To make this transformation the student must know the
relation between the hypotenuse and the two sides of a right
triangle, the values of the trigonometric functions of an angle
in terms of the sides of a right triangle, and the rules for
differentiation.
Differential expressions involving trinomial surds may be
rationalized in a similar maimer.
The expressions
-^^^^coHbx, (3)
^ = <^ sin bx, (4)
120 DISCUSSION ON MATHEMATICS BEPOBT.
may be integrated with great facility if complex quantities
are employed, because e^eosbx and e^Biabx are the rec-
tangular components of a vector whose modulus is e^ and
whose argument is bx. The integrals of (3) and (4) are found
from the integral of
in which 2; is a complex variable of the form y + iy. The
integral of (5) is readily found to be
' = (a + iby + <^,.-i«"-' +'-'+C,x+6^ (6)
• • • •
in which Cn_i, • • • Cj, Co, are constants of the form C = C + iC.
Equation (6) may be written
" = (a^-Tfey* '"''"' *""""" "^ ^-»'^' +..-+C,x+C,. (7)
The integral of (3) is the real part of (7) and the integral of
(4) is the imaginary part of (7) divided by ♦.
Again in differential equations we find the linear equations
^ + ay = 6 cos nx, (8)
dy _ . ._.
-^ -f ay = 6 sm nx, (d)
and their solutions can be obtained from the solution of the
equation
^ + 02 = be^, (10)
in which 2 = 1/ + iy»
The foregoing illustrates a few of the applications of com-
plex and imaginary quantities, and includes a first treatment
of hyperbolic functions as trigonometric functions of imagi-
nary quantities.
Some little consideration should also be given to the com-
plex and imaginary branches of certain curves, as for example,
DISCUSSION ON MATHEMATICS BEPOBT. 121
the circle, the ellipse, and the hyperbola. It should be noted
that the equation of the circle a^-{'y^ = a^ i& also the equation
of an imaginary hyperbola for values of a; > a and < — a.
This is important, since of the three forms of binomial surds
Va^—x% y/cF+Q^, y/ix^ — aS the first is obtained from the
equation of the circle x* + 2/^ = (i*, and the latter two from
the equation of the hyperbola x^ — 2/* = a*; but all three are
obtained from the equation of the circle if imaginary quanti-
ties are made use of.
Professor J. E. Boyd: I want to emphasize eveiything Pro-
fessor Gunther just said about the use of complex quantities.
We cannot derive a formula for an eccentrically loaded long
column without the use of them ; we cannot make alternating
current calculations without them. A student might as well
learn how to use them. I endorse what he says about the use of
integral tables in teaching calculus. Our professors in calcu-
lus last year adopted a book that advised the use of tables.
This year a book of the other type was selected. We did not
use the tables any more than was absolutely necessary and
found the result satisfactory. The student does not need
tables often, except to make use of the several transformations.
Professor P. L. Emory: The average student is vastly lack-
ing in a knowledge of the use of logarithms. He also lacks
the ability to read trigonometric formulas from the triangle.
The tendency of the report is to include more material than
can be covered in an engineering course. I would be satisfied
to have a little more training in a few principles which stu-
dents must know so well that they have confidence in their
knowledge. One of the most serious diflSculties that I encoun-
ter is with the constant of integration. This is largely the
fault of the text-books. I have a grievance against the text-
book writer who omits the constant in all cases, supplemented
by the remark that it should always be added. We cannot
expect the student to remember a footnote to be applied with
each operation.
Professor J. B. Webb: I am pleased to hear what Pro-
fessor Emory said about the constant of integration and his
122 DISCUSSION ON MATHEMATICS BEPOBT.
explanation of the difficulty, but I think the trouble is more
in the teaching than in the text-books. In reading Mr. F. W.
Taylor's book on his system, I was interested in one of the
illustrations which he uses. He takes the case of loading cars
with pig iron, where, by the application of his system he
about tripled the amount that a man could do in a day, and at
the same time enabled the man to earn more money. One of
the first things he did was to examine the men that were in the
gang, and he found that but one man in eight was suitable
for this work. He used only those who were fitted for it. We
have about the same proportion, perhaps, of the unfit in our
classes, and the ones fitted for engineering could do three
times the work and do it better if our classes were conducted
on the Taylor system.
I have had some interesting experiences with the complex
variable. Having studied the subject in Germany in 1878-
1880, on my return to this country I tried to teach its use.
Objections were made by those not acquainted with the sub-
ject, that it was too advanced and of little practical use, so
that it proved to be harder to convince the average American
teacher of its importance than to arouse the interest of intelli-
gent students. Some of the professors were convinced, but
that was where the trouble lay. If a student was conditioned
because he did not get through with his mathematics, some
said I taught " over his head " and gradually the standard
would be forced down. The trouble with the present schools
is that they want too many students and are going to hold
all they have and get more if they can. They do not call out
the seven and keep the one. After Dr. Steinmetz, a layman,
produced his book on the treatment of alternating currents,
using complex variables, there was less objection made to
them. Now I say it is a disgrace that it should be necessary
for a layman to show professional teachers that a certain part
of mathematics is needed. What we should do is to eliminate
the students who are not capable of profiting by what we know
should be taught, and then hold the others to a high standard.
We expect too much of the student who takes calculus. Of
DISCUSSION ON MATHEMATICS BEPOBT. 123
a semester in calculus at least one-half is spent in reviewing
previous mathematics. A course in calculus is an excellent
review of geometry, algebra and especially of trigonometry,
and at its close we should not expect the average student to
know much more about it than he did about trigonometry at
the start.
This committee was appointed to see what was the matter
with the teaching of mathematics. They imply that good text-
books are lacking. I cannot agree with this and would rather
have one of the old-fashioned text-books than those outlined
in their report. If they intend to give simply a list of subjects
that students should be drilled in, well and good; but if the
report intends to prescribe the methods of thought and of
logical deduction, to be used in those subjects, then I think it
is all wrong.
Professor Magruder: The introduction to the report states
clearly the purpose of the syllabus.
Professor W. J. Risley: The suggestions that have been
made here this afternoon are very good. I am in favor of a
section on imaginary quantities. When I approached some of
the Harvard professors of engineering subjects I found that
they wanted their students to perform vector addition analyt-
ically. They said that the teachers of mathematics were
teaching a lot of things of which little or no use was made
later. To a great extent Professor Webb was right in stating
that he had to teach the professors of engineering what they
ought to teach, in order that they might understand some of
the mathematics which he attempted to send to them. On the
other hand, sometimes the professors of engineering have to
teach the professors of mathematics some things that they
don't know that their students ought to know. Neither set is
to be criticized too severely unless they are unwilling to learn
when the right way is pointed out.
Principal Arthur L. Williston: I was very much inter-
ested in what Professor Webb said a moment ago, referring
to Mr. Taylor's work and his method of culling out one man
124 DISCUSSION ON MATHEMATICS BEPOBT.
of the eight who was especially adapted for a particular kind
of work, using him intensively on that kind of work, and find-
ing tasks for which the other seven are fitted. That idea is
really at the bottom of all of our difficulty in this discussion
of teaching mathematics to engineers, which we have had
almost since we began trying to teach engineers. As there are
few men of the naturally analytical kind that Professor Webb
describes it does not make much difference what sort of
methods we use with them. As a matter of fact a very small
proportion of the men who form the body of eminent engi-
neers have that type of mind. We all know the sort of fellow
who thrives on complex quantities. And I am sure the
majority of those here will bear me out in my statement that
a very small proportion of the successful, eminent engineers
of this country are of that kind. The ideal plan would be to
separate those fellows from the mass and give them a course
in real mathematics. They would like it and it would be a
pleasure to the instructors to teach them. But let us take the
other group. For the most part, the man who is going io be a
successful engineer in industrial work is a practical, concrete
man. He does not handle imaginary, complex, abstract quan-
tities easily. And yet that is the very type of mind that the
world wants in its important industrial activities. Those fel-
lows, who, by the way, constitute the great majority, want
mathematics not as an analytical light but simply as a neces-
sary evil, if you please, as a tool that they must use. If in
our talking and our thinking we could learn to talk of mathe-
matics as two subjects, one thing for the first type, another
for the second, it would simplify all our discussion. It is
absolutely futile l^o attempt to teach the first kind of mathe-
matics to three out of four young men who will be good engi-
neers whether the colleges turn them out as fitted to be engi-
neers or not. They are going to be engineers. As I under-
stand it, the work of this committee has been to some extent
a movement toward trying to get the teaching of mathematics
for engineers differentiated from the teaching of pure mathe-
matics. I am sorry that the difference is not more marked.
DISCUSSION ON MATHEMATICS BEPOBT. 125
Professor E. R. Manrer: I prefer to hear a teacher of
mathematics discuss this syllabus, because he can see it in the
light of his experience in teaching the subject. To be sure,
others have good ideas as to what knowledge and training
engineers ought to have in mathematics, but they fail to ap-
preciate the difficulties of teaching the subject. So, between
two criticisms, one offered by teachers bf mathematics and the
other by teachers who have never taught mathematics, I place
more confidence in the former. In estimating the value of
mathematical instruction we are apt to forget that, in many
schools, particularly the large ones, more or less inexperienced
men are employed as instructors in the departments of mathe-
matics. The results suffer on that account. In addition there
is the poor quality of the working material. I try to be chari-
table when I judge the students that come to me from the de-
partment of mathematics on those two accounts. Many of the
boys have had their training at the hands of inexperienced
men and many have very little mathematical talent. I think
the syllabus is good as a list of topics with which all engineer-
ing students ought to be familiar. I agree with Professor
Webb in that we ought not to set this up as a subject matter
for all teachers of mathematics to use and not depart from it
in any particular. The teacher of mathematics, or of any sub-
ject in an engineering school ought to understand his subjects
well enough to get up his own syllabus, if necessary.
The President: An informal committee of instructors in
the University of Illinois, formed of a dozen men representing
the department of mathematics, mechanics, civil engineering,
electrical engineering and mechanical engineering, made a
careful study of the report of the Mathematics Committee to
see whether the syllabi covered the ground which these pro-
fessors thought should be covered in class. In general I may
say that they approve almost wholly of the contents and in
general of the matters of emphasis as to what part should be
well understood, what other, only partly known. With your
permission I shall include this report in the discussion.
The committee of University of Illinois instructors selected
9
126 DISCUSSION ON MATHEMATICS REPOBT.
to discuss the preliminary report of the Committee on the
Teaching of Mathematics to Students of Engineering submit
the following recommendations :
Since the syllabi are meant to embody the minimum equip-
ment in mathematics of a good engineer, they have been dis-
cussed from that point of view. But it is the opinion of the
committee that much could be gained by publishing a list of
topics that should be included in the courses discussed, and
emphasizing by a star those which are ^' so essential that every
engineering student should have them so firmly 6xed in his
memory that he will never need to look them up in a book/'
The discussions of the committee were confined to the syllabi
which are printed in the Proceedings, i. e., Algebra, Trigo-
nometry, Analytic Geometry, and Calculus. Section numbers
refer to the sections as published in the syllabus.
Algebra,
1. Under factoring some mention should be made of the
important cases of collecting coefficients, and of quadratic
trinomials.
2. Important principles "and rules should be given in trans-
lated word form as well as in symbolic form (as is done once on
page 8 and in the differentiation rules in the calculus syl-
labus). Students often fail to get the fuU meaning of sym-
bolic forms. The operations with fractions and the definitions
and laws of exponents especially need statement in word form.
3. If algebra follows trigonometry, the three forms for
imaginaries should be included.
4. The notions equality, identity and equation should be
carefully differentiated.
5. The principles of equivalent equations should be in-
cluded, for a student should know what operations introduce
or take out roots.
6. More emphasis is needed on the ** completing the square"
process, for it is often needed later in integration and analytics
when no solution is required.
7. Harmonic progression should be omitted.
DISCUSSION ON MATHEMATICS BEPOKT. 1 27
Trigonometry.
1. The committee agrees that the syllabus is satisfactory
and probably is complete enough for the average engineer.
Some members expressed a desire for the memorization of
more formulas as particularly useful to electric engineers.
2. Some members desired greater stress on the visible hand-
ling of formulae. By visible is meant graphical so far as the
expression of relationship and formulae can be. For ex-
ample the student should not so much remember the six fun-
damental definitions as formulae as he should remember the
defining triangle and its ratios. The same idea should be
carried throughout.
Analytic Oeometry.
1. The syllabus states in the introduction, ''This syllabus
is confined mainly to the conic sections; but a satisfactory
course in analytic geometry should include also the study of
many other curves." This committee believes that the syl-
labus would be improved by including the most important of
these ''many other curves" including the so-called engineer-
ing curves.
2. The equation of a straight line passing through two given
points should be included.
3. The equation of a straight line should be written in such
a form and taught in such a maimer that all constants of the
line are readily determined.
4. The method of treating the conic sections in the syllabus
is commended. For obtaining a proper facility in handling
the practical applications of these curves, it is desirable to
study each form separately even at the expense of the addi-
tional time that is required when this method is employed.
The properties of these curves as given are amply suflScient.
5. The geometrical construction of the conies should be in-
cluded and given more than a mere reference.
6. In the transformation of coordinates the method rather
than the equations should be remembered.
7. The subject matter in articles 46-54 is not that which a
128 DISCUSSION ON MATHEMATICS BEPOBT.
student should remember, but belongs to that class of things
which can easily be referred to when required.
8. Much greater emphasis should be placed upon work in
polar coordinates.
9. It is desirable for the student to be familiar with cylin-
drical coordinates and the committee commends the inclusion
of these coordinates in the syllabus.
10. Great stress should be laid upon representation with
space coordinates. Any single equation in space coordinates
represents some surface. If the equation is in three variables
the surface may be any form, if in two variables the surface is
a cylinder, if in one variable the surface is a plane or a system
of planes parallel to one of the coordinate planes. Great
emphasis should be placed upon the fact that it requires a
pair of simultaneous equations to determine a line in space.
11. Article 71 should be omitted from the syllabus, though
included in a course in Analytic Geometry.
12. In the first sentence of the second paragraph of the
introduction the phrase **a course should consist chiefly of
problems'' should be changed to read **a large number of
problems should supplement the treatment of , general prin-
ciples."
Calculus.
The committee reports very favorably on the syllabus for
the first part of calculus. A subcommittee drew up a synop-
sis of a course in calculus before reading the syllabus as
printed in the Bulletin. The two did not differ in many
essential details. The main question that came up was whether
a topic was included under "those facts and methods which
every student should have so firmly fixed in his memory that
he will never need to look them up in a book," or simply under
** those topics included in an elementary course in calculus."
These two classes are referred to below as first and sec6nd
classes. The specific changes suggested in the syllabus are as
follows :
1. Section 5. Hyperbolic functions should be included in
the second of the above classes. Mnemonic rules for changing
DISCUSSION ON MATHEMATICS KEPOBT. 129
a trigonometric formula to the corresponding formula in
hyperbolic functions should be included.
2. Use arcs in x, arc cos a;, etc., instead of sin~^ x, cos~^ x, etc.
3. Section 21 (Formal work in integration). Tables of
integrals should not be used until the student has had con-
siderable practice in formal integration.
4. Include in section 22, integration by separation into
partial fractions.
5. Much practice in differentiation and integration with
respect to variables represented by symbols other than x, y, z
should be given.
6. In connection with differential equations (Sections 24,
25, 26) use cPy/dx^ instead of dy'/dx,
7. Include linear differential equations of first order in con-
nection with sections 25, 26.
8. Include sections 15 (Theorems on infinitesimals), 22
(Integration formulas), 35 (Theorem of Duhamel), 38 (Sub-
tangents, subnormals, etc.), 41 (Curvature), in the second of
the above classes.
9. Include angular velocity and acceleration in section 42.
10. We particularly commend sections 7, 12 (note) and 14.
Professor A. M. Buck: A good many people, and espe-
cially some who are mathematicians, forget that with the engi-
neering student mathematics is a subject that is taken not for
its own sake, but in order that problems can be solved after-
wards. If we take the view-point of the students we find that
they appreciate this point better than their teachers do. Stu-
dents have told me that they could not get along in mathe-
matics because they did not know what use they were going to
make of it. Had it been brought to their attention that the
mathematics would have some application to their engineering
work they would have gone into it with good spirit and would
have obtained more benefit from the work. Taking it as an
abstract study they simply would not give it the necessary
time. If the teacher of mathematics will look at his subject
from an engineering view-point and see that those things
which he teaches are to be used as tools and that the better the
130 DISCUSSION ON MATHEMATICS BEPOBT.
student has his tools in hand the better work he can do, then
there will be an improyement in the teaching.
Professor H. B. Thayer: I am going to state my opinion
from the view-point of the engineer. I have spent more time
outside in practice than I have in teaching. All of the latter
has been along the line of structural design, where I have
been using the work of the mathematical department. In the
first place, in my experience as a student, mathematics came
fairly easy to me. I found that when an examination was
imminent, I could cram up for it the night before and forget
it afterwards. That is about what nineteen out of twenty
students will do. Complicated notation tends to discourage
the student from getting what is extremely important to get,
namely, fundamental principles. I find that students know
their mathematics fairly well but they don't know how to
apply it. This, it seems to me, is far more important for them
to learn than such extremely complicated mathematical prob-
lems as are often given them. In actual engineering experi-
ence the applications of any but these fundamental formulae
are few and far apart. In the very infrequent cases where
the more complicated formulae are used it is only necessary to
refer to tables in the text-books, as the majority of successful
engineers do today. In my opinion, the ideal engineer need
not have an extremely mathematical training. In running a
railroad, it is far less important to get the line exactly curved
and mathematically accurate, than it is to run it where it will
cut least into expenses, which is the main point involved.
Imaginary quantities do not teach this. The student must be
taught to use efficiency engineering in handling his mathe-
matics. If this can be taught well, we shall have better engi-
neering students than if we attempt to teach them to handle
their problems by imaginary complex quantities.
Professor G. H. Morse: A previous speaker has referred
to alternating currents and to lack of familiarity with the
mathematics needed for this subject. I wish to emphasize the
absolute necessity for a certain amount of study of complex
quantities in this connection. I recently made a tour of a
DISCUSSION ON MATHEMATICS BEPORT. 131
number of western institutions — Illinois^ Purdue, Armour
Institute, and Wisconsin — ^with the object of discovering how
the professors were teaching electrical engineering. At Illi-
nois I found Professor Berg, who spent a great many years at
the General Electric works developing their many products.
I learned that he has given up entirely all methods of teaching
alternating currents except that involving the use of complex
quantities illustrated by graphics, of course. He insists upon
this method, both for himself and his assistants. The so-called
trigonometrical methods have no standing with him whatever.
At Purdue I found Professor Harding, and his attitude, while
not as radical as that of Professor Berg, was very similar.
In my own case I find that the use of complex quantities in
teaching alternating currents is wonderfully elucidating in
certain parts of the subject.
Some years ago I had the notion that there was mathematics
for engineers to use, the kind that is a necessary evil, and that
there was mathematics for mathematicians, in which they had
great pleasure in soaring, and which they jealously guarded
from use, preferring not to have any practical applications
made of it. Since I have been associated with the mathe-
maticians at the University of Nebraska my ideas have entirely
changed. I now find that every stage of these flights in pure
mathematics is a ''short cut.'' The higher the flight the
shorter and more useful the cut. If only the engineers can
appreciate these flights their work will be greatly simplified.
Professor 8. B. Gharters, Jr.: I wish to emphasize the
fact that we are dealing in engineering with two totally differ-
ent classes of students. In every group, in the proportion of
about one to fifteen or twenty, there is one engineer. Such a
man should have and will take and enjoy the fullest mathe-
matical training. On the other hand, the comparatively larger
number are not engineers at all. They are simply men who
are getting a certain amount of engineering training; and
these men fill the bulk of the positions. From the colleges of
the west a great many must go out into practical work as
mining superintendents, superintendents of construction in
132 DISCUSSION ON MATHEMATICS BEPOBT.
the installation of plants, etc. Now that class of work absorbs
the bulk of our men, and these have no use for higher mathe-
matics whatever. It might be a help to them and it might not.
Among those graduates whom I have observed, the ones who
have had the best success have not been great mathematicians.
The highest paid man we have among our alumni today, is one
who could not pass any mathematical examination, I am
reasonably certain. We have a few men who graduate every
year who should be given higher mathematics. We have a
feeling that, if it were possible, engineering should be divided
into two courses; the longer course of, say, five or six years,
with adequate mathematical training, for the man who shows
special aptitude on those lines. Those men should be the
leaders in the designing branch of the profession. A second
class of men need not have the higher mathematics, but should
have the proper training in handling men. These must do the
bulk of the work. We need a certain number of men to do the
designing and hand down formulas which these other men can
follow. We need more men to take those mathematical
formulae and from them get the results. That was illustrated
to me by a friend who stated that in the American Bell Tele-
phone Company there is one man who does the principal
mathematical work for the system. In each division they have
mathematicians to interpret this work to the rank and file.
Probably twenty-five or thirty experts do the mathematical
work for this large company and the rest of it is done by the
engineers who need have only the ordinary mathematical
training.
Professor H. 8. Jacoby: Allow me to call attention to the
fact that this report deals with minimum requirements, and
that we should express our appreciation of the splendid work
done by the committee. The report may not be perfect in
every part, but it will be worth a great deal to have it adopted,
printed and made available to the teachers whose work is
affected by it. It may be made a starting point for definite
recommendations ; changes may be made later as the necessity
for them appears. If in any institution the mathematical
DISCUSSION ON MATHEMATICS BEPOBT. 133
courses are of such a character as to require enlargements to
conform to the recommendations, it is very likely that they
will be modified in time. The report ought not to be a hin-
drance to any teacher of mathematics, or to any course of
study which is now more extensive in its scope.
Professor Webb: It occurs to me that there is something
else that can be said about the cause of the trouble between
engineers and mathematicians. An engineer very often has
a problem that he does not see through. He has a general
idea that mathematics is a powerful instrument, which needs
a mathematician to solve the problem ; and he thinks that if
he knew a little more mathematics he could solve it himself.
As a matter of fact, the problem may be very simple as to its
mathematics, and it may be only that he does not see through
its practical or engineering side. A school teacher came to
me with a problem a few days ago and said she had given it
to different people to solve, and some advocated one solution
and some another. One said that its solution heeded calculus ;
I thought it could be solved quite simply, but she thought not.
This was a problem of the so-called practical variety. A barn
forty feet square has a horse tethered to one corner of it by a
rope one hundred feet long. How much grass can the horse
graze over without going over the same grass twice ? The solu-
tion of this is very simple, but one should not expect mathe-
matics to solve it before the problem has been thoroughly
analyzed. Problems of this nature are constantly met with in
engineering work. Very little mathematics may be needed
after they are properly analyzed, but if this calls for more
common engineering sense and ingenuity than the engineer
has, one must not expect the average mathematician, much less
the recruit graduate, to make good the deficiency.
SYLLABUS ON COMPLEX QUANTITIES.*
BY CHAS. O. GUNTHEE,
Professor of Mathematics, Stevens Institute of Technology.
1. Derivation of f ormulce :
6<^=cos tf + 1 fidn tf, e-*^=cos tf — t sin tf,
costf= ^ , tsintf= s •
2. Definition and graphical representation of a complex
quantity. Polar trigonometric and polar exponential equiva-
lents of = X + ty, that is,
z=p{eosO-{-imn0)y polar trigonometric;
z=pe*(f, polar exponential;
in which p =y/x^ + y* is the modulus (the positive sign being
always associated with it) ; and Oy given by the relation
tan$=y/x, is the argument of z. Any multiple of 2v may
be added to the argument without altering the complex
quantity.
3. Graphical addition, subtraction, multiplication and divi-
sion of complex quantities. Graphical solution of the equa-
tion «» db 1 = 0. Logarithms of complex quantities.
Applications to Integration.
4. The expression
dy = taxL^p sec^'**^ OdO,
in which p and r are positive integers or zero, is by the substi-
* This syllabus was prepared as an appendix to the report of the
Committee on the Teaching of Mathematics to Engineering Students at
the request of the members of the Society present at the Pittsburgh
meeting.
134
SYLLABUS ON COMPLEX QUANTITIES. 135
tution tan0=f sina (a being an imaginary quantity) trans-
formed into
dy=i{ — 1) 'sin*' a cos*'' ada.
This latter expression can be integrated by doubling a as many
times as necessary.
The foregoing includes the integration of the expression
dy = cot*' csc****^ Ode,
since the latter expression may be written
As found in text-books, the integration of the expression
dy = cos*tfsin*tfcW (1)
is readily accomplished in three cases, namely :
(a) When either 7i or ft is an odd positive integer.
(&) When h-{-ki8 aji even negative integer.
(c) When both h and k are even positive integers, or zero.
The first two of these cases include fractional values for
h and k.
By means of the substitution given above, all the other cases
in which h and k are integers can be brought under one or
more of the three cases just mentioned.
In the above are also included all the cases for which the
general binomial differential expression
dy=x^{a-{- bx^)p/9dx
can be integrated without resorting to infinite series. This
expression is only another form of (1), since V* + ft^** can
always be represented by one of the three sides of a right
triangle and therefore expressed as a trigonometric function
of one of the acute angles of the triangle.
In determining the value of a definite integral, if the
variable is changed the limits should be changed to correspond.
For example, in finding the length of the arc of the parabola.
136 SYLLABUS OK COMPLEX QUANTinES.
y^^sssiax^ from the vertex to the point (a, 2a) , we have
1 /•««
/
•/Ui
£Bln «al aH sin «sl
008^ ada^ ai I (1 + oos 2a)(2a
. -in ssO t/< Bin «sO
ss a log^ (oos a + { sin a) + ai sin a eos a + C I
= a[log.(i/2 + l) + i/2].
Farther applications of complex quantities to integration
will be found in the author's discussion on p. 119.
<iln«sl
■ln«KO
c
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11