n
I
University of California Berkeley
THE THEODORE P. HILL COLLECTION
of
EARLY AMERICAN MATHEMATICS BOOKS
PRACTICAL TREATISE
ON THE
DIFFERENTIAL AND INTEGRAL
CALCULUS,
WITH SOME OF ITS
APPLICATIONS TO MECHANICS AND
ASTRONOMY.
BY
WILLIAM G. PECK, PH.D., LL.D.,
Profeswr of Mathematics and Astronomy in Columbia. Colleen
and of Mechanics in tht School of Mines.
A. S. BARNES & COMPANY,
NEW YORK AND CHICAGO.
PUBLISHERS' NOTICE.
PECK'S MATHEMATICAL SERIES.
CONCISE, CONSECUTIVE, AND COMPLETE.
I. FIRST LESSONS IN NUMBERS.
II. MANUAL OF PRACTICAL ARITHMETIC.
III. COMPLETE ARITHMETIC.
IV. MANUAL OF ALGEBRA.
V. MANUAL OF GEOMETRY.
VI. TREATISE ON ANALYTICAL GEOMETRY.
VII. DIFFERENTIAL AND INTEGRAL CALCULUS.
VIII. ELEMENTARY MECHANICS (without the Calculus).
XL ELEMENTS OF MECHANICS (with the Calculus).
NOTE. Teachers and others, discovering errors in any of
the above works, will confer a favor by communicating them
to us.
Entered according to Act of Congress, in the year 1870, by
WILLIAM G. PECK,
In the Office of the Librarian of Congress, at Washington.
PREFACE.
THE following pages are designed to embody the course
of " Calculus and its Applications," as now taught in
Columbia College. This course being purely optional, is
selected by those only who have a taste for mathematical
studies, and it is pursued by them more with reference to
its utility than as a means of mental discipline. These
circumstances have given to the present work a practical
character that can hardly fail to commend it to those who
study the Calculus for the advantages it gives them in
the solution of scientific problems. To meet the wants
of this class of students, great care has been taken to avoid
superfluous matter; the definitions have been revised and
abbreviated; the demonstrations have been simplified and
condensed ; the rules and principles have been illustrated
and enforced by numerous examples, so chosen as to famil-
iarize the student with the use of radical and transcen-
dental quantities; and finally, the manner of applying the
Calculus has been exemplified by the solution of a variety
of problems in Mechanics and Astronomy.
The method employed in developing the principles of
the science is essentially that of Leibnitz. This method,
4: PKEFACE.
generally known as the method of infinitesimals, has been
adopted for several reasons: first, it is the method adopted
in all practical investigations; second, it is the method
most easily explained and most readily comprehended ;
and third, it is a method, as will be shown in the final
note, identical in results with the more commonly adopted
method of limits, differing from it chiefly in its phraseology
and in the simplicity of its results.
The author cannot conclude this prefatory note with-
out acknowledging his obligations to those students who,
from year to year, have been willing to turn aside from
the attractive pursuit of classical learning to engage in the
sterner study of those processes that have contributed so
much to the progress of modern science. Without their
interest and co-operation this book would never have been
written. He would also take this opportunity to express
his thanks to his distinguished colleague, Professor J. H.
VAN AMRISTGE, not only for many valuable suggestions
made during the progress of the work, but also for much
effective labor in reading and correcting the proofs as they
came from the press.
COLUMBIA COLLEGE,
Nwember 24&, 1870.
CONTENTS.
PART 1.-DIFFERENTIAL CALCULUS.
I. DEFINITIONS AND INTRODUCTORY REMARKS.
Art.
1. Classification of Quantities 9
2. Functions of one or more Variables 9
3. Geometrical Representation of a Function 10
4. Differentials and Differentiation 10
5. Geometrical Illustration 13
6. Infinites and Infinitesimals 13
7. General Method of Differentiation 14
II. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS.
8. Definition of an Algebraic Function 15
9. Differential of a Polynomial 15
10. Differential of a Product 16
11. Differential of a Fraction 17
12. Differential of a Power 18
13. Differential of a Radical 18
III. DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS.
14. Definition of a Transcendental Function 24
15. Differential of a Logarithm 25
16. Differential of an Exponential Function 26
17. Differentials of Circular Functions 29
18. Differentials of Inverse Circular Functions 32
IV. SUCCESSIVE DIFFERENTIATION AND DEVELOPMENT OF
FUNCTIONS.
1 9. Successive Differentials 35
20. Successive Differential Coefficients 35
21. McLaurin's Formula 37
M. Taylor's Formula 41
CONTEXTS.
V. DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES AND OF
IMPLICIT FUNCTIONS.
Art. PACK
23. Geometrical Representation of a Function of two Variables. 45
24. Differential of a Function of two Variables 4G
25. Notation for Partial Differentials 47
26. Successive Differentiation of Functions of two Variables ... 48
27. Extension to three or more Variables 50
28. Definition of Explicit and Implicit Functions 50
29. Differentials of Implicit Functions 50
PART II APPLICATIONS OF THE DIFFERENTIAL
CALCULUS.
I. TANGENTS AND ASYMPTOTES.
30. Geometrical Representation of first Differential Coefficient. . 53
31 . Applications 54
32. Equations of the Tangent and Normal 55
33. Asymptotes 57
34. Order of Contact 59
II. CURVATURE.
35. Direction of Curvature GO
36. Amount of Curvature 62
d7. Osculatory Circle 63
38. Radius of Curvature 64
39. Co-ordinates of Centre of Curvature 65
40. Locus of the Centre of Curvature 66
41. Equation of the Evolute of a Curve 67
III. SINGULAR POINTS OF CURVES.
42. Definition of a Singular Point 08
43. Points of Inflexion 68
44. Cusps 70
45. Multiple Points 71
46. Conjugate Points 73
IV. MAXIMA AND MINIMA.
47. Definitions of Maximum and Minimum 74
48. Analytical Characteristics 75
49. Maxima and Minima of a Function of two Variables 86
CONTENTS. 7
V. SINGULAR VALUES OP FUNCTIONS.
Art. PAOR
50. Definition and Method of Evaluation 88
VI. ELEMENTS OF GEOMETRICAL MAGNITUDES.
61. Differentials of Lines, Surfaces, and Volumes 91
VII. APPLICATION TO POLAR CO-ORDINATES.
52. General Notions, and Definitions 93
53. Useful Formulas 94
54. Spirals 97
55. Spiral of Archimedes 98
VIII. TRANSCENDENTAL CURVES.
56. Definition 99
57. TheCycloid 99
58. The Logarithmic Curve 101
PART III. INTEGRAL CALCULUS.
59. Object of the Integral Calculus 102
60. Nature of an Integral 1 02
61. Methods of Integration ; Simplifications 104
62. Fundamental Formulas 104
63. Integration by Parts 114
04. Additional Formulas ' . 115
65. Rational and Entire Differentials 120
06. Rational Fractions 121
67. Integration by Substitution, and Rationalization 128
68. When the only Irrational Parts are Monomial 128
69. Binomial Differentials 130
70. Integration by Successive Reduction 133
71. Certain Trinomial Differentials 141
72. Integration by Series 144
73. Integration of Transcendental Differentials. 145
74. Logarithmic Differentials 145
75. Exponential Differentials 147
70. Circular Differentials 148
77. Integration of Differential Functions of two Variables 153
PART IV. APPLICATIONS OF THE INTEGRAL CAL-
CULUS.
I. LENGTHS OF PLANE CURVES.
78. Rectification. . 156
CONTENTS.
II. AREAS OP PLANE CURVES.
A rt ' PAOl
79. Quadrature 101
III. AREAS OF SURFACES OF REVOLUTION.
80. Surfaces Generated by the Revolution of Plane Curves 105
IV. VOLUMES OF SOLIDS OF REVOLUTION.
81. Cubature 1G7
PART V. APPLICATIONS TO MECHANICS AND
ASTRONOMY.
I. CENTRE OF GRAVITY.
82. Principles Employed 171
83. Centre of Gravity of a Circular Arc 173
84. Centre of Gravity of a Parabolic Area 174
85. Centre of Gravity of a Semi-Ellipsoid of Revolution 175
86. Centre of Gravity of a Cone 176
87. Centre of Gravity of a Paraboloid of Revolution 176
II. MOMENT OF INERTIA.
88. Definitions and Preliminary Principles 176
89. Moment of Inertia of a Straight Line 177
90. Moment of Inertia of a Circle 178
01. Moment of Inertia of a Cylinder 179
92. Moment of Inertia of a Sphere 180
III. MOTION OF A MATERIAL POINT.
93. General Formulas 181
94. Uniformly Varied Motion 182
95. Bodies Falling under influence of Constant Force 183
96. Bodies Falling under action of Variable Force 186
97. Vibration of a Particle of an Elastic Medium 188
98. Curvilinear Motion of a Point 190
99. Velocity of a Point rolling down a Curve 192
100. The Simple Pendulum 193
101. Attraction of Homogeneous Spheres 195
102. Oibital Motion 199
103. Law of Force 204
104. Note on the Methods of the Calculus 205
PART I.
DIFFERENTIAL CALCULUS.
I. DEFINITIONS AND INTRODUCTORY REMARKS,
Classification of Quantities.
I. THE quantities considered in Calculus are of two
kinds: constants, which retain a fixed value throughout
the same discussion, and variables, which admit of all pos-
sible values that will satisfy the equations into which they
enter. The former are usually denoted by leading letters
of the alphabet, as a, I, c, etc.; and the latter by final
letters, as x, y, z, etc. ; particular values of variable quan-
tities are denoted by writing them with one or more dashes,
as x', y", z'", etc.
Functions of one or more Variables.
2. Relations between variables are expressed by equa-
tions. In an equation between two variables, values may
be assigned to one at pleasure; the resulting equation de-
termines the corresponding values of the other. The one
to which arbitrary values are assigned is called the inde-
pendent variable, and the remaining one is said to be a
function of the former. If an equation contain more than
two variables, all but one are independent, and that one is
a function of all the others. The fact that a quantity
10 DIFFEEENTIAL CALCULUS.
depends on one or more variables may be expressed as
follows :
y =f(z) ; z - <p(x, y) ; F(x, y, z) = 0.
The first shows that y is a function of x, the second that
z is a function of x and y, and the third that x, y, and z,
depend on each other, without pointing out which is u
function of the other two.
Geometrical representation of a Function.
3. Every function of one variable may be represented b}
the ordinate of a curve, of which the variable is the cor-
responding abscissa. For, let y be a function of x, and
suppose x to increase by insensible gradations from oo
to + oo . For each value of x there will be one or more
values of y, and these, if real, will determine the position
of a point with respect to two rectangular axes. These
points make up a curve, at every point of which the rela-
tion between the ordinate and abscissa is the same as that
between the function and independent variable. This
curve is called the curve of the function.
For values of x that give imaginary values of y there are
no points; for those that give more than one real value of
y there is a corresponding number of points.
In a similar manner -it may be shown that a function
of two variables represents the ordinate of a surface of
which the variables are corresponding abscissas.
Differentials and Differentiation.
4. Of two quantities, that is the less whose value is
nearer to oo, and that is the greater whose value is
nearer to + oo. A quantity is said to increase when it
DEFINITIONS AND INTRODUCTORY REMARKS. 11
approaches + GO, and to decrease when it approaches oo.
The ordi nates of a curve originate from the axis of X and
of any two, that is the greater whose extremity is nearer
4- oo, and that is the less whose extremity is nearer co;
in like manner of two abscissas, that is the greater whose
extremity is nearer -f- oo , and that the less whose extremity
is nearer oo .
In w r hat follows we shall suppose the independent varia-
ble to increase by the continued addition of a constant but
infinitely small increment. For every change in the value
of the variable there is a corresponding change in the value
of the function. In some cases, as the variable increases,
the function increases ; it is then said to be an increasing
function : in other cases the function decreases as the
variable increases ; it is then said to be a decreasing func-
tion. In all cases, the change in value is called an incre-
ment ; for increasing functions the increment is positive,
and for decreasing functions it is negative. The increment
of the function is always infinitely small, but it is not con-
stant, except in particular cases.
The infinitely small increment of the independent varia-
ble is called the differential of the variable, and the cor-
responding increment of the function is called the differ-
ential of the function. Hence, the differential of a quantity
is the difference between two consecutive values of that
quantify. It is to be observed that the difference is always
found by taking the first value from the second.
The operation of finding a differential is called differen-
tiation. The object of the differential calculus is to ex-
plain the methods of differentiating functions.
DIFFERENTIAL CALCULUS.
Geometrical Illustratiou.
5. Let KL be a curve in the plane of the rectangular
axes OX and OY 9 and let OA and OB be two abscissas
differing from each other by an infi-
nitely small quantity A B. Through
A and B draw ordinates to the
curve, and let PR be parallel to
OX. OA and OB are consecutive
abscissas, AP and BQ are consecu-
tive ordinates, and P and Q are
consecutive points of the curve.
The part of the curve PQ does
not differ sensibly from a straight line, and if it be pro
longed toward T, the line PT is tangent to the curve at
P. If we denote any abscissa OA by x, and the corre-
sponding ordinate by y, we have,
Fig. ].
The line AB is the differential of the independent varia-
ble, denoted by the symbol dx; RQ is the differential of
the function, denoted by dy\ and PQ is the differential
of the curve KL, denoted by ds.
The right angled triangle, RPQ, gives the relation
ds = Vdx* + dy\ Denoting the angle RPQ by 6, we
have, from trigonometry,
dy
~ ;
dx
_dy_
~ ds~
dy
dx
and, cosd =-=- =
cfe
dx
-f- dy*
DEFINITIONS AND INTRODUCTORY REMARKS. 13
Infinites and Infinitesimals.
6. A quantity is infinitely great with respect to another,
when the quotient of the former by the latter is greater
than any assignable number, and infinitely small with re-
spect to it, when the quotient is less than any assignable,
number. If the term of comparison infinite, quantities of
the former class are called infinites, and those of the latter
infinitesimals.
Infinites and infinitesimals are of different orders. Let
us assume the series,
a a a
' ' ?' a 5 ' "? a ' aX) ' '
In which a is a finite constant and x variable. If we
suppose x to increase, the terms preceding a will diminish,
and those following it will increase ; when x becomes greater
than any assignable quantity, becomes infinitely small
30
with respect to a, and because each term bears the same
relation to the one that follows it, every term in the series
is infinitely small with respect to the following one, and
infinitely great with respect to the preceding one. The
quantity ax being infinitely great with respect to a finite
quantity, is called an infinite of i\\.Q first order; ax*, ax*,
etc., are infinites of the second, third, etc., orders. The
quantity - being infinitely small with respect to a finite
x
quantity is called an infinitesimal of the first order; -^ -3,
x x
etc, are infinitesimals of the second, third, etc., orders.
It is to be observed that the product of two infinitesi-
mals of the first order, is an infinitesimal of the second
1* D1FFEKEXT1AJ. CALCULUS.
order. For, let x and y be infinitely small with respect
to 1, we shall have,
1 : x : : y : xy.
Hence, xy bears the same relation to y that x does to 1,
that is, it is infinitely small with respect to an infinitesimal
of the first order; it is therefore an infinitesimal of the
second order. The product of three infinitesimals of the
first order is an infinitesimal of the third order, and so on.
In general, the product of an infinitesimal of the ?/? th order
by one of the n th order, is an infinitesimal of the (m + n) tb
order. The product of a finite quantity by an infinitesimal
of the w th order, is an infinitesimal of the w th order.
From the nature of an infinite quantity, its value will
not be sensibly changed by the addition or subtraction of
a finite quantity. A finite quantity may therefore be dis-
regarded in comparison with an infinite quantity. For
a like reason an infinitesimal may be disregarded in com-
parison with a finite quantity, or with an infinitesimal of a
lower order. Hence, whenever an infinitesimal is con-
nected, by the sign of addition, or subtraction, with a finite
quantity, or with an infinitesimal of a lower order, it may
be suppressed without affecting the value of the expression
into which it enters.
General method of Differentiation.
7. In order to find the differential of a function, we give
to the independent variable its infinitely small increment,
and find the corresponding value of the function; from
this we subtract the preceding value and reduce the result
to its simplest form; we then suppress all infinitesimals
which are added to, or subtracted from, those of a lowei
order, and the result is the differential required.
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 15
This method of proceeding is too long for general use,
and is only employed in deducing rules for differentiation.
II. DIFFERENTIATION OF ALGEBRAIC FUNCTIONS.
Definition of an Algebraic Function. '
8. An algebraic function is one in which the relation
between the function and its variable can be expressed by
the ordinary operations of algebra, that is, addition, sub-
traction, multiplication, division, formation of powers de-
noted by constant exponents, and extraction of roots indicated
by constant indices. Thus,
y* = 2px, y = ax 3 ^/bx, and ^/y = *^a*x bx* %
are algebraic functions.
Differential of a Polynomial.
9. Let a and c be constants, and r, s, t, functions of x\
assume
y = ar + s t + c\ (1)
in which y denotes the polynomial in the second member,
and is therefore a function of x.
If we give to x the increment dx, the functions y, r, s*
and t, receive corresponding increments dy, dr, ds, dt, and
we have,
y + dy = a(r + dr) + (s 4- ds) - (t + dt) + c (2)
subtracting (1) from (2), we have,
dy = adr + ds dt (3)
Hence, to differentiate a polynomial, differentiate each
term separately and take the algebraic sum of the results.
Comparing (1) and (3) we see, first, that a constant factor
remains unchanged, and secondly, that a constant term
disappears by differentiation.
16 DIFFERENTIAL CALCULUS.
Differential of a Product.
10. Let r and s be functions of x. Placing their pro-
duct equal to y, we have,
y = rs (1)
Giving to x the increment dx, we have, as before,
y 4- dy = (r -f dr) (s 4- ds) = rs + rds + sdr + drds. . (2)
Subtracting (1) from (2) and suppressing ^TY?S, which is an
infinitesimal of tbe second order, we have, after replacing
y by its value rs,
d(rs) = rds + sdr (3)
Hence, to differentiate the product of two functions,
multiply each by the differential of the other, and take the
algebraic sum of the results.
If we suppose r = tw, we have, from the rule,
dr = tdw + wdt ;
which substituted in (3), gives,
d(stiu) = twds + stdiv + swdt (4)
In like manner the principle may be extended to the
product of any number of functions. Hence, to differen-
tiate the product of any number of functions, multiply the
differential of each ~by the continued product of all the
others, and take the algebraic sum of the results.
COR. If we divide both members of (4) bystw, we have,
d(sfw) _ ds dt dw ..
1 fJ I
stw s t w
and similarly, where there are a greater number of factors.
Hence, the differential of a product divided ly that pro-
duct, is equal to the sum of the quotients obtained ly
dividing the differential of each factor by that factor.
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 17
Differential of a Fraction.
11. Let 6 1 and t be functions of x, and assume,
Giving to x the increment dx, we have,
Subtracting (1) from (2),
+ ds s ids sdt
dy =
t + dt t f + tdt '
s
Replacing y by its value -, and suppressing tdt in compari-
son with f , we have,
ids - sdt
= jr .(8)
Hence, the differential of a fraction is equal to the denom-
inator into the differential of the numerator, minus the
numerator into the differential of the denominator, divided
by the square of the denominator.
If either term of the fraction be constant, its differential
will be, 0. When the denominator is constant, formula (3)
becomes,
'6) = 7'
... (4)
when the numerator is constant it becomes,
18 DIFFERENTIAL CALCULUS.
Differential of a Power.
12. Let s be a function of x, and m a constant. Assume
y = * m ..... (i)
Giving to x the increment clx, we have,
y + dy = (s + ds) m = s m + ms m " l ds
(etc.) ..... (2)
In which all the terms of the development, after the second,
contain the square, or some higher power, of ds. Subtract-
ing (1) from (2), we have,
dy = ms m ~^ds + (etc.) ds 2 .
Suppressing all the terms of the second member, after the
first, in accordance with the principle laid down in Art. 6,
and replacing y by its value, we have,
d(s m ) = ms m ~^ds ..... (3)
Hence, to differentiate any power of a function, diminish
the exponent oj 'the function by 1, and multiply the result by
the primitive exponent and the differential of the function.
Differential cf a Radical.
13. Let s be a function of x, and assume,
y = ft ..... (i)
DIFFERENTIATION" OF ALGEBRAIC FUNCTIONS. 19
We may write (1) in the form,
1
Differentiating (2), as in the last article, we have,
1-1 1-rc
dy = -s n ds = -s n ds ..... (3)
J n n
Replacing y by its value, and remembering that
s \n W e have.
.
(4)
That is, the differential of a radical of the ^ th degree
5.3 equal to the differential of the quantity under the
radical sign, divided by n times the (n l) th poiver of the
radical (ifcz *ra,
uuj -nC
,
^~s = f- ..... (5)
oT? " ^ 1&
That is, the differential of the square root of a quantity
is equal to the differential of the quantity under the radical
sign, divided by twice the radical. & b*-^
The preceding rules are sufficient to differentiate any
algebraic function whatever.
20 DIFFERENTIAL CALCULUS.
EXAMPLES.
1 1. Let y = 5z 3 .
This is the product of the function cc 3 by the constant 5 ;
hence, (Art. 9),
dy = 5d(x*) = 5X 3x*dx = 15x*dx (Art. 12).
2. Let y = x 3 + 2x 2 + 3x + 4.
This is a polynomial. Hence, from Art. 9,
dy = 3x 2 dx + ^xdx + MX.
3. Lety = ( + J) s .
Considering a + Jo; as a single quantity, we have,
(Art. 12),
dy = 3(+ te) 8 d(a + lx) = 3(a + lx)*bdx = 3b(a + bx)*dx.
4. Let y = (a + lx*) n .
dy = n(a
5. Let y = A/ 2 - 2 = (a 2 - z 2 ) 2 .
dy = J(a 2 - z 2 )"^ x - Zxdx = -
6. Let y = (a + x)(b + 2a?).
By Art. 10, we have,
dy= (a + x)d(b + 2x 2 ) + (b + 2x 2 )d(a -\-
i- (b + 2x*)dx ; or, fy
B + b)dx.
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 21
By Art. 11, we have,
x 2 )d(ax) axd(a* + a 2 ) __ a(a 2 -x 2 )
~~
,
8. Let y=^ x +
dy = d[x + (a 2
X
W,t, d\x + (a*
10. y=a to <i
Am. dy = -
11. y = (a 2 - x*)*. Ans. dy = - x(a* - z*)dx.
12. y = (%ax x 2 ) 3 . Ans. dy = 6(a - x)(2ax x*)'dx,
DIFFERENTIAL CALCLLUS.
13. y = (a+bx n ). Ans.dy = mn1)x n - l (
14. y=(2ax + x*) n . Ans.dy=2n(a + x)(2ax
15. y = x(a + x)(a z
Ans. dy = (a 3 -f 2a*x + Sax 2 + x*)dx.
16. y = (a + x) m (b + x} n .
17. v =
a x
a + x'
18. y =
x + 3
-x + l
19. y = fi
20. y =
21. y =
22. y =
23. y = (a a;)
24. y= (a + x)Va
+OX
Ans. dv =
(x + 2)(x
(
. dy = 7-
-
- 2
Ans. dy =
Ans. dy = xdx -
'
(2rcg + b)dx
~ 2V ax 2 + bx + o ;
( + 3:?;)^
-4ft*. tfy = - ; .
%Va + x
(a - 3x)dx
Ans. dy = ; -
2Va - x
DIFFERENTIATION OF ALGEBRAIC FUNCTIONS. 23
* - 3)dx
. ,
25. y \l x* -{- X s )*. Ans. dy = -
26. y (a 2 x 2 )Va + x.
Ans. dy
87. y = -/=
va
23. =
. dy =
(3a -
adx
20. =x-
Ans. dy =
2v / tf*-^ i (-rVat*-
30. y = x(l 4 x
31. y= (1
32. v =
_*). Ans. dy =
_
v 1
a + VI
33. y = (1 + x)Vl - x. Ans. dy =
- a;
34. =
DIFFERENTIAL CALCULUS.
36 - y = 2x -1- V%x - 1 - V%z -I- etc., ad inf.
Squaring,^ 8 =2a;-l-V2aj- 1 - V&e - 1 - etc., ad inf.
Hence, y* = 2x 1 y, or y 9 + y = Zx 1.
Solving, y = - 5
/.<& = 5
III. DIFFERENTIATION OF TRANSCENDENTAL FUNCTIONS.
Definition of a Transcendental Function.
14. A transcendental function is one in which the rela-
tion between the function and its variable cannot be ex-
pressed by the ordinary operations of algebra.
Transcendental functions are divided into three classes*
Logarithmic, in which the relation is expressed by loga-
rithmic symbols; Exponential, in which the variable enters
an exponent; and Circular, in which the relation is ex-
pressed by means of trigonometric symbols. Thus,
y = log (ax* + #), is a logarithmic function,
y = (ax 2 + cY x ~ c , is an exponential function,
and y = sin 2 # 2tan#, is a trigonometric function.
TRANSCENDENTAL FUNCTIONS. 25
Differential of a Logarithm.
15. It is shown in Algebra that
/ ?/2 -yS ? /4 \
log (1 + y) = M\JJ - J + ^ '-- + elc.J .... (1)
In which M is the modulus of the system and y any quan-
tity whatever.
Substitute for y the quantity , s being a function of #,
s
and we have, after reduction,
4- ds ,ds ds* ds*
But the logarithm of a quotient, is equal to the logarithm
of the dividend, diminished by the logarithm of the divi-
sor; changing the form of the first member and suppress-
ing all the terms in the second member, after the first,
(Art. 6), we have,
tLt
log (s + ds) - log s = M- ..... (3)
s
But the first member is the difference between two con-
secutive values of log s; it is therefore the differential of
the logarithm of s; hence,
fj~
d(logs) = M- ..... (4)
s
That is, the differential of the logarithm of a quantity is
equal to the modulus into the differential of the quantity,
divided ly the quantity.
In analysis, Napierian logarithms are almost always used.
26 DIFFERENTIAL CALCULUS.
Denoting the Napierian logarithm by I, and remembering
that the modulus of this system is 1, we have,
The base of the Napierian system is represented by the
letter e, and its logarithm in that system is equal to 1.
Differential of an Exponential Function.
16. Let a be a constant quantity, s any function of z>
and assume
y = <f (i)
Taking the logarithms of both members of (1), we have,
ly sla (2)
Differentiating (2), we have,
dy
-^- la X ds (3)
y
"Replacing y by its value, and reducing, we have,
d(a s ) = triads (4)
Hence, to differentiate a quantity formed by raising a con-
stant quantity to a power denoted by a variable exponent,
multiply the quantity ly the logarithm of the root and the
differential of the exponent.
Again, let us have,
y=t s (1)
in which both t and s are functions of x. Taking the
logarithm of both members, we have,
ly=slt (2)
TRANSCENDENTAL FUNCTIONS. 27
\\ r heiice, by differentiating both members,
f=W? ..... (3)
Replacing y by its value, clearing of fractions and reducing,
we have,
(4)
That is, to differentiate a quantity formed by raising a
variable quantity to a power whose exponent is variable,
differentiate first as though the root alone were variable,
then as though the exponent alone were variable, and take
the sur.i of the results.
EXAMPLES.
1. Let y = l = l(a + x)-l(a-z);
(t x
d(a + cc) d(a x) dx dx
: ail = - -- -- 1 --
a + x a x a + x a x
9, Let y = I A/ 1 = ^(1 + x) - \l(l - x);
1 dx 1 dx dx
.-. dy = - - + a = r ~~
9 21 + x 21 x l x
3. Txjt y ~ (a) e ; /. ly = e x la, and = lae^ledx.
But, le 1, /. dy = (a} e e*ladx. Ans.
28 DIFFERENTIAL CALCULUS.
4. Let y = l(lx) = T'x.
*
d(lx] dx
By the rule, dy = \ = r-
Ix xlx
5. y = #Z(a 4- #). ^4^5. tZ^ /( 4- #)(/ 4-
xdx
a + x
6. y = l(a + xY = %l(a 4- x). Ans. dy
l.y = <?(x-l).
8. y = e%T 2 - 2z + 2).
a 4- x
Ans. dy = e^atffa;.
. dy - e x x z dx.
,
Ans. dy ---- dx.
Ans. dy = e x \lx + a k
11. ?/ = / + a 4-
12. =
13. y = I
+ 1 + a;
15. y=aT'
16. y=a lr .
Ans. dy =
dx dx
Ans. dy --- ._
,2-
1 _ ^2
-4w*. Jy
Ans. dy a
I + x .
dx.
TRANSCENDENTAL FUNCTIONS. 29
Differentials of Circular Functions.
17. Assume the equation,
y = siux (1)
Giving to x the increment dx, and developing, we
have,
y + dy = sin (a; + dx)
= smxcosdx + cosxsmdx . . . . (2)
Because the arc dx is infinitely small, its cosine is equal
to 1, and its sine is equal to the arc itself. Hence.
y -f dy = smx + cosxdx (3)
Subtracting (1) from (3), and replacing y by its value,
we have,
d(smx) = cosxdx (a)
Equation (a) is true for all values of x\ we may there-
fore replace x by (90 x) ; this gives,
c/[sin(90 - a;)] = cos(90 - a;)c?(90 - x) (4)
Reducing, we have,
d(cosx) = siuxdz ()
We have, from trigonometry, the relation,
sinrc
tana; = - - (5)
cosz
Differentiating both members, we have,
,,, , cosxd (sinx) sinxd (cosx)
d(t&nx) = - s .',
v 3
SO DIFFERENTIAL CALCULUS.
Performing indicated operations, and reducing by the
relation, sin 2 a; -f cos 8 z 1, we have,
dx
c?(tanz) = - - ..... (c)
1 2
Replacing x, in formula (c), by (90 x) and reducing,
we have,
rlv
d(cotx) = - -^- ..... (d)
Sill 2 X t
We have, from trigonometry,
ver-sinz = 1 cosx ..... (G)
co-versinz = 1 sinz ..... (7)
Differentiating (6) and (7), we have,
c?(ver-siiic) = siuxdx ..... (e)
d(co-\GYS\nx) = cosxdx ..... (/)
Wo have, from trigonometry,
secz = - ..... (8)
cosz
cosecx = -: ..... (0)
suix
Diflerentiating (8) and (9), we have, after reduction,
d(secx) = tsiiixsecxdx .
d(cosQCx) == cota; cosecxdx
The lettered formulas, from (a) to (/*) inclusive, are
sufficient for the differentiation of any direct circular
function.
TRANSCENDENTAL FUNCTIONS. 31
EXAMPLES.
1. Let y = sin mo:.
dy = cosmx d(?)ix) = m cosmx . dx t
2. Let y = sm m x.
dy = m sin m ~ x dsmx = m &m m xcosxdx.
3. Let y = sm2x cosx.
dy = (d sm2x) (cosx) + (dcosx)(sm2x)
= 2cos2x cosx dx sin2x smx dx.
4. Let ?/ = l(sm z x) 2lsinx.
. dsinx 2cosx 7
dy = 3 : = ~. dx = 2cotx dx,
smx smx
5. Let y = j = ^(i + cosx) - %l(l - cosx).
V 1 cosay
_ 1 smxdx 1 sinxdx sinxdx dx
2 I + cosx 21 cosx ~ 1 cos*x ~ siuz*
6. Let y = e x cosx.
dy = e x d(cosx) + cosx d(e x ) e x (cosx smx) dx.
7. y = z-sinx + cos x. Ans. dy = xcosxdx.
8. y = 2xsinx + (2 x z )cosx. Ans. dy = x 2 sinxdx.
9. y = tana x. Ans. dy = ttm*xdx.
10. y = e cosx smx. Ans. dy = e cosx (cosx - sin*x)dx.
f cot^\
11. y = 2l(&in.x) + coseca:. Ans. dy = [ 2cotx -- : -jdx,
32 DIFFERENTIAL CALCULUS.
12. y = l(cosx + V 1 sinz). Ans. dy = V idx
. dx
- -) Ans.dy = -- .
1 smxj J cosx
14. y =- /[tan (45 + Jaj)].
17
15. y sin(fe). Ans. dy = -cos(lx)dx.
Differentials of Inverse Circular Functions.
18. It is often convenient to regard an arc as a function
of one of its trigonometrical lines. Such functions are
called inverse circular functions, and are expressed by
such symbols as the following :
sin~~V> cos V> tan V*^,
which are read the arc ivliose sine is y> the arc whose cosine
is y, the arc whose tangent is y, etc.
Formulas for the differentiation of inverse circular func-
tions may be deduced from the lettered formulas of the last
article. Thus, from formula (), we find,
cosx
Tf we make sins = y, we have,
x = sin" 1 ?/, and cosx = Vl
Substituting these in (1), we have,
TRANSCENDENTAL FUNCTIONS. 33
In like manner, from formula (b), we find,
smx
Making cosa; = y, we have,
x
= cos~ y, and sinz = Vl y* ;
Substituting these in (2), we have,
vr^
By a similar course of reasoning we find from the re-
maining lettered formulas of Art. 17, the corresponding
formulas lor inverse circular functions, as given below:
_1 dy
dy
1 + y*
1 dy
d (versin y) =
-I,A *y
d(co versin 1 y) = (/"*)
dv
,// 1..\ _ y /~i\
yVy* - i
n "">
2*
34
DIFFERENTIAL CALCULUS.
EXAMPLES.
Let y cos~~ 1 # / v/l # 8 = cos"
- x * =
and A/I _
/. dy
(1 -
Let = sin
1
~
(
vrrP
dx
3. Let y = sin" 1 2x\/l %*
2(1 -
-*
_
= tan
2dx
DEVELOPMENT OF FUNCTIONS. 35
l /y //7 -^ / F\ &
6. y = V# 2 a; 2 4- flrsin" . Ans. dy = I I dx.
a \a + x/
. _
7. = versm A .
Vd^
8. y = sec -1 2a;.
IV. SUCCESSIVE DIFFERENTIATION AND DEVELOPMENT
OF FUNCTIONS.
Successive Differentials.
19. The differential obtained immediately from the func-
tion is the first differential of the function ; the differential
of the first differential is the second differential of the func-
tion ; the differential of the second differential is the third
differential of the function, and so on ; differentials thus
obtained are called successive differentials, and the opera-
tion of obtaining them is called successive differentiation.
If a function be denoted by y, its successive differentials
will be denoted by the symbols dy, d*y, d*y, etc. Thus,
if y = ax 3 , we have, by successive differentiation, dx be-
ing constant, dy = 3ax 2 dx, d z y = Gaxdx 2 , d*y = 6adx 3 ,
fry = 0.
Successive Differential Coefficients
20. If the differential of a function be divided by the
differential of the variable, the quotient is the first differ-
ential coefficient of the function ; the differential coefficient
of the first differential coefficient is the second differential
coefficient of the function, and so on. Differential coeffi-
36 DIFFERENTIAL CALCULUS.
cients, derived in this manner, tire called successive differ-
ential coefficients. If a function of x be denoted by y,
its successive differential coefficients are denoted by the
symbols, -r, ~, -, etc. Thus, if we have, as before,
y ax 3 , we have, from the principles just explained,
It is to be observed that the successive differential
coefficients are entirely independent of the differential of
the variable ; so long therefore as this is infinitesimal, the
differential coefficients will in no way be affected by any
change in its absolute value.
EXAMPLES.
J^ind the successive differential coefficients of the follow-
ing functions:
1 . v = ax n .
. ( = nax^; = n(n -
n (n - l)(n - 2)ax n ~ ; etc
If n is a positive whole number, there will be a finite
number of successive differential coefficients; otherwise
their number is infinite.
2. y = ax 3 f bx*.
Ans. ^ =
dx
DEVELOPMENT OF FUNCTIONS. 3?
3. = fl *.
4. y = since.
Ans. -/ cosx: -=-f- = sin ;
dx dx*
2. _ cosx: -5- smz: etc.
c?a; 3 '
5. y = Z(a? + 1).
6. y = xe x .
McLaurin's Formula.
21. McLaurin's formula, is a formula for developing a
function of one variable into a series arranged according
to the ascending powers of that variable, the coefficients
being constant. Let y be any function of x, and assume
the development,
= A+Bx+ Cx* + Dx 9 + Ex* + etc ...... (1)
38 DIFFERENTIAL CALCULUS.
It is required to find such values for A, B, C, etc., as will
make the assumed development true for all values of x.
Differentiating, and finding the successive differential co-
efficients of y, we have,
2Cx + Wtf + 4^ 3 + etc. ... (2)
+ l.Z.Wx + 3.4^ 2 + etc. . . . (3)
ax
4- etc ....... (4)
etc., etc., etc.
But, by hypothesis, the value of y, and consequently the
values of its successive differential coefficients, are to be
true for all values of x\ hence, they must be so for x 0.
Making x ~ 0, in equations (1), (2), (3), etc., and denoting
what y becomes under that hypothesis by (y} ; what
~ becomes by ( ) ; what =-? becomes by ( -7-^ j, and so
dx ' \clxJ dx z J \dx 2 J
on ; we have,
(y) =A; .:A = (y);
123-D- ' Z?=
T23\3&
etc., etc., etc.
DEVELOPMENT OF FUNCTIONS. 39
Substituting these values in (I), we have,
dy\ x
i 1 T n I -* ,TK i 1 T o I - ^ r\ I ^ tt-/
which is the formula required. Hence, to develop a
function of one variable in terms of that variable, find its
successive differential coefficients ; then make the variable
equal to in the function and its successive differential
coefficients, and substitute the results for (y), (~f\ \~di
etc., in formula (5).
Thus, let it be required to develop sinz into a series
arranged with reference to x. We have,
y = sinz, .-. (y) =
dy fdy
= cosx, .: [ -?-
dx \dx
= smz
dx~* = *y- i^J =
etc., etc., etc.
Hence,
In like manner we find,
SX =l - + - + etc
It is to be observed that (7) may be found from (6) by
differentiating and then dividing by dx.
40 DIFFEKENTIAL CALCULUS.
McLau rin's Formula enables us to develop any function
of one variable when it can be developed in accordance
with the assumed law. But there are functions which
cannot be developed according to the ascending powers of
the variable ; in this case the function, or some of its suc-
cessive differential coefficients, become oo,when the variable
is made equal to 0.
As a general rule, when the application of a formula
gives an infinite result, the formula is inapplicable in that
particular case.
EXAMPLES.
1. y = (a + X) U .
Ans
n , n n-1 , n (n 1) n -2 .
. y = a +y# x+ 12
n (-!)(- 2) g .- 8gi
JL,</,{)
X 2 , X 3 X 4 * X 6
^ + ^-- + -
3. y = a*.
4. y =
Make x* 2;, and devoir p; then replace z by its value.
x 2 x* x 6 5x*
DEVELOPMENT OF FUNCTIONS. 41
5. y = e* x .
, x* 3x* Sx*
'= [+ * + -
1.2.3.4.5.6
6. ^
r- n + etc.
Taylor's Formula.
22. Taylor's Formula is a formula for developing a
function of the sum of two variables into a series arranged
according to the ascending powers of one, with coefficients
that are functions of the other.
LEMMA.
If u' f(x -\- y), the differential coefficient of u' will be
the same, whether we suppose x to remain constant and y
to vary, or y to remain constant and x to vary, for the
form of the function is the same, whichever we suppose to
vary ; and it has been shown, in Art. 20, that the value of
the differential coefficient is independent of the value of
the differential of the variable. Hence, if x + y be in-
creased either by dx or by dy, and the differential coeffi-
cient taken, the result will be the same, which was to be
shown.
Let u' be a function of (x + y), and assume the devel-
opment,
u' = P+Qx + Rx* + Sx* + Tx* + etc ...... (1)
in which P, Q, R, etc., are functions of y. It is required
to find such values of P, Q, R, etc., as will make equation
42 DIFFERENTIAL CALCULUS.
(1) true for all values of both x and y. Since the assumed
development is to be true for all values of x and t/, it must
be so for x = 0. Making x = in (1), and denoting what
u' becomes under this hypothesis, by u, we find,
That is, P is what the original function becomes, when
the leading variable is made equal to 0.
Finding the differential coefficient of u', under the sup-
position that x is constant and y variable, we have,
du' dP dQ t dR dS .
-= = -=- + -T^X +^-x* + -= x* + etc ...... (2)
dy dy dy dy ay
Again, finding the differential coefficient of u\ on the
supposition that y is constant and x variable, we have,
- = Q + 2l?x + 3Sx* + Tx 3 + etc ...... (3)
ux
But, the first members of (2) and (3) are equal, by the
lemma; hence their second members are also equal. If
we place them equal we have an identical equation, because
it is true for all values of x, and consequently the coeffi-
cients of the like powers of x in the two members are
equal to each other. Placing their coefficients equal, we
have the following results :
_dP . dn
V ~ dy ' - dy 9
~~
oo _ o_
= ~ "
etc., etc., etc.
DEVELOPMENT OF FUNCTIONS. 43
Substituting the values of P, Q, R, S, etc., in (1), we
have,
; _ du x
f dy ' I + dy* ' + * '
which is the required formula. Hence, to develop any
function of the sum of two variables, we make the leading
variable equal to 0, and find the successive differential
coefficients of the result; then substitute them in the
formula.
Thus, let it be required to develop (x + y} n into a
series arranged according to the ascending powers of y.
Making y = 0, we have,
u = x n ,
etc., etc., etc.
Substituting these in (4), we have,
(x + y) n = z n + ~x n -\j + '-i M
n(n 1 (n 2) n *
- -- -- L x n d3 + etc.
44 DIFFERENTIAL CALCULUS.
This is the binomial formula, in which n is any
constant.
The formula is applicable to every function of the sum
of two variables; but it sometimes happens that certain
values of the variable entering the coefficients make the
first term or some of its successive differential coefficients
infinite ; for these particular values, the function cannot
be expressed by a series of the proposed form.
EXAMPLES.
Develop the following functions in terms of y.
1. u' = sm(x + y).
Making y = 0, we have, u = sinz ;
du d z u
./.-=- = cosx-j -j 9 smx: etc.;
dx dx 2
it y* y 3
nence, u = sin# -f cosx^- since ~- cosx-^-r-
1 l./O l.-C.d
Making x = 0, whence since = 0, and cosx = + 1, we have,
n - siny = y - ^ + 1 ^ 4 , + etc., as already shown.
2. u =Z(a? +y).
Ans. u' = Ix + - |^ 4- ~ - - + etc,
x 2x* 3x 3 4o; 4
3. w' =
Ans. u' = a*(l + (la)y + ( ^/ 2 +^V + etc.).
I M T L1C IT F U X CT 1 XS.
4. u' = cos(x -f ?/)
/ y v 2
Ans. u = COBX smar cosx
+ SUM r-n + COSX
V. DIFFERENTIATION OF FUNCTIONS OF TWO VARIABLES,
AND OF IMPLICIT FUNCTIONS.
Geometrical Representation of a Function of two Variables.
23. It may be shown, as in Art. 3, that every function
of two variables represents the ordinate of a surface of
which those variables are the corresponding abscissas. If
the vertical ordinate be taken as the function, it may be
expressed by the equation,
In this equation, x and y are independent variables, and
each may vary precisely as though the other were constant.
To illustrate, let PQR
be the surface whose
equation is (1).
If, for a given value
of y as OS, we suppose
x to vary, equation (1)
will represent a section
of the surface parallel
to the plane xz and at
the distance OB from
it. If for a given value
of x, as OA 9 we sup- Fig>2 .
pose y to vary, equa-
tion (1) will represent a section parallel to the plane yi
46 DIFFERENTIAL CALCULUS.
and at the distance OA from it. These sections will be
called parallel sections, and the corresponding planes will
be called planes of parallel section. The planes A CD
and BEF determine the ordinate KK' = z. Let A CD
A'C'D', be consecutive planes of parallel section, at a dis-
tance from each other equal to dx, and BEF, B'E'F', also
consecutive planes of parallel section, at a distance from
each other equal to dy. These planes determine four
ordinates KK', LL', NN', and MM', which may be de-
noted by the symbols z, z', z", and z'". Of these, z and z'
are consecutive when x alone varies, z and z" are consecu-
tive when y alone varies, and z and z'" are consecutive when
both x and y vary.
Differentials of a Function of two Variables.
24. The difference between two consecutive states of the
function when x alone varies is called the partial differ-
ential with respect to x, and is denoted by the symbol
(dz) x ; the difference between two consecutive states when
y alone varies is called the partial differential with respect
to y, and is denoted by the symbol (dz) ; the difference
between two consecutive states when both x and y v#ry is
called the total differential, and is denoted by the ordinary
symbol, dz without a subscript letter. Let us assume the
figure and notation of Art. 23. We have, from the defini-
tion,
z' = z + (dz) x ; whence, by differentiation,
= (*),+ (<*'*),,,, ..... (2)
The symbol (d z z) x y indicates the result obtained by dif-
ferentiating z as though x were the only variable, and then
IMPLICIT FUNCTIONS. 47
differentiating that result as though y were the only varia-
ble, the order of the subscript letters indicating the order
of differentiation. From what precedes, we also have,
z'" = z' + (dz') y ;
substituting for z' and (dz') , their values taken from (2),
we have
transposing z to the first member, replacing z'" z by its
value dz, and neglecting the part (d*z) x , because it is an
infinitesimal of the second order, we have, finally,
(3)
Hence, the total differential is equal to the sum of the par-
tial differentials.
EXAMPLES.
1. z = ax* + by 3 .
(dz) x 2axdx-, (dz) y = 3by 2 dy .*. dz = Zaxdx + 3by*dy.
2. z = ax z y 5 . Ans. dz = 2ay 3 xdx + 3ax 2 y z dy.
3. z = x&. Ans. dz yx y ~ 1 dx + x y lxdy.
Notation employed to designate Partial Differential
Coefficients.
25. From the nature of the case, there can be no such
thing as a differential coefficient of a function of two va-
riables; but the quotient of a partial differential by the
48 DIFFERENTIAL CALCULUS.
differential of the corresponding variable, is called a par-
tial differential coefficient The form of the symbol indi-
cates the variable with reference to which the function has
been differentiated, and no subscript letter is required.
Thus, in Example 2, Art. 24, we have,
|? = 2aij*x, and ^ = 3ax*y*.
A similar notation is employed for partial differential
coefficients of a higher order, as will be seen in the follow-
ing article.
Successive Differentiation of Functions of two Variables.
26. The second differential of a function of two varia-
bles is found by differentiating the first differential by the
rules already given. The third differential comes from the
second in the same manner that the second comes from
the first, and so on. In finding the higher partial differen-
tials, the result obtained by differentiating the function
first with respect to x, and that result with respect to y, is
the same as though we had differentiated the function first
with respect to y, and the result with respect to x. For,
from Art. 24, we have,
"' = z' + (dz') y = z + (dz) x + (dz} y + (dt)^
And in like manner, we have,
*'" = *" + (M\ = z + (dz) y + (dz) x + (<**),, .
Equating these values of z'", and reducing, we and,
IMPLICIT FUNCTIONS. 49
which was to be shown. From equation (4), by an exten-
sion of the notation in Art. 25, we have,
j(dz\ 7 (^\
d*z _ d*z Qr \dx) = C \dy)
dxdy dydx dy dx
From what precedes, we have,
\ J
* ** = (&*)* . + (<***)* + (**) + (**)
, y ,, y ,, ,
EXAMPLES.
1. Given, z = z*y* 9 to find d z z.
x
Given, z = y 3 ^, to find
50 DIFFERENTIAL CALCULUS.
Extension to three or more Variables
27. If we have a function of three or more independent
variables, we may find its differential by differentiating
separately with respect to each variable, and taking the
sum of the partial differentials thus obtained. The second
and higher differentials are found in an entirely analogous
manner.
Definition of Explicit and Implicit Functions.
28. An explicit function, is one in which the value of
the function is directly expressed in terms of the variable.
Thus, y = \/2rx x 2 , is an explicit function. An implicit
function, is one in which the value of the function ife
not directly given in terms of the variable. Thus, in the
equation,
ay z + bxy + ex 2 + d = ;
y is an implicit function of x. Implicit functions are
generally connected with their variables by one or more
equations. When these equations are solved the implicit
function becomes explicit.
Differentials of Implicit Functions.
29. The differential of an implicit function may be
found without first finding the function itself. For, if
we differentiate both members of the first equation, we
shall thus find a new equation, which, with the given one,
will enable us to find either the differential, or the differ-
ential coefficient of the function. For example, suppose
we have ihe equation,
I * -h 2xy 4- a" - a 9 = . . . . (1)
IMPLICIT FUNCTIONS. 51
to find the differential coefficient of y. Differentiating
(1), and dividing by 2, we have,
ydy + xdy + ydx + xdx = (2)
Finding the value of --. we have,
dx
4. i
dx
Again, let us have the relation,
xy = m (1)
to find the value of C -j-. Differentiating (1), we
xdy -f ydx = (2)
Whence,
iH=-l < 3 >
But from (1), we have,
~x ~ X*'
Substituting in (3), we find,
dv m
-
dx x*
(4)
^ '
EXAMPLES.
Find the first aau second differential coefficients of y
in the following implicit functions:
52 DIFFERENTIAL CALCULUS.
We have,
also,
+ 3y*d*y-3d*y=0,
y
_
' ~dx* ~ ly* ' dx* ~ 9 * (l-
2. ?2 - % mX + a;2 _ ^ = Q.
a; ,
and -- =
. - -- , -5- - -
dx y mx dx 9 (y mx)
PART II.
APPLICATIONS OF THE DIFFEKENTIAL
CALCULUS.
I. TANGENTS AND ASYMPTOTES.
Geometrical Representation of the First Differential
Coefficient.
30. It has been shown (Art. 3), that any function of
one variable maybe represented by the ordinate of a curve,
of which that variable is the cor-
responding abscissa. We have also
seen (Art. 5), that an element of
the curve, PQ, does not differ from
a straight line. This line, prolonged
toward T, is tangent to the curve
at P. The angle that the tangent
makes with the axis of X is equal
to RPQ; denoting it by 6, and
remembering that the tangent of the angle- ;it the base
of a right angled triangle is equal to the piTjH-ndicular
divided by the base, we have,
=
dx
The tangent of d is taken as the measure of tlu* slope,
not only of the tangent, but also of the curve at the point
P. Hence, the slope of a curve, at any point, in
54 DIFFERENTIAL CALCULUS.
by the first differential coefficient of the ordinate at that
point.
The slope is positive or negative, that is, the curve slopes
upward or downward, according as the first differential
coefficient is plus, or minus.
Applications.
31. The principle just demonstrated enables us to find
the point of a curve, at which the tangent makes a givm
angle with the axis of x.
Thus, let it be required to find the point on a given
parabola at which the tangent makes an angle of 45 with
the axis. Assume the equation of the parabola,
dii p
.'. -r = -
dx y
Placing this equal to 1, we have,
- = 1, or y = p.
y
But y p, is the ordinate through the focus. Hence, the
required point is at the upper extremity of the ordinate
through the focus.
Again, let it be required to find the point at which a tan-
gent to the ellipse is parallel to the axis of x. Assume the
equation,
> dy _ _ l*x_
'* Tx ~ ~ a*y
Placing this equal to 0, we have,
-'&*
TANGENTS AND ASYMPTOTES. 55
which can only be satisfied by making x 0; this, in the
equation of the curve, gives y = b. Hence, the tangent
at either vertex of the conjugate axis fulfills the given
condition.
Again, to find where the tangent to an hyperbola is per-
pendicular to the axis; assume the equation of the curve,
whence, by the rule,
d-tf l z x
-y- =-r- = co ; .*. y = 0, x = a.
dx a z y
Hence, the tangent at either extremity of the transverse
axis, fulfills the given condition.
Equations of the Tangent and Normal.
32. Let P be a point of the curve whose co-ordinates are
x", y" ; then will the equation of a straight line through
it, be of the form,
y - y" = a(x - x") . . . . (1)
in which a is the slope. If we make a equal the tangent
of RPQ, the line will be tangent to the curve. But the
tangent of RPQ is -r-fr\ hence, we have, for the equation
of a tangent line,
If we make a equal to minus the reciprocal of the tan-
gent of RPQ, the line will be perpendicular to the tangent,
56 DIFFERENTIAL CALCULUS.
that is, it will, be normal to the curve; hence, we have, for
the equation of a normal line,
(3)
Iii these equations x and y are general co-ordinates of
the lines, and x", y" are the co-ordinates of the point of
contact.
Let it be required to find the equation of a tangent to
an ellipse. We found, in the last article, the value of
-j- = -- j- ; substituting for x and y the particular values
u" and y", and putting the result in (2), we find,
l>*z" ', ,,x
y ~ y = ~~^/ f(x ~ x) >
whence, by reduction,
Making a = I, in this equation, we have,
yy" + xx" = a 2 ,
which is the equation of a tangent to a circle.
To find the equation of a normal to an ellipse we substi-
tute the value of yr in equation (3). which gives,
Mak ng a I, we have,
x
which is the equation of a normal to a circle.
TANGENTS AND ASYMPTOTES. 57
If, in equation (2), we make y 0, the coi responding
value of x x" will express the length of the subtangent,
counted from the foot of the ordinate through the point
of contact; denoting this by S.T., we have,
,T=-y" .(6)
9 dy"
If, in equation (3), we make y = 0, the corresponding
value of x x" will express the length of the subnormal,
counted from the foot of the ordinate through the point
of contact; denoting this by 8.N., we have,
(7)
Substituting in (6) and (7) the values of -y^7 taken from
Cv\b
the equation of the ellipse, we have,
Asymptotes.
33. An asymptote to a curve, is a line that continually
approaches the curve and becomes tangent to it at an
infinite distance. The characteristics of an asymptote
are, that it is tangent to the curve at a point that is infi-
nitely distant, and that it cuts one or both axes at a finite
distance from the origin.
To ascertain whether a curve has an asymptote, assume
the equation of the tangent (Art. 32), and in it make
x, and y, successively equal to 0.
3*
58
DIFFERENTIAL CALCULUS.
A
Fig. 4.
Jn this manner we find,
4J) = y" - *"3L and
dy"
We then find -77 from the _
dx
equation of the curve, substi-
tute it in (I) and make the hypothesis that places the
point (x", y' 1 ) at an infinite distance. If this supposition
make either AD or A T finite, the curve has an asymptote,
otherwise not.
Let it be required to find whether the hyperbola has an
dn" b z c"
asymptote. AYe have found (Art. 31), - = -r- Jfl which
J ' dx a 2 y
in (1) gives,
Whence, by reduction,
ff i
~-^, nu&AT =
(3)
or,
y
The only hypothesis that places a point of the hyperbola
at an infinite distance is y" GO , and x" = GO ; both
these sets of values, ir. (4), make AD and A T equal to 0.
Hence, the hyperbola has two asymptotes, both passing
through the centre.
TANGENTS AND ASYMPTOTES. 59
To find whether the parabola has an asymptote, make
' = in (1) ' whence >
AD = y" ~ ] ~r= and AT = -x".
\J \J
If we make y" GO , and x" = oo , to put the point of
contact at an infinite distance, we find both AD and AT
equal to oo , which shows that the parabola has no asymp-
tote.
Order of Contact.
34. Two lines have a contact of the first order, when
they have two consecutive points in common. The first of
these is the point of contact ; this point is common to both
lines, and, as we have just seen, the first differential coef-
ficients of the ordinates of the two lines at this -point are
equal. Conversely, if two lines have a common point, and
if the first differential coefficients of the ordinates of the
lines at that point are equal, the lines have a contact of the
first order.
Two lines have a contact of the second order, when they
have three consecutive points in common. The first of
the three is the point of contact, and because the lines have
three consecutives ordinates common, the first and second
differential coefficients of their ordinates at this point are
equal. Conversely, if two lines have a common point, and
the first and second differential coefficients of their ordi-
nates at that point equal, the lines have a contact of the
second order.
In like manner it may be shown, if two lines have a
point in common, and n successive differential coefficients
of their ordinates at that point equal, that the lines have
60 DIFFERENTIAL CALCULUS.
a contact of the w th order, n being any positive whole
number.
If a line oe applied so as to cut a given line in an even
number of points, it will, just before the first, and just
after the last, lie on the same side of the given line ; and
this is true when the points of secancy are consecutive, in
which case the lines have a contact of an odd order.
Hence, if two lines have a contact of an odd order, they do
not intersect at the point of contact. If the applied line cut
the given line in an odd number of points, it will, just
before the first, and just after the last, lie on opposite sides
of the given line; and this is also true when the points of
secancy are consecutive ; hence, if two lines have a contact
of an even order, they intersect at the point of contact.
If the applied line be straight, and the contact of the
n th order, the given curve will have n consecutive values
of dy, equal to each other. If each of these be taken from
the next in order, the differences will be 0, that is, the
curve will have n 1 consecutive values of d 2 y equal to
0. In like manner, we may show that it has n 2 consec-
utive values of d*y equal to 0, and so on, to the (n 4- l) lh
different al of y, which will not be 0, but will be plus or
minus, according as the (n 4- l) th value of dy, counting
from the point of contact, is greater or less than the n th
value of dy. Conversely, if the (n + l) th differential of y
is plus, the (n 4- l) th value of dy is greater than the w th ;
if minus the (n 4- l) tb value of dy is less than the n ih .
II. OURVATUKE.
Direction of Curvature.
35. Let KL be a curve, whose concavity is turned
downward, that is, in the direction of negative ordi-
CURVATURE.
Cl
nates ; let AP, BQ, and CQ' be consecutive ordinates, Alt
and BO being equal to dx ; prolong PQ toward T. Then
will RQ, and R Q', be consecutive values of dy, and con-
sequently their difference, R'Q' RQ, will be the value of
d*y at P. The right angled triangles, PRQ, and QR'S,
have their bases, and the angles at their bases, equal ; hence,
their altitudes, RQ,
and R 8, are equal ;
but R'Q' is less than
R'S, because Q' is
nearer QO than S ;
hence, R'Q' is less
than RQ, and con-
sequently the value
of d*y is negative.
Conversely, if d~i/
is negative, R'Q' is
less than H'S, and the curve in passing from P is concave
doivmvard. In like manner, it may be shown that the
curve is concave upward at P, when d z y is positive. Be-
cause the sign of the second differential coefficient is the
same as that of the second differential, we have the follow-
ing rule for determining the direction of curvature, in
passing from any point toward the right,
Find the second differential coefficient of the ordinate at-
the point ; if this is negative, the curve is concave down-
ward, if positive, it is concave upward.
We may regard y as the independent variable, and find
the second differential coefficient of x; if this be negative
at any point, the concavity is turned toward the left, if
positive, toward the right.
Example. Let it be required to determine the direction
of curvature at any point of a parabola.
62 DIFFERENTIAL CALCULUS.
Assume the equation, y 2 = 2px ; whence, by differentia-
tion and reduction,
p*
p
The first of these is negative for positive values of y, and
positive for negative values of y ; hence, the part of the
curve above the axis of X is concave downward, and the
part below is concave upward. The second is positive for
all values of x andy; hence, the curve is everywhere con-
cave toward the right.
If the tangent at P have a contact of the n ih order, n
jeing greater than 1, the second differential coefficient of
y at that point will be (Art. 34). In this case it may
be shown, as above, that the curve bends downward in
passing toward the right, when the (n + l) th value of dy,
counting from P, is less than the n th value of dy, and up-
ward when the (n + l) th value of dy is greater than the n th .
The former condition is satisfied when the (n + 1)"' differ-
ential coefficient of?/ is negative, and the latter when it is
positive (Art. 34).
Amount of Curvature.
36. The change of direction of a curve in passing from
an element to the one next in order, is the angle between
the second element and the prolongation of the first. This
angle is called the angle of contingence, it is negative when
the curve bends downward, and positive when it bends
upward. The total change of direction in passing over any
arc is equal to the algebraic sum of the angles of contin-
gence at every point of that arc.
The curvature of a curve is the rate at which it changes
direction. When the curvature is uniform, as in the
CURVATURE.
63
circle, it is measured by the total change of direction in
any arc. divided by the length of that arc. In any curve
whatever the curvature may be regarded as uniform for
an infinitely small arc; hence, the curvature at any point
of a curve, is equal to the angle of contingence at that
point, divided by the corresponding element of the curve.
Denoting the angle of contingence by dd, the correspond-
ing element by ds, and the curvature by c, we have,
Oscillatory Circle.
37. An oscillatory circle, to a given curve, is a circle that
has three consecutive points in common with that curve.
Thus, if the circle whose centre is C, pass through the
three consecutive points P, Q, and R, of the curve, KQ, it
is osculatory to that curve at P.
The circle and curve have two consecutive tangents, PT
and QT, and
two consecutive
normals, P C and
QC, in common;
they have also
the angle of con-
tingence at Q, in
common ; hence,
they have the
same curvature
at the point P.
Because P C and
QC are perpen- Fi e- 6 -
dicular to PT, anl QT, their included angle, is equal to
64 DIFFERENTIAL CALCULUS.
the angle of contingency dd; hence, PQ = CP X d\ or
denoting PQ by ds and CP by 7?, we have, ds = Held, or,
---
cfe 5
This equation ^shows that the curvature of a curve at
any point is equal to the reciprocal of the radius of the
oscillatory circle at that point; for this reason, the radius
of the oscillatory circle is called the radius of curvature.
Radius of Curvature.
38. If we denote the inclination of the tangent, FT, to
the axis of X by 6, the angle of contingence, TQT, equal to
PCQ, will be equal tod&; but we know that 4 = tan" 1 ^;
dx
differentiating, we have,
fg
j _ dx dxd z y
~ i 51 = d y* + ' dx * '
* dx*
Finding the value of R from equation (1), Art. 37, ana
replacing ds by its value, from Art. 5, we have,
_
~dxd*y
Dividing both terms of the second member by c7.c 3 , and
denoting the first differential coefficient of y by q', and its
second differential coefficient by q", we have,
(2)
CURVATURE. 65
When the curve bends downward, dd is negative, and
consequently R is negative; when the curve bends upward,
both arepositive.
It is sometimes convenient to regard R, or some other
quantity on which R depends, as the independent variable.
In this case both dy and dx are variable, and we have,
7 dxd 9 i/ dyd z x
dx* + rfy ~ ;
and this in (1), Art. 37, gives,
_
dxd*y-dyd*x ' ' ( }
Example. Required the value of R for the parabola.
Assume the equation, y 2 = 2px.
By differentiation and reduction, we have,
dy p d 2 ?/ p*
-r = <1 = -9 and -^ = q" = J
dx y dx 2 y s '
substituting the.e in (2), we have,
2 J P
The value of R is least possible when y = 0, increases
as y increases, and becomes oo when y = QO . Hence, the
curvature of the parabola is greatest at the vertex, and
diminishes as the curve recedes from the vertex.
Co-ordinates of the Centre of the Osculatory Circle.
39. In the right angled triangle, PHC, PH is perpen-
dicular to ED, and PC to EP ; hence, the angle, HPC, is
(16 DIFFERENTIAL CALCULUS.
equal to 6. Denoting the abscissa, AC, by a, and the ordi-
nate, DC, by /?, we have,
a = AH+ HO, and = BP - HP ..... (1)
All and BP are the co-ordinates of P, denoted by x and
y; IIP is equal to PCcosJ, and HC to PCsind; substi-
tuting for PC, or 7?, its value found in tlu last article, re-
membering that it is negative in the case under considera-
tion. (Art. 38), replacing sin, and cos4, by their values
taken from Art. 5, and reducing, we have,
dx* + dy* du
- -- ~- = x
\ ,
}q
J 1
'
.
d z y dx
' ft)
Locus of the Centre of Curvature.
40. A line drawn through the centres of the oscillatory
circles at every point of a curve is called the evolnte of that
curve, and the given curve, with respect to its evolute, is
called an involute. If we consider any radius of curvature
of a curve, we see that it intersects the one that precedes
and the one that follows it, and the distance between the
points of intersection is an element of the evolute ; hence,
the radius is normal to the involute, and tangent to the
evolute. We also see that the difference between two con-
secutive radii of curvature is equal to the corresponding
element of the evolute; hence, the difference between any
two radii of curvature of a given curve is equal to the cor-
responding arc of the evolute.
The evolute of a curve may be constructed by drawing
normals to the curve, and then drawing *.i curve tangent
CUKVATURE. 6?
to them all: the closer the normals, the more accurate will
be the construction. An involute may be constructed by
wrapping a thread round the evolute, holding it tense,
and then unwrapping it. Every point of the thread de-
scribes a curve, which is an involute of the given evolute.
Practically, the evolute is cut out of a board, or other solid
material. In this manner the outlines of the teeth of
wheels are sometimes marked out, the circle being taken
as an evolute.
Equation of the Evolute of a Curve.
41. The equation of the evolute of any curve may be
found by combining the equation of the curve with form-
ulas (2), Art. 39, and eliminating x and y. The method
will be best illustrated \>y an example; thus, let it be re-
quired to find the equation of the evolute of the common
parabola.
= ^.and n 1 ' ~
y y/ 3
substituting these in (2), Art. 39, we have, after reduction,
We have already found a' = .and n 1 ' ~ (Art. 38) ;
y y/ 3
a x + - -- = 3x + p, *. X -(a p),
6
*
substituting the values of x and y in the equation y 2 =
we have,
which is the equation required. It is the equation oi a
semi-cubic parabola.
68 DIFFERENTIAL CALCULUS.
EXAMPLES.
1. Find the equation of the evolute of an ellipse
Am. (aa)* + (?)* = (a* - b
2. Find the equation of the evolute of an hyperbola.
Ans. (aaft - (bpft = (a* + b
3. Find the radius of curvature of an ellipse.
r/4/,4
4. Find the radius of curvature of an equilateral hyper-
bola referred to its asymptotes, the equation being xy m.
(m* + ?; 4 )*
Ans. o = s -^-.
) , 1 1 .-i
III. SINGULAR POINTS OF CURVES.
Definition of a Singular Point.
42. A SINGULAR POINT of a curve, is a point at which the
curve presents some peculiarity not common to other
points. The most remarkable of these are, points of in-
flexion, cusps, multiple points, and conjugate points.
Points of Inflexion.
43. A POINT OF INFLEXION is a point at which the curva
tu re changes, from being concave downward to being con-
cave upward, or the reverse.
Inasmuch as the direction of curvature is determined by
the sign of the second differential coefficient of the ordi-
felXGULAK POINTS OF CURVES. Oi*
nate, it follows that this sign must change from to -f ,
or from + to , in passing a point of inflexion. But a
quantity can only change sign by passing through 0, or co' .
Hence, at a point of inflexion the second differential coef-
ficient of y must be either 0, or co . If the corresponding
values of x and y are such, that for values immediately
preceding and following them, the second differential coef-
ficient has contrary signs, then will each set of such values
correspond to a point of inflexion.
EXAMPLES.
1. To find the points of inflexion on the curve whose
equation is y = b + (x a) 3 .
We find, -=-| = G(x a) ; this, put equal to 0, gives
clx
x == a, which, in the equation of the curve, gives y = b.
When x < a, the second differential coefficient is negative,
and when x > a, it is positive. Hence, the point whose
co-ordinates are a and b, is a point of inflexion.
2. To find the points of inflexion on the curve whose
equation is y =. b -f- T V (x a) 5 >
d^u
We find ; 3^ = 2 (x <?) 3 ; this placed equal to 0, gives
x = a f whence, y b. When x < a, the second differen-
tial coefficient is negative, and when x > a, it is positive.
Hence, the point whose co-ordinates are a and b, is a point
of inflexion.
3. Find the co-ordinates of the points of inflexion on
the curve whose equation is y %r
o o _
Ans. x 7 r, and y ^r Vs>
70
DIFFERENTIAL CALCULUS.
1. Find the co-ordinates of the points of inflexion on
X 2 (a 2 x 2 }
the line whose equation is y = - ^ -.
,1/7 5
Ans. x = -a y t>, and y = #.
o oo
Cusps.
44. A CUSP is a point at which two branches terminate,
being tangential to each other. There are two species of
cusps; the ceratoid, named from its resemblance to the
horns of an animal, and the ramphoid, named from its
resemblance to the beak of a bird. The first is shown at
E, and the second at A.
The method of determining the position and nature of a
cusp point will be best shown by examples.
1. Let us take the curve whose equation isy=b (x a)i'
From this we find, *
du . , d*y
For x = a, -+ = 0, and -=-f =
' dx dx*
oo . For x < a, both are
imaginary, as is also the value of y. For x > a, both are
real, and each has two values, one plus and the other
minus. Hence, the point whose
co-ordinates are a and b is a cusp
of the first species.
2. Let us take the curve whose
equation is y x 2 #f. '
From it we find,
Fig. 7.
dy , 5 } ,d*y , 15 J
^ y/y _1_ _/* Qfir] -. 9 -+- T 6
r^ /CX m ~ J, , ctllU. : r /v -^- ~r Jj
///* > ' /A>>2
SINGULAR POINTS OF CURVES.
71
From the given equation we see that no part of the
curve lies to the left of the axis of y, and that there are
two infinite branches to the right of that axis.
The values of -- are both at the origin ; and at that
CLOu
d 2 ?/
point both values of -r-| are positive ; hence, the origin is
a cusp of the second species.
A discussion of the above equations shows that the
upper branch has its concavity always upward ; the second
64
branch has a point of inflexion for, x = ; its slope is
225
for x =
__
25
and it cuts the axis of x
at the distance 1 from the origin ; from
the origin to the point of inflexion,
it curves upward; after that point it
curves downward. The shape of the
curve is indicated in the figure.
For the ceratoid the values of the
second differential coefficient of y have
contrary signs, for the ramphoid they have the same sign.
Fie?. 8 .
\
Multiple Points.
45. A MULTIPLE POINT is a point where two or more
branches of a curve intersect, or
touch each other.
If the branches intersect, -p will
dx
have as many values at that point
as there are branches ; if they
are tangent, these values will be
equal.
Fig. a.
72 DIFFERENTIAL CALCULUS.
EXAMPLES.
1. Take the curve whose equation is y z = x 2 x*.
For every value of x there are two values of y equal
with contrary signs; hence, the curve is symmetrically
situated with respect to the axis of x. For x 0, we have
y zfc ; hence, both branches pass through the origin.
From the equation, we find,
c lli ----- ! 2^ 2
dx~y ' y 'vT^^^VT^^'
rl /it
For x = 0, -j- = 1, hence the origin is a multiple poinr,
by intersection. The curve is limited in both direc-
tions.
2. Take the curve whose equation is y = =fc \/x* + x*.
The two branches are. symmetrically placed with respect
to the axis of x, both pass through the origin, forming a
loop on the left and extending to infinity on the right.
We find, from the equation of the curve,
dy _ 5x* 4- 4r 3 _ 5x 2 + x
dx ~ ~ "
For x 0, we have -^ = ; hence, the origin is a mul-
dx
tiple point, the branches being tangent to each other and
to the axis of x.
/~a*~~x z
3. The curve whose equation is y = xA/ - ^
has a multiple point at the origin, and is composed of two
pointed loops, one on the right and the other on the left
of the origin.
SINGULAR POINTS OF CURVES. 73
4. The curve whose equation is y 2 = (x 2) (x 3) 2 v
symmetrical with respect to the axis of x and consists of a
looped branch which extends from x = 2 to x = 3 ; at the
latter point the upper part passes below the axis of x and
lower part passes above that axis, forming a multiple point ;
beyond x = 3 the two parts continually diverge, extending
tO X CO .
Conjugate Points.
46. A CONJUGATE, or ISOLATED POINT, is a point whose
co-ordinates satisfy the equation of a curve, but which has
no consecutive point. Because it has no consecutive point,
the value of the first differential coefficient of y at it, is
imaginary.
EXAMPLES.
1. Take the curve whose equation is y = x V% a.
In this case x and y = 0, satisfy the equation, but
no other value of x less than a gives a point of the curve.
Furthermore, we have,
dy_ _ 1 3x* - 2ax
dx~ 2 vz 3 ax 2 '
which for x = is imaginary, as also for all values of
x < a.
2. Take the curve whose equation is
y = ax 2 zl
For every positive value of x, there are two values of y,
and consequently two points, except when cosx = 1, when
the two points reduce to one. These points constitute a
series of loops like the links of a chain, having for a dia-
4
74 DIFFERENTIAL CALCULUS.
metral curve a parabola whose equation is y = ax 2 . For
every negative value of x the values of y are imaginary,
except when cosz = 1 ; in these cases we have a series of
isolated points situated on the diametral curve, whose
equation is y ax 2 . We have, by differentiating and
reducing,
dx
which is imaginary for negative values of x.
. IV. MAXIMA AND MINIMA.
Definitions of Maximum and Minimum.
47. A function of one variable is at a maximum state,
when it is greater than
the states that im-
mediately precede and
follow it; it is at a
minim urn state, when
it is less than the states
that immediately pre-
cede and follow it.
Thus, if KL be the
curve of the function,
BN will be a maxi-
mum, because it is
greater than AM and
(70, and EQ will be
a minimum, because it
is loss than DP and
FR.
1
[ S *"
M
/
\0
VI
^^/
x
Q
K
L,
) ABC DBF
Fig. 10.
MAXIMA AND MINIMA. 75
Analytical Characteristics of a Maximum or Minimum.
48. If we examine the figures given in the last article,
we see that the slope of the curve KL, or the differential
coefficient of the function, changes from + to in pass-
ing a maximum, and from to + in passing a minimum.
But a varying quantity can only change sign by passing
through 0, or oo ; hence, the first differential coefficient
of the function must be either 0, or oo, at either maximum
or minimum state.
If, therefore, we find the first differential coefficient of the
function, and set it equal to and oo, we shall have two equa-
tions which will give all the values of the variable that belong
to either a maximum or minimum state of the function.
They may also give other values ; hence the necessity
of testing each root separately. This" may be done by the
following rule :
Subtract from, and add to, the root to be tested, an infi-
nitely small quantity; substitute these successively in the
first differential coefficient ; if the first result is plus and
the second minus, the root corresponds to a maximum ; if
the first is minus and the second plus, it corresponds to a
minimum; if both hare the same sign, it corresponds to
neither a maximum nor a minimum.
The maximum or minimum value may be found by sub-
stituting the corresponding root in the given function.
EXAMPLES.
1. Let y- 3 + (x-2) 2 -
By the rule, we have, ^ = 2(x 2) = 0, :. x 2.
Substituting for x the values, 2 dx, and 2 + dx, we find
DIFFERENTIAL CALCULUS.
for the corresponding values of the differential coefficient
2dx, and + 2dx ; hence, x = 2, corresponds to a mini-
mum, which is y = 3,
2. Let y = 4 - (x - 3)f
We have,
For x = 3 ofe, and 3 -f do;, we have the differential coef-
ficient equal to
_J2 , _ _2_
a *
hence, x 3 corresponds to a maximum, y = 4.
3. Let*/ = 3 + 2(z-l) 3 -
We have,
Substituting 1 dx, and 1 + dx, for re, in the first differ-
ential coefficient, we have in both cases a positive result;
hence, x 1 does not correspond to either a maximum, or
minimum.
The preceding method is applicable in all cases ; but,
when the first differential coefficient of the function is 0,
as it is in most instances, there is an easier process for
testing the roots. In this case, the function being repre-
sented by the ordinate of a curve, the maximum and
minimum states correspond to points at which the tangent
is horizontal ; furthermore, the curve lies wholly above, or
wholly below, the tangent at the point of contact, that is,
the tangent does not cut the curve at that point; hence,
the contact is of an odd order, and consequently the first
of tit e successive differential coefficients of the function that
MAXIMA AND MINIMA. 77
does not reduce to must be of an even order (Art. 34).
If this is negative, the curve bends downward after passing
the point of contact, and the root corresponds to a maxi-
mum, if positive the curve bends upward, and the root cor-
responds to a minimum (Art. 35). Hence the following
practical rule for finding the values of the variables that
correspond to maxima and minima:
Place the first differential coefficient of the function equal
to 0, and solve the resulting equation; substitute each root
in the successive differential coefficients of the function,
until one is found that does not reduce to ; if this is of
an even order and negative, the root corresponds to a maxi-
mum, if of an even order and positive, to a minimum; but
if of an odd order, it corresponds to neither maximum nor
minimum.
EXAMPLES.
1. Let y = x z 3x + 2.
We have,
Also,
-2- -+2
fa* ' UcJ|
Hence, x = -, corresponds to a minimum state, which is,
The symbol [Taj 3 is use< * to denote wliat jr? hi-c
3
when the variable that enters it is made equal to '-.
78 DIFFERENTIAL CALCULUS.
2. y = 4- (z-3) 4 .
we also find,
=-24.
Hence, [?/] 3 4, is a maximum.
In applying the above rule, a positive constant factor
may be suppressed at any stage of the process; for it is
obvious that such suppression can in no way alter the
value of the roots to be found, or the signs of the successive
differential coefficients.
3. Let y = 3z 3 9z 8 27z + 30.
Rejecting the factor, + 3, and denoting the result by u, we
have,
du - z -<u 2 o q\ o-
dx~
Rejecting the factor, + 3, and denoting the resulting value
of the function by u' 9 we have,
= 4, and r- = 4.
Lax J 3 Lax i
The first corresponds to a minimum and the second to a
maximum. Substituting in the given function, we have,
for the maximum and minimum values,
y' = 45, and y" = 51.
When the first differential coefficient is composed of
MAXIMA AND MINIMA. 79
\**'tobk> factors, the method of finding the corresponding
vaiues ol the second differential coefficient may be simpli-
fied as follows: let us have,
'iR - p x o .
5 = l x ^>
P and Q being- functions of x, and suppose that [P] a = 0.
3j the rule for differentiating, we have,
,
dx* ' dx
but because P Decomes when x = a, we have,
dxJa
Hence, find }lie differential coefficient of the factor that
reduces to 0, ?), tiUiirty it by the other factor, and substitute
the root in th , product. The result is the same as that
found by substituting the root in the second differential
coefficient.
4. Let # = (x - 3) z (x 2).
We have,
= 2(x - 3) (x - '2) -H (x - 3) 2 = (x - 3)(3a? - 7) = 0;
By the principle just demonstrated,
4
Hence, [y] 0, is a minimum, and [/].? = , is a
maxmum.
SO DIFFERENTIAL CALCULUS.
If y represents any function of x 9 and if u = y n , we
have,
^_ ny n-\dy
dx ~ ny dx
If x = a reduces ~ to 0, but does not reduce y n ~* to U.
dx
we have, from what precedes,
If n is positive, any value that makes y a maximum, or
minimum, "also makes u a maximum, or minimum ; if w
is negative, any value that makes y a maximum, or i{m-
?w?/m, makes w a minimum, or maximum. The converse
is also true, if we except the values that make y n ~^ equal
to 0. Hence, if care be taken to reject the exceptional
values, we may throw off a radical sign in seeking for a
maximum or minimum.
5. Let y =
Throwing off the radical sign, suppressing the fact ',
+ 2, and denoting the result by u, we have,
The value, x = 0, makes y = 0, and is to be rejected.
The value x = -fa makes -^ 2 negative, an 1 gives
y = '-:=,& maximum.
3 A/3
MAXIMA AND MINIMA. 81
In like manner, if we have u ly, we have.
du 1 dy , . rdy^\
-j- = . ~, and if -r ~ > we nave > as before,
dx y dx LdxJ a
rd 2 u-\ _ r 1 d*y~\
Ldx*\ a ~ Ly ' dx 2 ^'
Hence, we infer, as before, that we may, with proper pre-
caution, treat the logarithm of a function instead of the
function itself, and the reverse.
x 2 3x + 2 (x i\( x 2)
6. Let y = ^^~^, or y = ^^^j.
Passing to logarithms, and denoting ly by u, we have.
u = l(x - 1) + l(x -2)- l(x + 1) - l(x + 2).
Whence,
___
dx x - 1 x 2 x + 1 x + 2
6(x* - 2)
= ; A -^>:
Both values of x make y negative. The first makes
dPu
-T-J negative, and the second makes it positive. Hence we
have, y = 12\/2 17, minimum, and y = 12\/2 17,
maximum.
PROBLEMS IN MAXIMA AND MINIMA.
1. Divide 21 into two parts, such that the less multiplied
by the square of the greater, shall be a maximum.
Solution. Let x he the greater, and 21 x the less. We
4*
8* DIFFERENTIAL CALCULUS.
have, for the equation of the problem, y (21 x)x 2 .
By the rule we find that x 14 makes y a maximum,
Hence, the parts are 14 and 7.
2. Find a cylinder whose total surface is equal to S, and
whose volume is a maximum.
Solution. Denote the radius of the baee by x, the alti-
tude by y, and the volume by V. From the geometrical
relations of the parts, we have,
S = 2x z + 2*x X y, and V vx 2 X y.
Combining, we have, for the equation of the problem,
,
V= x z - - - = - (Sx -
2ifx 2 ^
/~fi /~S
V is a maximum when x A / ; whence, y = 2 A / ~,
I/ 6tf I/ or
or y 2x. That is, the altitude is equal to twice the
radius of the base.
3. Find the maximum rectangle that can be inscribed
in an acute-angled triangle whose base is 14 feet, and alti-
tude 10 feet.
Ans. The base is 7 feet, and the altitude 5 feet.
4. Find the maximum triangle that can be constructed
on a given base, and having a given perimeter.
Solution. Denote the base by 5, a second side by x, the
third side by %p b x, the perimeter by 2p, and the
area by A. From the formula for the area in terms of
the three sides, we find the equation of the problem,
A = Vp(p - b) (p x)(b+x -p).
For x = p JZ>, A is a maximum. Hence, the triangle is
isosceles.
MAXIMA AND MINIMA. 83
5. To find the maximum isosceles triangle that can be
inscribed in a circle.
Ans. An equilateral triangle.
6. To find the minimum isosceles triangle that can be
circumscribed about a circle.
Ans. An equilateral triangle.
7. To find the maximum cone that can be cut from a
sphere whose radius is r.
Ans. The radius of the base is ^ , and the altitude is -r.
o o
8. To find the maximum cylinder that can be cut from
a cone whose altitude is h, and the radius of whose base
is r.
2 1
Ans. The radius of its base is -r, and its altitude -Ji.
o o
9. To find the altitude of the maximum cylinder that
can be cut from a sphere whose radius is r.
o
Ans. Altitude =
10. To find the maximum rectangle that can be inscribed
in an ellipse whose semi-axes are a and b.
Ans. The base is a\/2, and the altitude
11. To find the maximum segment of a parabola that
can be cut from a right cone whose altitude is h, and the
radius of whose base is r.
g __
Ans. The axis is equal to -V^ 2 + ?'*
12. To find the maximum parabola that can be inscribed
in a given isosceles triangle.
q
Ans. The axis is equal to - tbB the altitude.
84 DIFFERENTIAL CALCULUS.
13. To find the maximum cone whose surface is c\ a-
stant, and equal to S.
1 /S
Ans. The radius of the base is ^y
14. To find the maximum cylinder that can be cut from
& given oblate spheroid, whose semi-axes are a and b.
Ans. The radius of the base = a\ ,5,
o
2
and the altitude =
15. From the corners of a rectangle whose sides are a
and b f four squares are cut and the edges turned up to form
a rectangular box. Required the side of each square when
the box holds a maximum quantity.
a + b 1
. - a 7 o
Ans. TT -zVa* ab + b 2 ,
o o
16. To find the minimum parabola that will circum-
scribe a given circle.
Solution. Let x and y be the co-ordinates of the point
of contact, a and b the terminal co-ordinates, 2p the vary-
ing parameter, and r the radius of the circle. For any
circumscribed parabola, we have,
a = x + p + r, and b = v2pa = v%p(x + p + r) ..... (1)
But from the triar.gle, whose sides are the radius, the sub-
normal, and the ordinate of the point of contact, we have,
and b =p + r ..... (2)
It will be shown hereafter that the area of a parabola is
two-thirds the rectangle having the same base and altitude,
MAXIMA AND MINIMA. 85
Assuming this property, and denoting the area by A, we
have, for the equation of the problem,
Applying the rule, and remembering that p and A are
variable, we find that A is a minimum when %p = **.
17. Find the maximum difference between the sine and
versed-sine of a varying angle.
Ans. When the arc is 45.
18. Find the maximum value of y in the equation
y = -*>.
Ans. y' e*.
19. Divide the number 36 into two factors such that the
sum of their squares shall be a minimum.
Ans. Each factor is equal to G.
20. Divide a number m into such a number of equal
parts that their continued product shall be a maximum.
Tfl
Solution. Let x denote the number of parts, - - one
x
part, and ( J the continued product; hence, the equation
of the problem is,
fm\ x ,m
y = ( 1 ; or, ly = xl = xim xlx.
\x / x
m &
.'. x = , and y e
86 DIFFERENTIAL CALCULUS.
m m
~e e . ~e
3 X ', .* y == ^ ,
m 9
is a maximum.
.21. What value of x will make the expression ,
1 + tana:
& maximum?
Ans. x = 45.
22. Find the fraction that exceeds its square by the
greatest possible quantity.
Ans. \.
23. The illuminating power of a ray of light falling
obliquely on a plane varies inversely as the square of the
distance from the source, and directly as the sine of its
inclination to the plane.
How far above the centre of a horizontal circle must a
light be placed that the illumination of the circumference
may be a maximum ? r
Ans. H -.
~V2
Maxima and Minima of a Function of Two Variables,
49. A function of two variables is at its maximum state
when it is greater, and at its minimum state when it is less
than all its consecutive states.
If two parallel sections (Art. 23) be taken through a maxi-
mum or minimum ordinate of a surface, the ordinate will be a
maximum or minimum in each section, and in nearly
every practical case the reverse will hold true. The ex-
ceptional cases will be those in which the maximum or
minimum ordinate corresponds to a singular point of the
surface. Omitting these, the problem is reduced to find-
Ag an ordinate that shall be a maximum, or a minimum
MAXIMA AXD MINIMA. 87
ir both the parallel sections through it This requires
that and -7- be simultaneously equal to 0. The equa-
dy dx
tions thus obtained will determine all the values of x and
y that correspond to maxima or minima states, and each
set of such values may be tested in the manner explained
in the last article, observing that for the section parallel
to the plane xz, the successive differential coefficients of z,
are found by supposing y constant, and for the section
parallel to yz, they are found by supposing x constant.
EXAMPLES.
1. Let z = x s 3xy + y*.
We have,
^ = 3x* - 3y = 0, and -^ = 3y 2 - 3x = 0;
ctzc if
.: x = 0, y = 0; and x = 1, y 1.
Also,
d*z ,. d*z , . d*z a d*z
G% = b ; and -y-r = 6y, -=-= = 6.
dx z dx z dy z dy 3
The first set of values reduce the second differential
coefficients to 0, but not the third differential coefficients ;
hence, they are to be rejected. The second set of values
make both of the second differential coefficients positive;
hence, they correspond to a minimum, which is z = 1.
2. Let z =
Ill
For, a; = -, y = -, z = , a maximum.
3. Let z = Vp(p x)(p y}(x -f y p).
Dropping the radical sign, passing to logarithms, and
denoting the logarithm of z 2 by u, we have,
u = l(p) + l(p -x) + l(p - y} + l(x + y - p).
88 DIFFERENTIAL CALCULUS.
Hence,
du 1
dx p x x + yp
du_ 1
dy p y x -^- y p
2 2
x 3 p, y = 5 j9, make z a maximum.
o o
V. SINGULAR VALUES OF FUNCTIONS.
Definition and Method of Evaluation.
50. A singular value of a function is one that appears
under an indeterminate form, for a particular value of the
variable. Thus, the expression -- - reduces to -, for
x z
the particular value, x 0, whereas its real value is -.
Singular values that take the above form of indeter-
mination, may often be detected, and their real value
found by means of the calculus. Let u = -, in which 2
y
and y are functions of x that reduce to for x a. Clear-
ing of fractions and differentiating, we have,
udy + y du dz ;
if we make x a, the second term disappears, and we have,
If both dz and dy reduce to for x = a, we have, in like
manner,
and so on. Hence, the following rule:
SINGULAR VALUES OF FUNCTIONS. 89
Divide tlie successive differentials of the numerator ly the
corresponding differentials of the denominator; substitute
the particular value of the variable in the resulting fraction,
continuing the operation till one is reached that is not inde-
terminate; the value thus found is the value required.
Thus, in the example above, we have,
tx sinari rl cosan _ rsinari _ rcosari _ 1
S JO ~~ L 3x* Jo . L~6^ Jo ~~ L~6~Jo "" 6*
EXAMPLES.
1. Find the value of | -- z -J . Ans. "4.
%. Find the value of [ 4 _ ^ ~ ^ -^] . Ans. g.
p#3 (i%x OX^ -f- Ct>^~\
3. Find the value of '= . Ans. 0.
L x a -Jo,
rl x n ~\
4. Find the value of - . Ans. n.
\ J. X J 1
re x e~ x ~\
5. Find the value of ^7- ^- . Ans. 2.
L/(J + x) JO
6. Find the value of f" 1 ~ cos _5] . Ans. 0.
L i?/ JO
7. Find the value of
8. Find the value of jjrZj An *-
DIFFERENTIAL CALCULUS.
9. Find the value of - Ans. - ,
l -f tanaJtf 4
10. Find the value of [~-JjL ^~\ . Ans. 2.
L - * - z -* a
Singular values that appear under the forms X <*>,
, and GO oo , can be reduced to the form discussed, and
then treated by the rule. The method of proceeding in
each case will be illustrated by an example.
11. The expression (1 x) tan ~ 9 which reduces to x oo ,
M
1 _ %
when x = 1, can be placed under the form , which,
2
by the rule, reduces to - for the particular value, x = L
f ~ r/
12. The expression, tan-^- ~ j= -- , which reduces to
* ( x ~~ 1)
for x = 1, can be placed under the form - , and
.
this, by the rule, gives for x = 1, the value -- .
13. The expression, ztanz - sec.r, which reduces to
/c
if crsinz i r
& oo , for the value x = -, may be written - -=-,
2 cosx
which, by the rule, gives for x = -^, the value 1.
Al
ELEMENTS OF GEOMETRICAL MAGNITUDES.
91
VI. ELEMENTS OF GEOMETRICAL MAGNITUDES.
Differentials of Lines, Surfaces, and Volumes.
51. Let AP and BQ be two consectVive ordimites of the
curve, IfL, whose equation is y =f(x)', then will PQ be
the differential of the length of the
curve, and APQB will be the dif-
ferential of the area, bounded by
the curve, the axis of x, and any
two ordinates; if we suppose the
figure to revolve about the axis of
x, the curve will generate a surface
of revolution, and the area between
the curve and axis will generate a
volume of revolution ; the surface generated by PQ, is the
differential of the surface of revolution, and the volume
generated by APBQ, is the differential of the volume of
revolution. When the equation of KL is given, the values
of these differentials may always be found in terms of x
and dx, or of y and dy.
1. Denote PQ by dL ; we shall have, as in Art. 5,
Fig. 11.
dL =
To apply this formula, we differential the equation ol
the cnrve, combine the resulting equation with that of the
curve, so as to find the differential of one variable in terms
of the other variable and its differential, and substitute
this in the formula. Thus, let it be required to find the
differential of the arc of a parabola. Assuming the equa-
tion of tLc parabola, y z = Zpx, we find, by differentiation
and combination,
DIFFEKENTIAL CALCULUS.
dy = faA/jfr and dx = H-dy, or dy 9 =
and dx* = dy 2 .
p*
Substituting, and reducing, we have,
dL =
xJ. Denote APQB by dA ; this is made up of two parts,
the rectangle, AR = ydx, and the triangle, RPQ = -dydx;
&
but the latter is an infinitesimal of the second order, it may
be therefore neglected in comparison with the former;
hence, we have,
dA=ydx ..... (2)
This formula may be applied in a manner similar to that
just explained. Thus, if it be required to find the differ-
ential of the area of a parabola, we have,
_
dA = dy ; or, dA = (v%px)dx.
3. Denote the surface generated by PQ by dS ; this
surface is that of a frustum of a cone, in which the radius
of the upper base is y, the radius of the lower base, y + dy y
and the slant height, ydx 2 + dy*. Hence,
dS = [%y + 2<V + dy)] X Vdx z + dy*.
Neglecting dy in comparison with y, and reducing, we
have,
dS = 2y\/dx* + dy* (3)
APPLICATION TO' POLAR CO-ORDINATES.
93
To find the differential of a paraboloid, we proceed as
before. Substituting the values already found, and re-
ducing, we have,
dS =
P 2 ', or, dS =
4. Denote the volume generated by APQB, by dV ;
this volume is that of a frustum of a cone, in which the
radius of the upper base is y, of the lower base, y + dy^
and the altitude dx. Hence,
neglecting dy in comparison with y, we have,
dV=y*dx ..... (4)
Applying the formula to the paraboloid of revolution, we
have, as before,
z/ 3
3V = * dy, or, d V = Zepxdx.
VII. APPLICATION TO POLAR CO-ORDINATES.
General Notions, and Definitions.
52. In a polar system, the
radius vector is usually taken
as the function, the angle be-
ing the independent variable.
Let P be any point of the
curve, PL, in the plane of
the axes, OX, OY ; let
be the pole, and OX the ini-
tial line, of a system of polar
co-ordinates. Then will OP, Fig. 12.
YD X
94 DIFFERENTIAL CALCULUS.
denoted by r, and XOP, denoted by <p, be the polar co-
ordinates of P, and the equation of PL may be written
under the form,
r =
It will be convenient to express the values of 9 in terms
of it as a unit ; in this case, it is laid off on the directing
circle, GS, in the direction of the arrow when positive, and
in a- contrary direction when negative. Let GH be "the
measure of <p for the point P, and let HI be equal to the
constant infinitesimal, dq> ; draw OIQ, and with as a
centre describe the arc, PR ; then will OP and OQ be
consecutive radius vectors, P and Q will bo consecutive
points, RQ will be the differential of r, PQ will be the
differential of the arc, POQ will be the differential of the
area swept over by the radius vector, and PQ prolonged,
will be tangent to the curve at P.
The line, BC, perpendicular to the radius vector, OP, is
the movable axis, and the perpendicular distance, OA, is the
polar distance of the tangent. Prolong the tangent to meet
the movable axis at B ; at P draw a normal, and produce
it to meet the movable axis at G ; then is OB the subtan-
gent, PB the tangent, 00 the subnormal, and PC tht;
normal at the point P.
Useful Formulas.
53. 1. Differential of the arc. Denote the differential
of the arc, PQ, by dL ; RP being infinitesimal may be re-
garded as a straight line perpendicular to OQ ; it is equal
to rdv ; hence,
dL = Vdr* + r*d$* (1)
APPLICATION TO POLAR CO-ORDINATES. 95
2. Differential of the area. Denote the differential of
the area swept over by the radius vector by dA ; this is
made up of the sector, OPR, and the triangle, RPQ ;
but, RPQ is an infinitesimal of the second order, and the
sector is an infinitesimal of the first order ; neglecting the
former in comparison with the latter, and remembering that
the area of a sector is equal to half the product of its arc
and radius, we have,
dA = ir'dp ..... (2)
3. Angle letween the radius vector and tangent. Denote
the required angle by V; the angle, RQP, differs from the
required angle, OPB, by an infinitesimal, and may, there-
fore, be taken for it; but tan RQP, equals RP, divided by
RQ ; hence, we have,
= ..... (3)
dr
4. Polar distance of the tangent. Denote the distance
OA by p; the triangles, QPR and QOA, are similar:
hence,
QP : RP : : QO : OA.
But, PO differs from QO by an infinitesimal, and may,
therefore, be taken for it; making this change, and sub-
stituting for the quantities their values, we have,
+ r$ : ry \\r\p;
hence,
= _M*
9(5
DIFFERENTIAL CALCULUS.
5. Formula /or subtangenl. Denote the snbtangent by
S.T; in the triangle, POB, the perpendicular, OB, is
equal to the base, OP, multiplied by the tangent of the
angle at the base; hence,
S.T =
dr
(5)
G. Formula for subnormal. Denote the subnormal by
S.N; the triangles, OPB and OOP, are similar; hence,
tznOCP = t'diiOPB; but, OC, equals OP, divided by
hence,
7. Formula for the radius of curvature. Denote the
radius of curvature by p; let P and Q be two consecutive
points of the curve, KL, and let
PC and QC be normals at these
points, meeting at C ; then will
C be the centre of the oscillatory
circle ; draw OM perpendicular
to PC, it will be parallel to the
tangent at P, and, consequently,
the intercept, PM, will be equal
to the polar distance of the tan-
gent denoted by p ; draw C,
OP, and Q. From the figure, we have,
(7 2 = r z f p 2 - 2p P .
If we pass from P to Q, r will become r + dr, p will
become p + dp, and OC and p will remain unchanged.
If, therefore, we differentiate the preceding equation under
the supposition that r and p are variable, the resulting
Fig. 13.
APPLICATION TO EOLAR CO-ORDINATES. 07
equation will express a relation between p, r, and/*. Dif-
ferentiating, \ve have,
TflT
0; .: f = - ..... (7)
8. Formula for the chord of curvature. The chord ol
curvature is the chord of the oscillatory circle that passes
through the pole and the point of osculation. Denote it
by C ' ; prolong PC and PO till they meet the circle at R
and N, and draw RN. The right angled triangles, PNR
and PMO, are similar, hence,
PN : PR : : PM : PO,
or,
nence
C: 2 P up : r;
=, or t'=8p ..... (8)
r dp
Spirals.
54. If a straight line revolve uniformly about one of its
points as a centre, and if, at the same time, a second point
travel along the line, in accordance with any law, the lat-
ter point will describe a spiral. The part described in one
revolution of the straight line is a spire. The fixed point
is the pole. If we denote the distance from the pole to the
generating point by r, and the corresponding angle,
counted from the initial line, by 9, we have, for the gen-
eral equation of spirals,
r = /(?) (1)
When this relation is algebraic, the spiral is said to be
algebraic ; when this relation is transcendental, the spiral
is said to be transcendental.
98 DIFFERENTIAL CALCULUS.
Among algebraic spirals, the most important are the
spiral of Archimedes, the parabolic spiral, and the hyper-
bolic spiral, corresponding, respectively, to the right line,
the parabola, and the hyperbola.
Spiral of Archimedes.
55. The equation of this curve is,
r 09 ..... (2)
We see that the generating point is at the pole when
the variable angle is 0, and that the radius vector increases
uniformly with the variable angle, being always equal to o
times the arc of the directing circle that has been swept
over. Differentiating equation (2), we have,
dr = ady.
Substituting, in formulas (1) to (8), Art. 53, we find,
dL = acfyl + 9 2 ; S.T= a^
dA = 4-aV*; S.N=a;
~
If we take a straight line, whose equation is y = ax, and
lay off the abscissa of any point on the directing circle,
and the ordinate of that point on the corresponding
radius vector, the point thus determined is a point of th<!
spiral of Archimedes.
TRANSCENDENTAL CURVES. 99
In like manner, we may discuss and construct the para-
bolic spiral, whose equation is r 2 = 2p$, and the hyperbolic
spiral, whose equation is r$ m. The former corresponds
to the ordinary parabola, and the latter, to the ordinary
hyperbola referred to its asymptotes.
VIII. TRAXSCEXDENTAL CURVES.
Definition.
56. A transcendental curve, is a curve whose equation
can only be expressed by the aid of transcendental quanti-
ties. The cycloid, and the logarithmic curve are examples
of this class of lines.
The Cycloid.
57. The cycloid is a curve that may be generated by a
point in the circumference of a circle, when that circle
rolls along a straight line. The point is called the genera-
trix, the circle is called the generating circle, and the
straight line is the base of the cycloid. The curve has an
infinite number of branches, each corresponding to one
revolution of the generating circle.
To find the equation of one branch, A PM ; let A be tne
origin, the centre of the generating circle, and P the
place of the generatrix after the circle has rolled through
the angle KCP, denot-
ed by <p ; denote the co-
ordinates, AL and LP,
by x and ?/, and let the
radius of the generating AL
ci rcle be represen ted by r. Fig. 14.
From the figure, we have,
= AK- LK.
100 DIFFERENTIAL CALCULUS.
Bat, AK is equal to the arc /fP, and KP is the arc
whose versed sine is y, to the radius r, or r times the arc
7/
whose versed sine is , to the radius 1 ; LK is equal to
P/>, which, from a property of the circle, is equal to
=t V(2r y)y, the upper sign corresponding to the case
in which P is to the left of KE, and the lower sign to the
case in which P is to the right of KE ; and AL is equal
to x. Substituting these values, and reducing, we have,
j nt
x = rversin" -
which is the equation sought. From it we see that y can
never be less than 0, nor greater than 2r; we see, also, that
y is equal to 0, for x = 0, and for x = 2<7rr, and that for
every value of y, there are two values of x, one exceeding
xr as much as the other falls short of it.
Differentiating and reducing, we have,
dx -
(Mi -
which is the differential equation of the cycloid. From it
we deduce the equation,
~r ^ A/ -- 1> whence, -~ = -- -.
dx y y dx z y 2
From, the first, we see that the tangent to the curve is
vertical for x 0, and x = 2tfr, and that it is horizontal
for x = *r ; from the second, we see that the curve is always
concave downward.
Substituting, in formulas (6) and (7), Art. 32, we have.
=--===, and
TRANSCENDENTAL CURVES.
101
The Logarithmic Curve.
58. The logarithmic curve, is a curve in which any
ordinate is equal to the logarithm of the corresponding
abscissa. Its equation is, therefore,
y logs, or y = M.lx;
in which M is the modulus of the
system.
If M > 0, y will be negative when
x < 1, positive when x > 1, when
x = 1, and infinite when x 0.
If M < 0, y will be positive when
x < 1, negative when x > 1, when x = 1, and infinite
when x = 0.
In no case can a negative value of -a? correspond to a
point of the curve.
Differentiating, we find,
Fig. 15.
~i~
dx
- 7
dx 2
The first expression has the same sign as M y and varies
inversely as x. Hence, when M> 0, the slope is positive,
when M < 0, the slope is negative, and in both cases tin-
curve continually approaches parallelism with the axis
of x.
The second expression has a sign contrary to that of M,
and varies inversely as the square of x. Hence, when M > 0,
the curve is concave downward, as KDC, and when M < 0,
it is concave upward, as KDW.
PART III.
INTEGEAL CALCULUS.
Object of the Integral Calculus.
59. The object of the integral calculus is to pass from a
given differential to a function from which it may have
been derived. This function is called the integral of the
differential, and the operation of finding it is called inte-
gration. The operation of integration is indicated by this
sign, /, called the integral sign. Integration and differ
entiation are inverse operations, and their signs, when
placed before a quantity, neutralize each other; thus,
, %
dx = x.
A constant quantity connected with a function by the sign
of addition, or subtraction, disappears by differentiation,
hence, we must add a constant to the integral obtained;
and inasmuch as this constant may have any value, it is
said to be arbitrary. The integral, before the addition of
the constant, is called incomplete, after its addition it is
said to be complete. By means of the arbitrary constant,
as we shall see hereafter, the integral may be made to
satisfy any one reasonable condition.
Nature of an Integral.
60. The differential of a function is the difference
Ix'tweon two consecutive states of that function; hence,
NATURE OF AN INTEGRAL. 103
any state of the function whatever, is equal to some pre-
vious, or initial state, plus the algebraic sum of all the
intermediate values of the differential. But this state is
the integral, by definition ; hence, if the arbitrary constant
represents the initial state, the incomplete integral repre-
sents the algebraic sum of all the differentials from the
initial state up to any state whatever, and the complete
integral represents the initial state, plus the algebraic sum
of all the differentials from that state up to any state
whatever.
Before any value has been assigned to the constant, the
integral is said to be indefinite ; when the value of the
constant has been determined, so as to satisfy a particular
hypothesis, the integral is said to ^particular ; and when
a definite value has been given to the variable, this integral
is said to be definite.
The values of the variable that correspond to the initial
and terminal states of a definite integral are called limits,
the former being the inferior, and the latter the superior
limit ; the integral is said to lie between these limits. The
value of tlip definite integral may be found from the indefi-
nite integral by substituting for the variable the inferior
and superior limits separately, and then taking the first
result from the second. The first result is the integral
from any state up to the first limit, and the second is the
integral from the same state up to the last limit; hence,
the difference is the integral between the limits. The
C b ' T,' i,
symbol for integrating between the limits is I > m wnicn a
is the inferior, and I the superior limit.
This article will be better understood when we come to
its practical application in Articles 78 and following.
104 LNTEGKAL CALCULUS.
Methods of Integration 5 Simplifications.
61. The methods of integration are not founded on pro-
cesses of direct reasoning, but are mostly dependent on
formulas. The more elementary formulas are deduced by
reversing corresponding formulas in the differential ,cal-
culus; the more complex ones are deduced from these by
various transformations and devices, whereby the expres-
sions to be integrated are brought under some known
integrable form.
It has been shown (Art. 9), that a constant factor re-
mains unchanged by differentiation; hence, a constant
factor may be placed without the sign of integration. It
was also shown in the same article, that the differential of
the sum of any number of functions is equal to the sum
of their differentials; hence, the integral of the sum of
any number of differentials is equal to the sum of theii
integrals. These principles are of continued use in inte-
gration.
Fundamental Formulas.
62. If we differentiate the expression -, we have,
reversing the formula, and applying the integral sign to
both members,
Remembering that the signs of integration and lifferentia*
METHODS OF INTEGRATION. 105
tion neutralize each other (Art. 50), and adding a constant
to complete the integral, we have,
[1]
Hence, to integrate a monomial differential, drop the
differential of the variable, add 1 to the exponent of the
variable, divide the result ~by the new exponent, and add a
constant.
This rule fails when n = 1 ; for, if we apply it, we get
a result equal to oo ; in this case the integration may be
effected as follows :
From Art. 15, we have,
7/ 7 \ _ ,
d(alx) = a = ax dx ;
x
reversing and proceeding as before, we have,
fax~ l dx = of = alx+C ..... [2]
From Art. 16, we have,
reversing and proceeding as before, we have,
L3J
In like manner, reversing the formulas in Article 17, let-
tered from a to li t we have,
I cosxdx smx + C ............ . [4]
5*
106 INTEGRAL CALCULUS.
I Biaxdx cosz + O .......... [5]
r_dx = ............ [6 j
J cos z x
-^L = cot* + (7 ........... [7]
sm 2 a;
I siuxdx = versing + .......... [8]
/ cosa;^ = coversinz + C ....... [9]
Jt&nxsQcxdx = seai? + C ......... [10]
/ cota;coseca;^ = cosecx + ..... [11 J
In like manner, from the formulas in Art. 18, lettered from
a' to h', we deduce the following:
r-M= = S ^- l y + G ........ [12]
JVl-y*
-^= = co S -V+<7 ....... [13]
[15]
METHODS OF INTEGRATION. 107
/ /=== = versiu~V + C ...... [161
J ^y _ y 2
/- . ___ = coversin~V + O . . . [17]
V2 - 2
(7
-- -p- - = cosec-V + C ---- [19J
V* - 1
If in formulas (12), (13), (14), (15), (18), (19), we make
Ix , , ,
/ = , whence dy = , and reduce, we hare,
d
v*
dx 1 . _ bx
/cfo 1 _ ifz
^-^ = S5 ten +c
/__*= loot' 1 52+ (7. .[23]
/ a 2 4- ^ 2 a 2 aJ a
= l sec -l^ + a., .[241
a* a
108 INTEGKAL CALCULUS.
From (16) and (17), by substituting j- x for y, and re-
ducing, we have,
C
J
dx
= 1 versing* + C ..... [26J
*
1 . _i23 8
= T coversm x r- x + (7 . . [271
2
Formulas (1) to (19) are the elementary forms, to one
of which we endeavor to reduce every case of integration ;
the processes of the integral calculus are little else than a
succession of transformations and devices, by which this
reduction is affected.
In the following examples, and, as a general thing,
throughout the book, the incomplete integral is given, it
being understood that an arbitrary constant is to be added
in every case.
EXAMPLES.
Formulas (1) to (3).
1. dy = tx* dx. Ans. y = -j-,
i 24
2. dy = :z*dx. Ans. y = -lx*.
3. dy = 3x~*cfa. An*, y = x ,
METHODS OF INTEGRATION. 10!)
4. dy = Zx^dx. Ans. y = ~x *.
2 -4 10 -i
5. dy = - x dx. Ans. y = -x .
e. = - - fo
8. dy =
A
9. r??y = 3x~<lx. Ans. y - Six.
f* K
.cosa?
11. dy = e Binx cosxdx. Ans. y =
12. dy e cosx sinxdx. Ans. y =
Formulas (4) to (11).
14. dy = - lsin(o; 2 ) X xdx. Ans. y = T cos(z 2 ).
&
^ r\r\c* 2/1 nr* \
. y = lcot(3.r).
110
17. dy = &iu(ax)dx.
INTEGRAL CALCULUS.
Ans. y = versing).
18. dy cosf x 2 } X xdx. Ans. y coversinf x z Y
Formulas (12) to (19).
19. dy = . Ans. y = sin" 1 (2^).
20. dy =
Ans. y = sin i (x t )
a&dx
IF vT
Ans. y = ^sin (x
o
22. J =
cfa;
xdx
Zdx
y = cos" 1
,
= tan
= 2cot
25. < =
y 2
METHODS OF INTEGRATION.
Formulas (20) to (25).
; (a = v2j "b = 3).
1H
27. rf =
; (a =
Ans. y = -tan x ~.
= cot"
= - rrCOSCC
112 INTEGRAL CALCULUS.
32. dy - ?. Ans. y =
33. dy = . Ans. y = -cosec"
" /T^ n 9 . /-
V3
Formulas (26) and (27).
34. dy = = ; (a = 2, J = 3).
= versm
o
Zaxdx . ,
85. cfr= _ -; (=i, a=
EXAMPLES IN SUCCESSIVE INTEGRATION.
36.
Dividing by dx 2 ,
VT = axdx, or d( -~] = axdx.
dx 2 \dx z j
Integrating, we have,
d*y _ ax*
d& ~T
Multiplying by dx, and integrating again, we find,
dy ax 3
j. = _ + Cx+(7 .
METHODS OF INTEGRATION. 113
Multiplying again by dx, and integrating,
37. d*>y = x~*d
As before, we have,
-3
<&-*- + ux +
and finally,
1, Cx* C'x 2
= -fa + -g- + -y-
38. 6? 3 y = mdx*.
mx*
Ox
o /
39. t? 3 y = x* dx 3 .
105
40. J 2 ?/ = sx*dx*.
114 INTEGRAL CALCULUS.
Integration by Parts.
63. If u and v are functions of x, we have, from Art. 10,
d(uv) = udv + vdu.
Integrating and transposing, we have,
I udv = uv t vdu [28]
This is called the formula for integration by parts; it
enables us to integrate an expression of the form udv
whenever we can integrate an expression of the form vdu.
It is much used in reducing integrals to known forms.
EXAMPLES.
1. dy = xlx dx /
Let Ix = Uj and xdx = dv /
, dx , a*
/. du = , and v = -.
X A
Substituting in the formula, we have,
__ x 2 lx _ rxdx _ x*lx __ x 2
~2 J ~2~ ~2 T'
^ 2 = u, and - = dv;
, xdx 1
.". du -- - --- , and v =
METHODS OP INTEGRATION. 115
hence, we have, from the formula,
Vl - x z r dx Vl-x* .
y = / = sm" 1 ^.
X J A /i _ 3.8 iC
Additional Formulas.
64. 1. Formula (1) may be extended to cover the case
of a binomial differential of the form,
in which the exponent of the variable without the paren-
thesis is 1 less than the exponent of the variable within.
Let a + bx n = z; .-. bnx n ~ l dx = dz, or x n ~ l dx = ^.
bn
Substituting, and integrating, we have,
y =
Keplacing z by its equal, (a + &s n ), we have,
J '(. + fc^^i* = 1 + a
Hence, to integrate a binomial differential of the proposed
form, add 1 to the exponent of the parenthesis, divide the
result by the new exponent into the product of the coefficient
and the exponent of the variable within the parenthesis.
It is to be observed that a constant factor may be set
aside during the process of integration, and then intro-
duced, as explained in Article Gl.
116
CALCULUS.
EXAMPLES.
*
1. dy = (1 + x*)*xdx.
xdx
2. dv
4. dy (2
5. dy = -
Ans. y =
Ans. y =
Ans. y = (7
lo
. ;y = (2
2. The preceding formula fails when p = 1 ; in that
case, we have,
.
bnJ z In
Replacing z by its value,
EXAMPLES.
[30 1
1. dy = 3(3 -f
*fc
2. r/?x =
Ans. -=
re + a
7
3. r =
.
x + 4
. y = l(x + a),
Ans. y = 2 7a + 41
METHODS OF INTEGRATION. 11?
When the differential expression is a fraction in which
the numerator is equal to the differential of the denomi-
nator multiplied by a constant, its integral is equal to that
constant into the Napierian logarithm of the denominator.
4. dy = ^Ar Ans ' V = " 3 l ( a * ~ x ^'
3x2
Ans. y = l(x* + x 2 + x -f 1)
3. Let it be required to integrate the expression.
dx
\/x 2 dc. a 2
assume,
x z =t a z = z 2 , whence, xdx = zdz ;
adding zdx to both members and factoring, we have,
(x + z} dx = z(dx -f dz) ;
hence,
dx _ dx _ dx 4- dz _ d(x -\- z) >
applying the principle just deduced, we have,
Replacing z by its value, we have, finally,
/* ^ = i( x + V^io 5 ) + C [31]
J \/x 2 a 2
118
INTEGRAL CALCULUS.
EXAMPLES.
1 // - -
7 ~
- 5
4. Let it be required to integrate the expression,
dx
since, dx = d(a + #), and 2aa; + a; 2 = (a -f ^) 2 a*,
we have,
/(fa _ / d(a + x)
V^ctx + x*J >\/a +~z z a?
which can be integrated by Formula (31); hence,
/^= =l\a
V'2ax + x 2 (
... [32J
EXAMPLES.
dx
X
Ans. y = l(x + - + -v/|F+7 5
METHODS OF INTEGRATION-. 119
Vbx + 9z 2
5. Let it be required to integrate the expression,
dx
dy = -- -:
a z x 9
Factoring, we have,
dx 1 j dx dx )
a 2 x* ~2a( a + x + a x \ '
Integrating, we have,
But the difference of the logarithms of two quantities is
equal to the logarithm of their quotient; hence,
/V^- i = lj +0 . .[33|
J a* x z 2a a x
and in like manner,
..... [34|
a x + a
EXAMPLES.
dx 1 , 3 -f x
dx 1 x 2
120 INTEGRAL CALCULUS.
d z y
S. Given -j-^ = y, to find the relation between y and x.
Multiplying both members by
Integrating,
dt/ 2 dii
' '
T? - y
dx*
Integrating by Formula 31,
x =
d z s
4. Given -j-^ = n z s, to find tlie relation between a
and t, t being the independent variable.
Multiplying by 2ds,
dP
In tegrating,
^s_ 22 ds
dt^~ ~ V0 -*'
Integrating by Formula (20),
, 1 . _i ns ~.
t = - sin L ^ + C'.
n
Rational and Entire Differentials.
65. An algebraic function is rational and entire when it
contains no radical or fractional part that involves the
variable. If the indicated operations be performed, every
METHODS OF INTEGRATION. 121
rational and entire differential will be reduced to the form
of a monomial, or polynomial differential, each term of
which can be integrated by formulas (1) or (2). Thus,
if the indicated operations be performed, in the expression,
we have,
dy = If 8x - x* + 6z* - 3z 3 \dx.
Hence, by integration,
2 6 . 3 .
= * X + X ~ X
In like manner, any rational and entire differential may
be integrated.
Rational Fractions.
66. A rational fraction, is a fraction whose terms are
rational and entire. When a rational fraction is the differ-
ential of a function, and its numerator is of a higher degree
than its denominator, it may, by the process of division, be
separated into two differentials, one of which is rational
and entire and the other a rational fraction, in which the
numerator is of a lower degree than the denominator. The
former maybe integrated as in the last article; it remains
to be shown that the latter can be integrated whenever
the denominator can be resolved into binomial factors of
the first degree with respect to the variable. The method
G
122 INTEGRAL CALCULUS.
of integration consists in resolving the differential coeffi-
cients into partial fractions, by some of the known methods,
then multiplying each by the differential of the variable
and integrating the polynomial result. There are four
cases, depending on the method of resolving the differen-
tial coefficient into partial fractions. Each case will be
illustrated by examples.
First Case. When the binomial factors of the denomi-
nator are unequal.
Let us have,
.. \ dCC *~~" ) CCCC &Q5 ~~~
ay = = r-j ; -, dx.
Assume the identical equation,
2x-5 = A t B . V
( x - i) ( x _ 2) (x - 3) ~~ x - 1 x - 2 x-3 '
Clearing of fractions.
2a? 5 = A(x -2) (x 3) + B(x 1) (x - 3)
+ C(x - 1) (x - 2) (2)
In order to find A, B, and 0, we might perform the
operations indicated in (2), equate the coefficients of the
like powers of x, and solve the resulting equations; but
there is a simpler method in cases like this, depending on
the fact that (2) is true for all values of x.
Making x = 1, we find, 3 = 2.4;
Making x = 2, we find, 1 = B; .-. B = 1.
Making x = 3, we find, 1 = 2(7; /. C
METHODS OF INTEGRATION.
Substituting these in (1), multiplying by dx, and in
tegrating by Formula 30, we have,
(2x - 5)<fa _3 f dx f dx
jc -l)(x- 2) (x - 3) ~~ %Jz>-l Jx-%
Reducing the result to its simplest form, in accordance
with the elementary principles of logarithms, we have,
finally,
(x - 1)
EXAMPLES.
5x + l)dx _ (5x + l)dx
x - 2 " (x - 1) (a? + 2)*
a; + 2) = ?(a? - I) X (x
(a; - l)dx
y ~ x* + 6x + 8 " (x + 2) (x + 4)'
y =
x + 2)*
" x 3 +x 2 2x x(x 1) (x + 2)*
4. dv =
x(x-l)(x + 1)'
<, y l(x* x).
124 INTEGRAL CALCULUS.
Second Case. When some of the binomial factors are
equal.
In this case "there are as many partial fractions as
there are binomial factors in the denominator; but the
denominators of those corresponding to factors of the
form (x a) m y are respectively of the form (x a) m ,
(x - a) m ~ 1 ) (x - a) m ~ 2 , etc., down to x - a.
Let us assume,
*, - ( x * + *)<**
assuming the identical equation,
(x - 2p (x - 1) = (x^~2^ + x-2 + ~x~^\ W
and clearing of fractions, we have,
z 2 + x = A(x 1) + B(x 2) (x 1) 4- C(x 2) 2 ... (2)
Here again we might find the values of A, B, a id 0, by
the ordinary method of indeterminate coefficients ; but it
will be simpler to proceed as follows :
Making x = 2, we find, A = 6 ;
Making x = 1, we find, (7=2;
giving to A and C their values in (2), and then making
y> = 0, we find, = - 6 4- 2 + 8 .-. B = - 1.
Substituting in (1) and multiplying by dx, we have,
(x* 4- x)dx 6dx dx 2dx
_2)~2~(^ 1) ~ (x- 2) 2 ~ x 2 x
METHODS OF INTEGRATION. 125
The first partial fraction can be integrated by Formula
29, and the others by Formula 30 ; hence,
x 2 x Z x 2
EXAMPLES.
. -- - .
X 3 (X 1)
In this example the equal binomial factors are of the
form (x 0) or x, and the assumed identical equation is,
x* -2 _ A B_ C D
x*(x-l)~ x* + x 2 + z + x^l '
Clearing of fractions, and performing indicated operations,
we have,
x z - 2 = Ax A + ^ 2 - Bx + Cx 3 Cx* + Dx*.
Equating the coefficients of like powers in the two mem-
bers, and solving the resulting group, we have, A = ^,
J3 = 2, = 1, and D 1 ; substituting in (1), and
proceeding as before, we have,
x
j _ _ xdx
=
126 INTEGRAL CALCULUS.
Third Case. Wlien some of the factors are imaginary,
~but unequal.
In this case the product of each pair of imaginary
factors is of the form (x a) 2 + b z ; instead of assuming
a partial fraction for each imaginary factor, we assume
for each pair of such factors a fraction of the form,
A + Bx
xa *x
Assume the identical equation,
x A -f Bx O
(1)
(a. + i) (aj + l) x z + l a: + 1 '
Clearing of fractions, and reducing,
x = Bx* + (^ + 5)a; + A + Cx* + C (2)
Equating the coefficients of the like powers of x in the two
members, and solving the resulting equations, we have,
Substituting in (1), and multiplying by dx, we have,
xdx _ 1 (1 + x)dx __ 1 dx
1 / f/a; .T<:fe ^/./: \
2 U* + 1 + ^Tl "" a; + 17
Integrating by Formulas (14) and (30), and reducing, we
have,
METHODS OF INTEGRATION". 12?
Fourth Case. When some of the binomial factors are
imaginary and equal.
In this case, the denominator will contain one or more
factors of the form [(a; a) 9 + & 2 ] n ; in determining th<
partial fractions, we combine the methods used in the
second and third cases.
Let us assume,
x 3 dx
Assume the identical equation,
r 3 Ax + B Cx + D
Clearing of fractions, and proceeding as before, we find,
A = 2, B = - 4, C = 1, and D = 2.
Substituting these in (1), multiplying by dx, and separating
the fractions, we have,
. _ 2xdx kdx xdx
.J /-v.2 O
2) 2 (x 2 - 2.-C + 2) 2 x* - 2x + 2
(2)
'
Making (x I) 2 + 1 = z 2 + 1, whence x 1 = z, and
dx = dz, we have,
, _ Z(z + l)dz dz (z + 1)dz
or,
2^ zdz Sdz
y ~ (^T"i) 8 J^Ti 8 * + 1 2* + r
INTEGRAL CALCULUS.
, third, and fourth terms can be integrated by
known formulas ; the second is a particular case of the
form , which can be integrated by the aid of
(* + D
Formula D, yet to be deduced.
Integrating i\\Q first, third, and fourth, and indicating
the integration of the second, we have,
V = - W ~ T - a/TTl + \
From what precedes we infer that all rational differen-
tials are integrable; consequently, all differentials that
can be made rational in terms of a new variable are also
integrable.
Integration by Substitution, and Rationalization.
67. An irrational differential may sometimes be made
rational, by substituting for the variable some function of
an auxiliary variable ; when this can be done, the integra-
tion may be effected by the methods of Articles 65 and 66.
When the differential, can not be rationalized in terms of
an auxiliary variable, it may sometimes be reduced to one
of the elementary forms, and then integrated. The method
of proceeding is best illustrated by examples.
When the only Irrational Parts are Monomial.
68. When the only irrational parts are monomial, a dif-
ferential can be made rational by substituting for the
primitive variable a new variable raised to a power whose
METHODS OF INTEGRATION. 129
exponent is the least common multiple of all the indices
in the expression.
(** x*)dx , x
Let dy = * -- ..... (1)
The least common multiple of the indices is 6 ; making
x = z*, we have, x* = z 3 , x* = 2 4 , a?* = z 2 , and cfo =
these substituted in (I), give,
*9 _ *
Performing the division indicated, we have,
dy = 6 (zi z 6 z 5 + z 4 + 2 s z*
Integrating by known methods, and replacing z by its
value, a;*, we have, finally,
3464 6484
y = -gfi + y <> + a? - ^ - %*
6 - 31(1 + o)
When the only irrational parts are fractional powers
of a binomial of the first degree, the differential can be
rationalized by the same rule, and consequently can be
integrated.
Let us take the expression,
dy -= (x + Vx + 2 + ^aT+I) dx
130 INTEGRAL CALCULUS.
Assume,
x + 2 = z 9 , whence, x = z* 2, and dx v-u,*.
Substituting in (2), we have,
Integrating, and substituting for z its value, (a; -f 2) % we
have,
y = I (x + 2)* - *(x + 2) + | (x + 2)*+ J (z + 2)*.
Binomial Differentials.
69. Every binomial differential can be reduced to the
Form,
r
rfy = -4 (a + fa*)*x m ~ l dx ..... (1)
in which m, n, r, and 5 are whole numbers, and n positive,
Thus, the expression, (x~ * -f 8flT*)~*aAfe, can be re-
i_2 1
uu ^ u ^ uu * xv^*^ VJ . , /v^ ) %"^cfe by simply removing
the factor x~~k from under the radical sign. If, in this
result, we make x = z*, whence, dx = z 3 dz, it will be-
come, 4(1 + 2z)~*z 5 dz, which is of the required form. In
like manner, any similar expression may be reduced to
that form. The constant factor, A, may be omitted during
integration, and the exponent of the parenthesis, -, may
s
be represented by a single letter p, as is done in Article 70.
After integration, the factor may be replaced, and the value
of - may be substituted for p.
METHODS OF INTEGRATION. 131
FIRST CRITERION.
Form (1) may be integrated by some of the methods
T
explained in Articles Go and 66, when - is a whole number.
SECOND CRITERION.
Form (1) can be integrated when is a whole number.
1
For let us assume (a 4- bx n ) = z* 9 or, z = (a + lx n )*; we
have, by solution and differentiation,
= \-r) ;a
and,
m
m l , * /* a\~n sl-r
x m L dx = ^ ( 7 ) z dz.
bn\ 1} )
Substituting in (1), we find,
\ / ~ bn\ b ) V. /
Which expression is integrable, as was shown above, when
is a whole number.
n
THIRD CRITERION.
in T
Form (1) can be integrated when I- -, is a whole
number.
For let us assume,
132 IKTEGKAL CALCULUS.
Solving and differentiating, we have,
11 1 m m
x = (2" = a n (z* - &)"*; x m = a"(z 8 - Z>)~
and,
Substituting in (1), and reducing, we find,
r m r m r .
n~ m 1 ^ T+S 1
ti
(3)
Which can be integrated, as was shown above, when
m r .
1- -, is a whole number.
n s
EXAMPLES.
.... (4)
ITl 4
Here, = - = 2 ; hence, the second criterion is satis-
n &
fied.
Comparing (2) with (4), we find,
a = 1, I = 1, n = 2, m = 4, r = 1, s = 2, and z = (1 -f x 2 )*,
Hence,
y* /** K i
/ (1 + x z )*x s dx = I (z 9 l}z z dz =
/ / 53
15
METHODS OF INTEGKATIOK. 133
= dx(l + Z 2 )~ -* ..... (5)
Here, -+_ __ _ 2 hence, the third crite-
n s <i 2
rion is satisfied.
Comparing (3) and (5), we find,
a = 1, 5=1, n = 2, m = 3, r = 1, s = 2, and
_
FTence,
dx
Integration by Successive Reduction.
70. When an integral can be made to depend upon a
simpler integral of the same form, it is evident that by
successive repetitions of the process, we may ultimately
arrive at a form that can be integrated by one of the funda-
mental formulas. This method of integration is called
the method by successive reduction, and is effected by for-
mulas of reduction, sonic of which it is now proposed to
investigate.
1. If in the expression for a binomial differential, we
omit the factor A, represent the exponent of the paren-
thesis by p, and factor the result, we have,
fa I- b^x m "^dx = x m ~ n (a + bx n )W~*dx ..... (1)
134 INTEGRAL CALCULUS.
Let u = x m ~ n , and dv = (a + fa n ) p x n ~*dx.
Differentiating the first, and integrating the second
Formula (29), we have,
to = (m - n)x-^ d x; and , = ^
But, (a + lx n f + l = (a + lx n f (a + lx n ) ;
a(g + te*) p (a + &g n
=
Substituting in Formula (28), we have,
bn(p
n(p
Transposing the last term to the first member, and re
ducing, we have,
ln(p +
Dividing by the coefficient in the first member, we have,
METHODS OF INTEGRATION. 135
Formula A enables us to reduce the exponent of the
variable without the parenthesis, by the exponent of the
variable within the parenthesis, at each application. It
fails when pn + m = 0; but in this case, the third crite-
rion, (Art. 69), is satisfied, and the expression may be
integrated by a previous method.
2. The expression (a + bx n ) p may be placed under the
form,
or, a(a + lx n f~ l + b(a + bx n ) p ~ l x n ;
hence, we may write the equation,
f(a
= of (a + fo n ) p - l x m - l dx + lj\a +
The last term of this equation can be reduced by
Formula A. Eeplacing m by m -f n, p by p 1, and
multiplying by Z>, that formula becomes,
if (a + feY~ 1 z m+ "~ 1 <*
pn + m pn
Substituting in (2), and uniting similar terms, we have,
C i) m\
j (a -*- bx yx ~~ dx
J v
(ct -f- bx ) X pnCt f- , 7l\p 1 7711 7 r Tf\
:= + / (# ~r C>iC ) X uX . I Jj\
pn + m pn + m /
136 INTEGRAL CALCULUS.
Formula B enables us to reduce the exponent of the
parenthesis by 1 at each application. It fails in the same
case as Formula A. When m and p are negative, we
endeavor to increase instead of diminishing them. For
this purpose we reverse Formulas A and B.
3. Reversing Formula A, and reducing, we have,
J\a 4- lx n ) p x m - n - l dx
= (a + ftQP*V- _ (, + M) f lx n Px ^ dXt
a(m n) a(m n] / v
"Replacing m by m + n, we have,
f(a + WT.1& = - < fl +
./ v
m + n)r tofiyz
v
4. Reversing Formula B, and reducing, we have,
f(a + lx n f-^x m - l d X
= _ (a + fa")V
pna
In this, substituting p +1 for^?, we have,
J^
^
METHODS OF INTEGRATION. 137
The mode of applying Formulas A, B, C, and D, will
be illustrated by practical examples.
1. Let it be required to integrate the expression,
In Formula A, making
a =. r 2 , b 1, n 2, p = , and m = 3,
we find,
/' i /V2 _ ^2^7. r 2 /* i
( r t _ <*)*z*dx = - ( -f-^ + -^J (r 2 - x*ydx
..... (3)
In Formula B, making
a =r 2 , ft = 1, n = 2,p = $, and m = 1,
we find,
* ~
But
/ ON -i, / dx . \x
r* x z ) ^dx I sin -.
/ A/ r g _ ^2 r
Substituting this in (4), and that result in (3), we have,
Snally,
/(r 2 - x*y
4
r z( r z _ X 2\^
138
2. Let dy =
INTEGRAL CALCULUS.
= (- a* + x
In Formula G, making a = a 2 , b = 1, n = 2, p
and m 2, we have,
/
-a
But by Formula (24),
/_ x -l _i 7 / dk 1 _ i
a 2 + ic 2 ) % 1 c?a; = I - -- = - sec -.
J x^/ x z _ a z a , a
Substituting in (5), we have, for the value of y,
dx
**-
L_ __
SC
3. Let dn
In Formula D, making x = z, a = 1, ft = 1, n = 2,
= 2, and m = 1, we have,
But,
METHODS OF INTEGRATION. 139
Hence, we have, for the value of y,
r dz
J 7*TT-
h tan- 1 *
T 2 tf
This is the method of integration referred to under the
Fourth Case of rational fractions. By a continued ap-
plication of Formula D, any expression of the form,
7~^ -- 2\ n > can ^ e re duced t an integrable form.
(a + z )
The following additional examples can be reduced to
integrable forms by the application of Formulas A, B, C,
and D.
_
sin -.
tl
1 f/~ "if
Ans. y = -x(a* x*)~2+ sin" 1 -.
A 6 a
5. dy = (a*
6. dy=(l+ x*)x*dx. Ans. y = -
3 . _
140 LNTEGKAL CALCULUS.
5. Let it be required to simplify the expression,
x n dx
Changing the form and removing the factor x from under
the radical sign, we have,
in formula A, making,
a = 2a, b = - 1, n = 1, p = J, and m = n + $,
and reducing, we have,
l(pn + m) = n, and a(m n) = 2a(n i) = a(2n 1) ;
hence,
71-1 ,
a; V 2ax
/* X U dx
I =
/ ^\/2ax x 2
,9,,_1U/ ,-!& ^
Formula E enables us to reduce the exponent n by 1,
at each application ; if n is entire and positive, by a con-
tinued application of the formula we ultimately arrive at
the form, / ' , which is equal to versin" 1 -.
J v2axx 2 a
1. dy =
METHODS OF INTEGRATION. 141
EXAMPLES.
xdx
J /y
Ans. y = V%ax x 2 + a versin" -.
* a
Certain Trinomial Differentials.
71. Every trinomial differential that can be reduced to
the form
P
(a + lx x*fx m dx (1)
can be made rational in terms of an auxiliary variable,
and consequently can be integrated, when p and m are
whole numbers.
When p is even, the expression is already rational;
when p is odd, say of the form 2n + 1, the expression
becomes,
(a + lx x*) n (a + lx x^x m dx (2)
in which the only irrational part is Va 4- bx =fc x 2 . There
may be two cases; first, when x z is positive, and secondly,
when x 2 is negative.
14:2 INTEGRAL CALCULUS.
1. When x z is positive.
Assume,
Va + bx + x 2 = z x; /. a + "bx + x 2 = z 9 2zx + x* j
striking out the common term x 2 , and solving, we have,
z 2 a
; /. dx =
fyfi T
and
4^4^ (3)
Substituting these in (2), the result will be rational.
EXAMPLES.
~ (4)
Comparing (4) with (1), we find, a = 1, and 5 = 1.
Hence, from (3), we have,
+ 1
Substituting in (4), and making z = x + V 1 + x + x 2 ,
we hare,
y = l(%z + 1) = Z(2Z + 1 + 2 VI
^ 2 a; 1
Ans. y = Z(2a; - 1 + 2Vx 2 x - 1).
METHODS OF INTEGKATIOtf. 143
3 . rfy= __J|==.
Ans. y if -====\.
\ * "^ I V "^ r *v i ** X
S. IFAew x 9 is negative.
Let a and j3 be taken so as to satisfy the equation,
a + bx - x* = (x - a)(/J - x) (5)
Assume
Va + bx x 2 = V(x a)(# x) (x a)g;
squaring and reducing, we have,
P-x = (x a)z* ; or, z =
hence,
_ ff + ag8
: T+"^" ;
and
These values substituted in (1), will make it rational.
T , ,
Let e =
EXAMPLE.
dx dx
-*)
.-. a = - 2, and /S = 1 (7)
144 INTEGRAL CALCULUS.
From (C), we have,
6zdz
Substituting in (7), we find,
also z = A / - .
A
= - 2tan- 1
r + z z ' y - |/ jr+2
Integration by Series.
72. It is often convenient to develop a differential into
a series, arranged according to the ascending powers of
the variable, and then to integrate each term separately.
This method sometimes enables us to find an approxi-
mate value for an integral, which cannot be found in any
other manner; it also enables us to deduce many useful
formulas.
EXAMPLES.
1. dy =
Developing (1 x 2 ) * by the binomial formula, and
multiplying each term by x 2 dx, w r e have,
X 3 X 6 X
'y=w -TO -56-ifl - eta
METHODS OF INTEGRATION. 145
By Formula (14) y is equal to tan" 1 ^. Developing bj
the binomial formula, we have,
(1 + x*)~ l = 1 x 2 + ?; 4 x* + x 8 z 10 + etc.
Substituting, and integrating, we have the Formula,
__1 x 3 x 5 x' 1 x 9
tall z =a ,.__ + ___ + __etc.
= (1 x 4- x z x* + etc.)dx
y = 1(1 + x) = x - |- + f- - |- + etc.
3x s
Integration of Transcendental Differentials.
73. A transcendental differential is one that is expressed
in terms of some transcendental quantity. There are three
principal classes; viz., logarithmic, exponential, and cir-
cular.
Logarithmic Differentials.
74. A large ir'imber of cases come under the general
form,
dy = z m - l (lx) n dx (1)
146 INTEGRAL CALCULUS.
Assume,
u = (Ix) 71 , and dv = x m ~ l dx;
.: du = ^-^ ' -' and v = >
2; T/Z.
substituting in Formula (28), and reducing, we have.
y =
Formula ,F is a formula of reduction ; it reduces the
exponent of (Ix) by 1 at each application.
EXAMPLES.
1. dy = x(to)*dx.
2. rfy =
3. dti =
Am. y =
Eeversing Formula ^ and reducing, we have,
Replacing w 1 by w, and reducing, we have,
/^J_^ = __ x^ __ + m /?^~V*
(lx} n (n - 1) (Ix) 71 ' 1 + n- lJ\U)> 1 - 1
METHODS OF INTEGRATION. 147
The continued application of formula G gives as a final
/ v m lfl x
n ., which can be integrated
(ix)
by series. It fails when n = 1. Let m = 0, and n = I ;
we then have,
Exponential Differentials.
75. Let it be required to integrate an expression of the
form,
dij = x m a x dx (1)
Assume,
u x m , and dv a x dx\ .: du = mx m dx, and v = r .
la
0*
u = mx"" ~tlx, and v =
Substituting in Formula (28),
/ la la J
When m < 1, we have,
fifdx = <F la fifdx
** x m ~ (m-l}x m - 1 m-l^x- 1
INTEGRAL CALCULUb.
EXAMPLES.
1. dy = xa x da\ A us. .y ~j~\ x ~~ T~ )'
2. dy = x*a x dx.
a x f 3 3 2 (jx 6 \
AnS.y = T -\X -- : -- \- yy-r- I.
te\ / (/rt) 2 (/) v
e'Yfa ^ -/e^fo
3. dy - j- . Ans. y -- + I-
x 2 x J x
But by means of McLaurin's Formula, we find,
hence,
x , xdx
which, being substituted above, gives,
4. dy = x*e x dx. Ans. y = e x (x* - Zx + 2).
Circular Differentials.
76. Circular differentials may be integrated by the
methods of transformation and successive reduction, or
they may be reduced to algebraic forms by making
METHODS OF INTEGRATION. 149
EXAMPLES.
- T , , dx
1. Let av = - .
smx
Prom trigonometry, we have,
smx = 2sin^cosjx = 2tanJ#cos*
or,
dx dx
smx ~ 2tan %xcos 2 ^x ~ tanjz "
Hence,
- = Z(tanjz) ..... (1)
smx
T , 7
2. Let dy = - .
cosx
If we make x equal to 90 x in the preceding formula,
and reduce, we have,
3. Let dr =
From trigonometry, we have,
Hence,
sinz do? ^Ircosa;
tana; = -- ; or, - - = ; --- = ^
coso: taiue smx smx
=Z(sina;) ..... (3)
tuna;
150 INTEGRAL CALCULUS.
I T 1 7 $ X
4. Let ay = .
cote
Making x equal to 90 x in (3), and reducing, we have,
rdx = _ ..... (
J cotx
5. Let dy = -J^.
smftoosz
From trigonometry, we have,
sin^cosa; =
we also have,
dx =
hence,
dx _d(2x)
Applying Formula (1), we have,
r dx
J s
Let us have the expression,
dy sin m o;cos n ^^ ..... (6)
Making sinz = z, whence cosx *J\ z z ,
and dx = - .
Vl - z*
we have,
71-1
dy = z m (l - z 2 ) 3 dz ..... (7)
Form (7) can always be integrated when m and n are
whole numbers, because it will then satisfy one of the three
criterions (Art. 09).
METHODS OF INTEGRATION. 151
6. dy &'m z xcos & xdx.
Comparing with (6). w* find in = 2. and n -- 3; hence,
from (7), we have,
dy = (1 - z*)z*dz; /. y = - - ~ =
i
7. Let dy = sh\*xdx.
Here m = 3, and n 0; and dy = (1 z*)
making x = z, a = 1, = 1, n = 2, j9 = J, and m = 4,
in Formula A, we have,
But from Formula (29),
hence,
\ (1 - zrfz* - |(1 - z )t =' - i
8. Let dy = cos z xdx = dz^/\ ~z 2 .
Applying Formula (B), we find,
But,
152 INTEGRAL CALCULUS.
Hence,
y -(coszsince + x).
^
flnft (1 y,
9. Let dy = ^^ = ^ = (1 - z 8 )
Applying Formula (7, we have,
But,
Hence,
10. Let c?v = -,--%- = ,., ^ ,,, = (!-**)" Vfe,
22
By Formula Z), we have,
/(I _ g )
But,
sin.rcosa;
Hence,
?/ = i 4- /(tana;).
4/ 2cos 2 o;
METHODS OF INTEGRATION. 153
11. Let dy = tsm^xdx = (1 z 2 )'
y Jtan s o; tanjc 4- x.
EXAMPLES IN SUCCESSIVE INTEGRATION.
Making sinz = z, whence, d z y = z(dz) 2 , we have,
dy z 2 z 3
& = a" + ' y = 6~ + + '
or,
= ^_? + ^g^^, + @ m
o
13. c? 2 y = cosxsm*xdx 2 = cosa:(</coscc) 8 ;
^?y 1
**' /7f \ = o COS x + ^
and
y = - co& 3 x + Cfcosa; + 6*.
b
Integration of Differential Functions of two Variables.
77. A total differential of a function of two variables is
of the form,
<& = Pdx + Qdij (1)
7*
154 LNTECltAL CALCULUS.
In which P and Q are functions of x and y ; but differ-
ential expressions of that form are not necessarily total
differentials. That they may be so, they must (Eq. 5,
Art. 2G), satisfy the condition,
dP dQ
dy ^ dx~ W
When an expression of the form (1) is to be integrated,
we first apply the test expressed by (2) ; if that be satisfied,
the expression is integrable. To perform the integration,
integrate one of the partial differentials with reference to
its corresponding variable, that is, as though the other
variable were constant, and add such a function of that va-
riable as will satisfy equation (1). Thus, to integrate ex-
pression (1), we have,
z = Cpdx + R (3)
in which R is a function of y alone. The value of R may
be found by differentiating the second member of (3) with
respect to y, and placing its partial differential coefficient
equal to Q. Hence, we have,
d/Pdx
dy dy
Transposing, multiplying by dy, and integrating with
respect to y, that is, as though x were constant, we find,
METHODS OF INTEGRATION. 155
Substituting in (3), we find,
//Y dfpdx
Pdx+ J ( Q -
EXAMPLES.
1. dz = 3x z y 2 dx + 2x*ydy.
Here P = 3x*ij*, and Q = 2x*y, and (2) is satisfied.
Integrating by Formula (4), we hare,
z = z 3 y 2 + 2x 9 y 2x*y)dy = x*y* -f C.
2. dz = ydx + xdy.
The test (2) is satisfied. Integrating by (4), we have,
z = yx + i (xdy xdy)dy yx -f 0.
By Formula (4), we have,
z =
4. dz = (oxy y*)dx + (3a; 2 %xy}dy
+
y*x + f(3x z 2xy 3x 9
PART IV.
APPLICATIONS OF THE INTEGRAL CALCULUS.
I. LENGTHS OF PLANE CURVES.
Rectification.
78. Rectification is the operation of finding an expres-
sion for the length of a curve. This may be done by
finding an element of the length, as explained in Art. 51,
and then integrating it between proper limits. If we
apply the integral sign to Equation (1), Art. 51, we have,
L =Vdx* + dy* ..... (1)
1. To rectify the semi-cubic parabola, whose equation is,
Differentiating and substituting in (1), we have,
(2)
Integrating by Formula (29) , we have,
L = ^(4a* + 9y) + ..... (3)
As explained in Art. GO, this integral is indefinite, anti
LENGTHS OF PLANE CURVES. 157
expresses the length of an arc from any ordinate up to tiny
other ordinate.
If we estimate the length from a point whose ordinate
is 0, the length at that point is 0, and, from (3), making
L y and y 0, we have,
Substituting this value of C in (3), and denoting the cor-
responding value of L by L', we have,
This is a particular integral, and it expresses the length
of the arc from the particular ordinate 0, up to any
ordinate y.
If we wish to find the length from the particular ordi-
nate 0, up to an ordinate #, we make y b, in (4). Doing
so, and denoting the corresponding value of L' by JJ\
we have,
"= + )*- IT ..... ( g )
This is the definite integral, and expresses the length of
the arc from the ordinate 0, up to the ordinate b.
In this case the limits are and I ; the integral may be
found otherwise, as follows :
Making y = 0, in (3), and denote the corresponding
value of L by LQ, we have,
158 INTEGRAL CALCULUS.
Making y = #, in (3), and denoting the corresponding
value of L by L b , we have,
X 4 = ^(4t + 9S)* +0 ---- (7)
Equation (6) expresses the length of the curve from any
point up to the point whose ordinate is 0, and (7) ex-
presses the length of the arc from the same point up to a
point whose ordinate is b ; the first taken from the second
will therefore be the definite integral, and will express, as
before, the length of the arc from the point whose ordinate
is 0, up to the point whose ordinate is b. Making the
subtraction, and adopting the notation explained in Art. 60,
we have,
b
L " =
The same result as found in (5). In this case the initial
3
abscissa is and the terminal abscissa is - .
a
2. To rectify the cycloid, wJiose differential equation
(Art. 56), is
- y
Substituting, in (1), and reducing, we have,
L =
Integrating between the limits and 2r Formula (29),
we have,
2r
LENGTHS OF PLAHE CURVES. 159
This is the expression for half of one branch. Hence, the
whole branch is equal to eight times the radius of the
generating circle.
3. To rectify the circle, whose equation is,
y = (r* - xrf.
Differentiating, substituting in (1), and developing by
the binomial formula, we have,
L = rf(r* - x*)~tdx = f(dx + ^ . x*dx + ^
Performing the integration, making r = 1, and com-
mencing the arc at the point whose abscissa is 0, that is,
at the upper extremity of the vertical diameter, we have,
Formula (13) may be used for finding the length of an
arc, but the series is not very converging, and therefore
the process is tedious. We know, when x = J, that L' is
equal to an arc of 30, that is, to -J-tf. Making L' = $*,
and x = , we have,
5* = I + sis + ^75 + W&J + etc " = -
= 3.1415
4. A better method for deducing the value of if, is to
find the length of the arc, in terms of its tangent.
If L = tan a;, we have (Art. 18),
dL = = 1 + x '~ 1<lx -
160 INTEGRAL CALCULUS.
Developing by the binomial formula, we have,
dL = dx(l - x* -f x* - x 6 + x 8 - etc.) ..... (15)
Integrating, and determining (7, as before, we have,
x 3 x s x 11 x 9
i= *-T + 5--r + -- etc
To render (16) converging, x must be made small
Assume the trigonometrical formula,
tana + tanJ
and let a tan" 1 ???, b = tan" 1 ^, and a + I = tan l j
then will
m + n z m nff .
z - - ; or, n -= ..... (17)
1 mn mz + 1
1 2
[f z 1, and m = -, we have, n -\
o o
/. tan" 1 1 = tan" 1 ^ + tan" 1 !
o
21 7
If z = -, and wi = -, we have, n ^ ;
71 9
If 2 = , and m = -, we have, w = ;
17 4:0
.
91 1
If z = , and OT = -, we hare, n - - ^
AREAS OF PLANE CURVES. 1C]
Hence, "by successive substitution, we find,
- 4 = tan" 1 1 = 4tan- 1 i - tan" 1 -^- . . .\. . (18)
4 5 /coJ
If we make x -, in Equation (1C), we find the value of
5
tau 1 r, and. in like manner, we find the tan ^r; sub-
/vO J
stituting them in (18), and reducing, we have the value of
if. With very little labor, the value of if may be found to
8 or 10 places of decimals.
II. AREAS OF PL AXE CURVES.
Quadrature.
79. Quadrature is the operation of finding an expression
for the area of a portion of a plane bounded by a curve,
the axis of abscissas, and any two ordinates. This is done
by finding an expression for the elementary area, as ex-
plained in Art. 51, and then integrating the result between
proper limits. Applying the sign of integration to For-
mula (2), Art. 51, we have,
A =
1". To find the area of a parabola, whose equation is,
2/2 2px.
Finding the value of ?/, and substituting in (1), we have,
A = Vtyxdx = . a + ..... (2)
162 INTEGRAL CALCULUS.
If we commence the area at the ordinate 0, we have,,
C = ; hence,
(3)
That is, the area of any portion of a parabola, reckoned
from the vertex, is two-thirds the rectangle of its terminal
co-ordinates.
2. To find the area of a circle, whose equation is 9
y - V
tf ibstituting in (1), we find,
A
= C(r* -xrfdx ..... (4)
Reducing by Formula J3, and integrating the last term
by Formula (20), we have,
_*(r-*')*
Taking the integral uetween the limits, x = and x = r,
which gives the area of a quadrant, and remembering that
sin"" 1 1 sin"" 1 = Jtf, we have,
T" 1 1 cfT* 2
/I" = [sin 1 sin 0] = -: ; .'. Area = *7' 2 . . . (6)
A 4:
3. To find the area of the ellipse, whose equation is,
/ r-_//rt _ ^2\i
AREAS OF PLANE CURVES. 163
Substituting in (1), and integrating, as in the last example,
we have,
Integrating from x = to x = a, and multiplying by 4, we
Lave, for the area of the ellipse,
A" =*ab (8)
4. To find the area of the cycloid, whose diffeimtial
equation is,
Substituting in (1), we find,
f* y*dy
f
Applying Formula E twice, and Formula (26) once, wo.
find,
yV%ry - y*
~2~
+ <r ~ V%ry y* + rversin" 1 ^1 4- 0.
Taking the integral between the limits y = 0, and y = 2r,
and multiplying by 2, we find, for the area of the cycloid,
2A" = 3r* ..... (10)
That is, the area between one branch and the directing
line is three times the area of the generating circle.
5. To find the area of the logarithmic curve, whose
equation is, y = Ix.
164 INTEGRAL CALCULUS.
Substituting iu (1), we have,
A =f(lx)dx ---- (11)
Making m = 1 and n 1, in Formula F, and reducing,
we have,
A = xlx - x + ..... (12)
If we commence the arc at the ordinate corresponding to
<r. = 1, we have, (7=1, and
A' = x(to-l) + l ..... (13)
6. To find the area of a rectangular hyperbola^, bounded
by the curve, one asymptote, and any two ordinates to that
asymptote.
Assume the equation,
xy = m, and make m = 1, whence y -.
x
Substituting in (1), we have,
A = = lx + c
Commencing the area from the ordinate through the vertex,
where x = 1, we have, (7=0; hence,
A'=lx ..... (15)
That is, the area commencing from the ordinate through
the vertex is the Napierian logarithm of the terminal ab-
scissa. Had we not made m 1, we should, in like man-
ner, have found
A' = mlx.
In which the area is equal to the logarithm in a system
whose modulus is m.
AREAS OF SURFACES OF REVOLUTION. IG5
III. AREAS OF SURFACES OF REVOLUTION.
Surfaces Generated by the Revolution of Plane Curves.
8(fc The area of a portion of a surface of revolution,
bounded by two planes perpendicular to its axis, is de-
termined by finding an expression for an elementary zone,
as explained in Art. 51, and integrating the result between
proper limits. Applying the sign of integration to For-
mula (3), Art. 51, we have,
+ dy* ..... (1)
1. To find the surface of a sphere, the equation of the
generating circle being, y =. (r 2 x 2 )^.
Differentiating, we have, dy (r 2 x 2 )~ixdx; sub-
stituting the values of y and dy in (1), and integrating
from r to + r, we have,
= 2rjdx = 4r* (2)
8"
r
Hence, the area of the surface of a sphere is equal to four
great circles, or to two-thirds the surface of the circum-
scribed cylinder.
2. To find the surface of a right cone.
The equation of the generating line, the vertex of the
cone being at the origin, is
y = ax ; /. dy = adx.
1.66 INTEGRAL CALCULUS.
Substituting in (1), and reducing, we have,
S = ZiraVl + a z jxdx = ax 2 Vl + a 2 + C ..... (3)
If the initial plane pass through the vertex, we have, for
that plane, S = 0, and x = 0, whence (7=0; hence,
S r = *ax 2 Vl + a 2 = *y X #A/1 + a 2 ..... (4)
But a is the tangent of the semi-angle of the cone, and
consequently x^\ + a 2 is the slant height; 2*y is the
circumference of the cone's base; hence, the convex sur-
face is equal to half the circumference of the base into the
slant height.
3. To find the surface of the paraboloid of revolution.
Assume the equation y 2 = 2px, whence,
y = %px, and dy 2 dx 2 .
&X
Substituting in (1), and reducing, we have,
a = 2* fip* + 2px)ldx = - (p 2 + 2px)? + C ---- (5)
If the initial plane pass through the vertex, we have,
Which, in (5), gives,
4 C . To find the surface generated by revolving one branch
of a cycloid about its base.
VOLUMES OF SOLIDS OF REVOLUTION. 1G7
Assuming the equation, dx = db -, and substi-
tuting in (1), we have,
(7)
Applying Formula A, and integrating the last term by
Formula (29), we have,
(2r - y) + (2r - y) + 6'.
Integrating between the limits y and ?/ = 2r, we havo,
-r ..... (8)
This is the surface generated by half of one branch.
Hence, to find the whole surface, we multiply by 2. This
gives
IV. VOLUMES OF SOLIDS OF REVOLUTION.
Cubature.
81. Cubature is the operation of finding the volume of a
solid. When this is bounded by a surface of revolution
and two planes perpendicular to its axis, the volume may
be ascertained by finding the volume of an element
bounded by two such planes infinitely near to each
other, as explained in Art. 51, and then integrating the
108 INTEGRAL CALCULUS.
result between proper limits. Applying the sign of inte-
gration to Formula (4), Art. 51, we have,
V = *fy*dx (1)
1. To find the volume of a sphere.
Assume the equation, y z r 2 x 2 , and combine it with
(]) ; we have,
C.
Taking the integral between the limits, x = r, and
x = + r, we find,
V" = 2r* ~~ = T = 4*r 2 X r ..... (2)
That is, the volume is equal to the surface ~by one-third the
radius.
2. To find the volume of a spheroid of revolution.
There are two species of spheroids of revolution.
1st. The prolate spheroid, generated by revolving an
ellipse about its transverse axis.
2dly. The oblate spheroid, generated by revolving an
ellipse about its conjugate axis.
1st. The prolate spheroid. In this case the equation of
2
the meridian curve is, y 2 = (a 2 x z ). Substituting
this in (1), and integrating between the limits x = a,
and x = + a, we have,
- x*)dx = zb*a = !** X 2a ..... (3)
VOLUMES OF SOLIDS OF KEYOLUTIOJS". 1G9
That is, the volume is equal to two-thirds of the circum-
scribing cylinder.
2dly. The oblate spheroid. In this case, if the conjugate
axis coincide with the axis of x, the equation of the meri-
a 2
dian curve is, y 2 = (b 2 x 2 ). Substituting in (1), and
integrating from b to + I, we have,
V" = \a 2 b = la 2 X%b ..... (4)
o o
Hence, as before, we have, the volume equal to two-thirds
the circumscribing cylinder.^
In both cases, if a = b r, we have,
V" = ^3 ..... (5)
This hypothesis causes the ellipsoids to merge into tne
sphere.
3. To find the volume of a paraboloid of revolution.
The equation of the meridian curve is, y 2 = 2px. Hence,
from (1), we have,
V = %pixdx = px 2 + C ..... (6)
If the initial plane pass through the vertex, we have
(7=0, and
V = px 2 = y 2 X \x . . . . (7)
That is, the volume is equal to half the cylinder that has the
same base and the same altitude.
170 INTEGRAL CALCULUS.
4. To find the volume generated by revolving one branch
of the cycloid about its base.
The differential equation of the meridian curve is,
hence, from (1), we have,
Keducing by Formula E, integrating the last term by For-
mula (26), and taking the integral between the limits,
y = 0, and y 2r, we find,' for one-half the volume
required,
V" = *r 3 , or2F" = 5* 2 /- 3 = fl r(2r) X 2r ..... (8)
A o
Hence, the volume is equal tofive-eighklis the circumscribing
cylinder.
5. To find the volume generated by revolving the loj'triih-
mic curve about the axis of numbers.
The equation of the generatrix is y = Ix. Hence,
V= C(lxydx ..... (9)
Reducing by Formula F, we have,
V = [x(lx)* - 2(xlx - x)] + 0.
If the initial plane pass through the point whose
abscissa is 1, we have, C 2^. Hence,
2+1)] ..... (10)
PART V.
APPLICATIONS OF THE DIFFERENTIAL AND
INTEGRAL CALCULUS TO MECHANICS AND
ASTRONOMY.
I. CENTRE OF GRAVITY.
82. In what follows, bodies are supposed to be homoge-
neous ; the weight of any part of a body is therefore pro-
portional to its volume, and consequently the weight of
the unit of volume may be taken as the unit of weight.
Points, lines, and surfaces are supposed to be material:
A material point is a body whose length, breadth, and
thickness are infinitesimal ; a material line is a line
whose length is finite, and whose breadth and thickness
are infinitesimal; a material surface is a body whose
length and breadth are finite, and whose thickness is infi-
nitesimal; under this supposition a point is an elementary
portion of a line, a line is an elementary portion of a sur-
face, and a surface is an elementary portion of a solid.
The weights of the elements of a body are directed
toward the centre of the earth, and because the bodies
treated of are exceedingly small in comparison with the
earth, these weights may be regarded as a system of par-
allel forces; hence, the weight of a body is equal to the
sum of the weights of its elements and is parallel to
them.
172 DIFFERENTIAL AND INTEGRAL CALCULUS.
The centre of gravity of a body is a point through which
its weight always passes. This point may be found by the
principle of moments, which may be enunciated as follows:
the moment of the resultant of any number of forces, with
respect to an axis, is equal to the algebraic sum of the
moments of the forces, with respect to the same axis.
(Mechanics., Art. 35.)
In applying this principle, we assume the following re-
sults of demonstrations in mechanics : 1. The centre of
gravity of a straight line is at its middle point; 2. The
centre of gravity of a plane figure is in that plane ; 3. If
a plane figure have a line of symmetry, its centre of gravity
is on that line ; and 4. If a solid have a plane of symmetry,
its centre of gravity is in that plane. (Mechanics, Arts. 44,
45, 46.)
To deduce general formulas for finding the centre of
gravity of a body, assume a system of co-ordinate axes that
are to retain a fixed position with respect to the body, but
that change position when the body moves; denote the
volume, and consequently the weight, of any element of
the body by dv, and the co-ordinates of its centre of
gravity by x, y, and z; denote the weight of the body by
v = I dv, and the co-ordinates of its centre of gravity by
a,, y 1? and z r
If the body be placed in such a position that the plane
xy is horizontal, the weights of the elements and of the
body are parallel to the axis of z, the moment of the body,
with respect to the axis of y, is ^1 dv, the moment
of any element, with respect to the same axis, is
APPLICATIONS TO MECHANICS AND ASTRONOMY. 173
and the algebraic sum of the moments of all the elements
\alxdvj hence, from the principle of moments, we have,
/ fxdv
lv = Ixdv ; .: x 1 = -~ (1)
J Idv
In like manner, we have,
If the body be turned about so that the plane yz is
horizontal, we have, in like manner,
Cd V =r y dv; s.y !=?-%- (2)
/ / I dv
//*
!v=lz
J
I zdv
zdv; .: z l = J 7 (3)
/*
"When the body is in a plane, that plane may be taken
as the plane xy, in which case z l = ; if the body have an
axis of symmetry, that may be taken as the axis of x, in
which case z 1 = 0, and y l 0.
Centre of Gravity of a Circular Arc.
83. Let the radius perpendicular to the chord of the arc
be taken as the axis of x ; then will z lt and y } be equal fc
0. Denote the radius of the circle by r, ^ A
the chord by c y and the arc by A. The
origin being at the centre, the equation of
the arc is, y 2 = r 2 x 2 ; hence,
th =
=-TC
c
Fig. 16.
rdy __ rdy
x
12*
174 DIFFERENTIAL AND INTEGRAL CALCULUS.
Substituting in (1), and integrating between the limits,
y = j, and y = + \c, we "have, for the numerator,
Jrdy = rc,
and for the denominator,
/ / .2_ 2/ s L 2r 2rJ
-ic
Hence,
/*/
: : r : a:
That is, /Ae ce?^re of gravity of the arc of a circle is on the
diameter that bisects the chord, and its distance from the
centre is a fourth proportional to the arc, its chord, and
the radius.
Centre of Gravity of a Parabolic Area.
84. Let the area be limited by a double ordinate, and
denote the extreme abscissa by a. From the equation of
the curve, y* = 2px, we have,
y =
/. dv = ydx = V%p x*dx,
_ o
and xdv v%p - ofidx.
Substituting in (1). and integrating between
the limits and a, we have,
3
*i - 5 *
APPLICATIONS TO MECHANICS AND ASTRONOMY. 1?5
That is, the centre of gravity is on the axis, at a distance
from tlie vertex equal to three-fifths the altitude of Hie
segment.
Centre of Gravity of a Semi-ellipsoid of Revolution.
85. Let the axis of the ellipsoid be taken as the axis
of x. Then, if the origin be taken at the centre, the
equation of the generating curve is
In this case, we have,
B u
dv vy*dx = * (a 2 x z )dx,
b 2
and xdv = it - (a 2 x x s )dx.
a 2
Substituting in (1), and integrating between the limits
x 0, and x = a, we have,
That is, the centre of gravity of a semi-prolate spheroid
of revolution is on its axis of revolution, and at a distance
from the centre equal to three-sixteenths the major axis of
the generating ellipse.
If we change a to #, and b to a, we find for the semi-
oblate spheroid,
3. 3 _
176 DIFFERENTIAL AND INTEGRAL CALCULUS. .
Centre of Gravity of a Cone.
86. Let the axis of the cone be the axis of x, and the
vertex of the cone at the origin ; denote the altitude by h,
and the radius of the base by r ; then will the tangent of the
semi-angle of the cone be =-, the equation of the generating
li
T
line will be y x, and we have,
dv = *y z dx = if 7-r x*dx, and xdv =
li* if
Substituting in (1), and integrating between the limits
and h, we have,
_ 3
Centre of Gravity of a Paraboloid of Revolution.
87. Let the axis of the paraboloid be taken as the axis of
x. The equation of the parabola being y 2 = 2px, we have,
;, and xdv =
Substituting in (1), and integrating from x to x = a t
we have,
2
II. MOMENT OF INERTIA.
Definitions and Preliminary Principles.
88. The moment of inertia of a body with respect to an
axis, is equal to the algebraic sum of the products obtained
by multiplying the mass of each element of the body by
the square of its distance from the axis. If we take the
axis through the centre of gravity of the bodv. and denote
APPLICATIONS TO MECHANICS AND ASTRONOMY. 177
the mass of an element by dm, its distance from the axis
by z, and the moment of inertia by IY, we have,
K=J**dm (l)
If we take any parallel axis at a distance d from the
assumed axis, and denote the moment of inertia with
respect to it by K', we have (Mechanics, Art. 123).
K'= K + md* (2)
Moment of Inertia of a Straight Line.
89. Let the axis be taken through the centre of gravity
of the line and perpendicular to it. Let AB represent the
line, CD the axis, and E any ele-
ment. Denote the length of the | [
line by 21, its mass by m, the dis- ~~G jf
tance GE by x, and the length *"" JcT" 1 "
of the element by dx. From the Fi s- 19 -
principle of homogeneity, we have,
21 : dx : : m : dm ; /. dm = dx.
Substituting in (1), and integrating from x = I to
x = + I, we have,
K m-\ .'. K' = m ( + d* ) (3)
<J \O /
These formulas are entirely independent of the breadth
of AB in the direction of the axis CD ; they hold good,
therefore, when the filament is replaced by the rectangle
EF 9 the axis being parallel to one of its ends. In this
case m is the mass of the rectangle ; 21, its length ; and d,
the distance of the axis from the centre of gravity of the
rectangle.
178
DIFFERENTIAL AND INTEGRAL CALCULUS.
Moment of Inertia of a Circle.
90. First) let the axis be taken to coincide with one of
the diameters. Let ACB represent the circle, AB the
axis, and C'D' an element parallel to A
the axis. Denote 00 by r, OE by x,
EF by dx, C'D' by 2y, and the mass
of the circle by m. Then, because the
circle is homogeneous, we have,
BT
: 2ydx : : m : dm ;
= ^dx =
-TIT 2
Substituting in (1),
f(r* -
*J
Keducing by Formulas A and B, integrating by (20),
and taking the integral between the limits x = r and
r. = + r, we find,
(4)
Secondly, let the axis be taken through the centre and
perpendicular to the plane of the circle. Let KL be an
elementary ring, whose radius is x, and
whose breadth is dx ; then will its area
be Zifxdx, and from the principle, that the /
masses are proportional to the volumes,
we have,
m : dm ; .*. dm = j
APPLICATIONS TO MECHANICS AND ASTRONOMY.
Substituting in (1), and integrating from x = to x = r,
we have,
(5)
Thirdly, to find the moment of inertia of a circular
ring with respect to an axis perpendicular to its plane.
Let m denote the mass of the ring, r and r '
its extreme radii ;
Then, 7t(r 2 r' 2 ) : %nxdx ::m: dm ;
.\dm = ^ jj dx.
Substituting in (1) and integrating from
x = r' to x = r, we have
Fig. 22.
Formulas (5) and (6) are independent of the thickness
of the plate ; hence, they will be true whatever that thick-
ness may be. Hence, they hold good for a solid and hollow
cylinder, m being taken to represent the mass of the
cylinder.
Moment of Inertia of a Cylinder with respect to an Axis
perpendicular to the Axis of the Cylinder.
91. Let the axis be taken through the centre of gravity
of the cylinder, and let EF be an element perpendicular
to the axis of the cylinder. Denote the
length of the cylinder by 21, its radius
by r, its mass by m, the distance of EF
from the axis by x, and the thickness
by dx. Then, as before, we have,
dx : : m : dm ;
7 m
dm = r
180
DIFFERENTIAL AND INTEGRAL CALCULUS.
The moment of inertia of this element is given by For-
mula (4) ; but this is the differential of the moment of
inertia of the entire cylinder; hence, substituting dm for
m, and x for d, in (4), we have,
Integrating from I to + Z, we have,
( 7)
Moment of Inertia of a Sphere.
92. Let the axis pass through the centre of a sphere.
Let C'D' be an elementary segment perpendicular to
the axis, DC, whose volume is *y 2 dx
= *(r 2 x 2 )dx; its mass is found -- r ^- c '
from the proportion,
-
: if(r 2 x 2 )dx : : m : dm;
.: dm
3m
(r 2 -x 2 )dx.
Substituting this value of dm for m in equation (5), and
making r in that equation equal to y, or to Vr 2 x 2 , we
have, for the differential of the moment of inertia,
^
=(r 2 -x*) 2 dx.
Integrating between the limits x = r, and x = + r, we
have,
APPLICATIONS TO MECHANICS AND ASTRONOMY. 181
III. MOTION OF A MATERIAL POINT.
General Formulas.
93. Uniform, motion is that in which the moving point
passes over equal spaces in equal times ; variable motion
is that in which the moving point passes over unequal
spaces in equal times. The velocity of a point is its rate
of motion. The acceleration due to a force is the rate of
change that it produces in the velocity of a point. When
a force acts to increase the velocity, the acceleration is
positive) when it acts to diminish the velocity, the accelera-
tion is negative, and conversely.
Let us denote the mass of a material point by m, its
velocity at any time t, by v, the space moved over at the
time t, by s, and the acceleration due to the moving force
by <p.
When the motion is uniform, the velocity is constant, and
its measure is the space passed over in any time divided by
that time ; but we may, in all cases, regard the motion as
uniform for an infinitely small time dt ; denoting the
space described in that time by ds 9 we have,
ds
When the velocity varies uniformly the acceleration is
constant, and we take for its measure the change of
velocity in any time divided by that time; but we may, in
all cases, regard the velocity as varying uniformly for an
infinitely small time dt ; denoting the change of velocity
in that time by dv, we have,
182 DIFFERENTIAL AND INTEGRAL CALCULUS.
In the discussion of motion it is customary to regard the
time as the independent variable. Differentiating (1),
under that supposition, we find,
substituting this in (2), we have,
Equations (1), (2), and (3) are fundamental, and from
them we may deduce many laws of motion. If we multi-
ply both members of (3) by m, we have,
cl 2 s p _ d 2 s
In (4), F is the moving force. It will often be conve-
nient to regard the mass of the material point as the unit
of mass, in which case, we have, m = 1.
Uniformly Varied Motion.
94. Uniformly varied motion is that in which the ve-
locity increases, or diminishes, uniformly. In the former
case the motion is uniformly accelerated, in the latter it is
uniformly retarded, in both the acceleration is constant.
Denoting the constant value of <p by g, substituting in (3),
multiplying both meml jrs by dt, and integrating, we
have,
^ = fff. + C; or, v = gt f C (5)
Multiplying again by dt, and integrating, we find,
s = \gt* + Ct + C 1 (6)
APPLICATIONS TO MECHANICS AND ASTRONOMY. 183
The constant, (7, is what v becomes when t = ; this is
called the initial velocity, and may be denoted by v'. The
constant, 0', is what s becomes when t = ; this is called
the initial space, and may be denoted by s'. Making these
substitutions in (5) and (6), we have,
v = v'+fft (7)
* = s' 4- v't + \gt* (8)
Equations (7) and (8) enable us to discuss all the cir-
cumstances of uniformly varied motion (see Mechanics,
Arts. 103 to 108). If g represent the force of gravity, and
v' and s' be each equal to 0, (7) and (8) will express the
laws of motion when a body falls from rest under the in-
fluence of gravity, regarded as a constant force. Under
this supposition they become,
*=fft (9)
s = W (10)
That is, the velocities generated are proportional to tne
times, and the spaces fallen through are proportional to
the squares of the times.
Bodies Falling under the Influence of Gravity, regarded as
Variable.
95. In accordance with the Newtonian law, the attrac-
tion exerted by the earth on a body at different distances
varies inversely as the squares of their distances. Denoting
the radius of the earth, supposed a sphere, by r, the force
of gravity at the surface by g, any distance from the cen-
tre, greater than r, by s, and the force of gravity at that
distance by <p, we have,
3 * ~
184 DIFFERENTIAL AND INTEGRAL CALCULUS.
Substituting this value of 9 in (3), and at the same time
making it negative, because it acts in the direction of s
negative, we have,
Multiplying by 2ds, and integrating both members, we
have,
ds* 2r z Zr*
If we make v = 0, when s = li, we have,
aid, '=2<rr'j- ..... (13)
Equation (13) gives the velocity generated whilst the
body is falling from the height li to the height s. If we
make h co , and s = r, (13) becomes,
v* = 2gr; .-. v fyr ..... (14)
In this equation the resistance of the air is not con-
sidered.
If in (14) we make g = 32.088 feet, and r 20923590
feet, their equatorial values, we find,
v 3GG44 feet, or nearly 1 miles per second.
Equation (14) enables us to compute the velocity acquired
by a body in falling from an infinite distance to the sun.
If we make g = 890.16 feet, and r = 430854.5 miles, which
are their values corresponding to the sun, we find,
v = 381 miles per second.
APPLICATIONS TO MECHANICS AND ASTRONOMY. 185
If we make li equal to the distance of Neptune, and s
equal to the sun's radius in (13), we find the velocity that
a body would acquire in falling from Neptune to the sun,
under the influence of the sun's attraction.
To find the time required for a body to fall through any
ds
space, substitute for v in (13), and solve the results with
cct
respect to dt; this gives,
^ = V^'7^ = ~/^'vlb- (15)
The negative sign is taken because s decreases as t in-
creases. Reducing (15) by Formula E, and integrating
by Formula (20), we find,
If t when s = h, we have,
This, in (16), gives,
I ftversin- 1 J + fr*] . . . (17)
Making s = r in (17), we find,
IT, [(/<< - r-)* - I Avorsin- 1 }: + ^] . . . (18)
Which gives the time required for a body to fall from a
distance h, to the surface of the sun.
186 DIFFERENTIAL AND INTEGRAL CALCULUS.
Bodies Palling under the Influence of a Force that vaiies as
the Distance.
96. It will be shown hereafter, that if an opening were
made along one of the diameters of the earth, and a body
permitted to fall through it, the body would be urged
toward the centre by a force varying as the distance from
the centre, provided the earth were homogeneous. As-
suming that principle, and denoting the force at the sur-
face by g, the radius being r, and the force at the distance,
x, from the centre by <p, we have,
r : s : : g : <p ;
Substituting this value of <p in (3), and at the same time
giving it the minus sign, because it acts in the direction
of s negative, we have,
Multiplying by 2ds, and integrating,
|! = _i! + (7;oj ; )V . = (7-?,..
dt z r r
Making v 0, when s = r, we have, C ^ r 2 ; hence,
- = -,). -.(19)
cit* r
If s = 0, we find v = Vgr, which is the maximum velocity.
It is eqjal to the entire velocity generated by a body fall-
ing from an infinite distance to the surface of the earth
divided by A/2. If the body pass the centre, s becomes
amative, and we find the same values for v at equal dis-
APPLICATIONS TO MECHANICS AND ASTRONOMY. 18?
tances from the centre, whether s be positive or negative.
When s becomes equal to r, v reduces to 0, and the
body then falls toward the centre again, and so on con-
tinually.
Let us suppose A'C' to be the diameter along which the
body oscillates, and at the time the body starts from A'
let a second body start from the same
point and move around the semi-cir-
cumference, A' MO', with a constant
velocity equal to *fgr. At any point,
M, let this velocity be resolved into
two components, MQ and MN, the
former perpendicular and the latter
parallel to A'C'. The latter component will be equal to
y multiplied by cosTMN, or its equal, cosB'MII' ; but
COB&MH' = ^TJ,; denoting B'H' by s, whence H'M
we have, cosB'MH'
= ; hence, NN A/ ^- . (r 2 s*)* : but this
r I/ r
is equal to the velocity of the oscillating point when at IT.
Hence, the velocity of the vibrating point is everywhere
equal to the parallel component of the velocity of the
revolving point; they will, therefore, come together at the
points A and 6 r/ , and the position of the vibrating point
will always be found by projecting the corresponding posi-
tion of the revolving point on the path of the former. To
find the time for a complete vibration from A' to C", solve
(19) with respect to dt, whence,
9 Vr*-
( 20 )
188 DIFFERENTIAL AND INTEGRAL CALCL'LUS.
The negative sign is used because t is a decreasing func-
tion of s. Integrating (20) between the limits s = r and
s = r, we find.
t" =
This, we shall see hereafter, is the time of vibration of a
simple pendulum whose length is r.
The time maybe found otherwise, as follows: The space
passed over by the revolving point is ir-ry dividing this by
the velocity, V]/r, we have,
t T
The species of vibratory motion just discussed is some-
times called harmonic, being the same as that of a point
of a vibrating chord or spring.
Vibration of a Particle of an Elastic Medium.
97. It is assumed that if a particle of an elastic medium
be slightly disturbed from its place of rest, and then aban-
doned, it will be urged back by a force that varies directly
as the distance of the particle from its position of equilib-
rium ; on reaching this position, the particle, by virtue of
its inertia, will pass to the other side, again to be urged
back, and so on.
Let us denote the displacement, at any time t, by s, and
the acceleration due to the restoring force by <p ; then, from
the law of force, we have, <p = n*s, in which n is constant
for the same medium, under the same circumstances of
density, pressure, etc. Substituting for 9 its value from
Equation (3), and prefixing the negative sign, because it
APPLICATIONS TO MECHANICS AND ASTRONOMY. 189
acts in a direction contrary to that in which s is estimated,
we have,
Multiplying by 2ds, and integrating, we have,
-j = n*** + C=-V* ..... (23)
The Telocity is when the particle is at the greatest
distance from the position of equilibrium; denoting this
value of s by a, we have,
n*a* + (7=0; .-. C= -n*a*,
which, in (23), gives,
^ = n*(a* - *) ; or, ndt = J^= ..... (24)
dt* Vfl 2 - s 2
Integrating (24), we have.
nt + G' = sin" 1 - ..... (25)
Taking the sines of both members, and reducing, we have,
s = asui(nt + C) ..... (26)
If we make t = 0, when s = 0, we have (7 = 0, ana Equa-
tion (2G) becomes,
s = a sm(nt) ..... (27)
If (25) be taken between the limits a and + a, we find
for the time of a single vibration, denoted by \T,
190 DIFFERENTIAL AND INTEGRAL CALCULUS.
Substituting this in (27), we have, finally,
n n O1 ' 11 / / I /OC\
8 asiu\^~tj (28)
in which T is the time of a double vibration.
Solving (24), we have,
-1* (29)
Substituting for s, in (29), its value from (28), we find,
v = nA/ a* a 2 siii 2 f~tj = naA/1 sin 2 f>
Whence, we have,
v = nacos(~t\ (30)
This equation is used in discussing the laws of light,
and in many other cases.
Curvilinear Motion of a Point.
98. A point cannot move in a curve except under the
action of an incessant force, whose direction is inclined to
the direction of the motion. This force is called the de-
flecting force, and can be resolved into two components,
one in the direction of the motion, and the other at right
angles to it. The former acts simply to increase or di-
minish the velocity, and is called the tangential force ; the
latter acts to turn the point from its rectilinear direc-
tion, and being directed toward the centre of curvature
is called the centripetal force. The resistance offered by
the point to the centripetal force, in consequence of its
APPLICATIONS TO MECHANICS AND ASTRONOMY. 191
inertia, is equal and directly opposed to the centripetal
force, and is called the centrifugal force.
To find expressions for the tangential and centripetal
forces, let the acceleration due to the deflecting force in the
direction of the axis of x, at any
time t } be denoted by JT, and the
acceleration in the direction of
the axis of y by Y. Let these be
resolved into components acting
tangentially and normally. The
algebraic sum of the tangential
components is the tangential accel-
eration denoted by T, and the
algebraic sum of the normal components is the centripetal
acceleration denoted by JV. Assuming the notation of the
figure, we have,
T = JT
N JTsind I"cos0.
But from Articles (93) and (5), we have,
and,
-
dt*
dt*
dx dy
cosd = r ; smd = -/-.
ds d
Substituting in the preceding equations, and reducing oil
the supposition that t is the independent variable, and bv
means of Formula (3), Art. (38), we have,
_ _ d~x dx d 2 y dy
~ ~di* ' T* + W ' ts
(31)
192 DIFFERENTIAL AND INTEGRAL CALCULUS.
-_ _ d*xdy d*ydx
dt* . ds
ds z d*ydx d*xdy __ v*
~di*'~ ~ds*~ ~ ~R
But the centripetal acceleration is equal and directly op-
posed to the centrifugal acceleration. Denoting the lattei
by f, we have,
v 2
IVom (31) we see that the tangential force is independ
ent of the centripetal force, and from (32) we see that the
acceleration due to the centrifugal force at any point of the
trajectory, is equal to the square of the velocity divided by
the radius of curvature at that point.
Velocity of a Point rolling down a Curve in a Vertical Plane.
90. Let a point roll down a curve, situated in a vertical
plane, under the influence of gravity regarded as constant.
Let the origin of co-ordinates be taken at the starting
point, and let distances downward be positive. At any
point of the curve, whose ordinate is y, the force of gravity
being denoted by g, we have, for the tangential component
of gravity, #sin<3, or, g -f ; placing this equal to its value,
as
equation (31), we have,
dij d z s
Integrating and reducing, remembering that the constant
ife under the particular hypothesis, we have,
2 is* ; or? v =
APPLICATIONS TO MECHANICS AND ASTRONOMY. 193
The second member of (1) is the velocity due to the
height y. Hence, the velocity generated by a body rolling
down a curve, gravity being constant, is equal to that gen-
erated by falling freely through the same vertical height.
This principle is true so long as g is constant. But g
may be regarded as constant from element to element, no
matter what may be the law of variation. Hence, if a body
fall toward the sun, or earth, on a spiral line, the ultimate
velocity will be the same as though it had fallen on a right
line toward the centre of the attracting body. The direc-
tion of the motion, however, is not the same, for in the
former case it is tangential to the trajectory pursued, and
in the latter case it is normal to the attracting body.
The Simple Pendulum.
100. A simple pendulum is a material point suspended
from a horizontal axis, by a line without weight, and free
to vibrate about that axis.
Let ABC be the arc through which the vibration takes
place, and denote its radius by I. The angle CD A is the
amplitude of vibration ; half this angle,
ADB, denoted by a, is the angle of
deviation; and I is tlie length of tlie
pendulum. If the point start from
rest, at A, it will, on reaching any
point, //, of its path, have a velocity, v,
due to the height EK, denoted by y.
Hence,
v =
If we denote the angle HDB by 6, we have DK ' = 7cos0;
we also have DE=lcosa and since y is equal to DKDE,
we have,
y = J(cos0 - COSa),
194 DIFFERENTIAL AND INTEGRAL CALCULUS.
which, being substituted in the preceding formula,
gives,
v = A2#/(cos4 cosa).
Again, denoting the angular velocity at the time i by <p,
id rememberi
the pendulum,
and remembering that 9 = , we have, for the velocity of
Cit
Equating the two values of v, and reducing, we. have,
cosa
Developing cos$ and cosa by McLaurin's Formula, we have,
^ -?+'-**
a 3 *
~ 2~ f 24 ~
If we suppose the amplitude to be small, we may neglect
all the terms after the second ; doing so, and substituting
in (31), we have,
Integrating between the limits & == a, and A = + a } we
find,
t =
which is the formula for the time of vibration of a
pendulum.
APPLICATIONS TO MECHANICS AND ASTRONOMY. 195
Attraction of Homogeneous Spheres.
101. The Newtonian Law of universal gravitation may
be expressed as follows, viz. : every particle of matter
attracts every other particle, with a force that varies
directly as the mass of the attracting particle, and inversely
as the square of the distance between the particles. To
apply this law to the case of homogeneous spheres, let us
first consider the action of a spherical shell, of infinitesimal
thickness, on a material point within it.
Let D be the material point, APB a great circle through
it, and let the diameter ADB be taken as the axis of x.
If the circle be revolved
about AB, its circum-
ference will generate a
spherical shell, and any
element of the circnmfer-
ence, as P, will generate
an elementary zone whose
altitude is dx. For any
point of this zone, as P,
there, is another point, P',
symmetrical with it, and the resultant action of these points
on D is directed along AB, and is equal to the sum of the
forces into the cosine of the angle BDP. Hence, the re-
sultant attraction of the entire zone is directed along AB,
and is equal to the sum of the attractions of all its particles
into the cosine of the angle BDP. To find an expression
for this resultant, denote CP by r, CD by a, DP by z,
and C7? by x. Because the shell is infinitesimal in thick-
ness and homogeneous, the mass of the zone, is to the mass
of the shell, as dx, the altitude of the zone, is to 2r, the
altitude of the shell. Hence, if the mass of the shcil be
*9G DIFFERENTIAL AXD INTEGRAL CALCULUS.
denoted by m, the mass of the zone will be denoced by
- --. If the force exerted by the unit of muss at the unit
of distance be taken as the unit of force, the attraction
exerted by all the particles of the zone on the point D will
be equal to ~~ . , and the resultant action on D, in the
direction of AB, denoted by df, will be given by the equa-
tion,
From the triangle, PCD, we have,
DP* = z* = r 2 + 2 - 2ax = r 2 - a z - 2a(x - ),
and from the triangle, EDP, we have,
cos ED P = ^-^..
z
Substituting in (1), remembering that dx equals d(x rt),
we have,
m . (x - a] d(x - a) .
f= Wr' --- -- 5 ..... ( '
[ r _ a * - 2a(x - )]
Regarding (x a) as a single variable, reducing by For-
mula A, and integrating by Formula (29), we find,
_ a z _ 2a(x - a]
+
Taking the integral from x r to x .-= + r, we have,
APPLICATIONS TO MECHANICS AND ASfKOXOMY. 197
Hence, the effect of the attraction of the shell on any
point within it is null. If a sphere be described about
as a centre with a radius equal to #, we may call that paifc
which lies between it and the surface of the given sphere
the exterior sJicU, and the sphere itself may be called the
nucleus. From what precedes, we infer that any point
within a homogeneous sphere is acted on by the sphere
precisely as though the exterior shell did not exist. Hence,
a point at the centre of a sphere is not affected by the
attraction of the sphere.
If the point D' be taken without the shell, we have,
jyp 2 = z* = r* + fl 2 + 2ax = r 2 - a n + 2a(x + a),
and from the triangle, ED'P, we have,
cosED'P = ^f .
z
Substituting in (1), we have,
1 [r 2 - cr + 2a(x + a)]" 2 '
Reducing and integrating as before, we find,
- _ m ( . x -f a
r ( a\r z a z + %a(x + a^*
)T
,2 ..
+ C.
-a 8 -f 2a
Taking the integral between the -limits x = r and
= + ?, we have, when a > r
198 DIFFERENTIAL AND INTEGRAL CALCULUS.
But m is the mass of the shell, and a is the distance of D
from the centre; hence, the shell attracts a particle with-
out it as though its entire mass were concentrated at its
centre.
A homogeneous sphere may be regarded as made np of
spherical shells; hence, a sphere attracts any point without
it, as though the mass of the sphere were concentrated at its
centre. The same is true of a sphere made up of homoge-
neous strata, which vary in density in passing from the
surface to the centre. It is to be inferred that two homo-
geneous spheres, or two spheres made up of homogeneous
strata, attract each other as though both were concentrated
at their centres.
If an opening were made from the surface to the centre
of the earth, supposed homogeneous, and a body were to
move along it under the earth's attraction, it would every-
where be urged on by a force varying diretily as the dis-
tance of the body from the centre. For, denote the dis-
tance of the body from the centre at any instant by x.
The body will only be acted on by the nucleus whose radius
is x. If we take r to represent the radius of the earth,
and remember that the masses are proportional to these
volumes, we have,
4 4
M : m : : ^vr 3 : ^*x 3 : : r 3 : x 3 .
o o
Denoting the force of attraction at the surface by g, and
the force of attraction at the point whose distance from
the centre is x by/, we have, from the Newtonian law,
Hence, the proposition is proved. (See Art. 96.)
APPLICATIONS TO MECHANICS AND ASTRO N'O.MV. 199
Orbital Motion.
102. If a moving point, P 9 be continually acted on by a
deflecting force directed toward a fixed centre, it will de-
scribe a line or path, called an orbit. If the moving point
be undisturbed by the action of any other force, the orbit
will lie in a plane passing through the fixed centre and an
element of the curve. Let
this plane be taken as the
co-ordinate plane, let the
fixed point be the origin,
and let the orbit be repre-
sented by APB.
Denote the acceleration
due to the deflecting force
at any time t by/, its in-
Fig. 29.
clination to the axis by 9, and its components in the direc-
tions of the co-ordinate axes by A" and Y.
From the figure, we have,
X ~/cos?, and Y= /ship
If we regard / as the independent variable, dx and dy will
both be variable, and from Equation (3), Art 93, we have,
If we denote the co-ordinates of P by x and y, and ita
radius vector FP by r, we have, from the figure,
x . ii
cosp = -, sm<p = -;
T T
and
+ y* r 2 ; .-. xdx + ydy = rdr.
200 DIFFERENTIAL AND INTEGRAL CALCULUS*
Substituting in (1), we have,
Multiplying the first of Equations (2) by y, the second by
x } and subtracting the former from the latter, we have,
xd*y ycPx d (xdy ydx) A
w-V=' or M J =o-----(3)
Multiplying the first by dx, the second by dy, adding the
resulting equations and reducing, we have,
dxd*x dyd z y _ 4 .xdx + ydy _ fl
~W ~^dtT ~J F" ~ J
Integrating (3) and (4), Art. 77, we have,
and
1 /7r 2 4- (In* {*
I^TP* +:*--/>*'
or,
^ + 2 ffdr = C" (6)
Equations (5) and (G) make known the circumstances
of motion when the value of/ is given. It is found con-
venient to transform them to a system of polar co-o Mi nates,
whose pole is F, and whose initial line is FX. The for*
mulas for transformation are
x rcoep, and y = rsin<p (7)
Hence, by differentiating, we have,
du dr . dq>
and
dx dr . dq> , n *
= cosf-win^ (9)
APPLICATIONS TO MECHANICS AND ASTRONOMY. 201
\V*e also, have, Art. (53),
<&* = r*dq>* + dr* ...... (10)
Substituting in (5), and reducing by the relations,
%2 _j_ 2/2
aasiiKp 3, cos? = 0, and xcos? + f/sincp = --- f,
we have,
From equation (6), we have,
But, from (11), we have,
,, r z dq>
dt = ~C^
and this, in (12), gives,
Equations (11) and (13) are the equations required.
Multiplying (11) by dt, and integrating, we have,
= Ct + C'" (14)
But, by Art. 53, r z dy is twice the elementary area swept
over by the radius vector; hence, the first member of (14)
is an expression for twice the area swept over by the radiua
VHctor up to the time t. If we suppose the area to bo
reckoned from the initial line FX, and at the same time
9*
202 DIFFERENTIAL AND INTEGRAL CALCULUS.
suppose t = 0, we have C'" = 0. Substituting this in (14),
and in the resulting equation making t = I, we find
C equal to twice the area swept over in the first unit of
time ; denoting the area described in the unit of time bv
A, we have, C 2 A, and consequently,
= 2At (15)
From Equation (15) we infer that the areas described by
elae radius vector are proportional to the times of descrip-
tion, and this without reference to the nature of the de-
flecting force.
To find the equation of the orbit, let us assume as M
particular case, the Newtonian law of universal gravita-
tion. .Denoting the force exerted by the central body at a
unit's distance by k, we have, for the attraction at the
distance r,
Making r = -, whence dr = -- -, and substituting for (j
*nd/ their values in (13), we have,
or,
du 2 2k
_ +w2= __
(17)
V
To find the value of <7 V , let ~ = 0, when r = r', or
op
u = u r . But when - = 0, we also have - T = 0, that is.
APPLICATIONS TO MECHANICS AND ASTRONOMY. 203
the radius vector is perpendicular to the arc. If we denote
the corresponding velocity by v', we have,
and, consequently,
2k 2k
A* U ~ v'*r' 3 '
Making these substitutions in (18), we find,
fiV _ %k
775 ~~ _./-/
Denoting this value of O y by S, and solving Equation (18),
we have,
du* 2k \ k*
du* ( 2k \
W = 8 ' -QI*) =
Solving with reference to <p, and taking the negative sign
of the radical, we have,
VP 2 (u q) z vp z (u q) 2
Integrating, we have,
9 - a = cos- 1 -^^ ..... (20)
In which a is an arbitrary constant. If we suppose
9 0, when v = v\ r = r r , and u = u', the corresponding
value of a is the angular distance from the initial line to
the radius vector whose direction is perpendicular to the
element of the curve.
204: DIFFERENTIAL AND INTEGRAL CALCULUS.
Taking the cosines of both members of (20), we have,
i = cos(<p a), or, u = q + jocos(<p a)
and finally replacing u by its value, and solving, we have,
r =
q + jt?cos(<p a) k + l6SA* + k 2 X cos(9 a)
Equation (21) is the polar equation of a conic section.
Hence, the orbit of a particle about a central attracting
body is one of the conic sections. If the orbit is a closed
curve, it must be an ellipse. This corresponds to the case
of a planet revolving about the sun, or to that of a satellite
revolving about its primary.
Law of Force.
103. If the orbit of a revolving particle be an ellipse, it
must be attracted to the focal point by a force that
varies inversely as the square of the distance.
The polar equation of an ellipse is
.
1 4- ecos(<p a)
in which a is the semi- transverse axis, e is the eccentricity,
a is the angular distance from the fixed line to the radius
vector drawn to the nearest vertex, or the longitude of the
perihelion in the case of a planetary orbit. The angle,
p a, is the angle that is called in astronomy the true
anomaly.
Putting r = - , in (1), we get,
APPLICATIONS TO MECHANICS AND ASTRONOMY. 205
Differentiating twice, with respect to <p, we find,
du _ esin(cp a) .
cfy ~ " ~a(l - e z )
and,
d z u __ e cos (9 a) .
~ 2
Adding (2) and (4), we have,
d * u x
/"\
Resuming equation (13) of the last article, replacing r by
its value, -, and dr
'1C
reducing, we have,
its value, -, and dr by its value, -- r , differentiating and
'1C U
Combining (5) and (G), and solving with respect to /, we
Iiave,
C*u* C* 1_
/ - a (l - e*)' r/ ~ a(l - e*) ' r* '
Hence, f varies inversely as the square of r, which was to
be shown.
Note on the Methods of the Calculus.
104. All the rules and principles of the Calculus, in the.
present treatise, have been deduced in accordance with the
method of infinitesimals, as explained in Articles 6 and 7.
They might also have been deduced by the method of limits.
It remains to be shown that the results obtained by these
two methods are always identically the same.
206 DIFFERENTIAL AND INTEGRAL CALCULUS.
To explain the method of limits, let us denote any func-
tion of x by y ; that is, let us assume the equation,
y =/(*) (i)
If we increase x by a variable increment h, and denote the
corresponding value of?/ by y', we have,
y'=f(x + h) (2)
It is shown in Courteuay's Calculus, Article 4, that so long
as x retains its general value, the new state of the function
can be expressed by the formula,
y' =f(x + U) =f(x) + Ah + Bh* + Ch* + etc. . . (3)
in which A, B, C, etc., depend on x, but are independent
of h. If we subtract (1) from (3), member from member,
we have,
y' -y = Ah + 3h* + (etc.) h 3 (4)
Dividing both members of (4) by h, we have,
^-^ = A + Bh + (etc.) h* (5)
The first member of (5) is a symbol to express the ratio
of the increment of the variable) to the corresponding incre-
ment of the function } and the second member is the value
of that ratio.
It is shown in Algebra, that in an expression like the
second member of equation (5), it is always possible to
give to h a value small enough to make the first term
numerically greater than the algebraic sum of all the
others. If we assign such a value to h) and then suppose
h to go on diminishing, the second member will contin-
ually approach A, and when h becomes 0, the second mem-
APPLICATIONS TO MECHANICS AND ASTRONOMY. 20?
ber will reduce to A. Hence, A is a quantity towan 1 .
i/' ii
which the ratio, j , approaches as li is diminished, but
beyond which it cannot pass; it is therefore the limit of
that ratio. This limit is the differential coefficient of y,
and, as may easily be seen, it is entirely independent of dx.
The product of this by the differential of the variable is
Hie differential of y.
We may, therefore, enunciate the method of limits as
follows: viz., Give to the independent variable a variable
increment, and find the corresponding state of the function ;
from this subtract the primitive state, and divide the dif-
ference by the variable increment; then pass to the limit
of the quotient by making the increment of the variable
equal to 0; the result is the differential coefficient of the
function; if this be multiplied by the differential of the
variable, the product is the differential of the function.
In the case assumed, we have, by this method,
J = A; .:dy=Adx (6)
If we make h dx, in Equation (4), dx being infinitely
small, the first member will be the. differential of y, and
all the terms of the second member after the first may be
neglected. Hence, by the method of infinitesimals, we
have,
dy = Adx; .: -^ = A.
dx
This result is the same as that obtained by the method of
limits. But, by hypothesis, y represents any function of
x; hence, in all cases, the differential coefficient is iden-
tically the same whether found by the method of limits or
by the method of infinitesimals.
208 DIFFERENTIAL AND INTEGRAL CALCULUS.
Erery function, regarded as a primitive, is connected
with some other function, regarded as a derivative, by the
law of differentiation. This derivative is the differential
coefficient of the primitive. The object of the differential
calculus is to find the derivative from its primitive; the
object of the integral calculus is to find the primitive from
its derivative; every application of the calculus depends
on one of these processes, or on some discussion growing
out of one, or the other. Now, because the primitive and
the derivative are independent of the differentials of both
function and variable, 4 the relation between them is, of
necessity, independent of the methods employed in estab-
lishing that relation. But it has been shown that the
same relation is found between these functions, whether
we employ the method of infinitesimals or the method of
limits. Hence, these methods, and the results obtained bj
them, are in all cases logically identical.
BARNES'S POPULAR HISTORY
OF THE UNITED STATES.
By the author of Barnes's "Brief Histories for Schools." Complete in one superb
royal octavo volume of 800 pages. Illustrated with 320 wood engravings and 14 steel
plates, covering the period from the Discovery of America to the Accession of President
Arthur.
Part I. Colonial Settlement ; Exploration ; Conflict ; Manners ; Customs ; Educa-
tion ; Religion, &c., &c., until political differences with Great Britain threatened open
rupture.
Part II. Resistance to the Acts of Parliament ; Resentment of British Policy, and
the Succeeding War for American Independence.
Part III. From the Election of President Washington to that of President Lincoln,
with the expansion and growth of the Republic ; its Domestic Issues and its Foreign
Policy.
Part IV. The Civil War and the End of Slavery.
Part V. The New Era of the Restored Union ; with Measures of Reconstruction ;
the Decade of Centennial Jubilation, and the Accession of President Arthur to Office.
Appendix. Declaration of Independence ; The Constitution of the United States
and its Amendments ; Chronological Table and Index ; Illustrated History of the
Centennial Exhibition at Philadelphia.
The wood and steel engravings have been expressly chosen to illustrate the customs of
the periods reviewed in the text. Ancient houses of historic note, and many portraits of
early colonists, are thus preserved, while the elaborate plans of the Exposition of 1876
are fully given. The political characteristics of great leaders and great parties, which
had been shaped very largely by the issues which belonged to slavery and slave labor,
have been dealt with in so candid and impartial a manner as to meet the approval of
all sections of the American people. The progress of science, invention, literature, and
art is carefully noted, as well as that of the national physical growth, thus condensing
into one volume material which is distributed through several volumes in larger works.
Outline maps give the successive stages of national expansion , and special attention
has been given to those battles, by land and sea, which have marked the military growth
of the republic. JE1P" Specially valuable for reference in schools and households.
From Prof. F. F. BARROWS, Brown School,
Hartford, Conn.
" Barnes's Popular History has been in
onr reference library for two years. Its
concise and interesting presentation of
historical facts causes it to be so eagerly
read by our pupils, that we are obliged to
duplicate it to supply the demand for its
use."
From Hon. JOHN R. BUCK.
41 1 concur in the above."
From Hon. J. C. STOCK WELL.
"I heartily concur with Mr. Barrows in
the within commendation of ' Barnes's
Popular History,' as a very interesting and
instructiA'e book of reference."
From A. MORSE, Esq.
" I cordially concur in the above."
From Rev. WM. T. GAGE.
"I heartily agree with the opinions
above expressed."
From DAVID CRARY, Jr.
"The best work for the purpose pub-
lished."
From Prof. S. T. DUTTON, Superintendent
of Schools, New Haven, Conn.
" It seems to me to be one of the best
and most attractive works of the kind I
have ever seen, and it will be a decided
addition to the little libraries which we
have already started in our larger
schools."
From Prof. WM. MARTIN, of Beattystown.
N.J.
"This volume is well adapted to the
wants of the teacher. A concise, well-
arranged summary of events, and just the
supplement needed by every educator who
teaches American history."
From Prof. C. T. R. SMITH, Principal of
the Lansingburgh, N. F., Academy.
" In the spring I procured a copy of
' Barnes's Popular History of the United
States,' and have used it daily since, in
preparing my work with my class in Ameri-
can history, with constantly increasing
admiration at the clearness, fairness, and
vividness of its style and judicious selec-
tion of matter."
Prices. Cloth, plain edge, $5.00; cloth, richly embossed, gilt, edge, 86.00; sheep,
marble edge, $7.00 ; half calf, $8.00 ; half morocco, $8.00 ; full inorocto, gilt, $10.00.
SCHOOL AND COLLEGE TEXT-BOOKS.
The National Series Readers and Spellers,
THE NATIONAL READERS.
By PARKER and "WATSON.
No. I. National Primer 64 pp. 16
No. 2. National First Reader .... 128 " 16
No. 3. National Second Reader ... " 16
No. 4. National Third Reader . . . 288 " 12
No. 5. National Fourth Reader . . . 432 " 12
No. 6. National Fifth Reader. 600 " 12
National Elementary Speller .... 160 pp. 16
National Pronouncing Speller . . . . 188 " 12
THE INDEPENDENT READERS.
By J. MADISON WATSON.
The Independent First ( Pril n r ar y ) Reader . 80 pp. 16
The Independent Seoond Reader . . . 160 " 16
The Independent Third Reader . . . 240 " 16
The Independent Fourth Reader . . . 264 " 12
The Independent Fifth Reader . . . . 886 " 12
The Independent Sixth Reader . . . . 474 " '12
The Independent Complete Speller . . ' 162 " 16
The Independent Child's Speller (Script) 80 pp. 16
The Independent Youth's Speller (Script) 168 " 12
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
WATSON'S INDEPENDENT
READERS.
This Series is designed to meet a general demand for smaller and cheaper
books than the National Series proper, and to serve as well for intermediate
volumes of the National Readers in large graded schools requiring more books
than one ordinary series will supply.
Beauty. The most casual observer is at once impressed with the unpar-
alleled mechanical beauty of the Independent Readers. The Publishers be-
lieve that the aesthetic tastes of children may receive no small degree of
cultivation from their very earliest school-books, to say nothing of the impor-
tance of making study attractive by all such artificial aids that are legitimate.
In accordance with this view, not less than $25,000 was expended in their
preparation before publishing, with a result which entitles them to be con-
sidered "the perfection of common-school books."
Selections. They contain, of course, none but entirely new selections.
These are arranged according to a strictly progressive and novel method of
developing the elementary sounds in order in the lower numbers, and in all,
with a view to topics and general literary style. The mind is thus ledm fixed
channels to proficiency in every branch of good reading, and the evil results of
"scattering," as practised by most school-book authors, avoided.
The Illustrations, as may be inferred from what has been said, are ele-
gant beyond comparison. They are profuse in every number of the series, from
the lowest to the highest. This is the only series published of which this
is true.
The Type is semi-phonetic, the invention of Professor Watson. By it every
letter having more than one sound is clearly distinguished in all its variations
without in any way mutilating or disguising the normal form of the letter.
Elocution is taught by prefatory treatises of constantly advancing grade
and completeness in each volume, which are illustrated by woodcuts in the
lower books, and by blackboard diagrams in the higher. Professor Watson
is the first to introduce practical illustrations and blackboard diagrams for
teaching this branch.
Foot-Notes on every page afford all the incidental instruction which the
teacher is usually required to impart. Indices of words refer the pupil to the
place of their first use and definition. The biographies of authors and others
are in every sense excellent.
Economy. Although the number of pages in each volume is fixed at the
minimum, for the purpose recited above, the utmost amount of matter avail-
able without overcrowding is obtained -in the space. The pages %re much
wider and larger than those of any competitor and contain twenty per cent
more matter than any other series of the same type and number of pages.
All the Great Features. Besides the above all the popular features of
the National Readers are retained except the word-building system. The
latter gives place to an entirely new method of progressive development, based
upon some of the best features of the word system, phonetics, and object
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS,
PARKER & WATSON'S NATIONAL
READERS.
The salient features of these works which have combined to render them so popular
may be briefly recapitulated, as follows :
x. THE WORD-BUILDING SYSTEM. This famous progressive method
for young children originated and was copyrighted with these books. It constitutes a
process with which the beginner with words of one letter is gradually introduced to
additional lists formed by prefixing or affixing single letters, and is thus led almost
insensibly to the mastery of the more difficult constructions. This is one of the most
striking modern improvements in methods of teaching.
2. TREATMENT OF PRONUNCIATION. The wants of the youngest
scholars in this department are not overlooked. It may be said that from the first
lesson the student by this method need never be at a loss for a prompt and accurate
rendering of every word encountered.
3. ARTICULATION AND ORTHOEPY are considered of primary importance.
4. PUNCTUATION is inculcated by a series of interesting reading lessons, the
simple perusal of which suffices to fix its principles indelibly upon the mind.
5. ELOCUTION. Each of the higher Readers (3d, 4th, and 5th) contains elabo-
rate, scholarly, and thoroughly practical treatises on elocution. This feature alone lias
secured for the series many of its warmest friends.
6. THE SELECTIONS are the crowning glory of the series. Without excep-
tion it may be said that no volumes of the same size and character contain a collection
so diversified, judicious, and artistic as this. It embraces the choicest gems of Eng-
lish literature, so arranged as to afford the reader ample exercise in every department
of style. So acceptable has the taste of the authors in this department proved, not
only to the educational public but to the reading community at large, that thousands
of copies of the Fourth and Fifth Readers have found their way into public and private
libraries throughout the country, where they are in constant use as manuals of litera-
ture, for reference as well as perusal.
7. ARRANGEMENT. The exercises are so arranged as to present constantly
alternating practice in the different styles of composition, while observing a definite
plan of progression or gradation throughout the whole. In the higher books the
articles are placed in formal sections and classified topically, thus concentrating the
interest and inculcating a principle of association likely to prove valuable in subse-
quent general reading.
8. NOTES AND BIOGRAPHICAL SKETCHES. - These are full and ade-
quate to every want. The biographical sketches present in pleasing style the history of
every author laid under contribution.
9. ILLUSTRATIONS. These are plentiful, almost profuse, and of the highest
character or' art. They are found in every volume of the series as far as and including
the Third Reader.
ip. THE GRADATION is perfect. Each volume overlaps its companion pre-
ceding or Mlowing in the series, so that the scholar, in passing from one to another, is
only conscious, by the presence oi' the new book, of the transition.
11. THE PRICE is reasonable. The National Readers contain more matter than
any other series in the same number of volumes published. Considering their com-
pleteness and thoroughness, they are much the cheapest in the market.
12. BINDING. By the use of a material and process known only to themselves,
in common with all the publications of this house, the National Readers are warranted
to outlast any with which they may be compared, the ratio of relative durability
being in their favor as two to one.
NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
SUPPLEMENTARY READING.
Monteith's Popular Science Reader.
James Monteith, author of Monteith's Geographies, has here presented a Supple-
mentary Reading Book expressly for the work of instruction in leading and science at
on and the same time. It presents a number of easy and interesting lessons on Natural
Science and Natural History, interspersed with appropriate selections in prose and
poetry from standard authors, with blackboard drawing and written exercises. It
serves to instil the noblest qualities of soul and mind, without rehearsing stories of
moral and mental depravity, as is too often done in juvenile books. The book is elabo-
rately illustrated with fine engravings, and brief notes at the foot of each page add to
the value and teachableness of the volume. 12mo, half bound, 360 pages.
The Standard Supplementary Readers.
The Standard Supplementary Readers (formerly Swinton's Supplementary Readtrs),
edited by William Swinton and George R. Cathcart, have been received with marked
favor in representative quarters from Maine to California. They comprise a series of
carefully graduated reading books, designed to connect with any series of school Readers.
They are attractive in appearance, are bound in cloth, and the first four books are
profusely illustrated by Fredericks, White, Dielman, Church, and others. The six books,
which are closely co-ordinated with the several Readers of any regular eeries, are
1. Easy Steps for Little Feet. Supplementary to First Reader.
In this book the attractive is the chief aim, and the pieces have been written and
chosen with special reference to the feelings and fancies of early childhood. 128 pages,
bound in cloth and profusely illustrated.
2. Golden Book of Choice Reading. Supplementary to Second
Reader.
This book represents a ;reat variety of pleasing and instructive reading, consisting of
child-lore and poetry, nobU* examples and attractive object-reading, written specially for it.
1U2 pages, cloth, with numerous illustrations
3 Book of Tales. Being School Readings Imaginative and Emotional.
Supplementary to Third Reader.
In this book the youthful taste for imaginative and emotional is fed with pure and noble
creations drawn from the literature of all nations. 272 pages, cloth. Fully illustrated.
4. Readings in Nature's Book. Supplementary to Fourth Reader.
This book contains a varied collection of charming readings in natural history and
botany, drawn from the works of the great modern naturalists and travellers. 352 pages,,
tloth. Fully illustrated.
5. Seven American Classics.
6. Seven British Classics.
The " Classics " are suitable for reading in advanced grades, and aim to instil a
taste for the higher literature, by the presentation of gems of British and American
authorship. 220 pages each, cloth.
8
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
ORTHOGRAPHY.
Smith's Series.
Smith's Series supplies a Speller for every class in graded schools, and comprises
the most complete and excellent treatise on English Orthography and its companion
branches extant.
1. Smith's Little Speller.
First round in the ladder of learning.
2. Smith's Juvenile Definer.
Lessons composed of familiar words grouped with reference to similar significa-
tion or use, and correctly spelled, accented, and defined.
3. Smith's Grammar-School Speller.
Familiar words, grouped with reference to the sameness of sound of syllables dif-
ferently spelled. Also definitions, complete rules for spelling and formation of deriva-
tives, and exercises in false orthography.
4. Smith's Speller and Definer's Manual.
A complete School Dictionary, containing 14,000 wor**-, with various othei' useful
matter in the way of rules and exercises.
5. Smith's Etymology Small and Complete Editions.
The first and only Etymology to recognize the Anglo-Saxon our mother tongue;
containing also full lists of derivatives from the Latin, Greek, Gaelic, Swedish, Norman,
&c. , &c. ; being, in fact, a complete etymology of the language for schools.
Northend's Dictation Exercises.
Embracing valuable information on a- thousand topics, communicated in snch a
manner as at once to relieve the exercise of spelling of its usual tedium, and combine
it with instruction of a general character calculated to profit and amuse.
Phillip's Independent Writing Spellers
1. Primary. 2. Intermediate. 3. Advanced.
Unquestionably the best results can be attained in writing spelling exercises. This
series combines with written exercise a thorough and practical instruction in penman-
ship. Copies in capitals and small letters are set on every page. Spaces for twenty
words and definitions and errors are given in each lesson. In the advanced book there
is additional space for sentences. In practical life we spell only when we write.
Brown's Pencil Tablet for Written Spelling.
The cheapest i
64 lessons of 25 w
Pooler's Test Speller.
The best collection of " hard words " yet made. The more uncommon ones are fully
defined, and the whole are arrant/ed alphabetically for convenient reference. The book
is designed for Teachers' Institutes and " Spelling Schools," and is prepared by an
experienced and well-known conductor of Institutes.
Wright's Analytical Orthography.
This standard work is popular, because it teaches the elementary sounds ir a
plain and philosophical manner, and presents orthography and orthoepy in an easy.
uniform system of analysis or parsing.
The cheapest prepared pad of ruled blanks, with stiff board back, sufficient for
ords.
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
ORTHOGRAPHY Continued.
Barber's Complete Writing Speller.
"The Student's Own Hand-Book of Orthography, Definitions, ana Sentences, con-
sisting of \\ritten Exercises in the Proper Spelling, Meaning, and Use of Words."
(Published 1873.) This differs from Sherwood's and other writing spellers in its more
comprehensive character. Its blanks arc. adapted to writing whole sentences instead
of detached words, with the proper divisions for numbering corrections, &c. Such
aids as this, like Watson's Child's Speller and Phillip's Writing Speller, find their
raison d'etre in the postulate that the art of correct spelling is dependent upon written,
and not upon spoken language, for its utility, if not for its very existence. Hence
the indirectness of purely oral instruction.
ETYMOLOGY.
Smith's Complete Etymology.
Smith's Condensed Etymology.
Containing the Anglo-Saxon, French, Dutch, German, Welsh, Danish, Gothic, Swedish,
Gaelic, Italian, Latin, and Greek roots, and the English words derived therefrom
accurately spelled, accented, and defined.
From HON. JNO. G. McMvNN, late State
Superintendent of Wisconsin.
" I wish every teacher in the country
had a copy of this work."
From PROF. C. H. VERRILL, Pa. State
Normal School.
u The Etymology (Smith's) which we
procured of you we like much. It is the
best work for the class-rooin we have
seen."
From PRIN. WM. F. F HELPS, Minn. Slate
Normal.
"The book is superb just what is
needed in the department of etymology
and spelling."
From HON. EDWARD BALLARD, Supt. oj
Common Schools, State of Maine.
'' The author has furnished a manual of
singular utility for its purpose."
DICTIONARY.
Williams's Dictionary of Synonymes ;
Or, TOPICAL LEXICON. This work is a School Dictionary, an Etymology, a compilation
of Synonymes, and a manual of General Information. It differs from the ordinary lexicon
in being arranged by topics, instead of the letters of the alphabet, thus realizing the
apparent paradox of a " Readable Dictionary." An unusuaJ'y valuable school-book.
Kwong's Dictionary of English Phrases.
With Illustrative Sentences, collections of English and Chinese Proverbs, transla-
tions of Latin and French Phrases, historical sketch of the Chinese Empire, a chrono-
logical list of the Chinese Dynasties, brief biographical sketches of Confucius and
of Jesus, and complete index. By Kwong Ki Chin, late Member of the Chinese Edu-
cational Mission \n the United States, and formerly principal teacher of English in the
Government School at Shanghai, China. 9(0piges. Svo. Cloth.
From the Hartford Courant : " The volume is one of the most curious and interest-
ing of linguistic works."
From the New York Nation : " It will amaze the sand-lot gentry to bs informed that
this remarkable work will supplement our English dictionaries even for native Americans."
10
THE NATIONAL SERIES Or STANDARD SCHOOL-BOOKS.
ENGLISH GRAMMAR.
SILL'S SYSTEM.
Practical Lessons in English.
A brief course in Grammar and Composition. By J. M. B. SILL. This beautiful
hook, by a distinguished and experienced teacher, at once adopted for exclusive use
in the State of Oregon and the city of Detroit, simply releases English Grammar
from bondage to Latin and Greek formulas. Our language is worthy of being tanglit
as a distinct and indepei.dent science. It is almost destitute of inflections and yet
capable of being systematized, and its study may certainly be simplified if treated by
itself and for itsslf alone. Superintendent SILL has cut the Gordian knot and leads
the van of a new school of grammarians.
CLARK'S SYSTEM.
Clark's Easy Lessons in Language
Contains illustrated object-lessons of the most attractive character, and is couched
in language freed as much as possible from the dry technicalities of the science.
Clark's Brief English Grammar.
Part I. is adapted to youngest learners, and the whole forms a complete " brief
course " in one volume, adequate to the wants of the common school. There is no-
where published a superior text-book for learning thj English tongue than this.
Clark's Normal Grammar.
Designed to occupy the same grade as the author's veteran "Practical" Grammar,
though the latter is still furnished upon order. The Normal is an entirely new treatise.
It is a full exposition of the system as described below, with all the most recent im-
provements. Some of its peculiarities are, a happy blending of SYNTTHESES with
ANALYSES ; thorough criticisms of common errors in the use of our language ; and
important improvements in the syntax of sentences and of phrases.
Clark's Key to the Diagrams.
Clark's Analysis of the English Language.
Clark's Grammatical Chart.
The theory and practice of teaching grammar in American schools is meeting with a
thorough revolution from the use of this system. While the old methods offer profi-
ciency to the pupil only after much weary plodding and dull memorizing, this affords
from the inception the advantage of practical Object Teaching, addressing the eye by
means of illustrative figures ; furnishes association to the memory, its most powerful
aid, and diverts the pupil by taxjng his ingenuity. Teachers who are using Clark's
Grammar uniformly testify that they and their pupils find it the most interesting study
of the school course.
Like all great and radical improvements, the system naturally met at first with much
unreasonable opposition. It has not only outlived the greater part of this opposition,
bat finds many of its warmest admirers among those who could not at first tolerate so
radical an innovation. All it wants is an impartial trial to convince the most scep-
tical of its merit. No one who has fairly and intelligently tested it in the school-room
has ever been known to go back to the old method. A great success is nlre-idy
established, and it is easy to prophesy that the day is not far distant when it will be
the only system of teaching English Grammar. As the SYSTEM is copyrighted, no other
text-books can appropriate this obvious and great improvement.
Welch's Analysis of the English Sentence.
Remarkable for its new and simple classification, its method of treating connectives,
its explanations of the idioms and constructive laws of the language, &c.
11
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
GEOGRAPHY.
MONTEITH'S SYSTEM.
TWO-BOOK SERIES. INDEPENDENT COURSE.
Elementary Geography.
Comprehensive Geography (with 103 maps).
ESP" These volumes are not revisions of old works, not an addition to any series,
but are entirely new productions, each by itself complete, independent, compieheii-
sive, yet simple, brief, cheap, and popular ; or, taken together, the most admirable
" series " ever ottered for a common-school course. They present the following features,
skilfully interwoven, the student learning all about one country at a time. Always
revised to date of printing.
LOCAL GEOGRAPHY. Or, the Use of Maps. Important features of the maps
are the coloring of States as objects, and the ingenious system for laying down a much
larger number of names for reference than are lound on any other maps of same size,
and without crowdincr.
PHYSICAL GEOGRAPHY. Or, the Natural Features of the Earth; illus-
trated by the original and striking RELIEF MAPS, being bird's-eye views or photographic
pictures of the earth's surface.
DESCRIPTIVE GEOGRAPHY. Including the Physical; with some account
of Governments and Races, Animals, &e.
HISTORICAL GEOGRAPHY. Or, a brief summary of the salient points of
history, explaining the present distribution of nations, origin of geographical
names, &c.
MATHEMATICAL GEOGRAPHY. Including Astronomical, which describes
the Earth's position and character among planets ; also the Zones, Parallels, &c.
COMPARATIVE GEOGRAPHY. Or, a system of analogy, connecting new
lessons with the previous ones. Comparative sizes and latitudes are shown on the
margin of each map, and all countries are measured in the " frame of Kansas."
TOPICAL GEOGRAPHY. Consisting of questions for review, and testing
the student's general and specific knowledge of the subject, with suggestions for
geographical compositions.
ANCIENT GEOGRAPHY. A section devoted to this subject, with maps, will
be appreciated by teachers. It is seldom taught in our common schools, because it
has heretofore required the purchase of a separate book.
GRAPHIC GEOGRAPHY, or Map-Drawiug by Allen's "Unit of Measure-
ment" system (now almost universally recognized as without a rival), is introduced
throughout the lessons, and not as an appendix.
CONSTRUCTIVE GEOGRAPHY. Or, Globe-Making. With each book a set
of map segments is furnished, with which each student may make his own globe by
ibllowing the directions given.
RAILROAD GEOGRAPHY. With a grand commercial map of the United
States, illustrating steamer and railroad routes of travel in the United States, submarine
telegraph lines, &c. Also a " Practical Tour in Europe."
MONTEITH AND McNALLY'S SYSTEM.
THREE AND FIVE BOOKS. NATIONAL COURSE.
Monteith's First Lessons in Geography.
Monteith's New Manual of Geography.
McNally's System of Geography.
The new edition of McNally's Geography is now ready, rewritten throughout by
James Mouteitli and S. C. Frost. In its new dress, printed from new type, and illus-
trated with 100 new engravings, it is the latest, most attractive, as well as the most
thoroughly practical book on geography extant.
13
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
G EOG RAPH Y Continued.
INTERMEDIATE OR ALTERNATE VOLUMES IN THE FIVE BOOK SERIES.
Monteith's Introduction to Geography.
Monteith's Physical and Political Geography.
x. PRACTICAL OBJECT-TEACHING. The infant scholar is first introduced
to a picture whence he may derive notions of the shape of the earth, the phenomena of
day and night, the distribution of land and water, and the great natural divisions,
which mere words would fail entirely to convey to the untutored mind. Other pictures
follow on the same plan, and the child's mind is called upon to grasp no idea without
the aid of a pictorial illustration. Carried on to the higher books, this system culmi-
nates in Physical Geography, where such matters as climates, ocean currents, the
winds, peculiarities of the earth's crust, clouds and rain, are pictorially explained and
rendered apparent to the most obtuse. The illustrations used for this purpose belong
to the highest grade of art.
2. CLEAR, BEAUTIFUL, AND CORRECT MAPS. In the lower num-
bers the maps avoid unnecessary detail, while respectively progressive and affording
the pupil new matter for acquisition each time he approaches in the constantly en-
larging circle the point of coincidence with previous lessons in the more elementary
books. In the Physical and Political Geography the maps embrace many new and
striking features. One of the most effective of these is the new plan for displaying on
each map the relative sizes of countries not represented, thus obviating much confu-
sion which has arisen from the necessity of presenting maps in the same atlas drawn
on different scales. The maps of "McNally" have long been celebrated for their
superior beauty and completeness. This is the only school-book in which the attempt
to make a complete atlas also clear and distinct, has been successful. The map coloring
throughout the series is also noticeable. Delicate and subdued tints take the place of
the startling glare of inharmonious colors which too frequently in such treatises dazzle
the eyes, distract the attention, and serve to overwhelm the names of towns and the
natural features of the landscape.
3. THE VARIETY OF MAP-EXERCISE. Starting each time from a dif-
ferent basis, the pupil in many instances approaches the same fact no less than six
times, thus indelibly impressing it upon his memory. At the same time, this system is
not allowed to become wearisome, the extent of exercise on each subject being grad-
uated by its relative importance or difficulty of acquisition.
4. THE CHARACTER AND ARRANGEMENT OF THE DESCRIP-
TIVE TEXT. The cream of the science has been carefully culled, unimportant
matter rejected, elaboration avoided, and a brief and concise manner of presentation
cultivated. The orderly consideration of topics has contributed greatly to simplicity.
Due attention is paid to the facts in history and astronomy which are inseparably con-
nected with and important to the proper understanding of geography, and such only
are admitted on any terms. In a word, the National System teaches geography as a
science, pure, simple, and exhaustive.
5. ALWAYS UP TO THE TIMES. The authors of these books, editorially
speaking, never sleep. No change occurs in the boundaries of countries or of counties,
no new discovery is made, or railroad built, that is not at once noted and recorded, and
the next edition of each volume carries to every school-room the new order of thinvs.
6. FORM OF THE VOLUMES AND MECHANICAL EXECUTIpN,
The maps and text are no longer unnaturally divorced in accordance with the time-
honored practice of making text-books on this subject as inconvenient and expensive as
possible. On the contrary, all map questions are to be found on the page opposite tl.e
-map itself, and each book is complete in one volume. The mechanical execution is
unrivalled. Paper, printing, and binding are everything that could be desired.
7. MAP-DRAWING. In 1869 the system of map-drawing devised by Professor
Jerome Allen was secured exclusively for this series. It derives its claim to original-
ity and usefulness from the introduction of a fixed unit of measurement applicable to
every map. The principles being so few, simple, and comprehensive, the subject of
map-drawing is relieved of all practical difficulty. (In Nos. 2, 2*, and 3, and published
separately.)
8. ANALOGOUS OUTLINES. At the same time with map-drawing was also
introduced (in No. 2) a new and ingenious variety of Object Lessons, consisting of a
comparison of the outlines of countries with familiar objects pictorially represented.
14
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
GEOGRAPHY Continued.
9. SUPERIOR GRADATION. This is the only series which furnishes an avail-
able volume for every possible class in graded schools. It is not contemplated that a
pupil must necessarily go through every volume in succession to attain pn/i;:ieney.
Uu the contrary, two will suffice, but three are advised ; and, if the course will admit,
the whole series should be pursued. At all events, the books are at hand tor selection)
and every teacher, of every grade, can find among them one exactly suited to his class.
The best combination for those who wisli to abridge the course consists of Nos. 1, 2,
and 3 ; or, where children are somewhat advanced in other studies when they com-
mence geography, Nos. 1*, 2, and 3. Where but two books are admissible, Nos. 1* and
7 , or Nos. 2 and 3, are recommended.
A SHEEP RANCH IN MONTANA.
[Specimen Illustration from McNally's New Geography.]
13
THE NATIONAL SERIES OF STANDARD CCHOOL-BOOKS.
GEOGRAPHY Continued.
Monteith's Physical Geography.
This is a clear, brief statement of the pnysical attributes of the earth and their rela-
tions to the heavens. The illustrations ai.u maps are numerous and helpful. It pro-
vides full instruction in this important branch of study in an attractive way for the
youngest scholars. It contains i>4 pages in quarto form.
MAP-DRAWING.
Monteith's Map-Drawing Made Easy.
A neat little book of outlines and instructions, giving the "corners of States" in
suitable blanks, so that maps can be drawn by unskilful hands from any atlas ; with
instructions for written exercises or compositions on geographical subjects, and com-
parative geography.
Monteith's Manual of Map-Drawing (Allen's System).
The only consistent plan, by which all maps are drawn on one scale. By its use
much time may be saved, and much interest and accurate knowledge gained.
Monteith's Map-Drawing and Object Lessons.
The last-named treatise, bound with Mr. Monteith's ingenious system for commit-
ting outlines to memory by means of pictures of living creatures and familiar objects.
Thus, South America resembles a dog's head ; Cuba, a lizard ; Italy, a boot ; France, a
coifee-pot ; Turkey, a turkey, &c., &c.
Monteith's Colored Blanks for Map-Drawing.
A new aid in teaching geography, which will be found especially useful in recitations,
reviews, and examinations. The series comprises any section of the world required.
Monteith's Map-Drawing Scal^.
A ruler of wood, graduated to the "Allen fixed unit of measurement."
WALL MAPS.
Monteith's Pictorial Chart of Geography.
The original drawing for this beautiful and instructive chart was greatly admired in
the publisher's " exhibit " at the Centennial Exhibition of 1876. It is a picture of the
earth's surface with every natural feature displayed, teaching also physical geography,
and especially the mutations of water. Tke uses to which man puts the earth and its
treasures and forces, as Agriculture, Mining, Manufacturing, Commerce, and Transpor-
tation, are also graphically portayed, so that the young learner gets a realistic idea of
"the world we live in," which weeks of book study might fail to convey.
Monteith's School Maps, 8 Numbers.
- The "School Series" includes the Hemispheres (2 maps), United States, North
America, South America, Europe, Asia, Africa. Price, $2.50 each.
Each map is 28x34 inches, beautifully colored, has the names all laid down, and is
substantially mounted on canvas with rollers.
Monteith's Grand Maps, 8 Numbers.
The "Grand Series" includes the Hemispheres (1 map). North America, United
States, South America, Europe, Asia, Africa, the World on Mercator's Projection, and
Physical Map of the World. Price, 5.00 each. Size, 42 x 52 inches, names laid down,
colored, mounted, &c.
Monteith's Sunday-School Maps.
Including a map of Paul's Travels (5.00), one of Ancient Canaan (3. 00), and Mod-
ern Palestine (-?3.00), or Palestine and Canaan together (.$5.00).
16
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
MATHEMATICS
DAVIES'S COMPLETE
Davies
Davies
Davies
Davies' Primary Arithmetic.
Davies' Intellectual Arithmetic.
Elements of Written Arithmetic.
Practical Arithmetic.
University Arithmetic.
TWO-BOOK SERIES.
First Book in Arithmetic, Primary and Mental
Complete Arithmetic.
ALGEBRA.
Davies' New Elementary Algebra.
Davies' University Algebra.
Davies' New Bourdon's Algebra.
GEOMETRY.
Davies' Elementary Geometry and Trigonometry.
Davies' Legendre's Geometry.
Davies' Analytical Geometry and Calculus.
Davies' Descriptive Geometry.
Davies' New Calculus.
MENSURATION.
Davies' Practical Mathematics and Mensuration.
Davies' Elements of Surveying.
Davies' Shades, Shadows, and Perspective.
MATHEMATICAL SCIENCE.
Davies' Grammar of Arithmetic.
Davies' Outlines of Mathematical Science.
Davies' Nature and Utility of Mathematics.
Davies' Metric System.
Davies & Peck's Dictionary of Mathematics.
17
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DAVIES'S NATIONAL COURSE
OF MATHEMATICS.
ITS RECORD.
In claiming for this series the first place among American text-books, of whatever
class, the publishers appeal to the magnificent record which its volumes have earned
during the tkirty-jhe years of Dr. Charles Davies's mathematical labors. The unremit-
ting exertions of a l.fe-time have placed the modern series on tlie same proud eminence
among competitors that each of its predecessors had successively enjoyed in a course of
constantly improved editions, now rounded to their perfect fruition, for it seems
almost that this science is susceptible of no further demonstration.
During the period alluded to, many authors and editors in this department h?ve
started into public notice, and, by borrowing ideas and processes original with Dr. Davies,
have enjoyed a brief popularity, but are now almost unknown. Many of the series of
to-day, built upon a similar basis, and described as " modern books," are destined to a
similar fate ; while the most far-seeing eye will find it diflicult to fix the time, on the
basis of any data afforded by their past history, when these books will cease to increase
and prosper, and fix a still firmer hold on the affection of every educated American.
One cause of this unparalleled popularity is found in the fact that the eiiterfin.se of the
author did not cease with the original completion of his books. Always a praciicai
teacher, he has incorporated in his text-books from time to time tlie ad vantages or every
improvement in method* of teaching, and every advance in science. During ail the
years in which he has been laboring he constantly submitted his own theories and those
of others to the practical test of the class-room, approving, rejecting, or modify ing
them as the experience thus obtained might suggest. In this way he has been aiile
to produce an almost perfect series of class-books, in which every department of
mathematics has received minute and exhaustive attention.
Upon tlie death of Dr. Davies, which took place in 1876, his work was immediately
taken up by his former pupil and mathematical associate of many years, Prof. W. G.
Peck, LL.D., of Columbia College. By him, with Prof. J. H. Van Amringe, of Columbia
College, the original series is kept carefully revised and up to the times.
DAVIES'S SYSTEM is THE ACKNOWLEDGED NATIONAL STANDARD FOR THE UNITED
STATES, for the following reasons :
1st. It is the basis of instruction in the great national schools at West Point and
Annapolis.
2d. It has received the quasi indorsement of the National Congress.
3d. It is exclusively used in the public schools of the National Capital.
4th. The officials of the Government use it as authority in all cases involving mathe-
matical questions.
5th. Our great soldiers and sailors commanding the national armies and navies were
educated in this system. So have been a majority of eminent scientists in this country.
All these refer to " Davies " as authority.
6th. A larger number of American citizens have received their education from this
than from any other series.
7th. The scries has a larger circulation throughout the whole country than any other,
being extensive;/!/ used in every State in the Union.
It
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DAVIES AND PECK'S ARITHMETICS.
OPTIONAL OR CONSECUTIVE,
The best thoughts of these two illustrious mathematicians are combined in (he
Following beautiful works, which are the natural successors of Davies's Arithmetics,
sumptuously printed, and bound iu crimson, green, and gold:
Davies and Peck's Brief Arithmetic.
Also called the " Elementary Arithmetic." It is the shortest presentation of the sub-
ject, and is ndiqiiatt: for all grades in common schools, being a thorough introduction to
practical lite, except for the specialist.
At h'rst the authors play with the little learner for a few lessons, by object-teaching
and kindred allurements ; but he soon begins to realize that study is earnest, as he
becomes familiar with the simpler operations, and is delighted to find himself master of
important resuKs.
The second part reviews the Fundamental Operations on a scale proportioned iu
the enlarged intelligence of the learner. It establishes the General Principles and
Properties of Numbers, and then proceeds to Fractions. Currency and the Metric
(System are fully treated in connection with Decimals. Compound Numbers and Re-
duction follow, and finally Percentage with all its varied applications.
An Index of words and principles concludes the book, for which every scholar and
most teachers will be grateful. How much time has been spent in searching for a half-
forgotten definition or principle in a former lesson !
Davies and Peck's Complete Arithmetic.
This work c :rtainly deserves its name in the best sense. Though complete, it is not,
like most others which hear the same title, cumbersome. These authors excel in clear,
lucid demonstrations, teaching tl.e science pure and simple, yet not ignoring convenient
methods ;:i.d practical applications.
For turning out a thorough business man no other work is so well adapted. He will
have a clear comprehension, of the science as a whole, and a working acquaintance
with detai.s which must serve him well in all emergencies. Distinguishing features of
the book are the logical progression of the subjects and the great variety of practical
problems, not. puzzle*, which are beneath the dignity of educational science. A clear-
minded critic Las said of Dr. Peck's work that it is free from that juggling with
numbers which fome authors falsely call " Analysis." A series of Tables for converting
ordinal y weights ahd measures into the Metric .System appear in the later editions.
PECK'S ARITHMETICS.
Peck's First Lessons in Numbers.
This book begins with pictorial illustrations, and unfolds gradually the science of
numbers. It noticeably simplifies the subject by developing the principles of addition
and subtraction simultaneously ; as it does, also, those of multiplication and division.
Peck's Manual of Arithmetic.
This hook is designed especially 'or those who seek sufficient instruction to carry
them successfully through practical life, but have not time for extended study.
Peck's Complete Arithmetic.
This completes the series but is a much briefer book than most of the complete
arithmetics, and is recommended not only for what it contains, but also for what is
omitted.
It may be said of Dr. Peck's books more truly than of any other series published, that
they are clear and simple in definition and rule, and that superfluous matter of every
Kind has been faithfully eliminated, thus magnifying the working value of the book
ud saving unnecessary expense of time and labor.
19
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
BARNES'S NEW MATHEMATICS.
In this series JOSEPH FICKLIN, Ph. D., Proi'essor of Mathematics and Astronomy
in the University of Missouri, lias combined all the best and latest results of practical
and experimental teaching of arithmetic with the assistance of many distinguished
mathematical authors.
Barnes's Elementary Arithmetic.
Barnes's National Arithmetic.
These two works constitute a complete arithmetical course in two books.
They meet the demand for text-books that will help students to acquire the greatest
amount of useful and practical knowledge of Arithmetic by the smallest expenditure of
time, labor, and money. Nearly every topic in Written Arithmetic is introduced, and its
principles illustrated, by exercises in Oral Arithmetic. The free use of Equations ; the
concise method of combining and treating Properties of Numbers; the treatment of
Multiplication and Division of Fractions in two cases, and then reduceO-to one; Can-
cellation by the use of the vertical line, especially in Fractions, Interest, and Proportion ;
the brief, simple, and greatly superior method of working Partial Payments by the
41 Time Table " and Cancellation ; the substitution of formulas to a great extent for
rules; the full and practical treatment of the Metric System, &c., indicate their com-
pleteness. A variety of methods and processes for the same topic, which deprive the
pupil of the great benefit of doing a part of the thinking and labor for himself, have
beeu discarded. The statement of principles, definitions, rules, &c., is brief and simple.
The illustrations and methods are explicit, direct, and practical. The great number
and variety of Examples embody the actual business of the day. The very large
amount of matter condensed in so small a compass has beeu accomplished by econo-
mizing every line of space, by rejecting superfluous matter and obsolete terms, and by
avoiding the repetition of analyses, explanations, and operations in the advanced topics
which have been used in the more elementary parts of these books.
AUXJLI ARIES.
For use in district schools, and for supplying a text-book in advanced work for
{lasses having finished the course as given in the ordinary Practical Arithmetics, the
National Arithmetic has been divided and boutd separately, as follows :
Barnes's Practical Arithmetic.
' liarnes's Advanced Arithmetic.
In many schools there are classes that for various reasons never reach beyond
Percentage. It is just such cases where Barnes's Practical Arithmetic will answer a
good purpose, at a price to the jwpil much less than to buy the complete book. On the
other hand, classes having finished the ordinary Practical Arithmetic can proceed
with the higher course by using Barnes's Advanced Arithmetic.
For primary schools requiring simply a table Ixwk, and the earliest rudiments
forcibly presented through object-teaching and copious illustiations, we have
prepared
Barnes's First Lessons in Arithmetic,
which begins with the most elementary notions of numbers, and proceeds, by simple
steps, to develop all the fundamental principles of Arithmetic.
Barnes's Elements of Algebra.
This work, as its title indicates, is elementary in its character and suitable for use,
(1) in such public schools as give instruction in the Elements of Algebra : (2) in institu-
tions of learning whose courses of study do not include Higher Algebra ; (3) in schools
whose object is to prepare students for entrance into our colleges and universities.
This book will also meet the wants of students of Physics who require some knowledge ot
20
THE NATWAL SERIES OF STANDARD SCHOOL-BOOKS.
Algebra. The student's progress in Algebra depends very largely upon the proper treat-
ment of the four Fundamental Operations* The terms Addition, Subtraction, Multiplication,
and Division in Algebra have a wider meaning than in Arithmetic, and these operations
have been so denned as to include their arithmetical meaning ; so that the beginner
is sinrply called upon to enlarge his views of those fundamental operations. Much
attention has been given to the explanation of the negative sign, in order to remove the
well-known difficulties in the use and interpretation of that sign. Special attention is
here called to " A Short Method of Removing Symbols of Aggregation," Art. 76. On
account of their importance, the subjects of Factoring, Greatest Cowman Dirisor, and
Least Common Multiple have been treated at greater length than Is usual in elementary
works. In the treatment of Fractions, a method is used which is quite simple, and',
at tl/e same time, more general than that usually employed. In connection with R<idicl
Quantities the roots are expressed by fractional exponents, for the principles and rules
applicable to integral exponents may then be used without modification. The Equation
is made the chief subject of thought in this work. It is defined near the beginning,
and used extensively in every chapter. In addition to this, four chapters are devoted
exclusively to the subject of Equations. All Proportions are equations, and in their
treatment as such all the difficulty commonly connected with the subject of Proportion
disappears. The chapter on Logarithms will doubtless be acceptable to many teachers
who do not require the student to master Higher Algebra before entering upon the
study of Trigonometry.
HIGHER MATHEMATICS.
Peck's Manual of Algebra.
Bringing the methods of Bourdon within the range of the Academic Course.
Peck's Manual of Geometry.
By a method purely practical, and unembarrassed by the details which rather confuse
than simplify science.
Peck's Practical Calculus.
Peck's Analytical Geometry.
Peck's Elementary Mechanics.
Peck's Mechanics, with Calculus.
The briefest treatises on these subjects now published. Adopted by the great Univer-
sities : Yale, Harvard, Columbia, Princeton, Cornell, &c.
Macnie's Algebraical Equations.
Serving as a complement to the more advanced treatises on Algebra, giving special
attention to the analysis and solution of equations with numerical coefficients.
Church's Elements of Calculus.
Church's Analytical Geometry.
Church's Descriptive Geometry. With plates. % vols.
These volumes constitute the "West Point Course "in their several deparments.
Prof. Church was long the eminent professor of mathematics at West Point Military
Academy, and his works are standard in all the leading colleges.
Courtenay's Elements of Calculus.
A standard work of the very highest grade, presenting the most elaborate attainable
survey of the subject.
Hackley's Trigonometry.
With applications to Navigation and Surveying, Nautical and Practical Geometry,
and Geodesy.
21
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
GENERAL HISTORY.
Monteith's Youth's History of the United States.
A History of the United States for beginners. It is arranged upon the catechetical plan,
with illustrative maps and engravings, re view questions, dates in parentheses (that their
study may be optional with the younger class of learners), and interesting biographical
sketches of all persons who have been prominently identified with the history of our
country.
Willard's United States. School and University Editions.
The plan of this standard work is chronologically exhibited in front of the titlepage.
The maps and sketches are found useful assistants to the memory ; and dates, usually
so difficult to remember, are so systematically arranged as in a great degree to obviate
the difficulty. Candor, impartiality, and accuracy are the distinguishing features of
the narrative portion.
Willard's Universal History. New Edition.
The most valuable features of the " United States " are reproduced in this. The
peculiarities of the work are its great conciseness and the prominence given to the
chronological order of events. The margin marks each successive era with great dis-
tinctness, so that the pupil retains not only the event but its time, and thus fixes the
order of history firmly and usefully in his mind. Mrs. Willard's books are constantly
revised, and at all times written up to embrace important historical events of recent
date. Professor Arthur Oilman has edited the last twenty-five years to 1882.
Lancaster's English History.
By the Master of the Stoughton Grammar School, Boston. The most practical of the
" brief books." Though short, it is not a bare and uninteresting outline, but contains
enough of explanation and detai 1 to make intelligible the cause and effect of events.
Their relations to the history and development of the American people is made specially
prominent.
Willis's Historical Reader.
Being Collier's Great Events of History adapted to American schools. This rare
epitome of general history, remarkable for its charming style and judicious selection of
events on which the destinies of nations have turned, has been skilfully manipulated
by Professor Willis, with as few changes as would bring the United States into its proper
position in the historical perspective. As reader or text-book it has few equals and no .
superior.
Berard's History of England.
By an authoress well known for the success of her History of the United States.
The social life of the English people is felicitously interwoven, as in fact, with the civil
and military transactions of the realm.
Ricord's History of Rome.
Possesses the charm of an attractive romance. The fables with which this history
abounds are introduced in such a way as not to deceive the inexperienced, while adding
materially to the value of the work as a reliable index to the character and institutions,
as well as the historv of the Roman people.
22
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
HISTORY Continued.
Hanna's Bible History.
The only compendium of Bible narrative which affords a connected and chronological
view of the important events there recorded, divested of all superfluous detail.
Summary of History ; American, French, and English.
A well-proportioned outline of leading events, condensing the substance of the more
extensive text-books in common use into a series of statements so brief, that every
word may be committed to memory, and yet so comprehensive that it presents an
accurate though general view of the whole continuous life of nations.
Marsh's Ecclesiastical History.
Affording the History of the Church in all ages, with accounts of the pagan world
during the biblical periods, and the character, rise, and progress of all religions, as well
as the various sects of the worshippers of Christ. The work is entirely non-sectarian,
though strictly catholic. A separate volume contains carefully prepared questions for
class use.
Mill's History of the Ancient Hebrews.
With valuable Chronological Charts, prepared by Professor Edwards of N. Y. This
is a succinct account of the chosen people of God to the time of the destruction of
Jerusalem. Complete in one volume.
Topical History Chart Book.
By Miss Ida P. Whitcomb. To be used in connection with any History, Ancient or
Modern, instead of the ordinary blank book for summary. It embodies the names of
cnntfini>i>i-iu-ii 1-nlci-n from the earliest to the present time, with blanks under each, in
which the pupil may write the summary of the life of the ruler.
Oilman's First Steps in General History.
A "suggestive outline" of rare compactness. Each country is treated by itself, and
the United States receive special attention. Frequent maps, contemporary events in
tables, references to standard works for fuller details, and a minute Index' constitute
the " Illustrative Apparatus.' 1 From no other work that we know of can so succinct a
view of the world's history be obtained. Considering the necessary limitation of space,
the style is surprisingly vivid, and at times even ornate. In all respects a charming,
though not the less practical, text-book.
Baker's Brief History of Texas.
Dimitry's History of Louisana.
Alison's Napoleon First.
The history of Europe from 1788 to 1S15. By Archibald Alison. Abridged by Edward
S. Gould. One vol., Svo, with appendix, questions, and maps. 550 pages.
Lord's Points of History.
The salient points in the history of the world arranged catechetically for class use or
for review and examination of teacher or pupil. By John Lord, LL.D. 12mo, 300
pages.
Carrington's Battle Maps and Charts of the American
Revolution.
Topographical Maps and Chronological Charts of every battle, with 3 steel portraits
of Washington. Svo, cloth.
Condit's History of the English Bible.
For theological and historical students this book has an intrinsic value. It gives the
history of all the English translations down to the present time, together with a careful
review of their influence upon English literature and language.
23
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
BARNES'S ONE-TERM HISTORY
SERIES.
A Brief History of the United
States.
This is probably the MOST ORIGINAL SCHOOL-BOOK pub-
lished for many years, in any department. A few of it?
claims are the following :
1. Brevity. The text is complete for grammar school
or intermediate classes, in 290 12im> pages, large type.
It may readily be completed, if desired, in one term of
study.
2. Comprehensiveness. Though so brief, this book
contains the pith of all the wearying contents of the larger
manuals, and a great deal more than the memory usually
retains I'roin the latter.
3. Interest has been a prime consideration. Small
books have heretofore been bare, full of dry statistics, unattractive. This one is
charmingly written, replete with anecdote, and brilliant with illustration.
4. Proportion of Events. It is remarkable for the discrimination with which
the different portions of our history are presented according to their importance. Thus
the older works, being already large books when the Civil War took place, give it less
space than that accorded to the Revolution.
5. Arrangement. In six epochs, entitled respectively, Discovery and Settlement,
the Colonies, the Revolution, Growth of States, the Civil War. and Current Events.
6. Catch Words. Each paragraph is preceded by its leading thought in promi-
nent type, standing in the student's mind for the whole paragraph.
7:- Key Notes. Analogous with this is the idea of grouping battles, c.., about
some central event, which relieves the sameness so common in such descriptions, and
renders each distinct by some striking peculiarity of its own.
8. Foot-Notes. These are crowded with interesting matter that is not strictly a
part of history proper. They may be learned or not, at pleasure. They are certain
in any event to be read.
9. Biogiaphies of all the leading characters are given in full in foot-notes.
10. Maps. Elegant and distinct maps from engravings on copper-plate, and beauti-
fully colored, precede each epoch, and contain all the places named.
n. Questions are at the back of the book, to compel a more independent use of the
text. Both text and questions are so worded that the pupil must give intelligent
answers IN HIS OWN WORDS. " Yes" and "No " will not do.
24
THE NATIONAL SERIES OF STANDARD SCHOOL-COOKS.
HISTORY Continued.
12. Historical Recreations. These are additional questions to test the student's
knowledge, in review, ;is : "What trees are celebrated in our history?" "When
did a Jog save our army?" "What Presidents died in office?" "When was the
Mississippi our western boundary?" "Who said, 'I would rather be right than
President ' ? " &c.
13. The Illustrations, about seventy in number, are the work of our best artists
and engravers, produced at great expense. They are vivid and interesting, and mostly
upon subjects never before illustrated in a school-book.
14 Dates Only the leading dates are given in the text, and these are so associated
as to assist the memory, but at the head of each page is the date of the event imt
mentioned, and at the close of each epoch a summary of events and dates.
15. The Philosophy of History is studiously exhibited, the causes and effects
of events being distinctly traced and their inter-connection shown.
16. Impartiality. All sectional, partisan, or denominational views are avoided.
Facts are stated after a careful comparison of all authorities without the least prejudice
or favor.
17. Index. A verbal index at the close of the book perfects it as a work of reference.
It will be observed that the above are all particulars in which School Histories have
been signally defective, or altogether wanting. Many otlicr claims to favor it shares in
common with its predecessors.
TESTIMONIALS.
From PROF. WM. F. ALLEN, Slate Uni-
versity of H'isconsin.
"Two features that I like re.ry much
are the anecdotes at the foot of the page
and the 'Historical Recreations' in the
Appendix. The latter, I think, is quite
a n-w feature, and the other is very well
executed."
From HON. NEWTON BATEMAN, Superin-
tendent Public Instruction, Illinois.
" Barnes's One-Term History of the
United States is an exceedingly attrac-
tive and spirited little book. Its claim
to several new and valuable features seems
well founded. Under the form of six well-
defined epochs, the history of the United
States is traced tersely, yet pithily, from
the earliest times to the present day. A
good map precedes each epoch, whereby
the history and geography of the period
may be studied together, ns tttry nlwnys
sfionl I. be. The syllabus of each paragraph
is made to stand in such bold relief, by
the use of large, heavy type, as to be of
much mnemonic value to the student. The
book is written in a sprightly and pi-
Suant style, the interest never llag.-'ing
om beginning to end, a rare and diffi-
cult achievement in works of this kind."
From HON. ABNER J. PHIPPS, Sipe rill-
ten lent S-hools, Lwiston, Maine,.
4 - Barnes's History of the United States
has been used for severnl years in the
Lewiston schools, and has proved a very
satisfactory work. I have examined the
new edition of it."
From HON. R. K. BUCHET/L, City Superin-
tendent S hoofs, L-nicaster, Pa.
" It is the best history of the kind I have
ever seen.''
From T. J. CHARLTON, Superintendent
Public Schools, Vincmn-s, In I.
" We have used it here for six years,
and it has given almost per.ect satisfac-
tion. . . . The notes in fine print at the
bottom of the pages are of especial value."
From PROF. WM. A. MOWRY, E. ,j- C.
School, Providence, Jt. /.
" Permit me to express my high appre-
ciation of your book. I wish all text-
books for the young had equal merit."
From HON. A. M. K El LEY, City Attnrneij,
Lute Mayor, and President of the School
Board, City of RichmonJ, Va.
" I do not hesitate to volunteer to you
the opinion that Barnes 's History is en-
titled to the preference in almost every
respect that distinguishes a good school-
book. . . . The narrative generally exhibits
the temper of the jud^e ; rarely, if ever,
of the advocate."
25
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
A Brief History of An-
cient Peoples.
With an account of their monuments,
literature, and manners. 340 pages.
12mo. Profusely illustrated.
In this work 'the political history,
which occupies nearly, if not all,
the ordinary school text, is condensed
to the salient and essential facts, in
order to give room for a clear outline
of the literature, religion, architecture,
character, habits, &o., of each nation.
Surely it is as important to know some-
tking'abmit Plato as all about Caesar,
and to learn how the ancients wrote
their books as how they fought their
battles.
The chapters on Manners and Cus-
toms and the Scenes in Real Life repre-
sent the people of history as men and
women subject to the same wants, hopes
and fears as ourselves, and so bring the distant past near to us. The Scenes, which are
intended only for reading, are the result of a careful study of the unequalled collections of
monuments in the London and Berlin Museums, of the ruins in Rome and Pompeii, and
of the latest authorities on the domestic life of ancient peoples. Though intentionally
written in a semi-romantic style, they are accurate pictures of what might have occurred
and some of them are simple transcriptions of the details sculptured in Assyrian
alabaster or painted on Egyptian walls.
26
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
HISTORY Continued.
The extracts made from the sacred books of the East are not specimens of their style
and teachings, but only gems selected often from a mass of matter, much of which would
be absurd, ineaningless, and even revolting. It has not seemed best to cumber a booic
like this with selections conveying no moral lesson.
Ihe numerous cross-references, the abundant dates in parenthesis, the pronunciation
of the names in the Index, the choice reading references at the close of each general
subject, and the novel Historical Recreations in the Appendix, will be of service to
teacher and pupil alike.
Though designed primarily for a text-book, a large class of persons general readers,
who desire to know something about the progress of historic criticism and ,ne recent
discoveries made among the resurrected monuments of the East, but have no leisure to
read the ponderous volumes of Brugsch, Layard, Grote, Mommseu, and lime will lind
tli is volume just what they need.
Frnm HOMER B. SPRAGUE, Heal Master
Girls' II/f/h Schod, H'e.st Newton St., Bos-
t-n,M,ss.
" I Vug to recommend in strong terms
the adoption of Barnes's 'History of
Ancient Peoples' as a text-book. It is
about as nearly perfect as could be
hoped for. The adoption would give
great relish to the study of Ancient
History."
HE Brief History of France.
By the author of the " Brhf United States,"
^r with all the attractive features of that popu-
lar work (which see) and new ones of its own.
It is believed ti;at the History of France
has never before been presented in such
brief compass, and this is effected without
sacrificing one particle of interest. The book
reads like a romance, and, while drawing the
student by an irresistible fascination to his
task, impresses the great outlines indelibly upon the memory.
27
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DRAWING.
BARNES'S POPULAR DRAWING SERIES.
Based upon the experience of the most successful teachers of drawing in the United
States.
The Primary Course, consisting of a manual, ten cards, and three primary
drawing books. A, B, and C.
Intermediate Course. Four numbers and a manual.
Advanced Course. Four numbers and a manual.
Instrumental Course. Four numbers and a manual.
The Intenneuiate, Advanced, and Instrumental Courses are furnished either in book
or card form at the same prices. The books contain the usual blanks, with the unusual
advantage of opening Irom the pupil, placing the copy directly in front and above
the blank, thus occupying but little desk-room. The cards are in the end more econom-
ical than the books, if used in connection with the patent blank folios that accompany
this series.
The cards are arranged to be bound (or tied) in the folios and removed at pleasure.
The pupil at the end of each number has a complete book, containing only his own
work, while the copies are preserved and inserted in another fol.o reaily for use in the
next class.
Patent Blank Folios. No. 1. Adapted to Intermediate Course. No. 2. Adap'ed
to Advanced and Instrumental Courses.
ADVANTAGES OF THIS SERIES.
The Plan and Arrangement. The examples are so arranged that teachers and
pupils can see, at a glance, how they are to be treated and where they are to lie copied.
In this system, copying and designing do not receive all the attention. The plan is
broader in its aims, dealing with drawing as a branch of common-school instruction,
and giving it a wide educational value.
Correct Methods. In this system the pupil is led to rely upon himself, and not
upon delusive mechanical aids, as printed guide-marks, &c.
One of the principal objects of any good course in freehand drawing is to educate the
eye to estimate location, form, and size. A system which weakens the motive or re-
moves the necessity of thinking is false in theory and ruinous in practice. The object
should be to educate, not crani ; to develop the intelligence, not teach tricks.
Artistic Effect. The beauty of the examples is not destroyed by crowding the
pages with useless arid badly printed text. The Manuals contain all necessary
instruction.
Stages of Development. Many of the examples are accompanied by diagrams,
showing the different stages of development.
Lithographed Examples. The examples are printed in imitation of pencil
drawing (not in hard, black lines) that the pupil's work may resemble them.
One Term's Work. Each book contains what can be accomplished in an average
term, and no more. Thus a pupil finishes one book before beginning another.
Quality not Quantity. Success in drawing depends upon the amount of thovght
exercised by the pupil, and not upon the large number of examples drawn.
Designing. Elementary design is more skilfully taught in this system than by
any other. In addition to the instruction given in the books, the pupilwill find prii.ted
on the insides of the covers a variety of beautiful patterns.
Enlargement and Reduction. The practice of enlarging and reducing from
copies is not commenced until the pupil is well advanced in the course and therefore
better able to cope with this difficult feature in drawing.
Natural Forms. This is the only course that gives at convenient intervals easy
arid progressive exercises in the drawing of natural forms.
Economy. By the patent binding described above, the copies need not be thrown
aside when a book is filled out, but are preserved in perfect condition for future use.
The blank books, only, will have to be purchased after the first introduction, thus effect-
ing a saving of more than half in the usual cost of drawing-books.
Manuals for Teachers. The Manuals accompanying this series contain practical
instructions for conducting drawing in the class-room, with definite directions for draw-
ing each of the examples in the books, instructions for designing, model and object
drawing, drawing from natural forms, &c.
28
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DRAWING Continued.
Chapman's American Drawing-Book.
The standard American text-book and authority in all branches of art. A compilation
of art principles. A manual for the amateur, and basis of study for the professional
artist. Adapted for schools and private instruction.
CONTENTS. "Any one who can Learn to Write can Learn to Draw. " Primary In-
struction in Drawing. Rudiments of Drawing the Human ' Head. Rudiments in
Drawing the Human Figure. Rudiments of Drawing. The Elements of Geometry. -
Perspective. Of Studying and Sketching from Nature. Of Painting. Etching and
Engraving . Of Modelling. Of Composition. Advice to the American Art-Student.
The work is of course magnificently illustrated with all the original designs.
Chapman's Elementary Drawing-Book.
A progressive course of practical exercise.*, or a text-book for the training of the
eye and hand. It contains the elements from the larger work, and a copy should be in
the hands of every pupil ; while a copy of the " American Drawing-Book," named above,
should be at hand for reference by the class.
Clark's Elements of Drawing.
A complete course in this graceful art, from the first rudiments of outline to the
finished sketches of landscape and scenery.
Allen's Map-Drawing and Scale.
This method introduces a new era in map-drawing, for the following reasons : 1. It
is a system. This is its greatest merit. 2. It is easily understood and taught.
3. The eye is trained to exact measurement by the use of a scale. 4. By no special
effort of the memory, distance and comparative size are fixed in the mind. 5. It dis-
cards useless construction of lines. 6. It can be taught by any teacher, even though
there may have been no previous practice in map-drawing. 7. Any pupil old enough
to study geography can learn by this system, in a short time, to draw accurate maps.
8. The system is not the result of theory, but comes directly from the school-room.
It has been thoroughly and successfully tested there, with all grades of pupils. J>. It
is economical, as it requires no mapping plates. It gives the pupil the ability of rapidly
drawing accurate maps.
FINE ARTS.
Hamerton's Art Essays (Atlas Series) :
No. 1. The Practical "Work of Painting.
"With portrait of Rubens. 8vo. Paper covers.
No. 2. Modern Schools of Art-
Including American, English, and Continental Painting. 8vo. Paper covers.
Huntington's Manual of the Fine Arts.
A careful manual of instruction in the history of art, up to the present time.
Boyd's Kames' Elements of Criticism.
The best edition of the best work on art and literary criticism ever produced in
English.
Benedict's Tour Through Europe.
A valuable companion for any one wishing to visit the galleries and sights of the
continent of Europe, as well as a charming book of travels.
Dwight's Mythology.
A knowledge of mythology is necessary to an appreciation of ancient art
Walker's World's Fair.
The industrial and artistic display at the Centennial Exhibition.
29
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
BOOK-KEEPING TEXT.
Powers's Practical Book-keeping.
Powers's Blanks to Practical Book-keeping.
A Treatise on Book-keeping, for Public Schools and Academies. By Millard R.
Powers, M. A. This work is designed to impart instruction upon the science of accounts,
as applied to mercantile business, and it is Iwlieved that more knowledge, and thj.t, tooj
of a more practical nature, can be gained by the plan introduced in this work, than by
any other published.
Folsom's Logical Book-keeping.
Folsom's Blanks to Book-keeping.
This treatise embraces the interesting and important discoveries of Professor Folsom (of
the Albany " Bryant & Stratton College "), the partial enunciation of which in lectures
and otherwise has attracted so much attention in circles interested in commercial
education.
After studying business phenomena for many years, he has arrived at the positive
laws and principles that underlie the whole subject of accounts ; finds that the science
is based in value as a generic term ; that value divides into two clisses with varied
species ; that all the exchanges of values are reducible to nine equations ; and that all
the results of all these exchanges are limited to thirteen in number.
As accounts have been universally taught hitherto, without setting out from a radical
analysis or definition of values, the science has been kept in great obscurity, and been
made as difficult to impart as to acquire. On the new theory, however, these obstacles
are chiefly removed. In reading over the first part of it, in which the governing laws
and principles arc discussed, a person with ordinary intelligence will obtain a fair con-
ception of the double-entry process of accounts. But when he comes to study thoroughly
these laws and principles as there enunciated, and works out the examples and memo-
randa which elucidate the thirteen results of business, the student will neither fail in
readily acquiring the science as it is, uor in becoming able intelligently to apply it in
the interpretation of business
Smith and Martin's Book-keeping.
Smith and Martin's Blanks.
This work is by a practical teacher and a practical book-keeper. It is of a thoroughly
popular class, and will be welcomed by every one. who loves to see theory and practice
combined in an easy, concise, and methodical form.
The single-entry portion is well adapted to supply a want felt in nearly all other
treatises, which seem to be prepared mainly for the use of wholesale merchants ;
leaving retailers, mechanics, farmers, &c. , who transact the greater portion of the
business of the country, without a guide. The work is also commended, on this
account, for general use in young ladies' seminaries, where a thorough grounding
in the simpler form of accounts will be invaluable to the future housekeepers of the
nation.
The treatise on double-entry book-keeping combines all the advantages of the
most recent methods with the utmost simplicity of application, thus affording the
pupil all the advantages of actual experience in the counting-house, and giving a
clear comprehension of the entire subject through a judicious course of mercantile
transactions.
PRACTICAL BOOK-KEEPING.
Stone's Post-Office Account Book.
By Micah H. Stone. For record of Box Rents and Postages. Three sizes always in
stock. 64, 108, and 204 pages.
INTEREST TABLES.
Brooks's Circular Interest Tables.
To calculate simple and compound interest for any amount, from 1 cent to 1,000, at
current rates from 1 day to 7 years.
31
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
DR. STEELE'S ONE-TERM SERIES,
IN ALL THE SCIENCES.
Steele's i4-Weeks Course in Chemistry.
Steele's 14- Weeks Course in Astronomy.
Steele's i4-Weeks Course in Physics.
Steele's i4-Weeks Course in Geology.
Steele's i4-Weeks Course in Physiology.
Steele's i4-Weeks Course in Zoology.
Steele's i4-Weeks Course in Botany.
Our text-books in these studies are, as a general thing, dull and uninteresting.
They contain from 400 to 600 pages of dry facts and unconnected details. They abound
in that which the student cannot learn, much less remember. Tlie pupil commences
the study, is confused by the line print arid coarse print, and neither knowing exactly
what to learn nor what to hasten over, is crowded through the single term generally
assigned to each branch, and frequently comes to the close without a definite and exact
idea of a single scientific principle.
Steele's " Fourteen- Weeks Courses " contain only that which every well-informed per-
son should know, while all that which concerns only the professional scientist is omitted.
The language is clear, simple, and interesting, and the illustrations bring the subject
within the range of home life and daily experience. They give such of the general
principles and the prominent facts as a pupil can make familiar as household words
within a single term. The type is large and open ; there is no fine print to annoy ;
the cuts are copies of genuine experiments or natural phenomena, and are of fine
execution.
In fine, by a system of condensation peculiarly his own, the author reduces each
branch to the limits of a single term of study, while sacrificing nothing that is essential,
and nothing that is usually retained from tire study of the larger manuals in common
use. Thus the student has rare opportunity to economize his time, or rather to employ
that which he has to the best advantage.
A notable feature is the author's charming " style," fortified by an enthusiasm over
his subject in which the student will not fail to partake. Believing that Natural
Science is full of fascination, he has moulded it into a form that attracts the attention
and kindles the enthusiasm of the pupil.
The recent editions contain the author's " Practical Questions " on a plan never
before attempted in scientific, text-books. These are questions as to the nature and
cause of common phenomena, and are not directly answered in the text, the design
being to test and promote an intelligent use of the student's knowledge of the foregoing
principles.
Steele's Key to all His Werks.
This work is mainly composed of answers to the Practical Questions, and solutions of the
problems, in the author's celebrated " Fourteen-Weeks Courses " in the several sciences,
with many hints to teachers, minor tables, &c. Should be on every teacher's desk.
Prof. J. Dorman Steele is an indefatigable student, as well as author, and his books
have reached a fabulous circulation. It is sate to say of his books that they have
accomplished more tangible and better results in the class-room than any other ever
offered to American schools, and have been translated into more languages for foreign
schools. They are even produced in raised type for the blind.
32
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
THE NEW GANOT.
Introductory Course of Natural Philosophy.
This book was originally edited from Ganot's " Popular Physics," by William G.
Peck, LL.D., Professor of Mathematics and Astronomy, Columbia College, and of
Mechanics in the School of Mines. It has recently been revised by Levi S. Bur-
bank, A. M., late Principal of Warren Academy, Woburn, Mass., arid James I. Hanson,
A.M., Principal of the High School, Woburn, Mass.
Of elementary works those of M. Ganot stand pre-eminent, not only as popular
treatises, but as thoroughly scientific expositions of the principles of Physics. His
" Traite de Physique " has not only met with unprecedented success in France, but has
been extensively used in the preparation of the best works on Physics that have been
Issued from the American press.
In addition to the "Traite de Physique," which is intended for the use of colleges
and higher institutions of learning, M. Ganot published this more elementary work,
adapted to the use of schools and academies, in which he faithfully preserved the
prominent features and all the scientific accuracy of the larger work. It is charcter-
ized by a well-balanced distribution of subjects, a logical development of scientific
principles, and a remarkable clearness of definition and explanation. In addition, it is
profusely illustrated with beautifully executed engravings, admirably calculated to
convey to the mind of the student a clear conception of the principles unfolded. Their
completeness and accuracy are such as to enable the teacher to dispense with much of
the apparatus usually employed in teaching the elements of Physical Science.
After several years of great popularity the American publishers have brought this
important book thoroughly up to the times. The death of the accomplished educator,
Professor Burbank, took place before he had completed his work, and it was then
taken in hand by his friend, Professor Hanson, who was familiar with his plans, and
lias ably and satisfactorily brought the work to completion.
The essential characteristics and general plan of the book have, so far as possible,
been retained, but at the same time many parts have been entirely rewritten, much
new matter added, a large number of new cuts introduced, and the whole treatise
thoroughly revised and brought into harmony with the present advanced stage of sci-
entific discovery.
Among the new features designed to aid in teaching the subject-matter are the
summaries of topics, which, it is thought, will be found very convenient in short
reviews.
As many teachers prefer to prepare their own questions on the text, and many do not
have time to spend in the solution of problems, it has been deemed expedient to insert
both the review questions and problems at the end of the volume, to be used or not at
the discretion of the instructor.
Fram the Churchman.
" No department of science has under-
gone so many improvements and changes
in the last quarter of a century as that of
natural philosophy. So many and so im-
portant have been the discoveries and
inventions in every branch of it that
everything seems changed but its funda-
mental principles. Ganot has chapter
upon chapter upon subjects that were not
so much as known by name to Olmsted ;
and here we have Ganot, first edited by
Professor Peck, and afterward revised by
the late Mr. Burbank and Mr. Hanson. No
elementary works upon philosophy have
been superior to those of Ganot, either as
popular treatises or as scientific exposi-
tions of the principles of physics, and
his ' Traite de Physique ' has not only had
a great success in France, but has been
freely used in this country in the prepa-
ration of American books upon the sub-
jects of- which it treats. That work was
intended for higher institutions of learn-
ing, and Mr. Ganot prepared a more
elementary work for schools and acade-
mies. It is as scientifically accurate as
the larger work, and is characterized by
a logical development of scientific princi-
ples, by clearness of definition and expla-
nation, by a proper distribution of sub-
jects, and by its admirable engravings.
We here have Ganot's work enhanced in
value by the labors of Professor Peck and of
Messrs. Burbank and Hanson, and brought
up to our own times. The essential char-
acteristics of Ganot's work have been re-
tained, but much of the book lias been
rewritten, and many new cuts have been
introduced, made necessary by the prog-
ress of scientific discovery. The short
reviews, the questions on the text, and
the problems given for solution are desir-
able additions to a work of this kind, and
will give the book increased popularity."
34
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
FAMILIAR SCIENCE.
Norton & Porter's First Book of Science.
Setsf>rth the principles of Natural Philosophy, Astronomy, Chemistry, Physic "o y.
anJ Geology, on the catechetical plan for primary classes an I beginners.
Chambers's Treasury of Knowledge.
Progressive lessons upon first, common things which lie most immediately around
us, and first attract the attention of the young mind ; second, common objects from ti.e
mineral, animal, and vegetable kingdoms, manufactured articles, and miscellaneous
substances ; third, a systematic view of nature under the various sciences. May be
used as a reader or text-book.
Monteith's Easy Lessons in Popular Science.
This book combines within its covers more attractive features for the study of science
by children than any other book published. It is a reading book, spelling book, com-
position book, drawing book, geography, history, book on botany, zoology, agricul-
ture, manufactures, commerce, and natural philosophy. All these subjects are presented
in a simple and effective style, such as would be adopted by a good teacher on an
excursion with a class. The class are supposed to be taking excursions, with the help
of a large pictorial chart of geography, which can be suspended before them in the
school-room. A key of the chart is inserted in every copy of the book. With this
book the science of common or familiar things can be taught to beginners.
NATURAL PHILOSOPHY.
Norton's First Book in Natural Philosophy.
Peck's Elements of Mechanics.
A suitable introduction to Bartlett's higher treatises on Mechanical Philosophy, and
adequate in itself tor a complete academical course.
Bartlett's Analytical Mechanics.
Bartlett's Acoustics and Optics.
A complete system of Collegiate Philosophy, by Prof. W. H. C. Bartlett, of West
Point Military Academy.
Steele's Physics.
Peck's Ganot.
GEOLOGY.
Page's Elements of Geology.
A volume of Chambers's Educational Course. Practical, simple, an/ 1 "tninently
calculated to make the study interesting.
Steele's Geology.
CHEMISTRY.
Porter's First Book of Chemistry.
Porter's Principles of Chemistry.
The above are widely known as the productions of one of the most eminent scientific
men of America. The extreme simplicity in the method of presenting the science, while
exhaustively treated, lias excited universal commendation.
Gregory's Chemistry (Organic and Inorganic). 2 vols,
The science exhaustively treated. For colleges and medical students.
Steele's Chemistry.
36
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
NATURAL SCIENCE Continued.
BOTANY.
Wood's Object-Lessons in Botany.
Wood's American Botanist and Florist.
Wood's New Class-Book of Botany.
The standard text-hooks of the United States in this department. In style they are
simple, popular, and lively ; in arrangement, easy and natural ; in description, graphic
;:nd scientific. The Tables for Analysis are reduced to a perfect system. They include
the flora ol the whole United States east of the Rocky Mountains, and are well adapted
to the regions west.
Wood's Descriptive Botany.
A complete flora of all plants growing east of the Mississippi River.
Wood's Illustrated Plant Record.
A simple form of blanks for recording observations in the field.
Wood's Botanical Apparatus.
A portable trunk, containing drying pre.',s, knife, trowel, microscope, and tweezers,
and a copy of Wood's " Plant Record," the collector's complete outlit.
Willis's Flora of New Jersey.
The most useful book of reference e<cr published for collectors in all parts of the
country. It contains also a Botanical Directory, with addresses of living American
botanists.
Young's Familiar Lessons in Botany.
Combining simplicity of diction with some degree of technical and scientific knowl-
edge, lor intermediate classes. Specially adapted for the Southwest.
Wood & Steele's Botany.
See page 33.
AGRICULTURE.
Pendleton's Scientific Agriculture.
A text-book for colleges and schools ; treats of the following topics : Anatomy and
Physiology of Plants ; Agricultural Meteorology ; Soils as related to Physics ; Chemistry
of the Atmosphere ; of Plants ; of Soils ; Fertilizers and Natural Manures; Animal Nu-
trition, &c. I3y E. M. Pendleton, M. D., Professor of Agriculture in the University of
Georgia.
From PRESIDENT A. D. WHITE, Cornell
University.
" Dear Sir : I have examined your
' Text-book of Agricultural Science,' and it
seems to me excellent in view of the pur-
pose it is intended to serve. Many of
your chapters interested me especially,
and all parts of the work seem to combine
scientific instruction with practical infor-
mation in proportions dictated by sound
common sense."
From PRESIDENT ROBINSON, of Brown
University.
" It is scientific in method as well as in
matter, comprehensive in plan, natural
and logical in order, compart and lucid in
its statements, and must.be useful both as
a text-book in agricultural colleges, and
as a hand-book for intelligent planters and
farmers."
37
THE NATIONAL SER/ES OF STANDARD SCHOOL-BOOKS.
NATURAL SCIENCE Continued.
PHYSIOLOGY.
Jarvis's Elements of Physiology.
Jarvis's Physiology and Laws of Health.
The only books extant \vhicn approach this subject with a proper view of the true
object of teaching Physiology iu schools, viz., that scholars may know how to take care
of their own health. In bold contrast with the abstract Anatomies, which children
learn as they would Greek or Latin (and forget as soon), to discipline the mind, are these
text-books, using the science as a secondary consideration, and only so far as is neces-
sary for the comprehension of the laws o, tiealtk.
Steele's Physiology.
See page 33.
ASTRONOMY.
Willard's School Astronomy.
J3y means of clear and attractive illustrations, addressing the eye in many cases by
analogies, careful definitions of all necessary technical terms, a careful avoidance of ver-
biage, and unimportant matter, particular attention to analysis, and a general .adoption
of the simplest methods, Mrs. Willard has made the best and most attractive elemen-
tary Astronomy extant.
Mclntyre's Astronomy and the Globes.
A complete treatise for intermediate classes. Highly approved.
Bartlett's Spherical Astronomy.
The West Point Course, for advanced classes, with applications to the current wants
of Navigation, Geography, and Chronology.
Steele's Astronomy.
See page 33. .
NATURAL HISTORY.
Carll's Child's Book of Natural History.
Illustrating the animal, vegetable, and mineial kingdoms, witli application to the
arts. For beginners. Beautifully ai,d copiously illustrated.
Anatomical Technology. Wilder & Gage.
As applied to the domestic cat. For the use of students of medicine.
ZOOLOGY.
Chambers's Elements of Zoology.
A complete and comprehensive system of Zoology, adapted for academic instruction,
presenting a systematic view of the animal kingdom as a portion of external nature.
Steele's Zoology.
See page 33.
38
THE NATIONAL SERIES OF STANDARD SCHOOL-BOOKS.
LITERATURE.
Oilman's First Steps in English Literature.
The character and plan of this exquisite little text-book may be best understood irom
an analysis of its contents : Introduction. Historical Period of Immature English,
with Chart ; Definition of Terms ; Languages of Europe, with Chart ; Period of Mature
English, with Chart ; a Chart of Bible Translations, a Bibliography or Guide to General
Reading, and other aids to the student.
Cleveland's Compendiums. 3 vols. 12mo.
ENGLISH LITERATURE. AMERICAN LITERATURE.
ENGLISH LITERATURE OF THE XIXTii CENTURY.
In these volumes are gathered the cream of the literature, of the English-speaking
people for the school-room and the general reader. Their reputation is national. More
than 125,000 copies have been sold.
Boyd's English Classics. 6 vols. Cloth. 12mo.
MILTON'S PARADISE LOST. THOMSON'S SEASONS
YOUNG'S NIGHT THOUGHTS. FOLLOK'S COURSE OF TIME.
COWPER'S TASK, TABLE TALK, &c. LORD BACON'S ESSXYS.
This series of annotated editions of great English writers in prose and poetry is
designed for critical reading and parsing in schools. Prof. J. R. Boyd proves himself
an editor of high capacity, and the works themselves need no encomium. As auxiliary
to the study of belles-lettres, &e., these works have no equal.
Pope's Essay on Man. Ifimo. Paper.
Pope's Homer's Iliad. 32mo. Roan.
The metrical translation of the great poet of antiquity, and the matchless "Essay on
the Nature and State of Man," by Alexander Pope, afford superior exercise in literature
and parsing.
POLITICAL ECONOMY.
Champlin's Lessons on Political Economy.
An improvement on previous treatises, being shorter, yet containing evorythim>
essential, with a view of recent questions in finance, &c., which is not elsewhere
found.
39
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a.
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