LIBRARY
OF THB
University of California.
Class
OONTENTS,
I. Algebra.
Preliminary Notions
Definitions and Symbols
Addition
Subtraction ,
Brackets Vincula
Introductory Simple Equations
Questions in Simple Equations
Definitions continued
Multiplication
Division
Algebraic Fractions
Addition Subtraction
Multiplication Division
Greatest Common Measure
Least Common Multiple .
Simple Equations in general
Simultaneous Simple Equations
Theory of Exponents
Square Root of Polynomial
Cube Root of Polynomial
Surds Imaginary Quantities
Binomial Surds
Operations with Imaginary Quanti
ties ....
Quadratic Equations, Rule I
,, ,, Rule II.
Theory of Quadratic Equations
Simultaneous Quadratics
Theory of Proportion
Variation ...
Arithmetical Progression
Geometrical Progression .
Harmonical Progression .
Piling of Balls and Shells
Square Pile Rectangular Pile
General Rule for all Piles
The Binomial Theorem .
Limitations of the Theorem
The Exponential Theorem
Theory of Logarithms .
Construction of Logarithms
Exponental Equations
Compound Interest
Annuities
Increase of Population
Permutations
Combinations
Probabilities .
Life Annuities
Life Assurances
PAGE
1
2
4
6
7
8
10
12
13
16
19
20
21
22
24
25
30
36
37
39
40
42
45,
46
49
50
o2
58
60
61
62
64
67
68
69
70
77
78
79
80
84
85
86
87
88
91
93
99
102
Theory of Equations
Transformation of Equation
Limits of the Roots
Equal Roots .
Rule of Descartes .
Criteria of Imaginary Roots
Newton's Rule
Theorem for the Biquadratic ,
Solution of Equations .
Newton's Approximation
Approximation by Position
Cardan's Method for Cubics
Decomposition of a Biquadrat:
Recurring Equations
Binomial Equations
Vanishing Fractions
Maxima and Minima
Indeterminate Equations
Indeterminate Coefficients
Summation of Finite Series
The Differential Method
Construction of Tables .
Interpolation
Summation of Infinite Series
Recurring Series .
Reversion of Series
Convergency of Series
II. Plane Trigonometry,
Definitions, &c.
Measurement of Angles .
Sine, Cosine, Tangent, &c.
Fundamental Relations .
Solution of Right-angled Triangles
Oblique-angled Triangles
Miscellaneous Problems .
Quadrature of the Circle
Unit of Circular Measure
French and English Degrees .
Extension of Definitions
Sine and Cosine of (A+B)
Expressions involving Two Angles
Single Angles and their Halves
Multiple Angles Ambiguities
Applications of Formulai
Inverse Trigonometric Functions
Solution of a Quadratic by Tables
Solution of a Cubic by Tables
Construction of Trigonometric Tables
Developments of sin ^, cos ^ .
CONTENTS.
PAGE
Euler's Expressions for sin 6, cos 6 . 213
De Moivre's Theorem , . .213
Developments of cos"^, sin"^ . . 215
Developments of sinw^, co^n6 . .216
Development of 6 in powers of tan 6 216
Euler's Series : Machin's Series . 217
Developments of ^^, of sin 6, and of
cos ^ 218
Wallis's Expression for ^sr . . 218
Imaginary Logarithms . . .219
The Numbers of Bernoulli . . 220
Summation of Trigonometric Series 222
Imaginary Roots of Unity . . 225
Construction of Log Sines and Co-
226
III, Spherical Trigonometry,
227
231
233
235
Preliminary Theorems
Fundamental Formulae ,
Formulae for Sides .
Napier's Analogies
Right-angled Triangles
Rules 236
Quadrantal Triangles . , . 238
Solution of Oblique-angled Triangles 240
Examples of the Six Cases . 240-250
Area of Triangle : Spherical Excess . 252
Napier'
IV. Mensuration.
Area of Parallelogram and Triangle 255
,, ,, Triangle: three sides given 257
Areas of Quadrilaterals . . .258
Equidistant Ordinates . . . 260
Regular Polygons . . . .261
The Circle and its Sectors . . 262
Segment of a Circle . . , 264
Circular Ring .... 265
Inscribed and Circumscr. Triangle . 266
Prism and Cylinder . . . 267
Pyramid and Cone . . . .268
Frustum of Pyramid or Cone . . 269
Surface of a Frustum . . .270
The Sphere and its Surface . .271
Theorem of Archimedes . . . 273*
Volume of Segment and Zone . . 273
Equidistant Sections . . .274
Weight, &c., of Shot and Shells . 275
Weight of Powder in Shells . ..277
Applications of Maxima and Minima 279
V. Analytical Gteometry.
Equations of a Point . . . 285
Equation of a Straight Line . . 287
Straight Line subject to Conditions . 291
Problems on the Straight Line . 296
The Circle : Rectangular Axes . 300
The Circle : Oblique Axes . . 301
Tangent at a Point . . .302
Tangent from a Point . . . 302
Locus of an Equa, of Second Degree 304
Construction of Loci . . .305
Problems on the Circle . . .306
Polar Co-ordinates . . . 309
The Conic Sections . . .310
Equation of the Ellipse . . .311
The Principal Diameters . .313
Change of Co-ordinates . . .315
Conjugate Diameters . . . 316
Change from Oblique to Rectangular
Co-ordinates .... 318
Tangent to the Ellipse . . .319
Subtangent, Normal, Subnormal . 322
Polar Equation of the Ellipse . . 326
Radius of Curvature . . .327
Chord of Curvature . . .328
Area of the Ellipse . . .329
The Hyperbola . . . .330
Equation of the Hyperbola . . 331
Principal Diameters . . . 333
Conjugate Diameters . . . 334
Tangent to the Hyperbola . .336
The Asymptotes .... 336
Equation with Asymptotes for Axes 339
Conjugate Hyperbolas . . .340
The Parabola . . . .341
Parabola referred to Conjugate Axes 342
Tangent, Normal, &c. . . .344
Properties connected with Tangents . 345
Polar Equation of the Parabola . 348
Area of the Parabola . " . . 349
Radius of Curvature . . . 350
Locus of Equation of the Second De-
gree ...... 351
The Different Curves represented . 354
Determination of particular Loci . 355
Problems on Loci . . . .357
VI. Mechanics : Statics.
Conspiring and Opposing Forces
The Parallelogram of Forces
The Triangle of Forces .
Composition of Forces .
The Polygon of Forces .
Problems in Statics
The Principle of Moments
Parallel Forces acting in a Plane
Centre of Gravity of a Rigid Body
General Equations of Parallel Forces
Equilibrium in general .
Problems on Equilibrium
Mechanical Powers
Lever Balance Steelyard
Combination of Levers .
Wheel and Axle : Toothed Wheels
Pulley : Systems of Pulleys .
Inclined Plane : Screw .
The Wedge .
Mechanical Powers in Motion .
Principle of Virtual Velocities
Friction ....
361
362
364
365
367
367
372
375
377
383
387
390
396
396
401
402
404
407
410
412
418
419
CONTENTS.
XI
VII. Dynamics.
Uniform Motion ....
424
Variable Motion ....
425
Accelerated Motion
426
Falling Bodies : Gravity-
429
Motion on Inclined Planes
430
Parallelogram of Velocities
431
Motion of Projectiles
432
Circular Motion : Centrifugal Force .
439
Moving Forces ....
442
Principle of D'Alembert
449
Moment of Inertia
450
Impact of Bodies . . . .
451
VIII. Hydrostatics.
Transmission of Pressure
456
Hydrostatic Paradox
457
Levelling Instrument
460
Explanation of Symbols, &c. .
460
Centre of Pressure
465
Resultant of Fluid Pressures .
466
Equilibrium of a Floating Body
467
Equilibrium of a Rotating Fluid
469
Specific Gravities of Bodies .
470
Hydrostatic Balance . .
472
Common Hydrometer
472
Nicholson's Hydrometer .
473
Elastic Fluids : the Atmosphere
476
Law of Mariotte and Boyle
477
Altitudes by the Barometer .
479
The Wheel Barometer .
482
The Thermometer . . . .
482
The Syphon
482
The Common and Forcing Pumps .
483
The Diving Bell . . . .
485
Air Pumps . . . . .
486
The Condenser . . . .
487
IX.
Differential and Integral
Calculus.
Diflferentiation
Investigation of Rules .
Algebraic Functions
Applications to Geometry
Log. and Exp. Functions
Trigonometrical Functions
Inverse Functions ,
Integration of Particular Forms
Areas of Curves : Definite Integrals
Volumes of Revolution .
489
494
495
497
498
600
500
503
510
613
Lengths of Curve Lines .
Surfaces of Revolution .
Successive Differentiation
Theorem of Leibnitz
Maclaurin's Theorem
Taylor's Theorem ....
Limits of Taylor's Theorem
,, ,, Maclaurin's Theorem
Compound Functions
Implicit Functions
Vanishing Fractions
Maxima and Minima
Max. and Min. of Implicit Functions
Functions of Two Variables .
Change of Independent Variable
Failure of Taylor's Theorem .
,, ,, Maclaurin's Theorem
Asymptotes to Curves
Spiral Curves
Circular Asymptotes
Sectorial Areas
Contact : Osculation
Rad. of Curv. in Rect. Co-ordinates
Rad. of Curvature in Polar Curves .
Chord of Curvature
Consecutive Lines and Curves
Enveloping Curves
Singular Points ....
Integ. of Rational Fractions .
,, ,, Irrational Functions
Integration by Reduction
,, ,, Parts
,, ,, Series
Successive Integration .
11
516
519
620
621
625
527
529
530
633
634
641
645
548
652
553
555
655
558
560
561
663
565
668
569
573
575
576
58L
585
587
690
694
696
X. Applications to Mechanics.
The Centre of Gravity . . .597
Theorems of Guldinus . . .699
Moment of Inertia : Gyration . 600
Centre of Oscillation . . .603
Centre of Percussion . . . 604
Attraction of Bodies . . . 605
Velocity : Acceleration . . . 608
Equations of Motion . . .609
Cycloidal Pendulum . . .613
Simple Pendulum . , . .614
Problems on the Pendulum . . 615
The Ballistic Pendulum . . .618
The Cycloid 619
The Catenary . . . .620
Note on Interpolation . . . 622
Answers to Examples . . . 624
ERRATA.
Page 163, Ex. 8, for = read +. P. 183, line 2, for 88 read 80 ; and line 4, for 33
read 25. P. 206, Ex. 17, for tan read sin ; and Ex. 19, for tan h read tan B. P. 381
in the diagram, the point e should be on OF. P. 446, line 18, for exactly read very
nearly. P. 449, line 22, after denominator put -^g. P. 671, last line but one, omit
comma after C.
Q
Types fallen out or defective. Page 27, Ex. 10, -. P. 64, line 6, the exponent n-j-p.
fp 25
P. 72, Hne 8, -. P. 77, line 19, . P. 169, line 29, the minus sign.
q_ 2
P. 224, line 1, the exponent %. P. 225, line 16, the exponent .
w
P. 297, Prob. II., |. P. 499, Ex. 5, exponents of ar, z-1.
P. 515, line 8, %. P. 533, Ex. 1, -^^,
\^'
A COURSE
OP
ELEMENTARY MATHEMATICS,
ETC., ETC.
The preparation necessary for the profitable study of the following
course of Mathematics is a knowledge of common x\rithmetic, and some
acquaintance with the principles of Geometry, as taught in Euclid's
Elements. A student ignorant of these initiatory, but most important
departments of elementary science, would scarcely seek his first lessons
therein from a book such as this. The Elements of Euclid is a work
by itself ; universally known and esteemed, and everywhere to be easily
procured : to transfer its pages to the present performance, could be of
no possible advantage to the learner. And the same may be said of
common Arithmetic : both this and Euclid are more conveniently studied
from the ordinary manuals in popular use. We shall therefore commence
the volume now in the hands of the reader, with a treatise on Algebea
the indispensable foundation of the entire fabric of modern analytical
science.
I. ALGEBRA.
1. Preliminary Notions. Algebra may be regarded simply as
an extension of the principles of Arithmetic. In the latter science the
symbols of quantity, to which its rules and operations are applied, are
limited to the nine digits or figures 1, 2, 3, 4, 5, 6, 7, 8, 9, together with
the cypJier or zero, 0. And not only is the notation of Arithmetic limited
to these ten symbols, but each symbol is employed by every computer in
the same sense : the character or symbol 4, for instance, stands for four,
always ; 6 for six; 8 for eight, and so on : the symbols of Arithmetic are
thus fixed in meaning, as well as limited in number.
It is otherwise in Algebra: in this science the symbols of quantity
comprehend not only the figures of arithmetic, but also the letters of the
alphabet : the figures being, as in arithmetic, of invariable signification,
but the letters admitting of arbitrary interpretation. It is this latter
circumstance namely, the possession of a set of symbols which we may
employ to represent anything we please that gives to Algebra its pecu-
liarity and its power. In Arithmetic, known quantities only can be
denoted by symbols : in Algebra a quantity altogether unknown, in value,
at the outset of an inquiry, may be represented an alphabetical letter
serving this purpose, and then the rules of the science, to be hereafter
developed, will enable us ultimately to interpret its meaning, consistently
-84592
2 DEFINITIONS SYMBOLS OF QUANTITY SIGNS OF OPERATION.
with the conditions which connect it, in that inquiry, with the known
quantities concerned.
2. Definitions Symbols of QuantitySigns of Opera-
tion. As noticed above, the symbols by which the quantities operated
upon in algebra are represented, are the figures of ordinary arithmetic,
and the letters of the alphabet : the marks or signs by which these opera-
tions are indicated, are called signs of operation : the principal of these
are the following :
+ , plus, the sign of addition, implying that the quantity to which it is
prefixed is to be added.
, minus, the sign of subtraction, denoting that the quantity to which
it is prefixed is to be subtracted.
Thus 5 + 2, which is read 5 plus 2, signifies that 2 is to be added to
5 ; and 52, which is read 5 minus 2, indicates that 2 is to be subtracted
from 5. In like manner a+b, or a plus 6, implies that b is to be added
to a, that is, that the quantity represented by b is to be added to that
represented by a. And ab, or a minus b, implies that b is to be sub-
tracted from a. Of course we cannot actually perforin the addition and
subtraction operations thus indicated, till we know what numbers or
quantities a and b stand for.
It may be remarked here, that although the letters a, 6, &c. are but
the representatives of quantities or numerical values, yet, for brevity of
expression, we refer to them as the quantities themselves.
The crooked mark ~ placed between two quantities denotes the dif-
ference between those quantities : thus a~6 means the difference between
a and b, whether that difference be the result of subtracting b from a, or
a from b.
X , the sign of multiplication, when placed between two quantities,
implies that those quantities are to be multiplied together : thus 4x6,
or 6 X 4 means that 4 and 6 are to be multiplied together, and axb,or
bxa, implies in like manner the product of a and b.
Instead of the sign x , a dot placed between the factors is often used
for the sign of multiplication: thus 4.6, or 6.4, and a.b, or b.a, each
implies the product of the quantities between which the dot is placed.
It must be observed, however, that the dot should range with the lower
part of the figures or letters, and not with the upper part, to avoid con-
founding it with the decimal point, as, in the case of figures, might
otherwise happen: thus 6.4 means 24, but 6-4 means 6 and 4 tenths.
In the case of letters however, the dot is usually dispensed with alto-
gether, and the factors simply written side by side, without any inter-
vening sign at all : thus, ab, ex, bxy, abxz, &c. mean the same as a x ft.
cxx, bxxxy, axbxxxz; or as a.b, c.x, b.x.y, a.b.x.z, &c. This
suppression of the intervening sign of multiplication between the fac-
tors is not allowable when those factors are numbers, as is obvious : if
6 X 4, or 6.4 were written 64, sixty-four would be implied, and not 24,
as intended. But when a single numerical factor enters with the letters,
then the multiplying sign may be omitted, since no ambiguity can arise :
thus, 6xax6 or Q.a.b, may be more conveniently written Qdb, which
means 6 times the product of a and h, or as it is' more briefly read,
6 times a, b. It is proper, as here, always to place the numerical factor
first, and the literal factors afterwards ; and also to arrange these latter
in the order in which they succeed each other in the alphabet. The
numerical factor, thus placed first, is called the coefficient of the quantity
DEFINITIONS SYMBOLS OF QUANTITY SIGNS OF OPERATION. 8
multiplied by it: thus 6 is the coefiBcient of ah in 6a6, and 15 is the
coefficient of xyz in Ibccyz.
-T-, the sign of division, when placed before a quantity, supplies the
place of the words '< divided by," so that 8-r-2 means 8 divided by 2,
12-r3 means 12 divided by 3, a-^-b means a divided by b, and so on;
but, as in common arithmetic, division is more frequently indicated by
writing the dividend above and the divisor below a horizontal bar of
separation, thus:
is the same as a-r6, and ~ is the same as Zopy-r-2db.
zao
8. The four signs now explained indicating the four fundamental
operations both of arithmetic and algebra are, of course, those of most
frequent occurrence in calculation. Algebraists, however, economize their
signs of operation as much as possible, and never introduce them need-
lessly. This has been already exemplified in the case of multiplication :
the absence of sign between letters, placed side by side, as much implies
the multiplication together of the numbers those letters represent, as if
each were separated from the others by an oblique cross, or a dot. In
like manner, when a row of additive and subtractive quantities are con-
nected together by the proper signs, if the Jlrst of these quantities be
additive, or plus, the sign + is suppressed as superfluous; thus : a 6 +
c-f-rf 4 is the same as -f-a & + c+<? 4, and implies that a, c, and d
are additive ; or, as they are more frequently called, positive quantities,
and that b, and 4, are subtractive, or negative quantities. If the letters
a, b, c, d stood respectively for 2, 4, 3, 8, the interpretation of the
expression just written would be 5.
4. The term coefficient has been already defined : it is the numerical
multiplier of the algebraic quantity to which it is prefixed : when this
numerical multiplier is simply 1, it is not inserted : it is superfluous to
introduce unit-factors; a-\-h,is as well understood to be once a plus once
b, as \a-\-\h', but if the question were asked What is the coefficient
of <* or of 6 in the expression a -ffe? the answer would be, not nothing y
but 1.
6. =, equal to, is the sign of equality: it implies that what is written
on one side of it is equal to what is written on the other, thus :
7-f-4=ll, 7-4=3, 7-4-1-1=4, Bx-\-2x-x=ix, &c.
6. Any quantity of how many letters soever it may consist is called
a simple quantity, or a quantity of but one term, provided it be not sepa-
rated into distinct parts by the interposition of a plus or minus sign ;
thus, each of the following is a simple quantity, or a quantity of but one
term :
^ _ ^ _ 2ax 14 ,
Sahx, labay, ^, , &c.
7. Each of the following, however, is a compound quantity : the first
consists of two terms, the second of three terms, and the third of four
terms :
4
2a-i-86, 6a2b-\-c, 5ah+2cd~-Bm-\ - .
8. We shall now add a few exercises by which the learner may satisfy
himself as to whether he correctly understands what has already been
explained or not.
B ti
4 ADDITION OF ALGEBRA.
Exercises on the Definitions. In the following exercises we shall
suppose a=4:, 6^3, c=5, and d=7.
Find the values in numbers of the following expressions :
Expression. Interpretation. Value.
3a-|-26-c 12+6-5 13.
5b-^a-\-M
4c?+2c-6a+l
8c c^-36+4a
2ah-\-5cd-lQ
Consequently 3fl^ + 26 c= 13, hh ^a-\-^d=ll, 4Z + 2c 6^ + 1 = 15,
8c -cZ 36 + 4a =40, and 2a6 + 5cc? 16=183 ; and the values of the fol-
lowing expressions are to be obtained in a similar manner :
(1) 25+3a-2d
(2) i%-U-\-Zd.
(3) 4:ac-\-2cd-ab.
(4) Zahc+4J)cdQQ.
(5) '^+M-ia.
15-12+14
17.
28+10-24+1
15.
40-7-9+16
40.
24+175-16
183.
(6)
6+c c+cZ
a ' 26
alc 2a
(7)
d 6+c*
(8)
Zcd-iab-1
ah
2a6+4 "^
c-\-d'
^ For the proper values, see answers at the end.
9. Addition of Algebra. In Arithmetic, addition means the
collection into one sum of a set of quantities, all of which are additive or
positive : in Algebra the term is extended to the finding the aggregate
or balance of a set of quantities, some of which only may be positive, and
the others negative ; the result of such addition being plus or minus
according as the sum of the positive, or the sum of the negative terms,
preponderate. The sum of a set of algebraic quantities in the absence
of all interpretation of the symbols can be exhibited only when the
quantities are all alike; that is, when, so far as the letters are concerned,
they do not differ from one another. That a set of like quantities may be
added together, without our requiring to know what those quantities are,
is plain: thus, it is clear that 2a + 3a + 5a=10(2, whatever a may stand
for, and that 66 + 3a6 4a6=5a6, whatever ah may stand for. If how-
ever the quantities are not all like quantities, then to incorporate the
entire set into a single term as here, would be impossible. For instance,
if we had the set of quantities 4ax + 2a^ 3a^ + 26, all we could do would
be to actually collect the first three into one term, and then to annex to
the sum the fourth term 26 with its proper sign ; we should thus say that
4aa; + 2aic 3aa; + 26=3a2' + 26, an expression which we cannot further
reduce or simplify till we know something about the values of the letters.
10. Addition therefore divides itself into two cases : 1. When the
quantities to be added are all like quantities, and 2. When they are not
all like quantities.
Case I, When the quantities are all like quantities, that is, when
they differ in nothing but in their coefficients.
Rule I. Find the sum of the positive coefficients. 2. Find the sum
of the negative coefficients. 3. Take the difference of these two sums,
prefix to it the sign of the greater sum, and then annex the letters
common to all the quantities : the correct sum will thus be exhibited.
Note. When there are two or more columns to be added up, we
always commence with the first column on the left, and not, as in
arithmetic, with the first on the right, as it is more convenient to write
ADDITION OF ALGEBRA.
the several results, with their signs, from left to right, than from right
to left.
The following four examples are worked : the learner should clearly see
how the results are obtained, and satisfy himself of their accuracy, before
attempting the exercises below.
(1)
(2)
(3)
(4)
Zx
2ax-\-b
Gbx-Aa
Sabx-^Qlac
5x
dax+2b
-2bx+Sa
Zabx-2\aG
9x
ax-Zh
5bx2a
5abx-\-4:ac
-
-X
5ax-{-ih
12bx-6a
~-2abx-\-Zac
7x
ISx
\ax-bh
8bx-^9a
4:abx-\-ac
Sum i
IV^ax-h
l%x -
-Ubx-\-12ac
ExEECiS
3ES TO BE WOEKED.
(1)
(2)
(3)
(4)
5a
362/+ 4a
7ax-\-2byScz '
2lmx7ny-{-Z%.
Q
-la
2hy-Za
2ax-Zby+Acz
5imx-{-2ny^Q~.
ft
2a
- %+9a
Z^ax-hj+h^cz
-9|wa; Sny 6-.
9a
~t>hy-\- a
llax-{-5by2cz
15a
8by-2a
ax-\-Sby 7cz
13lmx-dny-7^.
(5) Add the following expressions: laxy2ih, Aaxy^b, 8axy-\-6b.
(6) 5ax+l, Bax-2, 6ax+i, -aa;+3, -7ax-5, 4ax-\-Z. (7) 8cz+2x-4, -
Zcz7x, 5czix-\-l, _9c2+3a;-8, 6x-\-2, 7. (8) Zaxy+2bz-ic, 6axyZbz-{-
7c, laxy-\-5bz~Zc, 2axy~bzQc, A\axybz-12c^ lZaxy-\-\bz-\-6Cj axy-{-ibz+
13c.
11. Case II. When the quantities are not all like quantities.
EuLE. Collect the like quantities from the several expressions, and
add them together : to the sum connect, with their proper signs, those of
the quantities which have no like.
Note. Athough it is of no consequence which set of like quantities
be added first, yet the custom is to commence with the quantity at the
top of the Jirst column on the left, and to put down, under that column,
the sum of all the quantities like it; then to collect the quantities like that
at the top of the second column, and so on, as in the following examples :
(1)
2x7y-{- iz
8z-\-2x y
2y-{-5z-\- a
4x-Zz+ 7y
8a;-3y+142+a
(2)
7ax2by-{-z
Zby -\-9z ax
62 -\-2ax-\-by
5by-Zax-\-Q
(3)
AocyzZocy-\- 2yz
5xy -\-Syz 7xyz
-9ax +xyz+ll
-iyz -\-7xy 8
5ax+7by+4z-^e 2xyz-{-9xy-\- 6yz9ax-\-d
12. ExEECisEs TO BE WORKED. In the following examples the
several expressions may be taken as they are, and placed one under
another as above ; or the arrangement of the terms may be changed, so