Definition of trigonometry, 1; undulating magnitude, 1; periodic
magnitude, 2; suggested by angular magnitude, 2; grādual mea-
surement of angle, 3; factors of 360, 3; circumference of circle, 4;
π, 4; multiplication and division by π, 5; arc÷radius, 5; arcual
measurement of angle, 5, 6; gradual and arcual comparisons, 6;
gradual measurement of arc, 6.
CHAPTER II.
On the Trigonometrical Functions, and on Formulæ of One Angle.
Axes, origin, projections, co-ordinates, abscissa, ordinate, 7;
r, 0, x, y, 7; sign of r, x, y, 8; four quarters and their signs, 8;
℗ and 2mπ + 0, 9; sine, cosine, tangent, cotangent, secant, cosecant,
versed sine, coversed sine, 9; complement, supplement, opponent,
completion, 10; trigonometrical functions as abstract numbers and
multipliers, 10; curve of sines, &c., 11; fundamental equations,
11, 12; limits of value, 12; signs, 12, 13; negative sign of r, 13;
initial and terminal values, 13; cosine even, sine odd, 13; tangent
odd, 14; 1⁄2mπ ±0 and its rules, 14, 15; double value of functions,
16; 15°, 18°, 30°, 45°, 60°, 72°, 75°, 16, 17; sin 0 : 0, (1 cos 0) ÷ 0,
tan 0 ÷ 0, 17, 18; 0 and 1 – 102, 18; older system of definitions,
18, 19; area of circle and sector, 20.
Extended notion of projection, 21; distinction of AB and BA,
and consequences, 21; similar distinction as to angles, 21; signs
of projections, 22; general investigation of cos (±0) and sin (4±0),
23; connexion of the formulæ, 24; cases of limited demonstration,
25, 26; collection of formulæ, 26, 27; remarks on the formulæ,
28, 29; cosine and sine of the sum of any number of angles, 29, 30;
cos no and sin n0, 30, 31; trisection of an angle, 31; series for cos 0
and sin 0, 32, 33; ditto for tan 0, 34; algebraic definition of trigo--
nometry, 34; cos" and sin"0, 34, 35, 36.
On the Inverse Trigonometrical Functions.
Functional notation, direct and inverse, 37; inverse trigono- metrical functions, 37, 38; examples in the use of inverse symbols, 38, 39, 40.
CHAPTER V.
Introduction of the unexplained Symbol √ — 1.
−
Remarks on the evidence of √ - 1 in this chapter, 41; connecting
formulæ of trigonometricals and exponentials, 42; De Moivre's
theorem, 42; multiplicity of directions in 0÷n, 43, 44; roots,
particularly of unity, 45, 46; transformations of a+b√ 1, and
selection of meaning in tan-1 (b÷ a), 46; extension of logarithms
with Naperian base, 47; extension of the Naperian base, 48;
isolated case of coincidence of logarithms in different systems, 48;
the negative quantities which have real logarithms, 49; equivalents
of De Moivre, 49; deduction of ordinary formulæ from them, 49, 50;
reduction of sine cos" to a linear form, 50, 51; mode of finding
Σan cos no x” and Σa, sin n0 x2, 51, 52; connexion of ø (x+y √ − 1)
and (x - y √ − 1), 52, 53; examples, 53, 54, 55; series for
tan-1x, 55; calculation of π, 55, 56; inverse connexion of trigo-
nometricals and exponentials, 57; use of multiplicity of value
of logarithms, 57, 58; resolution of sin into factors, 58, 59, 60;
Wallis's form of π, 61: deduction of approximate form for 1.2.3...~,
61, 62; factors of cos 0; logarithms of sin e, cos 0, tan 0, 63; Vieta's
expression for T, 63; Bernoulli's numbers, 64; series for tan x,
cotx, and (− 1) ̄1, 65.
CHAPTER VII.
On the Trigonometrical Tables.
Arrangement and extent of the tables, 71, 72; distinction of real
and tabular logarithms, 72; tables recommended, 73; argument,
interval, function, difference, interpolation, 73; method of inter-
polating, 74; choice of functions for accuracy, 75; tangents of
angles near to 90°, 75, 76; first notions of the construction of tri-
gonometrical tables, 76, 77, 78.
CHAPTER VIII.
On the Solution of Triangles.
Meaning of solution, 79; formulæ for right-angled triangles,
79, 80; cases of ditto ditto, 80; tabulated example, 30; formulæ
for oblique triangles, 81, 82, 83; cases of ditto ditto, 84, 85; mode
of entrance of double solution, 85; tabulated example, 86, 87;
mention of occasional rules, 87; reduction of triangular formulæ
to identities, 88.
CHAPTER I.
Description of a Symbolic Calculus.
Object, 89; peculiar symbols, meanings, rules of operation, 89;
possible deficiencies of either, 90, 91; complete absence of either,
91, 92; symbolic calculus, what, 92; recovery of meaning, significant
calculus, 93; slight example, with illustration of defects, 93, 94;
possibility of more than one mode of restoration to significance, 94;
step from specific to universal arithmetic, and thence to ordinary algebra, 95, 96; necessity for other than numerical distinction, and mode in which distinction is suggested by algebra, 95, 96; single algebra, phrase whence derived, 96, 97; twofold use of its signs, directive and conjunctive, 97; remarks on the progress of algebra, 98, 99, 100.
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CHAPTER IV.
Preliminary Remarks on Double Algebra.
Suggestion on √ −1 which gave rise to it, 109; the old algebra to be incorporated, 109; examination of magnitude of one dimension, time, gain and loss, 109, 110; a wider basis of significance in length affected by direction, 110, 111; geometrical introduction extension, not restriction, 111, 112; mode of making the application to problems of one dimension, 112, 113, 114; separation of subject-matter and. operative direction in arithmetical addition and multiplication, -15, 116.
CHAPTER V.
Signification of Symbols in Double Algebra.
Reason of the term double algebra, drawn from the meaning
of an isolated symbol, 117; meaning of =, 117; origin, 117; unit-
line, 118; axes of length and direction, 118; meaning of A + B,
A - B, B, 118; coincidences of common and extended addition,
&c., 118; addition really junction and its result joint effect, 118;
meaning of A× B and A ÷ B, 119; symbolic representation of
double-meaning symbol, (a, a), 119; coincidences of common and
extended multiplication, &c., 120; roots and powers, without refer-
ence to exponents, 120; developed expression of an algebraical
theorem, 121; construction of all double symbols by single ones
and √ −1, 122; re-introduction of trigonometry, 122; demonstration
of the symbolic rules, 123, 124, 125; deduction of fundamental
trigonometrical formulæ, 126; proof of the validity of that deduction,
and of its coincidence with ordinary proofs, 127, 128.
CHAPTER VI.
On the Exponential Symbol.
Assumption of arithmetical logarithms, definition of logometer,
129; choice of logometer, 129; multiple values of logometers, 130;
definition of AB, 130; limitation of &, 130; logometric equations,
131; proof of symbolic rules, and limitations, 131; proof of
g@√-1 — cos@ + sinë. √ — 1, 132; connexion of ɛ and the arcual unit,
132, 133, 134; transformation of Rs, 134; example of reduction
to significance, 135; fallacy exposed, 136; formulæ which supply
those of p. 131 when the limitations are removed, 136, 137.
CHAPTER VII.
Miscellaneous Remarks and Applications.
Extension of logarithms, 138; forms of ø (a + b √√ − 1), 138, 139;
extension of trigonometrical terms, 139, 140; discontinuity of pas-
sage from to in single algebra, 140; interpretation of a problem
impossible in single algebra, 140, 141; Cauchy's theorem on the
limits of imaginary roots of equations, 141, 142, 143, 144; Paradox
of single algebra which disappears in double algebra, 144, 145;
change of geometrical representation in passing from real to ima-
ginary, 145, 146.
CHAPTER VIII.
On the Roots of Unity.
Power of considering (± 1)" as either quantitative or directive,
147, 148; properties of the roots of + 1, 148, 149, 150, 151, 152, 153,
154; an solution of x2 = 1 when n is prime and 2-1 = 1 has been
solved, 155, 156, 157; Gauss's accession to Euclid's geometry, 157;
properties of the roots of -1, 158; formation of recurring expressions,
158, 159; Thomas Simpson's method of section of series, 159, 160.
CHAPTER IX.
Scalar View of Algebraical Symbols.
Law of ascent of algebraical operations, 161; illustrative notation,
162; scalar function, 162; ascent of algebraical operations, 162;
imperfection of ordinary notation for scalar representation, 163;
other law which this notation follows, 164; general law of scalar
ascent, 164; limitation of the scalar function, 164; inverse operation,
165; fundamental basis of algebra, 166; scalar notation in extension
of that of ordinary algebra, 166, 167.
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