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Bhaskara .... Binomial theorem. Bombelli .... Bounded variable Boyle Briggs Briggsian logarithms Biirgi 88, 282 186 11 53 493 53 448 246 282 282 303 Cantor 19 Cardano 63, Cardan's formulas Case Cauchy 208, Changing sign of roots . .. Characteristic Cof actor 337, Combination Common difference Common ratio Commutative law, of addition. of multiplication Comparison tests Complex numbers addition of conjugate division of equality of multiplication of , 53 201 201 367 264 154 282 384 362 81 86 1 2 471 34 38 61 47 36 44 501 502 INDEX Complex numbers — Continued polar form of 44 subtraction of 40 Complex roots of unity. 189 Components, of a directed line segment 33 of a displacement .... 37 Composite, number .... 3 quadratic function .... 392 Compound events 368 Compound interest ... 306, 309 Compound interest law. 311 Conditional equation .... 211 Conditionally convergent series 484 Conies 407 Conjugate complex numbers. 51 Conjugate of a quadratic surd. 114 Constant 55 Construction of regular poly- gons 193 Continuity, of a function. 166, 457 of an integral rational func- tion 156, 458 of a fractional function. 458 Convergent series 466 Cooling bodies 315 Coordinates 70 Cubic equation 197 reduced form of 197 Cyclotomic equations .... 193 D'Alembert 209 Dalton 347 Dampened vibrations .... 312 Dedekind 19, 53 Defective 244 Degree of integral rational func- tion 129, 391 Delian problem 186 De Moivres formula .... 189 Denominator 8 Dense 17 Density 62 variation of density in atmos- phere 313 Dependent events 368 Dependent variable .... 55 Depressed equation .... 166 Derivative, of a function. 139 of an integral rational func- tion 146 of higher order 146 Descartes 53, 168 Descartes's rule 168 Determinant, of second oi'der. 325 of third order 334 of ni\\ order 377 Dickson 180, 202 Difference 1, 2 Dimension 61, 391 Dimensional symbols .... QQ Diminishing roots of an equa- tion 149 Directed line, positive sense of. 21 Directed line-segment of a directed line 21 absolute value of 22 measure of 22 numerical value of ... 22 origin of 21 standard position of ... 22 terminus of 21 Directed line-segment of a plane 31 absolute value of 32 amplitude of 32 argument of 32 components of 33 modulus of 32 origin of 32 standard position of ... 32 terminus of ... .32 Discriminant, of quadratic equation 106 of quadratic function of two variables 395 Displacement 36 components of 37 resultant of two 37 Distributive law 2 Divergent series 467 Dividend 4 Division, by a positive or nega- tive divisor 28 by zero 29 INDEX 503 Division, of complex numbers. 47 of integers 4, n of positive rational numbers. Equivalence, of fractional equa- tions 243 of in-ational equations . . . 262 of quadratic equations . . . 00 of systems of simultaneous equations 411 Errors of observation .... 375 Euclid 6 Euclid's algorithm .... 6 Euler .... 207, 209, 435, 494 Events 366 compound 368 dependent 368 exclusive • 3(i8 independent 3()8 Exclusive events 368 Expansion, binomial .... 493 Taylor's 148 Expectation 374 Exponential, equations . . . 301 functions 274 series 405 Exponents, fractional, negative. and vanishing 265 irrational 270 Factor, highest common. 229 of an integer 3 of proportionality .... 67 prime 6 Factored form of rational func- tion 2;S3 Factorial 353 Factor theorem 132 Falling bodies 66 False position, mt'thod of . . 158 Fermat 435 Ferrari 207 Ferro 201 Field 114 Floating spheres 182 Follows  terms . . 10 Fractional equations .... 241 Fractional exponents .... 267 Fractional rational functions . 226 improper 226 in lowest terms 228 proper 226 reduction to lowest terms . . 229 Function 55 linear, quadratic, etc. See adjectives. Functional notation .... 131 Fundamental laws . .1.2. .], 12, 13 for coiiiplex imiiibers ... 51 Fundamental theorem of algebra 208 Galois 208 Gauss 5:], 104, 209 Gauss plane 63 Gay-Lussac 246 General term of a sequence . . 429 Geometric means . . . . 87, 88 Geometric progression ... 86 common ratio <if 86 .sum of 87 with infinitely many terms . 89 504 INDEX Girard 203, 209 Greater 11 Greatest common divisor . . 5 Gunter 290 Gunter's scale 289 Harmonic means 85 Harmonic progression .... 85 Hermite 208 Higher progressions .... 431 Highest common factor . . . 229 Homogeneous linear equations, with two unknowns .... 326 with three unknowns . . . 345 with n unknowns .... 387 Horner 164 Horner's method 164 Hyperbola 232, 405 asymptotes of rectangular . 232 i 35, 50 Identity 211 Imaginary numbers .... 50 Independent events .... 368 Independent variable .... 55 Indeterminate forms .... 460 Index laws 265 Indirect analysis 348 Induction, mathematical . 134, 434 Infinite roots of an equation . 224 Infinite series 466 convergent 466 divergent 467 oscillating • 467 Infinitesimal 448 Infinity 447 Inflection, point of 218 Integer 1 Integral rational function, of one variable 129 of two variables 391 Interest, simple 306 compound ,306, 309 Interpolation 285, 431 Inversion 359 Irrational, equations .... 261 exponents 270 functions 249 numbers 17 roots of quadratic equation . 112 Jacobi 264 Kelvin 312 Klein 186 Lagrange 208, 299 Legendre 435 Leibnitz . 139 Lemma 133 Length 61 Leonardo of Pisa 53 Less 11 Light, transmission of . . . 314 Limit 444 Linear equations, with one un- known 78 with two unknowns .... 321 with three unknowns . . . 342 with n unknowns .... 387 Linear function, of one variable 77 of two variables 320 Logarithm 277 common 281 natural 296, 299 Logarithmic, decrement . . . 312 increment 312 paper . 317 scale 289 series 496 tables 498 Lowest terms 10 Mannheim 292 Mantissa 282 Mariotte 246 Mass 62 Mathematical Induction . 134, 434 Maximum or Minimum, of an integral rational function . 173 of a quadratic function ... 97 Measure of a directed line-seg- ment 22 INDEX 505 Method, Horner's 164 Newton's 157 of false position 158 of mathematical induction 134, 434 of small corrections .... 420 of UQcletermined coefficients . 236 Minimum. See .Maximum. Minor of a determinant, of the third order .... 335 of the Jith order 382 Minuend 2 Modulus 32, 39 Monotonic laws 2 Mortality, table of . . . 373, 500 Motion, uniform 120 of a projectile 126 Multiple, lowest common . . 6 poles of a rational function . 231 roots of an equation .... 175 zeros of a rational function . 230 Multiplication, of complex num- bers 44 of positive integers .... 1 of positive and negative num- bers 26 of rational numbers .... 13 of the roots of an equation by ?» 153 Multiplicity of a zero or pole 230, 231 Napier 303 Napierian base 298 Natural logarithms . . 296, 298 Negative, exponents .... 268 numbers 20 Newton 139, 141 Newton's laws of motion 121, 123 Newton's method 157 Normal form for expressions which involve (luadratic surds 260 Normalization of a system of simultaneous quadratics . 413 Numbers, negative, complex, etc. See adjectives. Numerator 8 Numerical value 22 NuSez 59 Order, notion of 80, 352 of an integral rational func- tion of one variable. 129 of an integral rational fimc- tion of two variables. 391 Ordinate 70 Origin, of coordinates .... 70 of line-segment 11 of directed line-segment on a directed line 21 of directed line-segment in a plane 32 of scale of integers .... 7 Oughtred 290 Parabola 94 vertex and axis of ... 98 Partial fractions 234 Periodic decimal 91 Permanence, principle of. 268 Permutation 362 even or odd 359 principal 359 Plane, of the complex variable. 53 Gauss 53 Point of inflection 218 Polar form of a complex number 44 Pole of a rational function. 231 Polygons, construction of regu- lar. 193 Pound of force 125 Poundal 125 Power function 272 Power series 486 Precedes 11 Premium 373 Pressure, in the atmosphere. 313 of gases 245 Prime, factors 6 number. .- 3 relatively 6 Principal 306 Principal value, of Vx. 250 of y/x^ 263 506 INDEX Principle of permanence. 268 Probability 366 Product 1, 2 geometric construction of. 13 Progressions, arithmetic ... 81 geometric 86 harmonic 85 Ptolemy 52 Pythagoras 19, 53 Pythagoreans 19 Quadratic equations .... 99 complex roots of 104 discriminant of 100 equivalence of .... 99, 100 pure 109 rational and irrational roots of 111 roots of 101, 103 with zero roots 109 Quadratic function of one vari- able 94 graph of 95 maximum or minimum of. 97 standard form of 94 zeros of 99 Quadratic functions of two vari- ables 392 composite or non-composite. 392 discriminant of 395 Quadratic surds 112 Quartic equation 204 Radicals, properties of. 254 Rate of interest 306 Rational function 226 Rational numbers 8 Rational roots, of quadratic equation. Ill of equations of higher degree 179 Rationalizing the denominator 114, 260 Ratio test 474 Real numbers 50 Reciprocal 15 Redundant 244 Reflection 252 Regula falsi 158 Regular polygons, construction of 193 Relatively prime 6 Remainder 4 Remainder theorem .... 134 Resultant, of two displacements 37 of two equations 389 Riemann 264 Rolle's theorem 175 Roots of quadratic equation 101, 103 Scale, notion of 292 of positive and negative num- bers 20 Second derivative 148 Semi-logarithmic paper. 316 Sequence •. 80, 429 Series 433 Simultaneous equations 322, 407, 409 Skinner 374 Slide rule 290 Slope 73 of tangent of a curve. 136 Smith 186, 303 Solution of an equation. ^ ■■ ■ -I li- /t If s t ..:f ^ if ;«'r ^^■. '•=# : < , >1 Vf V ^^ ^ i ifi *t .•V, «%,■«!,. if. V. ^ .■■■'^^("^^Jl' ^' -4' If >f ,. 