INDEX.
Ahmes, 151.
Algebra, definition of, 2; history of, 54 et seq., 151; essence of, 55; the name of, 59; among the Arabs, 59 et seq., 152; in Europe, 60; in Italy, 64; in India, 152; the generality of, 69; hand-writing of, 69; application of geometry to, 100 et seq , 127 et seq.
Algebraical resolution of equations, limits of the, 96.
Alligation, generally, 44 et seq.; alternate, 47.
Analysis, indeterminate, 47 et seq., 55.
Angle, trisection of an, 62, 81.
Angular sections, theory of, 80.
Annuities, 16.
Arabs, Algebra among the, 59 et seq., 152.
Archimedes, 54, 58 footnote, 151.
Arithmetic, universal, 2 et seq.; operations of, 24 et seq.
Arithmetical progression revealing the roots, 112 et seq., 120.
Arithmetical proportion, 12.
Astronomy, mechanics, and physics, curves of errors in, 136.
Average life, 45 et seq.
Bachet de Méziriac, 58.
Ball, 152.
Binomial theorem, 115.
Binomials, extraction of the square roots of two imaginary, 77.
Biquadratic equations, 63, 88, 94, 133.
Bret, M., 93 footnote.
Briggs, 20.
Buteo, 61.
Cantor, 54 footnote; 60, footnote, 152.
Checks on multiplication and division, 39.
Circle, 144; squaring of the, 62; and inscribed polygon, problem of the, 138.
Coefficients, indeterminate, 89; greatest negative, 107 et seq., 117.
Common divisor of two equations, 121.
Complements, subtraction by, 26.
Constantinople, 58.
Continued fractions, solution of alligation by, 50 et seq.
Convergents, 7.
Cube, duplication of the, 62.
Cube roots of a quantity, the three, 70.
Cubic radicals, 75.
Curves, representation of equations by, 101 et seq; employment of in the solution of problems, 127–149; method of, submitted to analysis, 143 et seq.; advantages of the method of, 135, 144.
Decimal, fractions, 9; numbers, 27 et seq.
Decimals, multiplication of, 30; division of, 31.
DeMorgan, v.
Descartes, viii, 60, 65, 89, 93, 127.
Differences, the equation of, 114 et seq., 123.
Differential Calculus, 131.
Diophantine problems, 55.
Division, by nine, 34; by eight, 34; by seven, 34 et seq.; of decimals, 31.
Divisor, greatest common, 2 et seq.
Dühring, E. v.
Duodecimal system, 32.
Economy of thought, vii.
Efflux, law of, 42.
Eleven, the number, test of divisibility by, 37.
Elimination, method of, 121; general formulæ for, 122.
Equations, of the second degree, 56; of the third degree, 60, 66, 82; of the fourth degree, 63, 87, 133; of the fifth degree, 64; theory of, 65, 84; biquadratic, 88; limits of the algebraical resolution of, 96; of the fifth degree, 96; of the mth degree. 96; general remarks upon the roots of, 102 et seq.; graphic resolution of, 102; of an odd degree, roots of, 105; of an even degree, roots of, 106; real roots of, limits of the, 107 et seq.; common divisor of two, 121; constructions for solving, 100 et seq., 124; a machine for solving, 126.
Equi-different numbers, 13.
Errors, curve of, 136 et seq.
Europe, algebra in 60.
Experiments, average of, 46; an expedient for supplying lacunae in a series of, 148.
Falling stone, spaces traversed by a, 42.
False, rule of, 137.
Fermat, 58.
Ferrari, Louis, 64.
Ferreus, Scipio, 60 et seq.
Fifth degree, equations of the, 96.
Fink, 152.
Fourth degree, equations of the, 133.
Fractional expressions in equations, 134.
Fractions. 2 et seq.; continued, 3 et seq.; converging, 6; decimal, 9; origin of continued, 10.
Galileo, ix.
Geometers, ancient, 54 et seq., 58, 59.
Geometrical, proportion, 13; calculus, 24.
Geometry, 24, 60; application of to algebra, 100 et seq., 127 et seq.
Germany, 61.
Girard, Albert, 62.
Grain, of different prices, 44.
Greeks, mathematics of the, vii, 54 et seq., 151.
Hand-writing of algebra, 69.
Harriot, 65.
Horses, 43.
Hypsicles, 151.
Imaginary binomials, square roots of, 77.
Imaginary expressions, 79 et seq., 83.
Imaginary quantities, office of the, 87.
Imaginary roots, occur in pairs, 99.
Indeterminate analysis, 47 et seq., 55.
Indeterminate coefficients, 89.
Indeterminates, the method of, 83.
Ingredients, 48.
Interest, 15,
Intersections, with the axis give roots, 102 et seq , 113.
Inventors, great, 22.
Involution and evolution, 11.
Irreducible case, 61, 65, 69, 73, 82.
Italy, cradle of algebra in Europe, 61, 64.
Laborers, work of, 41.
Lagrange, J. L., v, vii et seq.
Lavoisier, xii.
Leibnitz, viii.
Life insurance, 45 et seq.
Life, probability of, 46.
Light, law of the intensity of, 129.
Lights, problem of the two, 129 et seq.
Logarithms, 16 et seq., 40; advantages in calculating by, 28; origin of, 19; tables of, 20.
Machine for solving equations, 124–126.
Mathematics, wings of, 24; exactness of, 43; evolution of, vii.
Mean values, 45 et seq.
Mechanics, astronomy, and physics, curves of errors in, 136.
Metals, mingling of, by fusion, 44.
Méziriac, Bachet de, 58.
Minimal values, 132.
Mixtures, rule of, 44 et seq., 49.
Mortality, tables of, 45.
Moving bodies, two, 98.
Multiple roots, 105.
Multiplication, abridged methods of, 26 et seq.; inverted, 28; approximate, 29; of decimals, 30.
Music, 22.
Napier, 17 et seq.
Napoleon, xii.
Negative roots, 60.
Newton, his problem, 145; viii.
Nine, property of the number, 31 et seq.; property of the number generalised, 33.
Nizze, 58 footnote.
Numeration, systems of, 1.
Numerical equations, resolution of, 96–126; conditions of the resolution of, 97; position of the roots of, 98. See Equations,
Observations, expedient for supplying lacunae in series of, 148.
Observer, problem of the, and three objects, 140.
Oughtred, 30.
Pappus, 59.
Parabolic curves, 144 et seq.
Peletier, 61.
Peyrard, 58.
Physics, astronomy, and mechanics, curves of errors in, 136.
Planetarium, 9.
Point in space, position of a, 139.
Polygon, problem of the circle and inscribed, 138.
Positive roots, superior and inferior limits of the, 109.
Powers, 10 et seq.
Practice, theory and, 43.
Present value, 15.
Printing, invention of, 59.
Probabilities, calculus of, 45 et seq.
Problems, 110; for solution, 62; employment of curves in the solution of, 127–149.
Proclus, 59.
Progressions, theory of, 12, 14.
Proportion, 11 et seq.
Ptolemy, 59.
Radical expressions in equations, 134
Radicals, cubic, 75.
Ratios, constant, 42; 2, 11 et seq.
Reality of roots, 76, 83, 85, 93.
Remainders, theory of, 34 et seq., 38. negative, 35 et seq.
Romans, mathematics of the, 54.
Roots, negative, 60; of equations of the third degree, 71; the reality of the, 74, 76, 79, 83, 85, 93; of a biquadratic equation, 94; multiple, 105; superior and inferior limits of the positive, 109; method for finding the limits of, 110; separation of the, 112; the arithmetical progression revealing the, 112 et seq., 120; quantity less than the difference between any two, 113; smallest, 116 et seq.; limits of the positive and negative, 119.
Rule, Cardan’s, 68; of false, 137; of mixtures, 44 et seq.; of three, 11 et seq,, 40 et seq.
Science, history of, 22; development of, vii et seq.
Seven, tests of divisibility by, 35.
Short-mind symbols, vii et seq.
Signs + and —, 57.
Squaring of the circle, 62.
Stenophrenic symbols, vii et seq.
Straight line, 144.
Substitutions, 111 et seq., 123.
Subtraction, new method of, 25 et seq.
Sum and difference, of two numbers, 56.
Supposition, rule of, 137, 148.
Symbols, vii et seq.
Tables, 137; expedient for supplying lacunæ in, 148.
Tannery, M. Paul, 58 footnote, 151.
Temperament, theory of, 23.
Theon, 59.
Theory and practice, 43.
Theory of remainders, utility of the, 38.
Third degree, equations of the, 71, 82.
Three roots, reality of the, 93,
Trial and error, rule of, 137, 148.
Trisection of an angle, 62, 81.
Turks, 58.
Undetermined quantities, 82.
Unity, three cubic roots of, 72.
Unknown quantity, 55.
Values, mean, 45 et seq.; minimal, 132.
Variations, calculus of, x.
Vatican library, 58.
Vlacq, 20.
Wallis, viii.
Wertheim, G., 58 footnote.
Woodhouse, x.
Xylander, 58.