Cubic equation

“ So in the cubic equation x3 − px2 + qx − r = 0, if the roots are a, b, c, and if ω is used to denote an imaginary cube root of unity, ω2 + ω + 1 = 0, then writing for shortness p = a + b + c, L = a + ωb + ω2c, M = a + ω2b + ωc, it is at once seen that LM, L3 + M3, and therefore also 718 (L3 − M3) 2 are symmetrical functions of the roots, and consequently rational functions of the coefficients ”

“ We knew beforehand that it had to be a cubic equation. Each cubic equation has three roots for every value of the independent variable, which is in space. It also has three roots for every value of the time. Basically, that means that if any primal unit exists in our space, it exists in three forms, the positive, the negative and the neutral. These units are the positron, the negatron, and the neutron. Those three are the three solutions in space to the co-ordinates of the existential primal point. ”

“ A quartic equation is solvable by radicals, and it is to be remarked that the existence of such a solution depends on the existence of 3-valued functions such as ab + cd of the four roots (a, b, c, d) : by what precedes ab + cd is the root of a cubic equation, which equation is solvable by radicals: hence ab + cd can be found by radicals; and since abcd is a given function, ab and cd can then be found by radicals. But by what precedes, if ab be known then any similar function, say a + b, is obtainable rationally ”

“ To fit a curve to the given data, after the constants have been determined it is necessary to find, by solving a cubic equation for each value, the values of t corresponding to the x’s of the respective classes. ”

“ Hence 417 is an exact root of the given equation.
Since the extraction of a cube root is merely the solution of a pure cubic equation, x3=n, the process given above may be utilized in finding cube roots. This is [45] precisely what Oughtred does in chap. xiv of his Clavis. If the foregoing computation is modified by putting Cq=0, the process will yield the approximate cube root of 247651713.
Oughtred solves 16 examples by the process of approximation here explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In all cases he finds only one or two real roots. ”

“ Hence it follows that the roots of the proposed equation are generally expressed by the formula
y = √a + √b + √c; where a, b, c denote the roots of this cubic equation,
But to find each particular root, we must consider, that as the square root of a number may be either positive or negative, so each of the quantities √a, √b, √c may have either the sign + or − prefixed to it; and hence our formula will give eight different expressions for the root. It is, however, to be observed, that as the product of the three quantities √a, √b, √c must be equal to √R or to −1/8 q ”

“ When a biquadratic equation has all its terms, its resolution may be always reduced to that of a cubic equation. There are various methods by which such a reduction may be effected. The following was first given by Leonhard Euler in the Petersburg Commentaries, and afterwards explained more fully in his Elements of Algebra. ”

“ The whole theory of the inflections of a cubic curve is discussed in a very interesting manner by means of the canonical form of the equation x3 + y3 + z3 + 6lxyz = 0; and in particular a proof is given of Plücker’s theorem that the nine points of inflection of a cubic curve lie by threes in twelve lines.
It may be noticed that the nine inflections of a cubic curve represented by an equation with real coefficients are three real, six imaginary; the three real inflections lie in a line, as was known to Newton and Maclaurin. ”

“ We therefore write 7^ = + sin (equation will be susceptible of indefinite increase, which is contrary to our hypothesis. We are therefore entitled to assume that = 0. The two roots of the cubic for h must, therefore, be pure imaginaries, and thus we have the condition ffl + &! + c 2 = . Finally we have for the determination of x, y, z the following threo equations : I x + m y + n z = const. ”

“ The quantity b is the volume of a molecule, which he supposes to be an impenetrable body, and all the virtue of the equation lies in this term which makes the equation a cubic in V, which is required to account for the shape of certain isothermal curves. ”

“ If the value M3 had been taken instead of L3, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x3 + qx − r = 0, it will readily be seen that the two solutions are identical, and that the function r2 − 4⁄27 q3 under the radical sign must (by aid of the relation p = 0 which subsists in this case) reduce itself to (L3 − M3) 2; it is only by each radical being equal to a rational function of the roots that the final expression can become equal to the roots a, b, c respectively. ”

“ They are parallel to the edges of the cube, and in the different classes coincide either with tetrad or dyad axes of symmetry. Five classes are included in this system, in all of which there are, besides other elements of symmetry, four triad axes.
In crystals of this system the angle between any two faces P and Q with the indices (hkl) and (pqr) is given by the equation
The angles between faces with the same indices are thus the same in all substances which crystallize in the cubic system: in other systems the angles vary with the substance and are characteristic of it. ”

“ The name was also given by Arthur Cayley to the conoidal cubic surface which has for its equation z (x2 + y2) = 2mxy; every point on this surface lies on the line given by the intersection of the planes y = x tan θ, z = m sin 2θ, for by eliminating θ we obtain the equation to the surface. ”

“ It gives squares, cubes, square roots, and cube roots (to 7 places) , reciprocals to 9 or 10 places, and resolutions into their prime factors of all numbers from 1 to 10,000, the first ten powers of numbers to 100, fourth and fifth powers of numbers from 100 to 1000, prime numbers from 1 to 100,102, eight-place hyperbolic logarithms to 10,000, tables for the solution of the irreducible case in cubic equations, &c. In the stereotyped reprint of 1840 only the squares, cubes, square roots, cube roots, and reciprocals are retained. ”

“ Cubic equations, 112, 113, 142–145, 149, 152, 153. See Algebra.
Cubic residues, 366.
Culmann, 299, 300.
Curtze, M., 299.
Curvature, measure of, 314.
Curve of swiftest descent, 234, 238.
Curves, osculating, 226; quadrature of, 42, 49, 177, 190, 193, 202, 220; theory of, 226, 240, 242, 243, 292, 321. See Cubic curves, Rectification, Geometry, Conic sections.
Cusanus, 154.
Cyclic method, 96, 97.
Cycloid, 171, 173, 176, 187, 190, 191, 225, 234, 240.
Cyzicenus, 33.
Czuber, 340.
D'Alembert, 254–256; ref. to, 254, 258, 262, 265, 268–270, 386. ”

“ Jerome Cardan is, as it were, the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan’s name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. |Cubic equations.| The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; ”

“ It is a form of the theorem for the case D = 1, that the co-ordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be √1 − θ2·1 − k2θ2, and writing θ = snu, the radical becomes = cnu, dnu ”

“ It can then be shown that if a variable curve, φ, of order m be drawn, passing through the double points of the curve, the values of the argument u at the remaining intersections of φ with ƒ, have a sum which is unaffected by variation of the coefficients of φ, save for additive aggregates of the periods. In virtue of the birational transformation this theorem can be deduced from the theorem that if any straight line cut the cubic y2 = 4x3 − g2x − g3, in points (u1) , (u2) , (u3) , the sum u1 + u2 + u3 is zero, or a period ”

“ We may now state in a more convenient form the fundamental distinction of the kinds of cubic curve. A non-singular cubic is simplex, consisting of one odd circuit, or it is complex, consisting of one odd circuit and one even circuit. It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection ”

“ There is no limit to the number of different dimensions that will give two cubes whose sum shall be exactly seventeen cubic inches. Here is the answer in the smallest possible numbers. One of the silver cubes must measure 2-23278/40831 inches along each edge, and the other must measure 11663/40831 inch. If the reader likes to undertake the task of cubing each number (that is, multiply each number twice by itself) , he will find that when added together the contents exactly equal seventeen cubic inches. See also No. 20, "The Puzzle of the Doctor of Physic." ”

“ If a1 is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by a1 will pass through Q. There will therefore be one line of the congruence passing through Q, and only one. For if two such lines pass through Q, then the lines S1Q and S2Q will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it. Hence:—
Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn. ”

“ Again, when there is in question only one pair of points or lines, these, if coincident, must be real; thus, a line meets a cubic curve in three points, one of them real, and other two real or imaginary; but if two of the intersections coincide they must be real, and we have a line cutting a cubic in one real point and touching it in another real point. It may be remarked that this is a limit separating the two cases where the intersections are all real, and where they are one real, two imaginary.
Considering always real curves, we obtain the notion of a branch ”

“ Its theoretical measure varied.—The actual measure of the coffer has varied in the hands of all its twenty-six measurers. But even its theoretical measure is varied also; for the size which the old coffer really ought to have as "a grand capacity standard," is, strangely enough, not a determined quantity. In his last work (1867) , Professor Smyth declares, as just stated, its proper theoretical cubic capacity to be 71,250 pyramidal cubic inches. ”

“ Playfair, in his second dissertation in the Encyclopædia Britannica, though he cannot but condemn Cardan, seems to think Tartaglia rightly treated for having concealed his discovery; and others have echoed this strain. Tartaglia himself says in a passage I have read in Cossali, that he meant to have divulged it ultimately; but in that age money as well as credit was to be got by keeping the secret; and those who censure him wholly forget, that the solution of cubic equations was, in the actual state of algebra, perfectly devoid of any utility to the world. ”

“ Thus if a body were subjected to three strains, at right angles to each other, the total change in the volume of the body would be approximately equal to the sum of the changes produced by the separate strains, provided that these are very small. In like manner not only is the expansion of every solid and liquid substance by heat approximately proportional to the change of temperature, when this change is very small in amount, but the cubic expansion may also be considered as being three times as great as the linear expansion. ”

“ The full symmetry of the cubic system is not realised, however, by a study of the cube alone; 39we only appreciate it when we come to examine the general form of the cubic system, that which is produced by starting with a face oblique to all three axes, and with different amounts of obliquity to each, and seeing how many repetitions of the face the symmetry demands. ”

“ And it then appears that there are two kinds of non-singular cubic cones, viz. the simplex, consisting of a single sheet, and the complex, consisting of a single sheet and a twin-pair sheet; and we thence obtain (as for cubic curves) the crunodal, the acnodal and the cuspidal kinds of cubic cones. It may be mentioned that the single sheet is a sort of wavy form, having upon it three lines of inflection, and which is met by any plane through the vertex in one or in three lines; the twin-pair sheet has no lines of inflection, and resembles in its form a cone on an oval base. ”

“ A sphere cut from a uniaxial crystal at one temperature will be a spheroid at another temperature. In biaxial crystals there are different coefficients of expansion along three rectangular axes, and a sphere at one temperature will be an ellipsoid at another. A result of this is that for all crystals, except those belonging to the cubic system, the angles between the faces will vary, though only slightly, with changes of temperature. ”

“ In the same manner as a cubic body contains dimensions within its cubic form, so also a cubic place contains dimensions under its cubic surface; hence the place of a body has volume, the same volume as the body ”