A new work, by Professor Orson Pratt, has been laid on our table, entitled “Pratt’s Cubic and Biquadratic Equations.” From the limited examination of the work we have been able to make, we are pleased to adopt the language of the Author in the preface, as our own, in reference to this valuable acquisition to the science of mathematics.

“1. The Author’s discovery of the Equation of differences, together with several other kindred discoveries, resulting from the properties of this equation, has enabled him to entirely dispense with every process for finding the limits of the roots; to dispense with the theorem of the Strum, and all similar theorems, having for their object the determination of the number of real roots and their situation in the arithmetical scale; to dispense with all processes for finding the first figure of a root by trial or successive substitution; and to dispense with the successive trial divisors used by Horner.

2. By this new method the first figure of a root is found in the same manner that the first figure of any quotient is obtained; and each new divisor is derived, by a simple formula, from the figures of the roots already developed; and instead of each divisor being disjointed from the dividend, and placed in a column far distant, as is the case in Horner’s method, it is made to occupy its usual place on the left of the dividend, as in common division; thus reducing the whole process into a more compact and simple form, more in accordance with the usual arithmetical form of extracting the square root, which it, in some respects, resembles.

3. All those cases in which the roots approximate each other in value, hitherto considered so difficult of solution, become, by this method, exceedingly simple; indeed, the nearer two roots approach equality the less is the labor in the operation of development.

4. A new process, simple and expeditious, has been devised for obtaining the remaining roots of a Cubic or Biquadratic equation, after one root has been found, without resorting to the common, or more tedious method of depressing the equation.

5. A new general formula has been discovered, by which the three roots of a Cubic equation, when they are all real, can be obtained in terms of the co-efficients, without resorting to the process of development figure by figure.

6. A new and simple method of extracting the cube root is given, by which the labor becomes several times less than by the usual methods. This very expeditious process requires only about the same number of figures as extracting the square root, and constantly maintains the divisors in the same horizontal lines with their respective dividends.

7. General Cubic and Biquadratic Equations which have, in all cases, two equal roots, are given, and considered by the Author of considerable importance in their relative bearings upon other equations.

8. A “General Solution” of the Biquadratic Equation is given, resembling in some respects Descartes’ Solution, but differing in other respects from all solutions with which the Author is acquainted, by obtaining a resulting auxiliary Cubic Equation whose second term is absent. These are some of the peculiarities in this little treatise; but the reader is referred to the propositions in the body of the work for further information.”

We trust the work will meet with the reception it justly merits.

[Deseret News, July 19, 1866]

[transcribed and proofread by David Grow, Sept. 2006]


Local and Other Matters.
From Friday’s Daily, April 14.
Solution of Cubic and Bi-Quadratic Equations; by Orson Pratt, Sen.

The above work issued from the London Press, contains some discussions which are unique and valuable. Its author, Orson Pratt, is one of the leaders in the Mormon Church at Salt Lake City. In the preface the author claims, by the discovery of the “Equation of Differences” and corollaries resulting therefrom, to have dispensed with the abstruse theorem of Sturm and the very laborious method of Horner, in determining the limits and approximate numerical values of the real roots of an equation. We have given some attention to the author’s claims. While many of his discussions, such as the composition, divisibility and reduction of equations, are by no means new and novel, being substantially the same as may be found in any “Higher Algebra,” there are some discussions which we have never seen elsewhere and in which we have taken much interest. Among these may be mentioned a very simple and elegant method of transforming an equation into another whose roots shall be equal to those of the given equation multiplied by a given quantity, the “Equation of Differences” and the “Numerical Solution of Cubics and Biquadratics.” To those who are interested in the discussions of higher equations this volume will be found both interesting and valuable.—Illinois Weslyan University Alumni Journal.

[Deseret News, Apr. 19, 1876]

[transcribed and proofread by David Grow, Sept. 2006]

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