solving a numerical equation, one a tedious method, discovered, according to the Preface, forty years before publication; and the other, printed as a supplement to the former, discovered, Mr. Holdred says, after the work had been announced for publication. Between this latter method and the one published by Mr. Horner, there is a remarkable resemblance. The name of this latter gentleman is not, however, mentioned in Mr. Holdred's Tract, nor is there any hint given as to the precise period when the alleged discovery was made, although from a remark in the last page of the Preface, it would seem that the publication of the work was delayed "some weeks," in consequence. This, however, is matter of but little moment as respects Mr. Horner's claims, for as this gentleman's paper had been six months before the public when that of Mr. Holdred first appeared, there can be no reasonable doubt as to whom the honor of the discovery belongs. Mr. Atkinson's method of solution is identical with that of Mr. Holdred's first method,—so completely identical that both papers appear like the work of one person. Mr. Atkinson, in the Preface to his "New Method of Extracting the Roots of Equations," published in 1831, at Newcastle, brings forward convincing proof that his method was publicly read at the Literary and Philosophical Society of Newcastle-upon-Tyne in August, 1809; and there seems to be no reason for doubting that this method was equally the independent discovery of Mr. Holdred and Mr. Atkinson; as, however, it is entirely superseded by that of Mr. Horner, there would be little interest created by either establishing or controverting this opinion. The present publication is intended to embrace all that is important in the researches to which I have just alluded, and thus to supply the English student with a treatise on the subject of numerical equations, adapted to the present improved state of that important branch of analysis. I shall rejoice if I be found to have succeeded in rendering these researches intelligible to the young mathematician; or if this little work shall be the means of facilitating their introduction into the mathematical courses of instruction prescribed in our public seminaries of education. Belfast College; J. R. YOUNG. PREPARING FOR THE PRESS, (Second edition, carefully corrected and improved throughout,) in one volume, 8vo. price 10s. 6d. cloth, ELEMENTS OF THE DIFFERENTIAL CALCULUS, With its application to the General Theory of Curve Surfaces and of Curves of double Curvature. BY J. R. YOUNG. On the 1st of October will be published, handsomely printed in quarto, price 7s. ON THE GENERAL SOLUTION OF NUMERICAL EQUATIONS, BY M. C. STURM. Translated from the Mémoires presentés par des Savans Etrangers à l'Académie Royale des Sciences. Tome vi. BY W. H. SPILLER. 4. The difference of any two powers always divisible by the dif- 5. If a be the root of an equation in x, the first member must be 6. Every equation of the nth degree has n roots 7. Determination of the equation from knowing the roots 8. An equation of the nth degree cannot have more than n roots 9. One root being known to determine the depressed equation 10. Rule for finding the coefficients of the depressed equation 11. To determine what functions the coefficients are of the roots 12. If the signs of the alternate terms of an equation be changed, the signs of all the roots will be changed 13. If the coefficient of the first term be unity, and the other coeffi- PAGE. 15. Useful deductions from the preceding proposition 16. Demonstration of the rule of Descartes 17. Use of the rule of signs in detecting imaginary roots 18. When the roots are all real, the rule of Descartes will discover 21. Numerical process for effecting the transformation 22. To transform an equation into another, whose roots are the 23. To transform an equation into another, whose roots shall be given multiples or submultiples of those of the former 24. To remove any proposed term from an equation 25. Formula for the solution of a quadratic, deduced from the pre- 27. When the second term is negative, and all the others positive, the coefficient of the second term taken positively, is a supe- 1 when they comprise an odd number 43 On Newton's Method of finding a Superior Limit, and on the 38. Newton's method of finding the limit 39. Another mode of obtaining the result of Newton's rule - Application of Newton's method, combined with the rule of 48. Directions for determining the nature and situation of the real b |