Elements of Algebra

CONTENTS. xxix Page Chap. IX. Of the Nature of Equations of the Second Degree - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 244 X. Of Pure Equations of the Third Degree - - - - - - - - 248 XI. Of the Resolution of Complete Equations of the Third Degree - - - - - - - - - - - - - - - - - - - - - - 253 XII. Of the Rule of Cardan, or that of Scipio Ferreo 262 XIII. Of the Resolution of Equations of the Fourth Degree - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 272 XIV. Of the Rule of Bombelli, for reducing the Re- solution of Equations of the Fourth Degree to that of Equations of the Third Degree - - - - 278 XV. Of a new Method of resolving Equations of the Fourth Degree - - - - - - - - - - - - - - - - - - - - - 282 XVI. Of the Resolution of Equations by Approxi- mation - - - - - - - - - - - - - - - - - - - - - - - - - - - - 289 PART II. PART II. Containing the Analysis of Indeterminate Quantities. Chap. I. Of the Resolution of Equations of the First De- gree, which contain more than one unknown Quantity - - - - - - - - - - - - - - - - - - - - - - - - - - - 299 II. Of the Rule which is called Regula Cæci, for de- termining, by means of two Equations, three or more Unknown Quantities - - - - - - - - - - - - - 312 III. Of Compound Indeterminate Equations, in which one of the Unknown Quantities does not ex- ceed the First Degree - - - - - - - - - - - - - - - - - - 317 IV. Of the Method of rendering Surd Quantities, of the form ($\sqrt{a + ax + cx^2}$), Rational - - - - - - - - - 322 V. Of the Cases in which the Formula $a + bx + cx^2$ can never become a Square - - - - - - - - - - - - - - 335 VI. Of the Cases in Integer Numbers, in which the Formula $ax^2 + b$ becomes a Square - - - - - - - - - 342 VII. Of a particular Method, by which the Formula $an^2 + 1$ becomes a Square in Integers - - - - - - - 351 VIII. Of the Method of rendering the Irrational Formula ($\sqrt{a + bx + cx^2 + dx^3}$) Rational - - - - - - - - - - 361 IX. Of the Method of rendering rational the incom- mensurable Formula ($\sqrt{a+bx+cx^2+dx^3+ex^4}$) 368 X. Of the Method of rendering rational the irrational Formula ($\sqrt[3]{a + bx + cx^2 + dx^3}$) - - - - - - - - - - 379 XI. Of the Resolution of the Formula $ax^2 + bxy + cy^2$ into its Factors - - - - - - - - - - - - - - - - - - - - - - - 387 XII. Of the Transformation of the Formula $ax^2 + cy^2$ into Squares and higher Powers - - - - - - - - - - - 396 XIII. Of some Expressions of the Form $ax^4 + by^4$, which are not reducible to Squares - - - - - - - - - 405