Notices respecting New Books, 371 



by Addition — Subtraction— Resolution of equations, Cases I. and II. 

 — Multiplication — Equations solvable by multiplication—Division — 

 Equations solvable by division — Resolution of equations, Case III. — 

 Algebraic functions— Resolution of equations, Case IV. — Involution 

 — Table of powers — Sir I. Newton's Rule for finding any power of a 

 binomial — Evolution — Table of roots — Resolution of equations, Case 

 V. — Proportion — Resolution of equations, Cases VI. VII. and VIII. — 

 Resolution of adfected quadratic equations — Resolution of equations 

 of all dimensions. 



Resolution of Equations of all Dimensions. 



" 1 80. Rule. — 1 . Arrange the terms of the given equation, whether 

 quadratic, cubic, biquadratic, or any higher dimension, in the order 

 of their powers, beginning with the highest, and place the numerical 

 or absolute terra on the right of the sign of equality, and all the other 

 terms on the left.— 2. Reduce the equation, if necessary, so that the 

 coefficient of the first term shall be unity j and supply the want of 

 any term in the regular series, putting zero for its coefficient. — 3. Di- 

 vide the absolute term into periods of as many figures each as there 

 are units in the index of the first term, if necessary ; and mark out 

 a place for the quotient on the right.— 4. Find by trial the first figure 

 of the required root of the equation, and place it in the quotient. — 

 5. Add this first figure to the coefficient of the second term and to each 

 successive sura, as often as there are units in the index of the first 

 term.— 6. Multiply each of these sums, except the last, by the first 

 figure, and add the products to the coefficient of the third terra and 

 to each successive sum. — 7. Proceed in this manner to the coefficient 

 of the last term, under which by this process will be found two sums -, 

 the first of which is the proper divisor for the first figure of the root, 

 and the second the trial divisor for the next. — 8. Multiply the first 

 divisor by the first figure of the root, and subtract the product from 

 the first period of the absolute term, bringing down the next period 

 to the remainder for a dividend. — 9. By means of the trial divisor, 

 find the second figure of the root, making some allowance for its in- 

 crement. — 1 0. Add this second figure to the last sum under the se- 

 cond term, and to each successive sum, in the same manner as was 

 done with the first figure j proceed to find the successive products 

 and sums as in finding the proper divisor for that figure, till the pro- 

 per divisor for the second figure of the root be found. — 11. Multiply 

 this divisor by the second figure in the root, and subtract the product 

 from the dividend, bringing down the third period, if any, to the re- 

 mainder for a new dividend.— 12. Proceed in the same manner till 

 the process terminate without a remainder, or till as many figures of 

 the root be found as are required. 



" 181. The method given in the preceding rule for the solution of 

 equations of all dimensions, supersedes the necessity of giving in this 

 short elementary treatise, the old methods of solving cubic and bi- 

 quadratic equations by the rules of Cardan, Tartaglia, Euler, and 

 others, as well as the various rules for approximating to the roots of 

 equations, which are to be found in all the larger works on algebra." 



"183. The solution of equations of all degrees bv the method from 

 3 B 2 which 



