SYNTHESIS. 542=(square of A's payment=) 4a3+3a2= (sum of the squares of B's and C's=) 25a2. Moreover A's payment (5a) exceeded B's (4 a) by a, and C's (3a) by 2 a. Q. E. D. PROBLEM 13. It is required to divide 11 into two such parts, that the product of their squares may be 784. ANALYSIS. Let a=11, b=784, x and y=the parts required; then by the problem, x+y=a, and xy2=b; from the first, y=ax; the square of this value substituted for y2 in the second, gives a-xxx2=b, whence by evolution a-x.x=b; that is, ax-x2= a2-4b b, or x2-ax=-b, x2-ax+ 4 4=±2, and x= Q. E. I. SYNTHESIS. First, a2 a+R 2 √b= -=a. 4 =) a+Ra2+2aR+R2 a2-2aR+R2 a*-2a2R2 + R+ restoring the value of R2= a2-4√b=) 16 X = (by b. Q. E. D. The solution of the problem in numbers, is x= a+a2-4b_11+√121-4784 PROBLEM 14. Given the sum of two numbers 24, and the product equal thirty-five times their difference, to find the numbers? ANALYSIS. Let x and y be the numbers required, s=24, m= 35; then by the problem, x+y=s, and xy=(m.x-у=) тх-ту. From the first, y=s-x; this value substituted in the second, gives sx-x2= (mx-ms+mx=)2 mx-ms, or x2+2m-s.x=ms; whence (putting a=2m-s) x2+ax=ms, SYNTHESIS. First, 2 s. R-a 2s-R+a 2s + 2 2 R-a 2s-R+a 2sR-R2+2aR-2 sa-a Secondly, 2 २ 2a+2s.R-R2-2 sa-a2 4 4mR-R-2 sa-a2 4 4mR-4 ma-4 ms 4 X 4 = (since a+s=2m) ; (which, because 4 ms+a3=R*,)= Q. E. D. R-a The answer to this problem in numbers is, x= 2 74-46 28 2 2 =14, and y= 2s-R-a 48-74+46 20 2 TO REGISTER THE STEPS OF AN ALGEBRAIC The register P is a method whereby the place from whence any step is derived, and the operation by which it is produced, are clearly pointed out, by means of symbols placed opposite the said step, in the margin. The symbols employed are + for addition, - for subtraction, × for multiplication, for division, for involution for evolution, for completing the square, = for equality, and tr. for transposition. When the register is used in the solution of any problem, it requires three columns; the right hand column contains the alge P The register will be found to be a very convenient mode of reference, where an ample detail of the work is required; but as modern algebraists prefer noting down results, and omit as much as possible particularizing those intermediate steps which are in a great degree evident, the register is now less in use than formerly. We are indebted to Dr. John Pell, an eminent Englislı mathematician, for the invention: it was first published in Rhonius's Algebra, translated out of the High Dutch into English by Thomas Brancker, altered and augmented by Dr. Pell, 4to. London, 1688. The learner will be enabled, by the specimen here given, to apply the method to other cases if he thinks proper; at least he should understand its use, as it is employed in the writings of Emerson, Ward, Carr, and some other books which are still read, braic operation, in the next the steps are numbered, and in the left hand column opposite to each step are placed, first the number of the step from whence it is derived, and then the symbol denoting the operation by which it is obtained. And here it must be noted, that the numbers 1, 2, 3, &e. in the register column, always denote the numbers of the steps, as first, second, third, &c. but when a figure has a dash over it, as 3, it denotes a number concerned in the operation. In the following example an additional column is placed on the left, for the purpose of explaining the process, Multiplying eq. 3. into b. Involving equation 2. Multiplying eq. 5. into c. Dividing equation 6. by y. Equating the 4th and 7th steps. 4-7 Multiplying eq. 8. into d. Multiplying eq. 9. into y. Dividing equation 11. by b. Comp. the square, &c. in eq. 12. 12 &c. 13 y2-dmy+ Evolving the root of eq. 13. 13 Adding to eq. 14. From the 7th and 15th eq. By restitution in the 15th eq. 15 restit. 17 y= |