2. The Indices laft found, are the two Extreams of four Numbers in Arithmetical Progreffion, the leffer Mean being the Index of the first Letter of the third Word; and the greater Mean is the Index of the fourth and laft Letter of the firft Word. Viz. 5.7.9. 11 are the four Terms in Arithmetical Progreffion. Whence it appears, that G (whofe Index is 7) is the first Letter of the third Word; and that i (whofe Index is 9) is the fourth or laft Letter of the firft Word; which being placed down, will ftand thus, 3. The fecond Letter of the third Word is the fame with the third Letter of the firft Word; and the fifth Letter of the third Word is the fame with the last Letter of the firft Word: whence the Letters will ftand thus, 4. The Sum of the Squares of the Indices of the first and second Letters of the firft Word is 520, and the Product of the fame Indices is seven Ninths of the Square of the greater Index, which is the Index of the faid firft Letter. Let a the greater, and the leffer Index. Then 14a+e=520according to the Data. 20 1- 4 6 + 49 a a 7130 8 w2 3 3 e 4ee = 3/2 a a 5aa520 ‡? a a 681aa42120 7 130 aa = 42120 8aa = 42120 130 49 a a 9a√32418, whofe Letter is s. and 9 110le=3a=14, whofe Letter is o. Hence the Letters will ftand thus, 5. The Difference between the two laft Indices, is the Index of the first Letter of the fecond Word, viz. 18—144 being the Index of the Letter D. Then the Letters will ftand thus, Soli. De** *• De ***. Gl. **i*. 6. The third and laft Letter of the fecond Word, alfo the third Letter of the third Word, are the fame with the fecond Letter of the firft Word; hence the Letters will stand thus, Soli Deo Glo * i *. 7. The Sum of the Indices of the fourth Letter of the third Word, and the fixth or laft Letter of the fame Word, being added to their Product is 35; and the Difference of their Squares is 288; the Index of the laft Letter being the leaft. Put a the greater, and the leffer Index, as before. 122570 a + aa aa+2a+1 a++2a3+aa288 a a +576a+288 6 xa a &c. 7 +1225 7+ 8 la++ 205 This laft Equation being refolved according to the Method which fhall be fhewed in the next Chapter, it will be a 17 it's 35- =1, the Index of Letter; and from the 4th Stepe= the Letter a. Then these two Letters being placed according to the Data above, are all that are required by the Enigma to compleat these Words. Soli Deo Gloria. Hh CHAP. CHAP. X. The Solution of Adfected Equations in Numbers. BEFORE we proceed to the Solution of Adfected Equations, it may not be amifs to fhew the Investigation (or Invention) of those Theorems or Rules for extracting the Roots of Simple Powers, made use of in Chapter 11. Part I. I fhall here make Choice of the fame Letters to reprefent the Numbers both given. and fought in my Compendium of Algebra. Viz. Let G, always denote the given Refolvend. r = e= { any Number taken as near the true Root as may be, whether it be greater or less. S the unknown Part of the Root fought by which r is to be either increased or decreased. Then if r be any Number lefs than the true Root, it will be rethe Root fought. But if r be taken greater than the true Root, it will then be re the Root fought. And put D for the Dividend that is produced from G, after it is leffened and divided by r, &c. (into the Co-efficients of Adfected Equations) according as the Nature of the Root requires. Thefe Things being premifed, we may proceed to raifing the Theorems. SECT. I. R the Square Root, viz. a a G. Quære a. I. FOR 'OR 2rr+zretee≈aa = G 2r 32re+ee=G-rr. Call it D, viz. D=G—rr. This fhews Ift the Method of =D. 2 D The Arithmetical Operations of both thefe Theorems, you have in the Examples of Section 2. Page 126, to which I refer the the Learner, fuppofing him by this Time to understand them without any more Words than what is there expreft. II. To extract the Cube Root; viz. aa a G. Quære a. Let rea, fuppofing r lefs than the true Root. I 2rrr+3rre+3ree+eee=aaa 33rre+3ree+eee G⋅ 33 4re+ee+ = eee 3r 3r = D = G Let be rejected or caft off, as being of small Value; then it 3r will be, re+ee=D, which gives this following By this Theorem or Rule, the 1ft and 2d Examples in Cafe 1. Page 132, are performed; the which being compared with this Theorem may be easily understood. Again, Suppose aaa=G, as before, and let r be taken greater than the true Root. Then I rea Sece being rejectI 3 2 rrr-3rre+3ree=a3=Gled as before. 2 ± 33rre—zree=rrr-G 33r 4 re-e rrr-G D 3r. D Which gives this Theorem By this Theorem the third Example in Cafe 2. Page 133, is performed. III. To extract the Biquadrate Root; viz. at G, Quære a. = Let 1rea fuppofing r lets than juft. 134 2—pt 2r++4rrre+6rree=a*=GS rejecting all the Pow3 4rrre + brree Gers of e above e e. 3÷2rr 42re +30e= By this Theorem the Biquadrate Root of any Number may be extracted. But, as I have already faid, Page 134, thofe Extractions may be very well performed by two Extractions of the Square Root. Vide Example, Page 135. IV. To extract the Surfolid Root, viz, a'= G. Quære a. -- G— 5r3 D = l. By =D, which gives this Theorem this Theorem the Surfolid Root, Example 1. Page 136, is ex tracted. But if r be taken greater than juft; then -e=a, and D, which gives this Theorem D = l. 583 this laft Theorem the Example in Page 137 is performed. By I prefume it needlefs to purfue the railing of thofe Theorems, for extracting the Roots of Simple Powers, any further; because the Method of doing it is general, how high foever they are; and therefore it may be easily understood by what is already done. SECT. 2. Notwithstanding I have already fhewed the Solution of Quadratick Equations, two feveral Ways, viz. by cafting off the lowcft Term; and by compleating the Square, vide Section 2. Page 195, &c. Yet it may not be amifs to fhew, how those Equations may be refolved into Numbers by this univerfal Method of continued Series; wherein, if the firft r be taken equal to the first true Root, or fingle Side of the Refolvend; and every fingle Value of e (as it becomes found) be still added to it, for a new r, then thofe Roots may be extracted without repeating a fecond Operation, as before in the fingle Powers. Cafe 1. Let aa+2ba G. It is required to find the Value of a. 4 12 2jrr +2re+ee=aa 1 x 2b 32b r + 2be2ba 2+3 -rr &c. 4rr+2br+2re+2be+ce=aa+2ba=G 52re+2be+ee—G—rr2br 5÷21 6 re+be + } e e = ¦ G — irr—br=D |