6. The third and laft Letter of the fecond Word, also the third Letter of the third Word, are the fame with the fecond Letter of the firft Word; hence the Letters will stand thus, 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. Then 1ae+a+e=35 And 2aa33by the Data. 122570 a + aa aa+2a+1 +2a3+aa=288 {a+ + 2 a3 + aa = 288 a a +576a+288 + 122570a + a a 8a+ + 2 a3-288 a a- 506a=1513 This laft Equation being refolved according to the Method which fhall be fhewed in the next Chapter, it will be a 17 it's Letter; and from the 4th Stepe= 35-a = 1, the Index of a+I the Letter a. Then these two Letters being placed according to the Data above, are all that are required by the Enigma to compleat thefe Words. Soli Deo Gloria. 1 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 1. I fhall here make Choice of the fame Letters to reprefent the Numbers both givea and fought in my Compendium of Algebra., Viz. Let G, always denote the given Refolvend. e= Sany Number taken as near the true Root as may be, whether it be greater or lefs. the unknown Part of the Root fought by which r is to be either increafed or decreased. Then if r be any Number lefs than the true Root, it will be + the 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. Thele Things being premifed, we may proceed to railing the Theorems. SECT. I. I. FOR the Square Root, viz. a a=G. Quære a. Let Ir+e=a 2rr+zretee=aa=G 2r 32re+ee=Grr. Call it D, viz. D=G-T. This fhews rft 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. aaa = G. Quære a. Let r+ea, fuppofing r lefs than the true Root, 12rrr +3rre+3ree+ece=aaa = G 2-rrr 33rre+3ree+cee=G=rrq I G =D 3r вее Let be rejected or caft off, as being of fmall Value; then it 3r will be, retee=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, Suppofe aaa=G, as before, and let r be taken greater than the true Root. Then Seee being reject2 rrr-3rre+3ree=a3=Gled as before. 2 ± 33rre-3ree = rrr-G rrr- -G 3+37 4re =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. a* = G, Quære a. Letra fuppofing r lets than juft. 142++4rrre+6rree=a*GS rejecting all the Pow34rrre+6rree G⋅ 2r+38 Hh 2 +ers of above e e. By 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 135.1 IV. To extract the Surfolid Root, viz. a3=G. Quære a. If r be taken less than juft, then r+e=a, as before, and =D, which gives this Theorem D 712e By 53 this Theorem the Surfolid Root, Example 1. Page 136, is extracted. But if r be taken greater than juft; then rea, and G 543 =D, which gives this Theorem D 2 e this laft Theorem the Example in Page 137 is performed. I prefume it needless to purfue the raifing 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 Qua dratick Equations, two feveral Ways, viz. by cafting off the loweft Term; and by compleating the Square, vide Section 2. Page 195, &c. Yet it may not be amifs to fhew, how thofe 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 + 2 ba = G. It is required to find the Value of a. Put 1 x 26 2+3 re=a 2rr + 2 re+ee=aa 4rr+2br+2re+2be+eeaa+2ba=G 4&c. 52 re+2be+ee = G-r2br 521 6 re+be+ } ee = & G — r r - br=D 61 Suppose Suppofe 364, and G= 38692865: If r6000, then r = 366000co, and 2 br4368000. But 36000000 + Ez 436800040368000 38692865 G. Therefore the first 6000. Let r=5000, then Cafe 2. If a a — 2 b a = G, then proceeding as above, there will arife this Theorem viz. 2 baaa &c. as above. Ꭰ r—btie e, &c. And in Cafe 3, I think it needless to trouble the Reader with the Work of these. two Theorems in Numbers; because if the laft Example of Cafe 1, be understood, the other will be eafy. Not but that the Method of compleating the Square is very ready and easy, as you may ob ferve by the Work in several Queftions of this Chapter. IN SECT. 3. N the Solution of all Adfected Equations, that are above (or higher than) Quadraticks, it will be the best Way to take r= the next nearest Root of the Equation: And then it will be re =a, if r be less than juft; or rea if r be greater than just (as at the Beginning of this Chapter). And all the Powers of the unknown Part of the Root, (viz. e) above it's Square (ee) are to be rejected or caft off, as before in raifing the Theorems for the |