which affords feveral pretty Queftions, the Solution whereof will difcover a certain Sentence confifting of three Words, which muft be found by the Help of Figures placed (or fuppofed to be placed) over the twenty-four Letters of the Alphabet. So that if the Index to that Letter be once found, the Letter to which it belongs is confequently known. The Enigma. 1. If the Difference between the Indices of the fecond Letter of the fecond Word, and the third Letter of the first Word, be multiplied into the Difference of their Squares, the Product will be 576; and if their Sum be multiplied into the Sum of their Squares, that Product will be 2336, the Index of the faid third Letter being the greatest. Let 1a the greater Index, or that of the 3d Letter. 6-57 = 1760 6+7 8 u 3 aaa + za ae + 3 aee + ece = 4096' a + c = 3 / 4096 = 16 9 3 Again 10a-exa~exa+e=576, foraa-ee = a + e 10 16 11 exa—e=36 II 2 12 a -e=6 12 +9 13 2a = 22 132 14 a = II 9+1415 e = 5 (xa-e From hence it appears, that the 3d Letter of the ft Word is 1, and the 2d Letter of the 2d Word is e. Note, In order to fet down the Letters (as they become found) in their proper Places, it may be convenient to supply the vacant Places with Stars. 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 laft Letter of the firft Word: whence the Letters will ftand thus, 4. The Sum of the Squares of the Indices of the firft and second Letters of the firft Word is 520, and the Product of the fame Indices is feven Ninths of the Square of the greater Index, which is the Index of the faid firft Letter. Let a the greater, and e = the leffer Index. Then I aa+ee = 520according to the Data. 3 And 5 x 8 a 1 2a e = Zaa 4 1 6.81aa = 42120 7 130 aa = 42120 8a a = 42120 49 a a 324 6 + 49 a a 5. The Difference between the two laft Indices, is the Index of the first Letter of the fecond Word, viz. 18 - 14 4 being the Index of the Letter D. Then the Letters will ftand thus, Soli. 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 second Letter of the firft Word; hence the Letters will ftand 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 least. aa+za+ 6 xaa &c. { 7 {a4 + 2 a 3 + aa=288 a a +576a+ 288 +1225-70 a + a a 7+ 8a4 + 2 a3 — 288 a a- 506a=1513 This laft Equation being refolved according to the Method which shall be thewed in the next Chapter, it will be a = 17 it's Letter; and from the 4th Stepe= 35 a = 1, the Index of 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: BE СНАР. Х. The Solution of Adfected Equations in Numbers. EFORE we proceed to the Solution of Adfected Equations, it may not be amifs to fhew the Investigation (or Invention) of thofe 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 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 lefs. the unknown Part of the Root fought, by which r is to be either increased or decreased. Then if be any Number lefs than the true Root, it will be re 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. Thefe Things being premifed, we may proceed to raifing the Theorems. SECT. I. I. FOR the Square Root, viz. aa = G. Quære a. Let 102 rr I r+e = a 2rr2re+ee=aa=G 32re+ee Grr. Call it D, viz. D=G-rr. This fhews 1ft the Method of 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. a a a = G. Quære a. Letrea, fuppofing r lefs than the true Root. 132rrr + 3rre +3ree+eeeaaa=G 2-rrr 33rre+3ree + eee= G-rrr eee G-rr 33r 14re+ee+ = ee 3r 3r =D Let be rejected or caft off, as being of small Value; then it 3r will be, rete e = D, which gives this following Theorem D r+e e By this Theorem or Rule, the rft 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 Ir -e=a Seee being reject132rrr-3rre+3ree=a3 = Ged as before. 2 ± 33rre- 3ree=rrr 33r 4ree Which gives this Theorem rrr-G = 3r = e By this Theorem the third Example in Cafe 2. Page 133, is performed. III. To extract the Biquadrate Root; viz. at G, Quære a. 2 Then re a fuppofing r less than just. I 2r4+4rrre+6rree=a+Grejecting all the Pow34rrre+6rree G-4 Sers of e above e e. G4 14 2rr 42re+ 3ee = 3÷2 Which gives this Theorem 2 rr D 2r-1-3e |