This Question being thus stated, it appears by Rule 1, Page 176, that it is capable of innumerable Anfwers; becaufe for any one of these three Letters, a, e, y, there may be taken any Number at Pleafure, provided it be less than 56. But although that may be truly done, yet there are feveral Ways of arguing about these Sort of Questions, which will limit or bound them to all their proper or poffible Answers in whole Numbers. Thus, 3 5 6 7 81 4e=336 — 16 a e = 84 — 4 a ; hence a 21 53a-28; hence aor 9} From the two laft Steps it appears, that the Quantity fignified by a, ought to be less than 21, and greater than 9; that is, any Number betwixt 9 3 and 21, may be taken for the Value of a: Confequently there may be eleven Answers to this Question in whole Numbers. Suppofe a 10, then e 8440 44, per 7th Step; and y=30282, per 8th Step. Again, if a=11, then e =8444 40, per 7th Step, and y = 33 -285, per 8th Step: and fo on for the reft, which will be as in the following Table. Thus it will be easy to find out and collect all the limited Anfwers to any Queftion (of this Kind) wherein there are only three Quantities propofed to be mixed: but when there are more than three, then the Work requires a little more Trouble; because the fingle Limits of all the Quantities above two must be found; that is, if there are four Quantities concerned in the Question, the Limits of two of them must be found; if five Quantities are concerned, then the Limits of three of them muft be found, &c. As in the following Queftion. Question Question 35. Suppose it were required to mix four Sorts of Wines together; viz. one Sort worth 7 s. 4 d. the Gallon, another Sort worth 4 s. 7 d. the Gallon, a third Sort worth 3 s. 8 d. the Gallon, and a fourth Sort worth 2 s. 9 d. the Gallon: How much of each Sort may be taken to make a Mixture of 63 Gallons, fo as that the whole Quantity may be fold for 5 s. 6 d. the Gallon, without Lofs, &c. First, let all these feveral Rates, and the mean Rate, be reduced to one Denomination, viz. into Pence. {$7 s. 4 d. = 8.8 d. 4s. 7 d. 55 d. and 5 s. 6d. = 66. Viz. { 13 s. 8 d. 44 d. Put a 2 s. 9 d. 33 d. the Quantity of that worth 88 d, the Gallon; e= that of 55 d. the Gallon, y that of 44 d. the Gallon, and u= that of 33 d. the Gallon. a+e+y+u=63 by the Question. 288a+ 55e+44y+ 33 u=415863 x 66 3 e+y+u=63 — a 455e44y+ 33 u = 4158 .88 a 3 x 335 33 et 33y+ 33 u 2079 33 a 16172e+y=1895a; hence a 2 or 374 3 x 558 55e+55y+55 u = 3465—55 a. 8 9÷1110 y2u=3a63; hence aor 21 From the 7th and 10th Steps it appears, that the Quantity of that Sort of Wine denoted by a, muft be lefs than 37 Gallons, and greater than 21 Gallons: that is, it may be a = any Number of Gallons betwixt 21 and 37. Whence it follows, that there may be collected 16 Anfwers to this Question from the Limits of a only. Next to find the Limits of e, y, and 2. Suppofe 11 a 22, then will 5 a 110, and 3 a = 66 But 122e+ y = 189 — 5 a = 79, per 7th Step. 12-2013 792; hence 2 or 39 Again 14 s+y+u=63—a=41, per 3d Step. 14e15y + u = 41-e 151316 lu - 38; hence 738 From the 13th and 16th Steps it appears, that if a 22, then =39, y=79-2e1, and ue — 381. Again, Suppofe 17 a 23, then 5 a 115, and 3 a 18 21 69 But 182e+y=1895474, per 7th Step. 2e19y=742e; hence == 37 Again 20 e+y+u=63—a = 40, per 3d Step. 2021y+u=40—e From the 19th and 22d Steps it appears, that if a 13, then e may be either 35 or 36. ་ Once more for a further Illuftration. Let|23|a24, then 5 a 120, and 3 a72 = y=69-2e; hence e 2 or 341 ue 30, hence e 3o. From hence it appears, that if a = 24, then e may be either 31, 32, 33, or 34, viz. it may be any Number betwixt 30 and 34 by the 25th and 28th Steps; from whence the Values of y and u may be easily found. Proceeding on in this manner with all the other fingle Values of a. there may be found above 120 Anfwers to this Question in whole Numbers and if you pleafe to put a Fractions, there may be found an innumerable Set of Anfwers; whereas the Rule of Alligation in Vulgar Arithmetick affords but only one Answer in Fractions; to wit, that of a 31, e10, y = 10, u = 102; as may be cafily tried per Rule Page 115, &c. Thefe two Examples being well understood (efpecially if the laft be thoroughly purfued) may fuffice to fhew the Method of limiting the Anfwers to all Sorts of Questions of this Kind. I fhall therefore conclude this Chapter of Queftions with giving a Solution to the Enigma (or Riddle) propofed (but not answered) by Mr John Kerfey, in the Clofe of the Appendix to his Arithmetick, which affords feveral pretty Queftions, the Solution whereof will discover a certain Sentence confifting of three Words, which must 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 firft 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 a the greater Index, or that of the 3d Letter. 3 а –ехаа — ее = 57 76. + a + e xa a + c e = 2 3 36} by the Question. 5 aaa-aae—a ee+ eee=576 7 2aae zee =1760 8aaa3aae + зa еe + eee = 4096 ww 9 +e10aa+ee= 2336 2336 = 146 ate 16 92114a+2ae + ee = 256 10-1213 aa—za e + e e 36 13 w214a -e=√ 36=6 9 +14 15 2α=22 From hence it appears, that the 38 15216] a=11 9-16 7 e=5 Letter of the ft Word is 1, and the 2d Letter of the 28 Word is e. Note, In order to fet down the Letters (as they become found) in their proper Places, it may be convenient to fupply the vacant Places with Stars. Thus {F1*/** First Word 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 firft Letter of the third Word; and that (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 first and second Letters of the first 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 the leffer Index. : Then aa+e=520 } according to the Data. And zae=a a 2a3e=&a 32 4 ee={a a I1 4 5 x 81 6+ 49 a a 7130 8 w2 8aa12324 9a =√32418, whofe Letter is s. 3 and 9 role= a=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 ***. Gl. * * i *. |