In serieses such as 3d and 4th, we observe the same rule, viz. raise the ratio to a power whose index is 1 less than the required term, and multiply the ratio raised to such power by the first term, and the thing is done. Suppose we desired to find the 8th terms of the 3d and 4th series, 2 x2 × 2 × 2 × 2×2×2×3 4 24 X 48 =27x3=384; or 3 3 = 384. 3x3x3x3x3x3x3x2=4374, or =4374, as before. Secondly, 543 x 1621 2 From which, it can be easily understood, the method of finding any assigned term of a geometrical series; and from the theorems given in the former part of this article, the sum, or any other part belonging to it, may be easily had. QUESTIONS TO EXERCISE THE PRECEDING RULES. 1. A person bought 50 yards of broad cloth, and was to give 1 cent for the first yard, 2 for the second, 4 for for the third, and so on, in geometrical proportion; how much will the last yard come to, and what the price of the whole? 1, 2, 4, 8, 16, 32, 64, &c. = = that 64 × 64 4096 that whose index is 12, and 4096 x 4096=16777216—that whose index is 24. 16777216 x 16777216=281474976710656 whose index is 48- the 49th term of the progression; and, 281474976710656 × 2=562949953421312= the last or 50th term, and $5629499534213.12= price of the 50th yard, and 250-1 2-1 X1= = 1125899906842621, sum of the whole series, which gives $11258999068426.21, price of the whole. 2. A man bargained for 26 horses; was to give 2 cents for the first, 6 for the second, 18 for the third, &c. in triple proportion geometrical; what was the price of the last horse, and of the whole? 0 1 2 3 4 5 1694577218886=index 25th= 26th term of the progression; and $16945772188.86 -price of the 26th horse. Again, 326-1 3-1 X2= Therefore, 5083731656657-sum of the reries. 1 1 3. Required, the sum of 1, 3, 4, 7, 1, &c. continued to 12 terms. Ans. 265220 177147' 4. If a man was to engage to pay another, i cent for the first month, 10 for the second, 100 for the third, and so on, in a tenfold ratio, for 12 month's service; how much would his wages amount to? Ans. 1012-1 10-1 X1=111111111111 cents, $1111111111.11, a sum sufficient to pay a thousand times the national debt. 5. A man agreed with his neighbour for a team of 4 horses; and was to give 10 cents for the first horse, 10 times as much for the second, increasing the price of each horse in a tenfold ratio; what was the price of the team? Ans. $1111.10. 6. If a man was to work 20 days for the following wages, viz. at 1 mill for the first day's work, 4 for the second, 16 for the third, and so on, increasing each day's wages in quadruple, or fourfold proportion; required, the amount of his wages. Ans. 366503875.921. HARMONICAL PROPORTION. Harmonical Proportion is that, which is between those numbers that assign the lengths of musical intervals, or the lengths of strings sounding musical notes; and of three numbers it is, when the first is to the third, as the difference between the first and second, is to the difference between the second and third, as the numbers 3, 4, and 6; for 36 :: 4—3 : 6—4. If the lengths of strings be as the numbers 3, 4, and 6, they will sound an octave 3 to 6, a fifth 2 to 3, and a fourth 3 to 4. Again, between four numbers, when the first is to the fourth, as the difference between the first and second, is to the difference between the third and fourth, as in the numbers 5, 6, 8, 10; for 5: 10 :: 6—5 : 10—8, and strings of the lengths 5, 6, 8, and 10 will sound an octave 5 to 10; a sixth greater, 6 to 10; a third greater, 8 to 10; a third less, 5 to 6; a sixth less, 5 to 8; and a fourth, 6 to 8. A series of numbers in Harmonical Proportion, is reciprocally, as another series in Arithmetical Proportion. SHarmonical, 10..12.. 15..20..30..60 Arithmetical, 6 60 In these 5 4 3 2 1 serieses, 10 12 5: 6; and 15 '} 12 15 4: 5; : 20: 3: 4; and so of the rest. Hence, it is perceivable, that those serieses have an obvious relation to, and dependence on each other. Let a, b, c, be three numbers in musical proportion: Then by what is shewn above, a : c :: a-b: b-c; but when bis greater than a, and c greater than b, then, a : c :: bɑ : c-b, by multiplying means and extremes ac-ab-bc-ac; or from the first, ab-ac-ac-bc but from the last but one, by the resolution of equations; ab 2ac-bc-ab and c=. <=1st formula; II. b=2ac• III.a= bc 2c-b 2a-b a+c; Hence, if the numbers 3, 4, and 6 were in question, and any two of them given to find the remaining one, it would only be necessary to substitute in the above formulas the figures answering to the place of the letters, and the thing is done; for in formula 1st, 3X4 12 2X3-4 2 = =6; 2d, 2×3×6 4x6 =4; and 3d, =3 2x6-4 EXAMPLE. Suppose it were required to find a musical mean proportional between the monochord 50=c, and the octave 2ac 2X25X50 25-a; then by theorem 2d, = a+c 25+50 2500 = 10°-333-b, the length of that chord called a fifth. = If there be four numbers in musical proportion, as a, b, c, d, then, since it is that a : d :: ab : cb, we have ac-ab-ad-db; but if d be greater than c, and b greater than a, then ad :: b-ad-c, and by multiplying means and extremes, ad-ac-db-da; from which, db 2ad-db 1. =a; 2. a× 2d—c=b; 3. α a 2d-c =d. Now it appears, that any three of the four quantities being given, the remaining one may be found. Let 10, 8, and 6 be given, to find a fourth harmonical proportion. By substitution of the figures in the first 10x6 formula, =5, the octave; for it must be notic2×10-8 ed, that the octave is always half the monochord, and so of all the rest. From what has been said, the harmonic divisions of the monochord, to sound the pure concords, will be as follow, viz. in the annexed table. The monochord is a vibrating string, say 10 inches long; all the other divisions of the same, as 9.439, 8.909, shew the lengths of other strings, sounding half and whole tones, one about. Such lengths as vibrate together are concords; where strings of different lengths fall in together, or as many of them as happen to vibrate together at once, form an unison. Much more might be said on this head, if necessary; but my object was only to give a few hints. Whoever would see the subject discussed at large, let him consult Martin's Philosophy. |