...146. Quantities are said to be in harmonic progression •when any 3 consecutive terms being taken ; the first is to the third as the difference between...second is to the difference between the second and the third. Thus, if a, b, с be the consecutive terms in a series, then if a : с :: a — b : b —...

...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...is to the difference between the second and third, as the numbers, 3, 4, 6. Thus, if the length of strings be as these numbers, the sound will be an octave...

...Harmonical Progression is a series of quantities of which, if any three consecutive terms be taken, the first is to the third as the difference between...is to the difference between the second and third. Thus, if a,, a.2, a3, a4, &c. be the consecutive terms of an harmonical progression, a,: 031:0, - a2:a2...

...a fifth 2 tp 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 the strings of such lengths will sound an octave...

...continual product equal to 576, to find the numbers. NB Three numbers are said to be in harmonical proportion, when the first is to the third, as the...is to the difference between the second and third. Thus a, x, b, are in harmonical proportion, when a ; b : : a — x : x — 6. And this proportion gives...

...extreme segments, as the other extreme segment is to the middle part. Three lines are in harmonical proportion, when the first is to the third, as the...the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion'...

...to the difference between the second and third ; or in four terms, when the first is to the fourth as the difference between the first and second is to the difference between the third and fourth ; or the reciprocals of numbers iu Arithmetical Proportion are in Harmonical Proportion....

...said to form a harmonica! series when of every three of its consecutive [following] terms the iirst is to the third, as the difference between the first...is to the difference between the second and third ; thus 12, 8, and 6 form a harmonica1 series, for 12 : 6 : : 12—8 : 8—6. 308. The middle one of...

...harmonical proportion. ,. 69. Four quantities are in harmonical proportion where the first is to the fourth as the difference between the first and second is to the difference between the third and fourth. Thus, if a : d : : a — b : с — d, a, b, c, and d are in harmonical proportion....

...b. (229.) Four quantities are said to be in harmonical proportion •when the first is to the fourth as the difference between the first and second is to the difference between the third and fourth. Thus, 2, 3, 4, 8 are in harmpnical proportion, for 2:8::3 — 2:8 — 4. Let a, b,...