PROPORTIONALS

, are the terms of a proportion; consisting of two extremes, which are the first| and last terms of the set, and the means, which are the rest of the terms. These Proportionals may be either arithmeticals, geometricals, or harmonicals, and in any number above two, and also either continued or discontinued.

Pappus gives this beautiful and simple comparison of the three kinds of Proportionals, arithmetical, geometrical, and harmonical, viz, a, b, c being the first, second and third terms in any such proportion, then : : a - b : b - c. In the arithmeticals, a a in the geometricals, a : b in the harmonicals, a : c

See Mean Proportional.

Continued Proportionals form what is called a progression; for the properties of which see PROGRESSION.

(For what respects Progressions and Mean Proportionals of all sorts, see Mean, and Progression.)

1. Four Arithmetical Proportionals, as 2, 3, 4, 5, are still Proportionals when inversely, 5, 4, 3, 2; or alternately, thus, 2, 4, 3, 5; or inversely and alternately, thus 5, 3, 4, 2.

2. If two Arithmeticals be added to the like terms of other two Arithmeticals, of the same difference or arithmetical ratio, the sums will have double the same difference or arithmetical ratio. So, to 3 and 5, whose difference is 2, add 7 and 9, whose difference is also 2, the sums 10 and 14 have a double diff. viz 4. And if to these sums be added two other numbers also in the same difference, the next sums will have a triple ratio or difference; and so on. Also, whatever be the ratios of the terms that are added, whether the same or different, the sums of the terms will have such arithmetical ratio as is composed of the sums of the others that are added.

So	3	,	5,	whose dif. is	2
and	7	,	10,	whose dif. is	3
and	12		16,	whose dif. is	4
make	22	,	31,	whose dif. is	9.

On the contrary, if from two Arithmeticals be subtracted others, the difference will have such arithmetical ratio as is equal to the differences of those.

So from	12	and	16,	whose dif. is	4
take	7	and	10,	whose dif. is	3
leaves	5	and	6,	whose dif. is	1
Also from	7	and	9,	whose dif. is	2
take	3	and	5,	whose dif. is	2
leaves	4	and	4,	whose dif. is	0

3. Hence, if Arithmetical Proportionals be multitiplied or divided by the same number, their difference, or arithmetical ratio, is also multiplied or divided by the same number.

The properties relating to mean Proportionals are given under the term Mean Proportional; some are also given under the article Proportion; and some additional ones are as below:

1. To find a 3d Proportional to two given numbers, or a 4th Proportional to three: In the former case, multiply the 2d term by itself, and divide the product by the 1st: and in the latter case, multiply the 2d term by the 3d, and divide the product by the 1st. So , the 3d prop. to 2 and 6: and , the 4th prop. to 2, 6, and 5.

2. If the terms of any geometrical ratio be augmented or diminished by any others in the same ratio, or proportion, the sums or differences will still be in the same ratio or proportion. So if , then is . And if the terms of a ratio, or proportion, be multiplied or divided by any one and the same number, the products and quotients will still be in the same ratio, or proportion. Thus, .

3. If a set of continued Proportionals be either augmented or diminished by the same part or parts of themselves, the sums or differences will also be Proportionals. Thus if a, b, c, d, &c be Propors. then are &c also Propors. where the common ratio is .

And if any single quantity be either augmented or diminished by some part of itself, and the result be also increased or diminished by the same part of itself, and this third quantity treated in the same manner, and so on; then shall all these quantities be continued Proportionals. So, beginning with the quantity a, and taking always the nth part, then shall , &c be Proportionals, or &c Propors. the common ratio being .

4. If one set of Proportionals be multiplied or divided by any other set of Proportionals, each term by each, the products or quotients will also be Proportionals. Thus, if , and ; then is , and .

5. If there be several continued Proportionals, then whatever ratio the 1st has to the 2d, the 1st to the 3d| shall have the duplicate of the ratio, the 1st to the 4th the triplicate of it, and so on. So in a, na, n²a, n³a, &c, the ratio being n; then a : n²a, or 1 to n², the duplicate ratio, and a : n³a, or 1 to n³, the triplicate ratio, and so on.

6. In three continued Proportionals, the difference between the 1st and 2d term, is a mean Proportional between the 1st term and the second difference of all the terms. Thus, in the three Propor. a, na, n²a;

Terms	1st difs.	2d dif.
n2a
	n2a - na
na
	na - a	n2a - 2na + a,
a

then . Or in the numbers 2, 6, 18;

18
	12
6		8 the 2d difference;
	4
2

then 2, 4, 8 are Proportionals.

7. When four quantities are in proportion, they are also in proportion by inversion, composition, division &c; thus, a, na, b, nb being in proportion, viz, 1 ; then by 2. Inversion ; 3. Alternation ; 4. Composition ; 5. Conversion ; 6. Division.

1. If three or four numbers in Harmonical Proportion, be either multiplied or divided by any number, the products or quotients will also be Harmonical Proportionals. Thus, 6, 3, 2 being harmon. Propor. then 12, 6, 4 are also harmon. Propor. and 6/2, 3/2, 2/2 are also harmon. Propor.

2. In the three Harmonical Proportionals a, b, c, when any two of these are given, the 3d can be found from the definition of them, viz, that ; for hence the harmonical mean, and the 3d harmon. to a and b.

3. And of the four Harmonicals, a, b, c, d, any three being given, the fourth can be found from the definition of them, viz, that for thence the three b, c, d, will be thus found, viz, .

4. If there be four numbers disposed in order, as 2, 3, 4, 6, of which one extreme and the two middle terms are in Arithmetical Proportion, and the other extreme and the same middle terms are in Harmonical Proportion; then are the four terms in Geometrical Proportion: so here the three 2, 3, 4 are arithmeticals, and the three 3, 4, 6 are harmonicals, then the four 2, 3, 4, 6 are geometricals.

5. If between any two numbers, as 2 and 6, there be interposed an arithmetical mean 4, and also a harmonical mean 3, the four will then be geometricals, viz, .

6. Between the three kinds of proportion, there is this remarkable difference; viz, that from any given number there can be raised a continued arithmetical series increasing ad infinitum, but not decreasing; while the harmonical can be decreased ad insinitum, but not increased; and the geometrical admits of both.