17. What is the square root of the cube root of 3261, correct to .1?

18. What is the cube root of the cube root of 421634, correct to .1?

PROPORTIONS.

343. Two quantities are either equal or unequal to one another. In the latter case, when one is greater than the other, we may consider their inequality in two different points of view; we may either inquire how much one of the quantities is greater than the other, or how many times the one is contained in the other. Thus, when I say that between 12 and 4 the difference is 8, these numbers are considered with regard to their difference; and when I say that 4 is contained three times in 12, they are considered with regard to their quotient.

344. We shall now make a few observations upon the first connexion. When between any two numbers, such as 15 and 10, the difference is the same as between any two other numbers, 11 and 6; we say that 15 stands to 10 in the same respect as 11 stands to 6. These four quantities form an arithmetical proportion, or an equidifference, which is written thus: 15-10= 11-6; the first and last terms are called the extremes, and the second and third the means. Now, were we to add to each difference the sum 10+6, or the subtracted parts, the results are 15+6 and 11+10. Therefore, in every arithmetical proportion, the sum of the extremes is equal to the sum of the means. And conversely. if 15+6=11+10, we have an arithmetical proportion: 15—10—11—6.

If three members or terms of an equidifference were given. the fourth could easily be found, for let the three first terms, 15, 10, 11 be given find the fourth x.

increased by 15=10+11, .. x=10+11—15=6, and thus we have the arithmetical proportion 15-10-11—6.

Hence, if three terms of an arithmetical proportion be given, the fourth (if an extreme) will be found by subtracting from the sum of the means the extreme given; and if it be a mean, by subtracting from the sum of the extremes, the mean given.

To find x in the following arithmetical proportion: 36—21=

·· x+36=21+49, it follows that x=21+49—36=34; .. 36-21-49-34.

Likewise, in the arithmetical proportion 35-17=x—8, we have x+17=35+8; .. x=35+8—17=26, and .. 35 — 17 =26-8.

If three quantities, as 25, 18, and 11, be in arithmetical proportion, then 25-18-18-11; hence it follows, from the equality between the sum of the means and that of the extremes, that 2×18=25+11, or twice the arithmetical mean, is equal to the sum of the extremes; and .. the arithmetical mean is equal 25+11 to half their sum, 18. 2

345. When we consider how many times one quantity is contained in another, or what part or parts one is of the other, such a relation is called a ratio; thus the ratio of 6 to 9 is written 6:9, or . These numbers, thus compared, are called terms of the ratio, the former being the antecedent and the latter the consequent.

When one antecedent is the same multiple, or part of its consequent, as another antecedent is of its consequent. The ratios are equal, thus & and are two equal ratios, since §=√31⁄2 =. The four quantities which form two equal ratios, are said to be proportionals, or to determine a geometrical proportion, or an equiquotient, which is usually read, 6 is to 9 as 8 is to 12, and written 6:98:12, or = Here 6 and 8 are the antecedents, and 9 and 12 the consequents; also, 6 and 12 are the extremes, and 9 and 8 the means.

If we were to multiply two equal ratios, and31⁄2, by 9×12, the product of the denominators or consequents, we have on one part 6 × 12, and on the other 8×9. Therefore, when four quantities are proportionals, the product of the extremes is equal to the product of the means.

Conversely, if four quantities, 6, 9, 8, 12, are such that 6 × 12 and 8×9 are equal, they are proportionals, 6:9=8 : 12, or $=12• Then a proportion may be formed with the factors of two equal products, the factors of one product being the extremes and those of the other the means.

If the first be to the second as the second to the third, the product of the extremes is equal to the square of the means; therefore, the mean proportional, or the geometrical mean, between two numbers is the square root of their product. The

mean proportional between 3 and 12 is 3x 12-6, .. £=&. Conversely, if we have 623 x 12, we infer a proportion 3:6=

6:12.

If a proportion contains one unknown term, as 6:98:x, then

8x9 6xx, and we find x=_ =12. ..6:9 8:12. Then

8×9
6

=

one of the extremes is found by dividing the product of the means by the other extreme, and one of the means is found by dividing the product of the extremes by the other mean.

346 Now, the four terms of a geometrical proportion may be transposed in several ways, without altering the proportion. The test of all these transformations will be that the product of the extremes is equal to that of the means.

347. The corresponding terms of two or more proportions can be multiplied together, and the products will also be in proportion.

Thus, if 30:15=6 : 3, or 3g=};
and 2:34:6, or

.. 30 × 2: 15×3=6×4:3 × 6, or

30 x 2

=

6x4

15x3 3x6

Therefore, the terms of a proportion may be raised to the square, cube, &c.; and also, the square root, cube root, &c., may be extracted, and the results will also be in proportion.

These properties of proportions are true for all numbers whatever.

348. In what has been said about the ratios of proportions, we have considered the quantities as abstract, and the ratios were likewise abstract; in the same manner do we find that the ratios of concrete quantities are abstract.

It is scarcely necessary to mention that the quantities forming a ratio must be of the same kind, for it would be too absurd to attempt to compare £6 to 12 lbs. of cheese, or 12 gallons of beer to 20 days, &c.

350. It has been shown that numbers connected thus:

3-8 8-13, or 3—8—13;

and 5:10=10: 20, or 5:10: 20,

are in proportion, viz, that the third term bears the same relation to the second as the second to the first. We can easily conceive a fourth term having the same relation to the third as the third has to the second; and also a fifth term having the same relation to the fourth as the fourth has to the third, &c; and the given proportions extended become :

58

3-8-13-18-23-28-33—38 — 43 — 48 - 53 5:10:20:40: 80: 160: 320: 640 : 1280: 2560 5120: 10240 Such continued proportions are termed progressions; the first row constitutes an arithmetical progression, or an equidifferent series, and the second a geometrical progression, or an equiquotient

series.

351. Since the terms of a progression may either increase or diminish by a common difference or by a common ratio, there are increasing and decreasing arithmetical progressions, and increasing and decreasing geometrical progressions.

Every increasing series may be converted into a decreasing one by inverting the terms, viz., by making the last term the first, &c.

It is evident that every progression can be supposed continued sine fine, for to every term a succeeding one may always be found.