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57. We will now show the application of the foregoing principles of continued fractions by the solution of several practical questions: 1. Express approximately the fractional part of 24 hours, by which the solar year of 365 days, 5 hours, 48 minutes, and 48 seconds, exceeds 365 days. 5 hours, 48 minutes, 48 seconds=20928 seconds. 24 hours=86400 seconds. Therefore, the true value of the fraction required is #####-###. Now, converting ### into a continued fraction, by

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and this, re-converted into its approximative values (by Rule under Art. 56,) gives {, }, +, +, or, or, ###. The fraction ; agrees with the correction introduced into the calendar by Julius Cæsar, by means of bisseatile or leap-year. The fraction so is the correction used by the Persian astronomers, who add 8 days in every 33 years, by having 7 regular leap-years, and then deferring the eighth until 5 years. 2. The French metre is 39°371 inches. Required the approximative ratio of the English foot to the metre. In this example, the true ratio is #####. Operating npon this fraction, as in the last example, we find some

of the first approximate values to be , so, I's, or, ##, *s. Hence, the foot is to the metre as 3 to 10, nearly; a more correct ratio is as 32 to 105. 3. The old Winchester bushel contains 2150-42 cubic inches, and the new Imperial bushel contains 2218 198 cubic inches. Required some of the approximative ratios of these numbers. In this case, we find some of the approximations to be #, #4, ##, ##, ###. Hence, the Winchester bushel is to the Imperial bushel as 32 to 33, nearly. Now, since in a bushel there are 32 quarts, it follows that the Imperial is a Winchester quart larger than the Winchester bushel, nearly. 4. What are some of the approximative values of the ratio of the diameter of a circle to its circumference 1 If we take the value of the circumference of the circle, whose diameter is 1, to 10 decimals, we have the vulgar fraction ###########, given to find its approximative values.

Proceeding with this, as in the former examples, we find some of the first approximative values to be #, #, ###, ###, &c.

58. Continued fractions have been the means of obtaining elegant approximations to the roots of surds.

As an example, let it be required to find the square root of #; or, what is the same thing, the ratio of the side of the square to its diagonal.

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2+ V2–1, and by thus continuing this process, we find I

I - - - - -
V2 to equal the following infinite continued fraction:

1 T+1 2T 2Ti 2-II 2+, &c. Some of the first approximative values of this fraction

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59. We will conclude this subject by pointing out some of the many remarkable properties which the approximative values of continued fractions possess. We will refer to the values just obtained for the ratio of the side of a square to its diagonal.

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II. Any of these values differ from the true value by a quantity which is less than the reciprocal of the square of its denominator. Thus, ##, which is the ratio much used by carpenters in cutting braces, differs from the true ratio by a quantity less than (P1)*=###.

III. Any two consecutive terms of these approximate values, when reduced to a common denominator, will differ by a unit in their numerators. Thus, #, and ##, when reduced to a common denominator, become or, and #.

IV. The numerator and denominator of all approarimative values of continued fractions are prime to each other; that is, they have no common measure.

[For a more extended developement of the properties of Continued Fractions, see my Treatise on ALGEBRA.]

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60. TheRE is another method of de-compounding fractions, which was first given by the celebrated LAMBERT.

As an example, we will resume our fraction ###, which may be made to take these forms:

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