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given to the nearest minute, which is sufficiently accurate for all ordinary practical purposes, and prevents the waste of time and labor in making closer determinations.

The logarithmic and trigonometric tables arranged specially for this work are based on the five-figure tables of Gauss, which have been modified to secure the convenience of compactness.

In preparing the book the author has consulted the best authorities both American and European. Some of the examples he has taken from these sources.

To his colleagues, Professors W. D. Taylor and T. W. Atkinson, who have rendered much assistance in reading proofsheets, he wishes to express his grateful acknowledgments. With their help he has hoped to eliminate all accidental errors; if any have been overlooked, he asks that those finding them will have the kindness to send him word.

April 18, 1898.

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1. Trigonometry treats of the measurement and properties of angles and triangles.

Plane figures only are considered in Plane Trigonometry.

2. To measure a quantity is to find how many times it contains a known quantity of the same kind, called the Unit of Measure.

The object of measuring a quantity is to determine its size or value. Trigonometry treats largely of measuring, or finding the sizes of, angles and lines by means of the relations they sustain to other known angles and lines.

3. The Size or Value of a quantity is expressed by prefixing to the name of the unit of measure the number which shows how many times that unit is contained in the quantity. As, 8 in., 10 ft.

4. In Degree-Measure of angles the unit of measure is of a right angle called a degree (°); the degree is divided into 60 equal parts called minutes ('), and the minute is divided into 60 equal parts called seconds (").


Thus, the angle which is of a right angle is written 200, 55° or 5.625°, 5° 37' or 5° 37.5', or 5° 37' 30".

5. The Relative Values of two or more quantities are the numbers which show how many times they contain a common measure. Thus, if the value of the line AB is m inches, and that of the line CD is n inches, the relative values of AB and CD are the pure and 1, CD being the unit of measure.

numbers m and n, or

m n

In general, mathematics treats only of the relative values of quantities, but in speaking of them the term “relative” is usually omitted. Thus, we say the values of the lines AB and CD are m and n.

Hence, in general, when letters are employed to represent the magnitudes of quantities, they stand for the relative, and not the concrete, values of the quantities; and, conversely, when lines are used to represent numbers, only the relative lengths of the lines are to be considered.

6. Two quantities are commensurable if there is a third quantity which is contained a whole number of times in each. The third quantity is called the common measure.

The ratio of any two commensurable quantities is equal to some simple common fraction, proper or improper.

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Regarding CD as the unit, the value of AB is tion showing the part which AB is of CD.

FIG. 2.

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7. Two quantities are incommensurable if there is no third quantity which is contained a whole number of times in each.


simple frac

There is no simple fraction equal to the ratio of two incommensurable quantities, but by taking the terms of the fraction sufficiently large there exists some simple fraction which comes as near to the ratio as we please.


For, let the lines AB and CD, Fig. 2, represent the quantities. Divide CD into n parts, each equal to CE, say; then, CD = n x CE. Since AB and CD are incommensurable, CE is not contained

in AB a whole number of times; let us suppose AB contains CE m times with a remainder FB less than CE. Then

ABmx CE + FB.

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