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1. TRIGONOMETRY was originally, as the name imports, the science which furnished methods for determining the magnitude of the sides and angles of triangles, but it has been extended to the treatment of all theorems involving the consideration of angular magnitudes.

2. By way of introduction to the subject we have to treat in this chapter, first, of the representation of lines by Algebraic Symbols, and, secondly, of the relation existing between the circumference of a circle and its diameter.

3. To measure a line AB we fix upon some line as a standard of linear measurement: then if AB contains the standard line p times, p is called the measure of AB, and the magnitude of AB is represented algebraically by the symbol p.

Since the standard contains itself once, its measure is unity, and it will be represented by 1.

4. Two lines are said to be commensurable when each contains a line taken as the standard of measurement (or, as it is commonly called, the unit of length) an exact number of times.

5. If the measures of two lines AB, CD be Ρ and ively, the ratio AB: CD is represented by the fraction

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6. Magnitudes of things cannot properly be made subject to the rules and operations of Algebra, as these rules and operations have only been proved for algebraical symbols. We must therefore find some algebraical representative for any magnitude before we subject it to algebraical operations: such a representative is the measure of that magnitude with the proper sign prefixed according to rules to be given hereafter.

7. In the application of Algebra to Geometry it is the practice of most writers to use the geometrical representative of a magnitude where the algebraical representative ought to be employed. Thus suppose p and q to be the measures of two lines AB, CD, we often find the fraction where we ought in

strictness to find the fraction



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is, however, sometimes less cumbersome, and we shall therefore retain it at the risk of a slight want of clearness.


(1) If the unit of length be an inch, by what number will 4 feet 6 inches be represented?

(2) If 7 inches be taken as the unit of length, by what number will 15 feet 2 inches be represented?

(3) If 192 square inches be represented by the number 12, what is the unit of linear measurement?

(4) If 1000 square inches be represented by the number 40, what is the unit of linear measurement?

(5) If 216 cubic inches be represented by the number 8, what is the unit of linear measurement?

(6) If 2000 cubic inches be represented by the number 16, what is the unit of linear measurement ?

(7) If a yards be the unit of length what is the measure of b feet?

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