EXPLANATION OF SOME SYMBOLS AND ABBREVIATIONS

EMPLOYED.

L, angle.
Zs, angles.

▲, triangle.

As, triangles.

or perp., perpendicular to, or at right angles to. ll, parallel to.

<placed between two quantities, implies that the one which comes before the symbol is less than the one which follows it. Thus, LA<▲ B is to be understood and read, "the angle A is less than the angle B."

> placed between two quantities denotes that the one which precedes is greater than the one which follows it. Thus, <B> ▲ A, is to be read, "the angle B is greater than the angle A.” may assist the memory to observe that the greater quantity is always on the side of the open part of the symbol, and the less quantity on the side of the closed part.

A', B', x', y', etc. are read, A prime, В prime, x prime, y prime, etc.

A', B'', x'', y', etc. are read, A second, B second, x second, second, etc.

A''' is read, A third; Aiv, A fourth; B', B fifth, etc.

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CASES IN PLANE TRIGONOMETRY AS REFERRED TO IN

THE FOLLOWING CALCULATIONS.

Case 1.

When the angles and one side are given.

Case 2.

When two sides and an angle opposite one of them are

When two sides and the included angle are given.*
When all the sides are given.

* In Cases 2 and 3, when the angle is a right one, the solution may be effected by the rules for right-angled triangles.

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REMARKS TO THE STUDENT.

1. A triangle is said to be determined in an analysis or a construction when a sufficient number of parts are known to solve it by any one of the preceding cases of plane trigonometry.

2. References are made to Davies's translation of Legendre's Geometry, the Roman numerals being put for the book, and common figures for the proposition. With these is an asterisk (*) referring to a marginal or foot-note, where the proposition is designated in like manner in which the same property is demonstrated in Playfair's edition of Euclid's Elements of Geometry.

3. In performing the trigonometrical calculation of a constructed problem, the rule manifestly must be, to begin the calculation where was begun the construction; that is, in the triangle first formed. For, if there was sufficient given to construct the triangle, there must be sufficient to calculate it. Then, with what is found in this triangle, proceed to the next one that was formed, and so on till what is desired is obtained.

4. When a circle, semicircle, or circular arc is used in construction, its radius, as a general rule, must be used in the calculation. An exception to this rule exists when a circular segment is described on a line merely to include an angle of a given magnitude.

5. In the analysis of geometrical problems the ingenuity and inventive powers of the student must be brought into close requisition. No definite rules to meet all cases can be given. We must, however, always draw a figure, supposing the problem constructed as required, and then, when practicable, work on those lines which are given in position or length, or both, and continue on until a triangle is obtained which has a sufficient number of parts given to construct it. Then recall, and observe carefully, the process that has been pursued, and construct the figure accordingly.

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An attempt will be made in the following problems to render this general advice familiar, and to lead the student on to the analysis of problems of considerable intricacy.

6. It will prove of great benefit to the student, in constructing +he problems, to assume definite quantities for the parts given, and then work with the scale and dividers with delicate precision, measuring the results accurately from the scale of equal parts, or, if angles, from the scale of chords, and compare these results with those obtained by trigonometrical calculations. This is an admirable preparation for "field work," architectural drafting, machinists, engineering, etc.

7. At the ends of the problems the limits are frequently specified within which the given quantities must be taken; and certain varied conditions of the problem are occasionally referred to, which cannot generally be comprehended to full advantage by the figure that is given. In such cases the student will derive decided benefit, and progress with much greater rapidity and satisfaction by drawing on paper, with pen or pencil, representations of these different conditions, so as clearly to comprehend the idea designed to be conveyed. In this way he will master all that is before him as he goes along, and acquire greater courage and power to overcome future difficulties as they arise.