centre, and that all the radii of a circle being right lines drawn from the centre to the circumference, must be equal to one another. And, by the same definition, since B is the centre of the circle ACE, BC, also, is equal to AB. But things which are equal to the same things, are equal to one another; therefore AC is equal to CB.

And since AC, CB, are equal to each other as well as to AB, the triangle ABC, is equilateral; and it is described upon the right line AB, as was required to be done.

PROBLEM.

To bisect a given finite right line; that is, to divide it into two equal parts.

Let AC be the given right line; it is required to divide it into two equal parts.

Upon that given line, AC, describe (as directed by the foregoing problem) the equilateral triangle ACB, and bisect the angle ABC, by the right line BD; then will AC be divided into two equal parts, as was required. For AB is equal to BC, because an equilateral triangle has all its sides equal to one another; BD is common to each of the triangles ADB, and CDB; and

the angle ABD is equal to the angle CBD. But when two sides, and the included angle of of one triangle, are equal to two sides and the included angle of another, each to each, their bases will also be equal. The base AD, is, therefore, equal to the base DC; and consequently, the right line AC is bisected in the point D; the operation required.

QUESTIONS.

What are propositions, and how are they divided?

is a theorem ?

What is a lemma?

sign,

==

-

What

What is a problem? What are corollaries? What is a scholium ? What does this signify? What does this sign < mean? What does this character > express? What is the signification of this sign + ? What does this sign denote? What does a figure, or number, prefixed to a letter expressive of a quantity, denote ? How is it demonstrated that the angles made by one right line standing upon another right line are, when taken together, equal to two right angles? How is it demonstrated that, if one side of a triangle be produced, the external angle will be equal to both the internal opposite angles? How is it proved that the three angles of any plane triangle, taken together, are equal to two right angles? How is an equilateral triangle to be described upon a given finite right line? How may a given finite right line be bisected.

CHAP. V.

TRIGONOMETRY.

TRIGONOMETRY, a term derived from the two Greek words, Trigonos, triangle, and Metron, measure, signifies, literally, the measure of triangles; but it denotes, generally, that science

which relates to the determination of the sides and angles of triangles, from certain parts which are given. It may be regarded as the applica tion of arithmetic to geometry.

It consists of two principal parts, namely,

Plane trigonometry, and spherical trigonometry.

Plane trigonometry treats of the application of numbers, to determine the relations of the sides and angles of any plane triangle, to one

another.

Spherical trigonometry treats of the application of numbers, in like manner, to spherical triangles.

From its numerous and important uses, trigonometry may be considered as one of the most interesting branches of pure mathematics, Practical and physical astronomy, navigation, surveying, mechanics; almost every branch of the pure and mixed mathematics, excepting geometry and arithmetic, are connected with the principles of trigonometry. It is very uncertain when trigonometry began to be culti vated as a science; yet it is traced to Hipparchus, who flourished about 150 years before the Christian era. At nearly the close of the eighth century after the birth of Christ, an alteration was introduced into this science by the Arabians, that of computing by sines, instead of chords; who likewise added to it several axioms and theorems, which are considered as the foundation of modern trigonometry. About the middle of the 15th century, some change, and great additions, were introduced into this science, by Purbach, and Regiomontanus, Werner

Copernicus, Rheinold, and Vieta, successively contributed to the advancement of the science, by useful alterations in the form of the tables for calculation, and by other improvements; and in more modern times, Napier, the inventor of logarithms, and Briggs.

In plane trigonometry, the circle is supposed to be divided into 360 equal parts, called degrees; every degree into 60 equal parts, called minutes; every minute into 60 equal parts, called seconds; and so on, into thirds, fourths, &c. These divisions are marked by the following characters, for degrees, minutes, "seconds. As 29° 16' 8"; that is, 29 degrees, sixteen minutes, eight seconds.

This division of the circle is, however, quite arbitrary, and any other number might have been employed instead of 360. The subdivisions, also, might have been formed upon any other scale, as well as the sexagesimal, or division by sixties; and accordingly the modern French mathematicians have adopted a different division. They suppose the circle to be divided into 400 degrees, or each quadrant, that is, each fourth part to consist of 100 degrees; the next subdivision is the 10th of a degree; the next, the 100th, and so on; and hence the measure of an angle is expressed by them, in the same manner as any other integral and decimal quantity.

Geometry demonstrates that any angles at the centre of a circle, have to one another, the same proportion, as the arches, or parts of the circumference intercepted between the lines which contain those angles, have to one another. Hence it is naturally inferred, that an angle at

the centre of a circle has the same ratio, or proportion to four right angles, which the arch, intercepted between the lines that contain the angle, has to the whole circumference. It also follows, that arches of a circle may be employed as measures of angles; and thus the comparison of angles, is reduced to the comparison of arches of a circle. From this principle, flow those definitions, upon which are constructed certain numerical tables; called trigonometrical tables, which are essential to the practice of both plain and spherical trigonometry.

TRIGONOMETRICAL DEFINITIONS.

1. If two straight lines intersect each other in the centre of a circle, the arch of the circumference intercepted between them, is called the measure of the angle which they contain. This explains what is meant by an angle subtending so many degrees. Thus, the arch AB is the measure of the angle contained by the lines CA and CH.