fourteen chapters present the geometrical part of Plane Trigonometry; they contain all that is necessary for practical purposes. The remaining chapters are of a more analytical character, and are important in the Theory of Mathematics. It will be found that the order of study may be varied at the discretion of the teacher, and the theoretical part taken at an earlier period. The range of matter included is such as seems required by the various examinations in Elementary Trigonometry which are now carried on in the country; it is hoped that nothing has been omitted which usually finds a place in such examinations. Any remarks relating to the book, and especially the indication of difficulties or omissions in the text or the examples, will be most thankfully received. CAMBRIDGE, I. TODHUNTER. TRIGONOMETRY FOR BEGINNERS. I. Measurement of Angles by Degrees or Grades. 1. THE word Trigonometry is derived from two Greek words, one signifying a triangle, and the other signifying I measure. Plane Trigonometry originally denoted the science in which the relations subsisting between the sides and the angles of a plane triangle were investigated, and the modes of investigation were almost entirely geometrical. But now the term Plane Trigonometry has a wider meaning, and comprises investigations with respect to plane angles whether forming a triangle or not, and the investigations are made by the aid of algebraical symbols and formulæ. Before beginning the present treatise the student should therefore become acquainted with Algebra, at least as far as the solution of simple equations. The parts of the elements of Euclid which are usually read are also necessary. 2. We have first to explain how angles are measured. Some angle is selected as the unit, and the measure of any other angle is the number of units which it contains. Any angle might be taken for the unit, as for example a right angle; but a smaller angle than a right angle is found more convenient. Accordingly a right angle is divided into 90 equal parts called degrees; and any angle may be estimated by ascertaining the number of degrees which it contains. If the angle does not contain an exact number of degrees we can express it in degrees and a fraction of a degree. A degree is divided into 60 equal parts called minutes, and a minute into 60 equal parts called seconds; and thus a 16 fraction of a degree may if we please be converted into minutes and seconds. 3. Thus, for example, half a right angle contains 45 degrees; a quarter of a right angle contains 223 degrees, which we may write in the decimal notation 22.5 degrees, or we may express it as 22 degrees 30 minutes; onesixteenth of a right angle contains 5ğ degrees, that is, 5.625 degrees, or 5 degrees, 37 minutes, 30 seconds. 4. Symbols are used as abbreviations of the words degrees, minutes, and seconds. Thus 5o 37' 30" is used to denote 5 degrees, 37 minutes, 30 seconds. 5. The method of estimating angles by degrees, minutes, and seconds, is almost universally adopted in practical calculations. Another method was proposed in France, towards the end of the last century, in connexion with a uniform system of decimal tables of weights and measures. In this method a right angle is divided into 100 equal parts called grades, a grade is divided into 100 equal parts called minutes, and a minute is divided into 100 equal parts called seconds. On account of the occurrence of the number one hundred in forming the subdivisions of a right angle, this method of estimating angles is called the centesimal method; and the common method is called the sexagesimal method, on account of the occurrence of the number sixty in forming the subdivisions of a degree. The centesimal method is also sometimes called the French method, and the sexagesimal method is called the English method. 6. Symbols are used as abbreviations of the words grades, minutes, and seconds, in the centesimal method. Thus, 5 37 30" is used to denote 5 grades, 37 minutes, 30 seconds in the centesimal method. A centesimal minute and second are not the same as a sexagesimal minute and second, and the accents or dashes which are used to denote minutes and seconds in the two methods are distinguished by sloping in different directions. 7. In the centesimal method any whole number of minutes and seconds may be expressed immediately as a decimal fraction of a grade. Thus, 37 centesimal minutes |