and partly upon its importance. But, since each chapter is nearly independent, it will be in the power of the teacher to abandon the order laid down in the book and to adopt another at his discretion. The examples have been selected with a view to illustrate every part of the subject, and, as the number of them exceeds fifteen hundred, I trust they will supply ample exercise for the student. Complicated and difficult problems have been excluded, because they consume time and energy which may be spent more profitably on other branches of mathematics. Each set of examples has been carefully arranged, commencing with very simple exercises and proceeding gradually to those which are less obvious; a few in each set may be omitted by the student who is reading the subject for the first time. The task of preparing an elementary treatise is far from easy, and I must therefore request the indulgence of teachers and students for the defects which they may discover in my plan or in the mode of executing it. I shall receive most thankfully notices of errors or omissions, or of unusual difficulties in the text or the examples, and also any suggestions respecting the work generally. ST JOHN'S COLLEGE, April, 1858. I. TODHUNTER. CONTENTS. 64 XII. Simultaneous Equations of the First Degree with more than XIII. Problems which lead to Simple Equations with more than XXI. Equations which may be solved like Quadratics XXII. Theory of Quadratic Equations and Quadratic Expressions XLV. Reduction of a Quadratic Surd to a continued Fraction XLVI. Indeterminate Equations of the First Degree XLIX. Recurring Series L. Summation of Series LI. Inequalities ALGEBRA. I. DEFINITIONS AND EXPLANATIONS OF SIGNS. 1. THE method of reasoning about numbers by means of letters and signs which are employed to represent both the numbers themselves and their relations is called Algebra. 2. Letters of the alphabet are used to represent numbers, which may be either known numbers, or numbers which have to be found and which are therefore called unknown numbers. It is usual to represent known numbers by the early letters of the alphabet a, b, c, &c., and unknown numbers by the final letters x, y, z; this is not however a necessary rule, and so need not be strictly obeyed. Numbers may be either whole or fractional. The word quantity is frequently used as synonymous with number. 3. The sign + signifies that the number to which it is prefixed must be added. Thus a+b signifies that the number represented by b must be added to the number represented by a. If a represent 9 and 6 represent 3, then a+b represents 12. The sign + is called the plus sign, and a + b is read thus “a plus b.” 4. The sign-signifies that the number to which it is prefixed must be subtracted. Thus a-b signifies that the number represented by b must be subtracted from the number represented by a. If a represent 9 and 6 represent 3, then ab represents 6. The sign - is called the minus sign, and a-b is read thus “ a minus b." 5. The sign × signifies that the numbers between which it stands must be multiplied together. Thus a xb signifies that the T. A. 1 number represented by a must be multiplied by the number represented by b. If a represent 9 and 6 represent 3, then a × b represents 27. The sign x is called the sign of multiplication, and a × b is read thus " a into b." Similarly a × 6 × c denotes the product of the numbers denoted by a, b and c. It should be observed that the sign of multiplication is often omitted for the sake of brevity; thus ab is used instead of a ×b, and has the same meaning; so abc is used for a xbx c. Sometimes a point is used instead of the sign x; thus a.b is used for axbor ab. The sign of multiplication must not be omitted when numbers are expressed by figures in the ordinary way. Thus 45 cannot be used to express the product of 4 and 5, because a different meaning has already been appropriated to 45, namely forty-five. We must therefore express the product of 4 and 5 thus 4 × 5, or thus 4.5. To prevent any confusion between the point thus used as a sign of multiplication and the point as used in the notation for decimal fractions, it is advisable to write the latter higher up; thus 4.5 may be kept to denote 4+1‰· 6. The sign signifies that the number which precedes it must be divided by the number which follows it. Thus a÷b signifies that the number represented by a must be divided by the number represented by b. If a represent 9 and b represent 3, then ab represents 3. The sign÷ is called the sign of division, and ab is read thus "" a by b." There is also another way of denoting that one number is to be divided by another; the dividend is placed over the divisor with a line between them. Thus is used instead of a÷b and has the same meaning. 7. The sign signifies that the numbers between which it is placed are equal. Thus ab signifies that the number represented by a is equal to the number represented by b, that is a and b represent the same number. The sign is called the sign of equality, and a=b is read thus "a equals b" or "a is equal to b." = |