is rarely an advantage; and in publishing the first edition of my Algebra I felt some apprehension that I had deviated too far from the ordinary methods. I have had great satisfaction in receiving from eminent teachers favourable opinions of the work generally and also of those parts which are peculiar to it. Several years have elapsed since I resolved to publish an Algebra and began to arrange the materials. Thus all the important chapters in the present work have been written and rewritten, and repeatedly revised by myself and my friends. With respect to some parts which were original at the time when they first occurred to me I have been anticipated in publication; this applies, for example, to Arts. 520, 611, and 677. I mention this, not as attaching any importance to such points, but merely because otherwise it might appear that I had been indebted for them to preceding authors. My manuscripts on these articles were in use among my pupils before the date in which, so far as I know, these articles were printed; indeed it was not until after my first edition was published that I saw the latter two articles in print elsewhere. Some portions of the present work were written long before I had any intention of publication; the chapter on the Multinomial Theorem, for example, was drawn up about fifteen years ago for the use of a fellow-student. The task of preparing an elementary treatise is far from easy, and I must therefore request the indulgence of teachers and students for any defects which they may discover either in my plan, or in the mode of executing it. I have to return my thanks to many able mathematicians who have favoured me with suggestions, which have been of great service to me in preparing the Second Edition; and I trust I shall still continue to receive similar valuable remarks. ST JOHN'S COLLEGE, I. TODHUNTER. CONTENTS. I XII. Simultaneous Equations of the First Degree with more than XIII. Problems which lead to Simple Equations with more than XIV. Discussion of some Problems which lead to Simple Equa- XLV. Reduction of a Quadratic Surd to a continued Fraction XLVI. Indeterminate Equations of the First Degree. XLVII. Indeterminate Equations of a Degree higher than the first. 367 • 466 475 ALGEBRA. I. DEFINITIONS AND EXPLANATIONS OF SIGNS. 1. THE method of reasoning about numbers by means of letters which are employed to represent the numbers and signs which are employed to represent 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 b represent 3, then a + b represents 12. called the plus sign, and a +b is read thus "a plus b.” The sign + is 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 a b represents 6. The sign-is called the minus sign, and a-b is read thus " a minus b." 5. The sign x signifies that the numbers between which it stands must be multiplied together. Thus ab signifies that the 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 × is called the sign of multiplication, and a × b is read thus "a into b." Similarly a × bx 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 × b× c. Sometimes a point is used instead of the sign ×; thus a.b is used for a × b or 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 advisible 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 ab 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 a÷b represents 3. The sign is called the sign of division, and ab is read thus "6 a by b". There is also another way of denoting that one number is to be divided by another; the divi a dend is placed over the divisor with a line between them. Thus b 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.” |