.5- 
 
 COf^NELL 
 
 UNIVERSITY 
 
 LIBRARIES 
 
 Mathematicf 
 
 Library 
 Whife Half 
 
CORNELL UNIVERSITY LIBRARY 
 
 058 531 736 
 
 
 DATE DUE 
 
 
 (^jlll^l. 
 
 
 
 
 BUL 2 
 
 9 199ft 
 
 
 
 nJ%fiX, ** 
 
 
 
 
 SEP ^ 
 
 " 2003 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CAVLORD 
 
 
 
 PRINTCOINU.S.A. 
 
The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/cletails/cu31924058531736 
 
Production Note 
 
 Cornell University Library 
 produced this volvime to replace 
 the irreparably deteriorated 
 original. It was scanned using 
 Xerox software and equipment at 
 600 dots per inch resolution 
 and compressed prior to storage 
 using CCITT Group 4 
 compression. The digital data 
 were used to create Cornell's 
 replacement volume on paper 
 that meets the ANSI Standard 
 Z39. 48-1984. The production of 
 this volume was supported in 
 part by the Commission on 
 Preservation and Access and the 
 Xerox Corporation. 1990. 
 
BOUGHT WITH THE INCOME 
 FROM THE 
 
 SAGE ENDOWMENT FUND 
 
 THE GIFT OF 
 
 Henrg W. Sage 
 
 1891 
 
 ^ KASiEMTICS / / 
 
 5901 
 
IRRATIONAL NUMBERS 
 
 AND THEIB REPBESENTATIOK BT 
 
 SEQUENCES AND SERIES 
 
 BT 
 
 HENRY PAEKER MANNING, Ph.D. 
 
 ASSISTANT PHOFESSOB OF PURE MATHEMATICS 
 IN BEOWN UNIVERSITT 
 
 FIRST EDITION 
 
 FIRST THOUSAND 
 
 NEW YORK 
 JOHN WILEY & SONS 
 London : CHAPMAN & HALL, Limited 
 1906 
 
Copyright, 1906 
 
 BT 
 
 HENRY PARKER MANNING 
 
 ROBEHT DRUMMOND. PRINTER NEW YORK 
 
PREFACE 
 
 This book is intended to explain the nature of irra- 
 tional numbers, and those parts of Algebra which depend 
 on what is usually called The Theory of Limits. 
 
 Many of our text-books define irrational numbers 
 by means of sequences; but to the author it has seemed 
 more natural to define a number, or at least to con- 
 sider a number as determined, by the place which it 
 occupies among rational numbers, and to assume 
 that a separation of all rational numbers into two 
 classes, those of one class less than those of the other, 
 always determines a number which occupies the point 
 of separation. Thus we have the definition of Dede- 
 kind, which is adopted by Weber in his Algebra. With- 
 out attempting to inquire too minutely into the sig- 
 nificance of this definition, we have endeavored to 
 show how the fundamental operations are to be per- 
 formed in the case of irrational numbers and to define 
 the irrational exponent and the logarithm. 
 
 Defining the irrational number by the place which it 
 occupies among rational numbers, we proceed to speak 
 of its representation by sequences; and when we have 
 proved that a sequence which represents a number is 
 regular and that a sequence which is regular repre- 
 
IV PREFACE 
 
 sents a number, we are in complete possession of the 
 theory of sequences and their relation to numbers. 
 
 The representation of a number by a sequence is 
 essentially the same as its representation as the limit of 
 a variable, and the notion of sequence seems to be 
 simpler than that of variable and limit. A section 
 has been added on Limits (Chap. II, IV) to bring out 
 the relation of the two points of view. But this sec- 
 tion may be omitted without any break in the con- 
 tinuity of the book, and the words "variable" and 
 " hmit " are used nowhere else. 
 
 The infinite series is defined as a sequence written in 
 a particular way. Theorems are given on the con- 
 vergency and use of infinite series sufficient to develop 
 the exponential, binomial, and logarithmic series. 
 
 The theory of irrational numbers given in Chapter I 
 has been adopted by Professor Fine, as stated else- 
 where (p. 56). Perhaps I may be permitted to add 
 that I did not see Professor Fine's book imtil after my 
 manuscript was in the hands of the printer. 
 
 In addition to the references on page 56, mention 
 may be made of two important articles on "The Con- 
 tinuum as a Type of Order," by Dr. E. V. Huntington, 
 in the Annals of Mathematics for July and October, 1905, 
 and "Introduction to the Real Infinitesimal Analysis 
 of One Variable," by Professors Oswald Veblen and 
 N. J. Lennes (John Wiley & Sons). 
 
 I am indebted to Mrs. Elsie Straffin Bronson, A.M., 
 of Providence, for many criticisms and suggestions. 
 
 Henry P. Manning. 
 
 Providence, February, 1906. 
 
CONTENTS 
 
 CHAPTER I 
 Irrational Numbers 
 
 PAGE 
 
 I. Infinite Sets of Objects 1 
 
 II. Definition of Irrational Numbers 5 
 
 III. Operations upon Irrational Numbers 14 
 
 rV. Exponents and Logarithms 25 
 
 CHAPTER II 
 
 Sequences 
 
 I. Representation of Numbers by Sequences 34 
 
 II. Regular Sequences 46 
 
 III. Operations upon Sequences 48 
 
 IV. The Theory of Limits 57 
 
 CHAPTER III 
 
 Series 
 
 I. Convergency of Series 65 
 
 II. Operations upon Series 81 
 
 III. Absolute Convergence 89 
 
 CHAPTER IV 
 Power Series 
 
 I. The Radius of Convergence 100 
 
 II. "Undetermined Coefficients" 108 
 
2 IRRATIONAL NUMBERS 
 
 With reference to a given set of objects the phrase is 
 used as defined above, to express the fact that any in- 
 teger n is the number of only a part of the objects of 
 the set. 
 
 Each integer except 1 is preceded by a finite number 
 of others in the order of counting, and each integer is 
 followed by an infinite number of others. 
 
 3. When we have two infinite sets of objects we can 
 often make them correspond; that is, we can pair them, 
 associating with each object of one set one and only 
 one of the other set. 
 
 Thus the even numbers can be paired with the odd 
 numbers. 
 
 The integers which are the squares of integers can 
 be paired with those which are not squares, although 
 the latter occur more frequently in counting. In this 
 way we have pairs of numbers as follows ; 
 
 1 and 2, 4 and 3, 9 and 5, 16 and 6, etc. 
 
 In this arrangement every number of either kind is 
 associated with one and only one of the other kind. 
 There is no place where the set of squares is exhausted 
 and the numbers which are not squares have to stand 
 alone.* 
 
 In Geometry the points of two circles may be asso- 
 ciated in this way, each point of one being associated 
 with one and only one of the other. We can do this, 
 
 * This illustration was used by Galileo. " Galileo and the Modern 
 Concept of Infinity," Dr. Edward Kasner, Bulletin of the American 
 Mathematical Society, June, 1905, p. 499. 
 
INFINITE SETS OF OBJECTS S 
 
 for example, by making the centres of the two circles 
 coincide and associating points which lie on the same 
 radius. 
 
 We may have a correspondence between two infinite 
 sets of objects when one set is a part of the other set. 
 Thus we can associate the set of even positive integers 
 with the set of all positive integers. In Geometry 
 we can associate the points on two segments of straight 
 lines even when one segment is longer than the other. 
 We can do this by making the two segments two sides 
 of a triangle and associating the points in which they 
 intersect any line parallel to the third side. 
 
 3. A rational number is any number which is a posi- 
 tive or negative integer or fraction, or zero. 
 
 We shall assume that we know how to add or mul- 
 tiply any two rational numbers, to subtract any rational 
 number from any other or the same rational number, 
 and to divide any rational number by any other or the 
 same rational number, with the single exception that 
 we cannot divide any number by zero. 
 
 The result of any of these operations will be a rational 
 number. 
 
 Between any two rational numbers there are others. 
 We get one such number by adding to the smaller some 
 part of their difference. Between this and each of the 
 other two another can be found, and so on. 
 
 Between any two rational numbers, therefore, there 
 is an infinite number of rational numbers. 
 
 If a is any positive rational number, there is an 
 integer n which is greater than a. For, if a is a posi- 
 tive integer any integer that comes after a in counting 
 
4 IRRATIONAL NUMBERS 
 
 will be greater than a, and if a =—, where p and q are 
 
 positive integers, any integer greater than p will be 
 greater. 
 
 If a and b are any two positive rational numbers, 
 there is an integer n such that n6>a;* namely, any 
 
 integer greater than the rational number r-. 
 
 If any particular integer n satisfies either of these 
 conditions, every integer beyond will satisfy the same 
 condition. 
 
 4. If we suppose all rational numbers arranged in 
 order of magnitude, then no one is followed by another 
 which comes next after it. For any two of them are 
 separated by others. 
 
 We cannot realize by our imagination this arrange- 
 ment of rational numbers. We can only reason about 
 it. Thus we can say of any two numbers in this 
 arrangement that one comes before the other, or of 
 any three that one comes between the other two. 
 
 It is possible, however, to arrange the set of all posi- 
 tive rational numbers so that one of these numbers 
 comes first and each of the others is preceded by only 
 a finite number of them, each one having a definite 
 numbered position. That is, it is possible to make 
 this set of numbers correspond to the set of positive 
 integers, one number of each set to be associated with 
 one and only one number of the other set. 
 
 One way of doing this is to arrange these numbers 
 
 * This is called the Law of Archimedes. 
 
DEFINITION OF IRRATIONAL NUMBERS 5 
 
 in the order of magnitude of the sum of the numerator 
 and denominator, those in which the sum is the same 
 being arranged in their own order of magnitude and 
 integers being regarded as fractions with 1 for denomi- 
 nator. This arrangement will be 
 
 il9l-^12 3,1.12 3^_5 
 
 2-"3T32'*5"'6 5"T3 2^---- 
 
 II. Definition of Irrational Numbers 
 
 5. Any rational number a separates all others into 
 two classes, those which are smaller being in one class 
 and those which are larger in the other, and every 
 number of the first class is less than every number of 
 the second class. We may put the number itself 
 into one of the two classes and then we have separated 
 all rational numbers into two classes, every number 
 of the first class less than every number of the second 
 class. 
 
 If we have put a into the first class, it is the largest 
 number in this class. In this case there is no number 
 in the second class which is the smallest number in 
 the second class. For any number b in the second class 
 is larger than a, and there are rational numbers between 
 a and b. These numbers are in the second class because 
 larger than a, and they are smaller than b, so that b is 
 not the smallest number in the second class. 
 
 If we have put a into the second class, it is the small- 
 est number in the second class and there is no number 
 in the first class which is the largest number in the 
 first class. 
 
6 IRRATIONAL NUMBERS 
 
 In either case a occupies the point of separation of the 
 two classes, and we may think of a as determined by 
 the separation. 
 
 Now there are ways in which we can separate all 
 rational numbers into two classes, those of the first 
 class less than those of the second, with no rational 
 number occupying the point of separation, that is, 
 with no number in the first class which is the largest 
 number in the first class, and no number in the sec- 
 ond class which is the smallest number in the second 
 class. 
 
 For example, there is no rational number whose 
 square is 2* If we separate all rational numbers 
 into two classes, putting into the first class all negative 
 rational numbers and all positive rational numbers 
 whose squares are less than 2, and into the second 
 class all positive rational numbers whose squares are 
 greater than 2, the numbers in the first class will be 
 less than those in the second class, and there is no 
 rational number which will be the largest number in 
 the first class or the smallest number in the second 
 class. 
 
 For, let a be any positive number in the first class, 
 that is, any positive rational number whose square 
 is less than 2. Let p be any other positive rational 
 number. 2 - a^ is positive, and (a + p)^ or a^ + p(2o + p) 
 will be less than 2 if 
 
 p(2a + p)<2-a2. 
 
 * This is a proposition of Euclid (Elements, X, 117). 
 
DEFINITION OF IRRATIONAL NUMBERS 7 
 
 This will be true if p is less than some positive number 
 ■p', and at the same time less than 
 
 2-a2 
 
 2a + p'" 
 
 Suppose we take p equal to a half of the smaller of 
 the two numbers 
 
 a-Vp will be a rational number greater than a and its 
 square will be less than 2. 
 
 Again, let a' be any number in the second class and 
 p some other positive rational number less than a'. 
 o'2— 2 is positive, and {a' —pf will be greater than 2 if 
 
 p(2a'-p)<a'2-2. 
 
 This will be true if 
 
 a'2-2 
 
 p< 
 
 2a' 
 
 If we take p equal to a half of this fraction, a' —p will 
 be a positive rational number less than a' and its square 
 will be greater than 2. 
 
 We assume: 
 
 In any separation of all rational numbers into two 
 classes, those of the first class less than those of the 
 second class, there is a number which occupies the point 
 of separation. 
 
8 IRRATIONAL NUMBERS 
 
 This is the definition of continuity. We assume that 
 the system of all numbers is continuous. 
 
 The number is determined by the separation. A 
 number thus determined, if not a rational number, 
 is called an irrational number. 
 
 6. An irrational number separating all rational 
 numbers into two classes is to be regarded as greater 
 than those of the first class and less than those, of the 
 second class. 
 
 Two irrational numbers are different if they do 
 not separate rational numbers into the same two 
 classes. There are numbers which are in the second 
 class in one case and in the first class in the other case. 
 They are greater than one and less than the other of the 
 two irrational numbers. 
 
 If a rational number c lies between two irrational 
 numbers, that irrational number which is less than c 
 is to be regarded as the smaller of the two and the 
 other as the larger of the two. 
 
 We have, then, the following theorem: 
 
 Theorem. — a, /3, and j being any three numbers, rational 
 or irrational, if a<,8 and /?<?■, then a<y. 
 
 Proof. — Any rational number c lying between a and 
 P is less than a rational number c' between /? and y, 
 c being in one of the two classes which determine /? 
 and c' in the other. But either of these numbers is 
 greater than a and less than y. Therefore, a<x. 
 
 In any separation of numbers into two classes deter- 
 mining a number, rational or irrational, we may include 
 
DEFINITION OF IRRATIOXAL NUMBERS 9 
 
 all irrational numbers in the two classes. An irrational 
 number less than a rational number of the first class 
 belongs in the first class, and an irrational number 
 greater than a rational nimaber of the second class 
 belongs in the second class. The number which occu- 
 pies the point of separation may be put into either 
 class. If it is in the first class, it will be the largest 
 number in this class and there will be no smallest 
 number in the second class, rational or irrational. If 
 it is in the second class, it will be the smallest num- 
 ber in the second class and there will be no largest 
 number in the first class. 
 
 7. In Geometry any line commensurable with a 
 given line a has a ratio to a which is a rational number, 
 and any positive rational number is the ratio of some 
 
 line to a. If the rational number is — , where m and n 
 
 n 
 
 are positive integers, we have but to divide a into n 
 equal parts and lay off one of these parts m times on 
 
 another line to get a line whose ratio to a is — . 
 
 If we take a for unit of length, the ratio of the other 
 line to a is called the length of the other line. 
 
 If a line h is not commensurable with a, we may 
 divide all lines which are commensurable with o into 
 two classes, putting those which are shorter than h in 
 the first class and those which are longer in the second. 
 All rational numbers are thus separated into two 
 classes and an irrational number is determined which 
 we call the ratio of h to a. 
 
 "We may think of all lines as laid off on some in- 
 
10 IRRATIONAL M UMBERS 
 
 definite line L from some fixed point on L. a being 
 the unit line, the distance of the end-point of any line 
 from will be the ratio of this line to a. We may let 
 points to the right of have a positive distance and 
 points to the left of a negative distance. The dis- 
 tance of itself will be zero. Every point, then, on 
 the line L will have a distance from which is a rational 
 or an irrational number. 
 
 A'oiy we assume in Geometry: 
 
 In any separation of the points of a line into two 
 classes, those of the first class being to the left of those 
 of the second class, there is a definite point of separa- 
 tion, a point whose distance from a given point on 
 the line, with respect to a given wnit of length, is either 
 a rational or an irrational number. 
 
 That is, we assume that the points of the line form a 
 continuous system. 
 
 Making this assumption, we have the following 
 theorem : 
 
 Theorem. — We can establish a correspondence (Art. 2) 
 between the points of a line and the set of all rational and 
 irrational numbers, associating each point with one and 
 only one number, and with each number one and only one 
 point. 
 
 Proof. — Taking a point on the line and a unit of 
 length, we have proved that every point on the Une has 
 a distance from which is a rational or an irrational 
 number, and that every rational number is the distance 
 from of some point on the line. 
 
 Any irrational number separating all numbers into 
 two classes determines a separation of the points of 
 
DEFINITION OP IRRATIONAL NUMBERS 11 
 
 the line into two classes, and so determines a point 
 whose distance from is this irrational number. There- 
 fore, every irrational number, also, is the distance from 
 of some point on the line. 
 
 If, then, we associate each point of the line with the 
 number which expresses its distance from 0, we have 
 a correspondence between the points and the system 
 of all rational and irrational numbers. 
 
 Every line has a ratio, rational or irrational, to a 
 given line a, and every rational or irrational number 
 is the ratio of some hne to a. 
 
 We make the same assumption of angles, of arcs on 
 the same or equal circles, and of many other magni- 
 tudes of Geometry. 
 
 8. There are several theorems in which geometrical 
 magnitudes are associated in such a way that any two 
 magnitudes of one set are in the same ratio as the 
 corresponding magnitudes of the other set. 
 
 Such a theorem is first proved when the two magni- 
 tudes of each set are commensurable. Then, when 
 the two magnitudes of each set are incommensurable, 
 their incommensurable ratios divide all rational num- 
 bers into the same two classes, and are, therefore, the 
 same irrational number. 
 
 For example, aasume it to have been proved that 
 when a line parallel to one side of a triangle cuts off 
 on the other two sides parts which are commensurable 
 with the whole sides, these parts have the same ratio to 
 the whole sides. When the line cuts off parts which 
 are not commensurable with the whole sides, these 
 parts also have the same ratio to the whole sides. 
 
12 IRRATIONAL NUMBERS 
 
 Let A'B' be parallel to the ade AB of the triangle 
 ABC and cut off on CA and CB parts CA' and CB' 
 not commensurable with CA and CB, to prove that the 
 incommensurable ratios of CA' to CA and of CB' to 
 CB are the same irrational number. 
 
 Any positive rational number which is less than the 
 ratio of CA' to CA is the ratio to CA of a commensurable 
 line shorter than CA', and a parallel to AB cutting 
 off this line, since it cannot cross the parallel A'B', 
 must cut off on CB a line shorter than CB'. But the 
 line so cut off has for ratio to CB the same rational 
 number. Therefore, any positive rational number less 
 than the ratio of CA' to CA, and so any rational number 
 less than the ratio of CA' to CA, is less than the ratio 
 of CB' to CB. In the same way we prove that any 
 r&tional number less than the ratio of CB' to CB is less 
 than the ratio of CA' to CA. 
 
 That is, these two ratios divide all rational numbers 
 into tlie same two classes, and are, therefore, the same 
 irrational number. 
 
DEFINITION OF IRRATIONAL NUMBERS 13 
 
 9. Theorem. — Given a separation of all rational num- 
 bers into two classes, those of the first class less than 
 those of the second, and any positive rational number s, 
 there are two numbers, one in each class, whose difference 
 is less than e. 
 
 Proof. — Let k and k' be two numbers in the first and 
 
 second class, respectively. There is a positive integer 
 
 k' —k 
 n greater than (Art. 3). If we divide k'-k into 
 
 n equal parts, each of these parts will be less than e. 
 Let p be one of these parts and add it to fc n times in 
 succession. We get the numbers 
 
 k k+p k+2p . . . k'. 
 
 Some of these numbers will be in the first class and the 
 rest in the second class. Let a be the last one in the first 
 class and let a' be the number which comes next after 
 a in this set of numbers, and which is, therefore, in 
 the second class. Then 
 
 a' — a=p< e. 
 
 There are, indeed, an infinite number of numbers in 
 each class, any one of those in one class differing from 
 any one of those in the other class by less than e. 
 
 10. Theorem. — Given a set of rational numbers sepa- 
 rated into two classes, those of the first class less than 
 those of the second, if for every positive rational number 
 e there are two numbers, one in each class, whose differ- 
 ence is less than e, ire can by means of this separation 
 determine a separation of all rational numbers, and a 
 
14 IRIIATIONAL NUMBERS 
 
 number, rational or irrational, which occupies the point 
 of this separation. 
 
 Proof. — To determine a separation of all rational 
 numbers that will include the given separation we must 
 put any rational number into the first class if it is less 
 than a number already in the first class, and into the 
 second class if it is greater than a number already in 
 the second class. There can be only one rational num- 
 ber whose position is not in this way determined. For 
 the difference between any two different rational num- 
 bers is a positive number, a value of e, and there are 
 two numbers in the given set, one in each class, whose 
 difference is less than this value of e. 
 
 If there is one rational number which is not less than 
 any nmnber already in the first class nor greater than 
 any mraiber already in the second class, we may put 
 it into either class. Every rational number is, then, 
 assigned to one or the other class, and a number, rational 
 or irrational, is determined by this separation. 
 
 Any separation of numbers into two classes, those of 
 the first class less than those of the second, is called 
 a cvi. We shall, however, use the phrase separation 
 into two classes. 
 
 III. Operations upon Irrational Numbers 
 
 11. Addition. — If, a, b, c, and d are rational num- 
 bers, and if a< b and c< d, then a + c<b+d. 
 
 Now let a and /? be two numbers, one or both irra- 
 tional. Let a be any number in the first and a' any 
 
OPERATIONS UPON IRRATIONAL NUMBERS 15 
 
 number in the second of two classes of rational numbers 
 determining the number a. Let b and b' in the same 
 way denote numbers of two classes determining ^. 
 
 Theorem. — We can determine a separation of all num- 
 bers into two classes by putting numbers of the form a+b 
 into the first class and numbers of the form a' +b' into 
 the second class. 
 
 Proof. a + b<a'+b'. 
 
 Now, e being any given positive rational number, there 
 are numbers a and a' whose difference is less than - , 
 
 and numbers b and b' whose difference is less than-. 
 
 Thus there are numbers a-Vb and a' +6' whose differ- 
 ence is less than s, and a separation of numbers is 
 determined. 
 
 The number determined by this separation is called 
 the sum of a and ^ and is written a +j3. 
 
 Cor. I. — c being a rational number and a irrational, 
 we can determine the sum a+c by the separation of 
 numbers of the form a-^c and ol -^-c. 
 
 Proof. — The separation of the numbers a+c and a' -Vc 
 is sufficient to determine a number, since for any given 
 positive rational number e we can determine numbers 
 a and a' so that 
 
 {a:^c)-{a^c)<s. 
 
16 IRRATIONAL NUMBERS 
 
 Now, given any number a, if k is the difference be- 
 tween a and some rational nimiber lying between a 
 and a, then 
 
 a + c = (a + k) + (c—k), 
 
 and by definition {a+k)+(c-k) is a number in the 
 first of two classes determining the sum a+c. Similarly 
 we prove that a' + c is a number in the second of two 
 classes determining a + c. 
 
 The number determined by the separation of the 
 rational numbers a+c and a' +c must, therefore, be the 
 same as the sum a+c. 
 
 Cor. 2. — ^The commutative and associative laws hold 
 in the addition of irrational numbers, since they hold 
 in the addition of the rational numbers used in defining 
 the sum of irrational nimibers. 
 
 That is, 
 
 (a+/?) + r=a + (/? + r). 
 
 12. Theorem. — a, /?, and y being any three numbers, 
 rational or irrational, if a<p, then a + j<p + j. 
 
 Proof. — Take a' and b so that a'< b, and let c and c' 
 be rational numbers of two classes determining y, c 
 and c' taken so that c' —c<b—a'. Then we have in 
 succession the inequahties 
 
 a + r<a'+c'<b + c<j3 + r; 
 whence a + r<^ + y. (Art. 6) 
 
OPERATIONS UPON IRRATIONAL NUMBERS 17 
 Cor.— If also r<^, then a + yK^ + d. 
 
 13. Subtraction. — We can define a—j3 as the number 
 which added to /3 will give a for sum. For all num- 
 bers may be separated into two classes, those which 
 added to /? give a sum less than a being in the first class 
 and those which give a sum not less than a in the second 
 class. 
 
 Theorem. — We can separate all numbers into two 
 classes by putting numbers of the form a — b' into the first 
 class and numbers of the form a' — b into the second class, 
 and this separation will determine the number a—jS. 
 
 Proof.— a-b'<a'-b, 
 
 and as in the proof of the theorem of Art. 11 we can 
 determine the numbers a, a', b, and b' so that the 
 difference of the numbers a — b' and a' — b shall be 
 less than e, this difference being the same as the differ- 
 ence of the numbers a + b and a'+b'. 
 
 Now, given any a and b', a-V + b' is a number in 
 the second of two classes determining the sum of the 
 rational number a-b' and /? (Art. 11, Cor. 1). But, 
 being equal to a, this number is in the first of two 
 classes determining a, and the sum of the rational 
 number a-b'. and /? is a number less than a. 
 
 Similarly we prove that all numbers of the form 
 a' —b added to /? give a sum which is greater than a. 
 
 The numbers a-b' and a' -h, therefore, determine 
 by their separation the number which added to /? 
 gives the sum a; they determine the number a-/?. 
 
18 IRRATIONAL NUMBERS 
 
 14. If a and b are any two rational numbers, and if 
 a<b, then —b< —a. 
 
 If we have all rational numbers separated into two 
 classes and we change the signs of all these numbers, 
 we shall get a new separation of all rational numbers, 
 those in either class in the first separation becoming 
 those in the other class in the second separation. Unless 
 these separations determine the number zero they will 
 determine two different numbers, one greater than zero, 
 or positive, and the other less than zero, or negative. 
 These two nimibers are said to be numerically equal 
 with opposite signs. Each is the negative of the other, 
 and that one which is positive is the numerical value of 
 both. 
 
 The numerical value of a number is also called its 
 modulus, and is often denoted by placing the number 
 between vertical lines. Thus, we write \a\ for the 
 numerical value of a. 
 
 a and /? being any two numbers, 
 
 a-^=a + (-/?); 
 
 for both members of this equation are determined by 
 a separation of rational numbers of the form 
 
 a — b' and a' —b, 
 which can also be written 
 
 a + {-b') and a' + i-b). 
 
 In a similar way we prove that the numbers a—/? 
 
OPERATIONS UPON IRRATIONAL NUMBERS 19 
 
 and /?— a are numerically equal with opposite signs, 
 so that we may write 
 
 /9-«=-(«-/8), 
 
 |/9-ai = |a-/?|. 
 
 ja— jS] is called the difference of the two numbers 
 a and ^. 
 
 The difference of two numbers may always be obtained 
 by changing the sign of one of them and adding, and 
 since the numerical value of the sum of two numbers 
 is equal to the svun or difference of their nimierical 
 values, we can say that the nimierical value of the 
 sum or difference of two numbers is equal to or less 
 than the sum and equal to or greater than the differ- 
 ence of their numerical values. 
 
 That is, we may write 
 
 \a±p\l\a\ + \^\, 
 and >l«|-l/?l. 
 
 and either letter in the last expression may be written 
 in the first term. 
 
 In particular, if |x-cl<a and \y-c\<p, then 
 
 \x — y\<a-\-p. 
 
 iS. Multiplication.— If o, h, c, and d are positive 
 rational numbers, a<h and c<d, then ac<bd. 
 
 Now let a and P be two positive numbers, one or 
 both irrational. Let a be any positive number in the 
 first and a' any number in the second of two classes 
 
20 IRRATIONAL NUMBERS 
 
 of rational numbers determining a, and let h and h' 
 represent in the same way two classes determining /?. 
 
 Theorem. — We can determine a separation of all num- 
 bers into two classes by putting numbers of the form ab 
 into the first class and numbers of the form a'V into the 
 second class. 
 
 Proof. ab<a'b'. 
 
 Now let e be any positive rational number, p being 
 any other positive rational number, there are numbers 
 a and o' whose difference is less than p, and numbers 
 h and b' whose difference is less than p. That is, there 
 are numbers for which 
 
 a' <a+p and 6'<&+p; 
 
 therefore, a'b' <ab+p{a+b-\-p). 
 
 If p is taken less than some particular positive 
 rational number pi and we let oi and b\ be two 
 particular rational numbers greater than a and p, 
 respectively, so that 
 
 a + 6 + p<oi+6i+pi, 
 
 and if we also take p less than 
 
 ai+bi+pi 
 we shall have a'b' <ab-\-e, 
 
 or a'b'-ab<u 
 
OPERATIONS UPON IRRATIONAL NUMBERS 21 
 
 The number determined by this separation is called 
 the product of a and ^ and is written a/3. 
 
 If one of the two numbers is zero, we shall say that 
 the product is zero. If one of the numbers is negative, 
 or if both are negative, the numerical value of the 
 product will be the product of their numerical values, 
 and the product itself will be positive or negative 
 according as they have like or unlike signs. 
 
 Cor. I. — c being a rational number and a irrational, 
 we can determine the product ac by the separation of 
 numbers of the form ac and a'c. 
 
 Proof. — ^The separation of the numbers ac and a'c 
 is sufficient to determine a number, since we can 
 make 
 
 a'c—ac<£ 
 
 by taking a and a' so that 
 
 a' — a<— . 
 c 
 
 Now, given any a, if A; is the ratio to a of some rational 
 number lying between a and a, so that the number 
 between a and a may' be written ak, then 
 
 ac=ak-j, 
 
 and by definition this is a number in the first of two 
 classes detei'mining the product ac. 
 
 Similarly we prove that a'c is a number in (he second 
 of two classes determining ac. 
 
22 IRRATIONAL NUMBERS 
 
 Therefore, the number determined by the separation 
 of the numbers ac and a'c must be the same as the 
 number ac. 
 
 Cor. 2. — The commutative, distributive, and associa- 
 tive laws hold in the multiplication of irrational num- 
 bers, since they hold in the multiplication of the rational 
 numbers used in defining the product of irrational 
 numbers. 
 
 That is, a^=pa, 
 
 (a/?)r=«(,5r), 
 
 16. Theorem. — a, /?, and y being any three positive 
 numbers, rational or irrational, if a<p, then ayK^y. 
 
 Proof, — Take a' and b so that a' <b; let ci be some 
 particular rational number less than y and take c and 
 c' rational numbers of two classes determining y, c and 
 c' taken so that 
 
 c>Ci and at the same time c'—c<— — -. — . 
 
 a 
 
 We shall then have 
 
 , c(b-a') 
 
 or a'c' < be. 
 
 That is, ay<a'c'<bc<^y. 
 
 Cor.— If also y<3. then ay < 1^3. 
 
OPERATIONS UPOX IRRATIONAL XUMBERS 23 
 
 17. Division. — a and /? being any two numbers, one 
 or both irrational and a not zero, we can define the 
 
 quotient - as the number which multiplied by a will 
 
 give /? for product. 
 
 For, if a and p are positive, we can separate all num- 
 bers into two classes by putting into the first class 
 those numbers which multiplied by a give a product 
 less than /?, and into the second class those which mul- 
 tiplied by a give a product not less than /?. Thi? 
 separation determines a nxmiber which multiplied by 
 a gives /? for product. 
 
 If ^ is zero, the quotient will be zero. 
 
 If one number is negative, or if both numbers arc 
 negative, the numerical value of the quotient wiU b( 
 the quotient of their numerical values, and the quo- 
 tient itself will be positive or negative according a? 
 they have like or unlike signs. 
 
 a and /? being positive, let k be some positive rational 
 number less than a, and for the moment restrict a to 
 rational numbers which are greater than k in the first 
 of the two classes determining a. a', b, and b' being 
 defined as before, we have the following theorem: 
 
 Theorem. — We can separate all numbers into two classes 
 by putting numbers of the form -,- into the first class and 
 
 numbers of the farm — into the second class, and this sep- 
 aration will determine the number -. 
 
24 IRRATIONAL NUMBERS 
 
 b h' 
 
 V b a'b'-ab 
 
 Now as in the proof of the theorem for multiplication 
 we have 
 
 a'b'-ab<p{ai+bi+pi), 
 
 and if we take p also less than 
 
 ai+bi+pi' 
 
 b' b 
 we shall have -7 < «• 
 
 a a 
 
 But, given any a' and b, —a' is a number in the 
 second of two classes determining the product of the 
 rational number -7 and a (Art. 15, Cor. 1); and, being 
 
 equal to b, this number is in the first of two classes 
 determining /?. Thus the product of the rational num- 
 ber -7 and a is a number less than 3. 
 a 
 
 Similarly we prove that all numbers of the form — 
 
 multiplied by a give a product which is greater than /?. 
 
 The numbers — and — , therefore, determine by their 
 
EXPONENTS AND LOGARITHMS 25 
 
 separation the number which multiplied by a gives the 
 
 g 
 product |8: they determine the number -. 
 
 IV. Exponents and Logarithms 
 
 18. We obtained an example of an irrational number 
 by separating all rational numbers into two classes, 
 those of the first class comprising all negative rational 
 numbers and all positive rational numbers whose 
 squares are less than 2, those of the second class com- 
 prising all positive rational numbers whose squares are 
 greater than 2. We can now say that the irrational 
 number so determined is a number whose square is 
 equal to 2, and that it may be called the square root 
 of 2. 
 
 In general we have the following theorem : 
 
 Theorem. — a being any positive rational or irrational 
 number and n any positive integer, there exists a positive 
 number /? whose nth power equals a. 
 
 Proof. — We can separate all positive numbers into 
 two classes according as their nth powers are less than 
 a or not less than a. Let a be any positive number 
 whose nth power is less than a, and b any positive 
 number whose nth power is not less than a. Then a 
 must be less than b; for, otherwise, the nth power of 
 a would not be less than the nth power of b (Art. 16). 
 Putting negative numbers into the first class we have, 
 therefore, a separation in which all numbers of the 
 
26 IRRATIONAL NUMBERS 
 
 first class are less than those of the second class, and 
 a number /? is determined by this separation. 
 
 Now a can be determined by the separation into two 
 classes of numbers of the form a" and &". Therefore, 
 the nth power of p is equal to a. 
 
 fi is called the nth root of a and is written y'a. 
 
 19. Theorem. — If a number a is greater than 1, and e 
 is any positive number, there is an integer n such that 
 
 a">£. 
 
 Proof.— Let a = l + d. 
 
 Then we can prove by induction 
 
 a"f l+n5. 
 
 For, assuming that this is true for n, we have 
 
 a"+'f{l+nd){l+d) 
 
 >l + in + l)d, 
 
 so that what we have assumed, if true for n, is true 
 for n + 1, and, being true for n = l, is true for all values 
 of n. 
 Therefore, for all values of n 
 
 a''>nd. 
 
 Now by Art. 3 there is an integer n for which n8> s 
 (if 5 or £ is irrational, n may be any integer greater 
 
EXPONENTS AND LOGARITHMS 27 
 
 than some rational number in the second of two classes 
 
 n 
 
 determining y\. For any such integer 
 
 Cor. — If a number a is greater than 1 and e is any 
 positive number, there is an integer n such that 
 
 Proof. — Let /? be a number lying between 1 and 
 1 + e. There is, then, an integer n such that 
 
 and for this number n 
 
 V'a</?<l + e; 
 
 that is, v^a - 1< e. 
 
 \/a is greater than 1 and differs from 1 by less than e. 
 
 If a particular integer n satisfies the theorem or the 
 
 corollary, the same will be true of any integer beyond. 
 
 20. Theorem. — I, r, and s being positive integers and a 
 a positive number, 
 
 {%)'' = (i/ay. 
 Proof. — Let a denote the kth root of a, so that a'' = a. 
 
 Then a = '\ a and a =(ya) . 
 
28 IRRATIOXAL NUMBERS 
 
 But a}" and a''' may be regarded, respectively, as the 
 product of s factors and the product of r factors each 
 equal to a'. Therefore, in the first place, 
 
 I »/ 
 a =ya, 
 
 and then, a'"" = (-/a)'". 
 
 That is, i%y' = {i/ay. 
 
 Using fractional exponents we write the last theorem 
 
 That is, if p denotes any positive rational number, we 
 have a definite meaning for the expression aP. 
 
 We shall understand that a negative exponent is 
 defined by writing 
 
 a" 
 and that a° = l. 
 
 21. Theorem. — o being a positive number, rational or 
 irrational, and p and q any rational numbers, 
 
 Proof. — First, when p and q are both positive, let 
 s be their common denominator, so that we can write 
 
 r , / 
 
 p=- and q = -. 
 
EXPONENTS AND LOGARITHMS 29 
 
 Then, if a' = a, 
 
 aP=a'; a^=a'", and aP+9=a'"+'"'. 
 But a''a'''=a'"+'"', 
 
 our theorem for positive integer exponents being a 
 particular case of the associative law. Therefore, 
 
 aPo«=aP+«. 
 
 Suppose one of these exponents negative and numer- 
 ically smaller than the other, say q=—q', where q' 
 is positive and less than p. Then 
 
 aPa,''=~r=aP~^; 
 a'' ' 
 
 for a^'aP~i'=aP. 
 
 If both exponents are negative, or if one exponent 
 is negative and numerically greater than the other, let 
 p== —p' and 5= —q". 
 
 Ill 
 aPfflS = — ;— , = , , . == aP"*"'. 
 ap'a''' aP'+«' 
 
 The formula reduces to an identity when either 
 exponent is zero. 
 
 22. Theorem. — a being a positive number, rational or 
 irrational, and p and q any rational numbers, p<q, 
 
 aP<a'' when a>l, 
 and aP>a'i " a<l. 
 
30 IRRATIONAL NUMBERS 
 
 Proof. — First, when o is greater than 1. 
 Using the same notation we have, when p and q are 
 positive, 
 
 aP=a^ and a' =«'"', r</. 
 
 But when o>l, a>\ and a'' KaJ" , the latter power 
 being obtained from the former by additional factors 
 a greater than 1. Therefore, 
 
 If p and q are negative, say 'p=—'p' and q=—q', 
 g' < p', then 
 
 a^'<aP' and aP<a^. 
 If p is negative and q positive, 
 
 aP<l and a''>l; hence, aP<a^. 
 If either exponent is zero, we have 
 l<a« or a''<l. 
 
 Second, when a<l put a=T, so that 6>1. Then 
 
 ?)P<6« and aP>a^. 
 
 Cor. — If p and g have the same sign, |p|<|g|, then 
 
 laP-l|<|a9-ll. 
 
 23, Theorem. — a being a positive number different 
 from 1, and p and q any rational numbers each numer- 
 
 V 
 
EXPONENTS AND LOGARITHMS 31 
 
 ically less than some given number M, to every positive 
 number e corresponds a positive number d such that 
 
 \a^—af\<E when \q — p\<d. 
 
 Proof. — ^Take p<q, say q-p=r. Then 
 
 oa-aP=aP(a''-l), 
 
 andifa>l, \a''-ap\<a^'ia'-l). 
 
 Now there is a positive integer n such that 
 
 fl^-l<i (Art. 19, Cor.) 
 
 and if we take 5 =— , we shall have, for values of r less 
 
 lb 
 
 than d, 
 
 \ai-ap\<e. (Art. 22, Cor.) 
 
 If a< 1, put a=-T so that 6> 1, and 
 \a9-aP\==\b-^-b-p\<e 
 when \-q + P\<^> 
 
 d bemg equal to the fraction — determined so that 
 
 1 £ 
 
32 IRRATIONAL NUMBERS 
 
 34. Theorem. — a being a positive number different 
 from 1, the separation of all rational numbers p into two 
 classes determining an irrational number X produces a 
 separation of all the numbers af, and so of all numbers, 
 into two classes, determining a number, rational or irra- 
 tional, which we may call a^. 
 
 Proof. — Let pi be any number in the first and p2 
 any number in the second of two classes of rational 
 numbers determining A. Then 
 
 aP'Ka"' ■ when a>l, 
 and a^>a^' " a<l. 
 
 Thus in either case the numbers a" are separated into 
 two classes, those of one class less than those of the 
 other class. 
 
 Moreover, there are numbers a''' and a*^ whose dif- 
 ference is less than any given positive number e, since 
 there are numbers pi and pz whose difference is less 
 than the number 8 of the last theorem. 
 
 Therefore, the separation of the numbers a^ into two 
 classes determines a separation of all numbers into two 
 classes, and a number, rational or irrational, which 
 occupies the point of this separation. 
 
 The preceding theorems, stated only for rational 
 exponents, can be proved true also of irrational expo- 
 nents. 
 
 For example, let pi and p2 be any two numbers of 
 two classes of rational numbers determining X, and gi 
 
EXPONENTS AND LOGARITHMS 33 
 
 and 52 rational numbers of two classes determining //, 
 X and /i being one or both irrational. 
 
 The separation of the numbers a^'a^^ and a^a^' into 
 two classes determines the same number, the product 
 a^a", as the separation of the same numbers written 
 aPi+«i and aP'+'"' into the same two classes, which de- 
 termines the number a^^". Therefore 
 
 35. Theorem. — a and b being any two positive num- 
 bers, a different from 1, there is a number A such that 
 
 a^ = b. 
 
 Proof. — We can separate all rational numbers into 
 two classes, putting a number p into one class if a^ < b, 
 and into the other class if aP>6. By the theorem of 
 Art. 22 the numbers p of one class will be less than 
 those of the other class. This separation determines a 
 number X, and b is the number denoted by a\ 
 
 X is called the logarithm of b to the base a. 
 
CHAPTER II 
 
 SEQUENCES 
 
 I. Representation of Numbers by Sequences 
 
 26. A sequence is an infinite set of numbers arranged 
 so that each one has a definite numbered position; 
 that is, one comes first, each is followed by one that 
 comes next after it, each except the first is preceded 
 by a finite number of others, and each is followed by 
 an infinite number of others. 
 
 A sequence is a set of numbers placed in correspond- 
 ence with the set of positive integers (Art. 2). 
 
 The numbers of a sequence are called its elements. 
 
 A sequence may be expressed by a formula which 
 gives the nth element for every value of n, or by a 
 statement which indicates in some way how each ele- 
 ment is determined. 
 
 A sequence is often indicated by a certain number of 
 the elements at the beginning followed by dots, but we 
 ought always to give a formula or to state the law by 
 which the elements are determined. The formula or 
 law need apply, however, only to those numbers not 
 given. Indeed, we may put for any finite number of 
 the elements any numbers we please, and give a for- 
 mula which applies only to the numbers not written. 
 
 34 
 
REPRESENTATION OF NUMBERS BY SEQUENCES 35 
 
 We will give the following examples of sequences: 
 
 (1) The sequence 
 
 
 1 
 
 11 1 
 
 2 3 ■ ■ ■ n ■ • • • 
 
 (2) The sequence of prime numbers 
 
 1 
 
 2 3 5 7 . . . . 
 
 (3) The sequence 
 
 
 1 
 
 2 n 
 
 3 
 
 5 • ■ ■ 2n + l • • ■ ■ 
 
 (4) The sequence 
 
 
 12 4 3 
 
 4 5 n-2 
 ^3 4 • • • n-3 
 
 n any integer greater than 5. 
 
 (5) A sequence of fractions whose denominators are 
 the successive integers beginning with 3, the fractions 
 corresponding to even integers all equal to ^ and the 
 fractions corresponding to odd integers alternately less 
 and greater than i, differing from i by the smallest 
 amount possible for fractions with the given denom- 
 inators; namely, the sequence 
 
 12 3 3 3 4 5 
 3 4 5 6 7 8 9 
 
 (6) The sequence whose first element is 1, and in 
 
36 SEQUENCES 
 
 which from any element a we get the next by adding 
 to a the largest integer not greater than y/a. The first 
 ten elements of this sequence will be 
 
 1 2 3 4 6 8 10 13 16 20. 
 
 (7) We may suppose a sequence to consist of one 
 number or certain numbers repeated. For example, 
 
 2 2 2 ... 2 ... . 
 
 27. It is possible to arrange a set of numbers in 
 a way that will form a sequence, and in another way 
 that will not form a sequence. We may, for example, 
 select an infinite set from the elements of a sequence 
 and let the remaining elements come after them. Thus, 
 taking the set of positive integers, we might suppose 
 all odd numbers to come first, in order, and then all 
 even numbers. In this arrangement there is a num- 
 ber, 1, which comes first, and each number is followed 
 by one which comes next after it; but any even num- 
 ber is preceded by an infinite number of other num- 
 bers. The numbers in this arrangement do not form 
 a sequence. 
 
 We have already seen that the set of all positive 
 rational numbers may be arranged so as to form a 
 sequence although these numbers do not form a se- 
 quence when arranged in order of magnitude (Art. 4). 
 Indeed, the set of positive integers, and consequently 
 the numbers of any sequence, may be arranged so that 
 between any two there are others, — so that no num- 
 ber of the set is followed by one which comes 
 
REPRESENTATION OF NUMBERS BY SEQUENCES 37 
 
 next after it. To do this we have but to establish 
 a correspondence between the set of positive integers 
 and the set of all positive rational numbers, as is done 
 on page 5, and then to arrange the set of positive 
 integers in the order of magnitude of the correspond- 
 ing rational numbers. In such an arrangement of 
 integers each integer is preceded by an infinite num- 
 ber of others, as well as followed by an infinite num- 
 ber of others. 
 
 38. An irrational number is determined by the place 
 which it occupies among rational numbers, but it is 
 often most conveniently represented by a sequence. 
 
 The sequence 
 
 01 Oz . . . On • • . 
 
 represents a nmnber a if for every positive number s 
 there is a place in the sequence beyond which all the 
 elements differ from a by less than e. 
 
 Thus the sequence 
 
 12 n 
 
 2 3 • ■ • ^m • * • 
 
 n 
 
 represents the number 1. 
 For, the difference between the number 1 and ^y^, 
 
 the nth element of this sequence, is — ^, and this 
 
 is less than s if n>--l. Now there is a positive 
 
38 SEQUEXCES 
 
 integer n which is greater than — 1, and all integers 
 
 beyond are greater still. Therefore, there is a place 
 in the given sequence beyond which all the elements 
 differ from 1 by less than e. 
 
 A sequence cannot represent two different numbers, 
 for no element could differ from both by less than 
 a half of their difference. 
 
 If a sequence represents a number a, we may speak 
 of a as the value of the sequence, or say that the sequence 
 is equal to a. 
 
 We sometimes say that a sequence is convergent 
 if it represents a number. 
 
 39. Theorem. — A sequence represents the number zero 
 if for every positive number s there is a place in the 
 sequence beyond which all its elements are numerically 
 less than e. 
 
 Proof. — The difference between a number and zero 
 is the numerical value of the number. 
 
 Thus the sequence 
 
 1 I 1 
 
 2 n 
 
 represents the number zero. For, -<s if n > -, and 
 
 n e 
 
 given any value of e, there is a point beyond which 
 this is true for all values of n. 
 
 30. We often use the expression " any given num- 
 ber", "any assigned number", "an arbitrary num- 
 ber", or "any number", meaning every number. 
 
REPRESENTATION OF NUMBERS BY SEQUENCES 39 
 
 That is, the theorem in which such an expression is 
 used is true for every number. 
 
 The expression " any number " may be used with 
 certain restrictions. In fact, we generally restrict it 
 to positive numbers, and we sometimes restrict it to 
 positive rational numbers. 
 
 We have made use of expressions of this kind in 
 Arts. 9 and 10 and in Arts. 19 and 23. 
 
 We shall generally use the letter e to represent 
 " any positive number ". 
 
 31. When a sequence represents a number a, the 
 place beyond which all the elements differ from a by 
 less than e will depend, in general, on the value of e. 
 
 We may say, however, 
 
 (1) If a particular place satisfies the condition for 
 a particular value of e, any place beyond will satisfy 
 the condition for the same value of e. 
 
 (2) If a particular place satisfies the condition for 
 a particular value of e, it will satisfy the condition 
 for any larger value of e. 
 
 Given two values of e, the first place which satisfies 
 the condition for the smaller value of e will generally, 
 though not always, be farther along than the first place 
 which satisfies the condition for the larger value of e. 
 
 There is no place which satisfies the condition 
 for all values of e unless there is a place beyond 
 which the elements of the sequence are all equal 
 to a. 
 
 A place in the sequence beyond which all the ele- 
 ments differ from a by less than e may be called a 
 place corresponding to e with respect to a. If a sequence 
 
40 SEQUENCES 
 
 represents a number a, then to every e corresponds 
 some place in the sequence with respect to a. 
 
 If a sequence does not represent o, there must be 
 at least one value of s for which the above condition 
 is not satisfied. By Aartue of (2) the same is true of 
 every smaller value of e. Let ei be a value of e to 
 which no place corresponds with respect to a. Then 
 there are in the sequence an infinite number of ele- 
 ments which differ from a by as much as £i. For, 
 beyond some particular place there is at least one, 
 say ai] beyond ai there is at least one, a2] beyond az 
 another, and so on without end. 
 
 33. Theorem. — // a is a number numerically less than 
 1, the sequence 
 
 a a^ . . . a" . . . 
 
 represents ike number zero. 
 
 Proof. — Since | « 1 < 1 , j 
 number e there is an integer n such that 
 
 Proof. — Since |n'l<l, j — r>l, and for any positive 
 
 (^)">^. 
 
 (Art. 19) 
 
 or \a\''<£. 
 
 This, being true for a particular integer n, is true for 
 all greater integers. 
 
 Again, since |a|" is the same as ja"] (Art. 15, just 
 before Cor. 1), there is a positive integer corresponding 
 
REPRESENTATION OP NUMBERS BY SEQUENC1':S 41 
 to e such that for all integers greater than this integer 
 
 la"|<s. 
 
 That is, for any positive number s there is a place 
 in the sequence of powers of a beyond which these 
 powers are numerically less than £. 
 
 33. Theorem. — Any number may be represented by a 
 sequence of rational numbers. 
 
 Proof. — Let 
 
 «! a2 . . . a„ . . . 
 
 be a sequence of rational numbers representing zero, 
 none of the numbers being themselves zero. To form 
 a sequence representing a number a we may take for 
 oi the greatest multiple of ai which is not greater than 
 a, for 02 the greatest multiple of a2 which is not greater 
 than a, and so on. Then the sequence 
 
 oi a2 . . . an ■ ■ . 
 
 will represent the number a. For, a„ differs from a by 
 less than a„, and for any e there is a place in the se- 
 quence of a's beyond which they are all numerically 
 less than e. 
 
 Suppose we wish to form a sequence representing the 
 square root of 2 and we take for the a's the sequence 
 
 1 '- '- .'- 
 2 3 n 
 
42 SEQUENCES 
 
 The largest multiple of - less than \/2 will have 
 
 for numerator the largest integer whose square is less 
 than 2n2. The largest squares less than the numbers 
 
 2 8 18 32 50 . . . 
 are 1 4 16 25 49 . . . 
 
 Therefore, the square root of 2 is repiesented by the 
 sequence 
 
 2 4 5 7 
 
 2 3 4 5 • • ■ 
 
 The diagram opposite will serve to show how these 
 numbers represent the square root of 2. 
 
 To represent a rational number we may take a 
 sequence all of whose elements are equal to this 
 number. 
 
 On the other hand, we may have sequences whose 
 elements are irrational numbers. 
 
 34. Theorem. — // a sequence represents a number a, the 
 numerical values of its elements form a sequence repre- 
 senting the numerical value of a. 
 
 Proof. — If ]a„ - fl ] < £, 
 
 then ] |a„l — |a|| < £, 
 
 the latter difference being equal to or less than the 
 former (Art. 14). Therefore, if any place in the origi- 
 
REPRESENTATION OF NUMBERS BY SEQUENCES 43 
 
 ■io 
 
 
 ■• 1 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 v7 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V7 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 
 4 
 
 « 
 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 i 
 
 t 
 
 / 
 
 \ 
 
 
 A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 « 
 
 
 / 
 
 \ 
 
 in 
 
 
 \ 
 
 / 
 
 10 S 
 
 \ 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 \, 
 
 
 " 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 / 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 3 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 The diagram is on the scale of 
 
 
 
 
 
 
 
 
 
 
 
 30 : 1 . Multiplying the numbers 
 
 
 
 
 
 
 
 
 
 
 
 of the sequence by 30, we have 
 
 
 
 30 
 
 30 
 
 
 
 
 
 
 
 30 30 40 37§ 42 40 38^ 41} 
 
 oO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 40 42 40^ J 40 41 t', . . 
 
 
 
 
 
 
 
 
 
 
 
 In other words, the divisions 
 
 
 
 
 
 
 
 
 
 
 
 represent 30ths. 
 
 (, 
 
 
 
 
 
 
 
 
 
 
 
 
 nal sequence corresponds to £ with respect to a, the 
 same place in the sequence of numerical values will 
 correspond to e with respect to the numerical value of o. 
 
 35. Theorem.—// a sequence represents a number a, 
 and M is any number different from a, there is a place 
 in the sequence beyond which all the elements differ from 
 a by less than the difference between a and M. 
 
 Proof —The difference between o and M is a positive 
 number not zero, a particular value of e. The theorem, 
 then, follows from definition. 
 
 Cor. I.— If M> a, there is a place beyond which all the 
 elements are less than M. 
 
^A SEQUENCES 
 
 Cor. 2. — If M<a, there is a place beyond which all 
 the elements are greater than M. 
 
 Cor. 3. — If a is not zero, there is a place beyond 
 which all the elements have the same sign as a. 
 
 For, if o is not zero, we can take zero for M, and any 
 number between zero and a has the same sign as a. 
 
 36 Theorem. — Given two convergent sequences, if there 
 is a place beyond which the elements of the first are greater 
 than the corresponding elements of the second, the num- 
 ber represented by the first is equal to or greater than the 
 number represented by the second. 
 
 Proof. — Let the two sequences be 
 
 ai a2 . . . a„ . . . representing a, 
 and bi b2 . . • bn • ■ • b, 
 
 and suppose for all values of n 
 
 If a<b, take ff = i(6-a). For all values of n beyond 
 a certain place we have 
 
 an<a + B and bn>b — £; 
 
 and, therefore, an—bn<a — b+2e=0; 
 that is, an < bn, 
 
 which is contrary to hypothesis. Therefore, 
 
 a^b. 
 
REPRESENTATION OF NUMBERS BY SEQUENCES -15 
 
 Cor. — If in a convergent sequence there is a place 
 beyond which all the elements are greater than a 
 certain fixed number b, then the number -which the 
 sequence represents is equal to or greater than b; or 
 if there is a place beyond which all the elements are 
 less than b, the number which the sequence represents 
 is equal to or less than b. 
 
 37. Theorem. — // each element of a sequence is greater 
 than the preceding and less than some number M, the 
 sequence represents a number, either M or some smaller 
 number. 
 
 Proof. — ^We can separate all numbers into two 
 classes, putting any number into the first class if there 
 is a place in the sequence beyond which the elements 
 are greater than this number, into the second class 
 if there is no place beyond which the elements are 
 greater than this number. The elements themselves 
 are numbers in the first class, each being less than 
 any that come after it. M is a number in the second 
 class. This separation of all nimibers determines a 
 number a, rational or irrational. 
 
 Now, e being any positive number, o — £ is a num- 
 ber in the first class, and there is a place in the sequence 
 beyond which the elements are all greater than a-e. 
 But no element is greater than a. Therefore, there 
 is a place beyond which the elements differ from a 
 by less than e, and hence the sequence represents a. 
 
 a belongs to the second class and is the smallest 
 number in the second class. Therefore, a is equal to 
 or less than M. 
 
46 SEQUENCES 
 
 In the same way we may prove 
 
 Theorem. — // each element of a sequence is less thai, 
 the preceding and greater than some number M, tht 
 sequence represents a number, either M or some larger 
 number. 
 
 II. Regular Sequences 
 
 38. The sequence 
 
 Oi 02 . . ■ a„ . . . 
 
 is regular if for every e there is a place in the sequence 
 beyond which all the elements differ from one another 
 by less than s. 
 
 39. Theorem. — // a sequence represents a number a, it 
 is regular. 
 
 Proof. — If a sequence represents a number a, there 
 is a place beyond which all the elements differ from a 
 
 by less than -, and beyond this place they will differ 
 
 from one another by less than e. 
 
 40. Theorem. — Conversely, a regular sequence repre- 
 sents a number. 
 
 Proof. — Given a regular sequence 
 
 Ql 02 ... On ... , 
 
 we can by means of this sequence determine a separa- 
 tion of numbers as follows: 
 
REGULAR SEQUENCES 47 
 
 Let any number c be in the first class if there is a 
 place in the sequence beyond which all the elements 
 are greater than c; otherwise let c be in the second 
 class. 
 
 There are numbers in each class. For, there is a 
 place in the sequence beyond which all the elements 
 differ from one another by less than e. a„ being an 
 element beyond this place, a„ - £ is a number in the 
 first class and a„ + £ is a number in the second class. 
 
 This separation of numbers into two classes deter- 
 mines a number a, rational or irrational. 
 
 Now a — e belongs to the first class, and there is a 
 place in the sequence beyond which all the elements 
 are greater than a-e. 
 
 Again, a + ^ belongs to the second class, and beyond 
 
 any place in the sequence there are elements not greater 
 than this number. But there is a place beyond which 
 all the elements differ from each other by less than 
 
 ■^, and if some of them are not greater than a+-x, they 
 
 must all be less than a + e. 
 
 We have, then, a place in the sequence beyond which 
 the elements are all greater than a-e, and a place 
 beyond which they are all less than a + e. Beyond 
 the farther of these two places they will all differ from 
 a by less than e. 
 
 Therefore, the sequence represents the number a. 
 
48 SEQUENCES 
 
 III. Operations upon Sequences 
 
 41. Theorem. — // a sequence represents a number a, 
 any sequence formed hy taking a part of its elements will 
 represent the same number a. 
 
 Proof. — Each element of the new sequence is at 
 least as far along in the original sequence, and a place 
 which corresponds to e in the original sequence will 
 certainly correspond to e in the new sequence. 
 
 In this theorem we suppose that the order of any 
 two elements in the new sequence is the same as in 
 the original sequence, that the elements taken form a 
 sequence and are not simply some finite number of 
 elements, and that no element taken is repeated except 
 as it may have been repeated in the original sequence. 
 
 The theorem may also be stated as follows: 
 
 Theorem. — // a sequence represents a number a, any 
 sequence which we get by leaving out a part of its elements 
 will represent the same number a. 
 
 Cor. I. The value of a regular sequence is not changed 
 by interpolating elements if the new sequence is regular. 
 
 Cor. 2. If two regular sequences have an infinite 
 number of elements in common, they represent the 
 same number. 
 
 42. Theorem. — // a sequence represents a number a, 
 we may change the order of its elements in any manner, 
 and it will still represent (lie number a if the elements in 
 
OPERATIOXS UPON SEQUENCES 49 
 
 the new arrangement actually form a sequence, each ele- 
 ment having a definite numbered position with only a 
 finite number of elements coming before it (see Art. 27). 
 
 Proof.— Let us suppose that all the elements of the 
 original sequence beyond the first n differ from a by 
 less than £. These first n elements will occupy certain 
 definite positions in the second sequence, and beyond 
 the farthest of them the elements of the new sequence 
 will all differ from a by less than s. Therefore, the 
 new sequence represents the same number a. 
 
 The number represented by a sequence does not 
 depend on the particular order of the elements. In 
 fact, we may represent a number a by an infinite set 
 of numbers not thought of as arranged in any order 
 whatever, the set being defined so that it is deter- 
 mined of every number that it is a number of the 
 set or that it is not a number of the set. 
 
 To do this we should have the following definition: 
 
 An infinite set of numbers represents a if for every e 
 it is true that all but a certain number of them differ 
 from a by less than e. 
 
 Thus we may dispense with the element of order, 
 but any set of numbers which represents a number a 
 may be arranged as a sequence,* and for us it will 
 be simpler and in many theorems necessary to con- 
 sider the set as so arranged. 
 
 * For example, when an infinite number of them are less than a 
 and an infinite number of them are greater than a, we might take 
 the numbers of the two sets alternately, those less than a in order 
 of magnitude and those greater than a in descending order of mag- 
 nitude. 
 
50 SEQUENCES 
 
 43. Theorem. — // the differences of corresponding ele- 
 ments of two sequences form a sequence equal to zero, and 
 one of the sequences represents a number a, the other 
 sequence represents the number a, and the two sequences 
 are equal. 
 
 Proof. — Let the sequences be 
 
 ai 02 . . . ttn . . . representing o, 
 
 and bi 62 ...&«.•• . 
 
 If there is a place beyond which the b's differ from the 
 corresponding o's by less than r-, and a place beyond 
 
 which the a's differ from o by less than ^, then beyond 
 
 the farther of these two places the b's will differ from 
 a by less than £. This being true for every e, the b's 
 must form a sequence representing a. 
 
 44. Theorem. — // two sequences represent two num- 
 bers a and b, the sums of their corresponding elements 
 form a sequence representing the sum a+b. 
 
 Proof. — There is a place in the first sequence beyond 
 which its elements differ from a by less than — and a 
 place in the second sequence beyond which its elements 
 differ from b by less than ■^. Beyond the farther of 
 these two places the sum of corresponding elements 
 
OPERATIONS UPON SEQUENCES 51 
 
 from the two sequences will differ from a + b by less 
 than £. 
 
 45. Theorem. — // the two sequences 
 fll 02 . . . fi„ . . . 
 and bi b2 . . . bn ■ . . 
 
 represent the numbers a and b, respectively, the sequence 
 
 a-i—bi 02-62 . . . ttn—bn . . . 
 mil represent the number a — b. 
 
 Proof.— (a„-6„)- (a- 6)= (o„- a) -(&„-&). 
 Therefore, 
 
 |(a„-6„)-(a-6)| <la„-a| + |6n-6|- (Art. 14) 
 
 Now there is a place in the first sequence beyond 
 which its elements differ from a by less than -, and a 
 place in the second sequence beyond which its elements 
 differ from b by less than ^. That is, beyond the 
 farther of these two places 
 
 |an-a|<2> |On-o. <2' 
 and, therefore, | (a„ - 6„) - (a - 6) ' < i. 
 
52 SEQUENCES 
 
 46. Theorem. — // two sequences represent two numbers 
 a and b, the products of their corresponding elements mil 
 form a sequence representing the product ab. 
 
 Proof. — Let the sequences be 
 
 fli 02 . ■ ■ a-n ■ ■ ■ representing a, 
 and bi b.2 . . . bn . . . " b. 
 
 We are to prove that the sequence 
 
 aibi 02^2 • • • ci„bn • . . 
 represents the product ab. 
 
 anbn —ab = a„6„ — a„b + a-nb — ab. 
 Therefore, 
 
 la„6„-o6|<|a„||6„-6| + |6||a„-a|. 
 
 Now if M is some number larger than \a\, there is a 
 place in the first sequence beyond which its elements 
 are numerically less than M. 
 
 Again, for any positive ntmiber e.' there is a place in 
 the first sequence beyond which its elements differ 
 from a by less than e' and a place in the second sequence 
 beyond which its elements differ from b by less than e' 
 
 Beyond the farthest of all these places we have 
 
 \an\<M, |a„-a|<e', and \bn-b\<s'. 
 
OPERATIONS UPON SEQUENCES 53 
 
 Hence there is a place beyond which 
 
 \anbn-ab\<M£' + \b\e', 
 
 and if e' is taken so that 
 
 Me' + \b\e'<c, 
 
 that is, if ''<Jf^l' 
 
 we shall have a place in the product sequence beyond 
 which its elements differ from ab by less than e. 
 
 47. Theorem. — If the two sequences 
 
 Qi 0,2 . . . Cln • • • 
 
 and bi b2 ■ • • bn ■ • • 
 
 represent the numbers a and b, respectively, and if the 
 number a is not zero and none of the elements of the first 
 sequence is zero, then the sequence 
 
 bi bi bn 
 
 Ol 02 an 
 
 represents the quotient — . 
 
 bn b (abn — ab) — (anb—ab) 
 
 Proof. = ■ 
 
 On CI aan 
 
 Therefore 
 
 bn_b_ 
 
 a,, a 
 
 = \a\\bn-b\ +\b\\an-a\ 
 < 
 
 a Un 
 
54 
 
 SEQUENCES 
 
 a is not zero, and if M is a number lying between zero 
 and \a\, tiiere is a place in the sequence of a's beyond 
 which all the elements are numerically greater than 
 M. That is, there is a place beyond which 
 
 _L JL 
 
 Again, for any positive number e' there is a place in 
 the first sequence beyond which its elements differ from 
 a by less than e', and a place in the second sequence 
 beyond which its elements differ from b by less than e'. 
 
 Beyond the farthest of these three places 
 
 bn_b_ 
 
 an a 
 
 <- 
 
 \a\B' + \b\e' 
 
 \a\M 
 
 Now if e' is taken so that the last expression shall be 
 less than e, that is, if 
 
 e'< 
 
 \a\Me 
 a\ + \b\' 
 
 we shall have a place beyond which 
 
 bn_b_ 
 an a 
 
 <s. 
 
 48. In other words, the operations of addition, sub- 
 traction, multiplication, and division of numbers repre- 
 sented by sequences, can be performed by performing 
 these operations on corresponding elements of the 
 sequences. 
 
OPERATIONS rrON SEQUENCES 55 
 
 Any sequence may first be changed in accordance with 
 the preceding theorems (Arts. 41 and 42), and thus the 
 result may be presented in a great variety of forms. 
 In particular, if we wish to divide by a number a which is 
 not zero, and a is represented by a sequence some of 
 whose elements are supposed to be zero, there can be 
 only a finite nimiber of these elements zero, and these 
 may be omitted or changed into other numbers differ- 
 ent from zero. 
 
 We can sometimes combine the corresponding ele- 
 ments of two sequences by addition, subtraction, 
 multiplication, or division and produce a sequence 
 which is regular, when the two given sequences are not 
 regular. 
 
 Or we may divide the elements of a sequence by the 
 corresponding elements of a sequence representing zero, 
 if zero does not itself occur among the elements of 
 the latter sequence, and when both sequences repre- 
 sent zero the resulting sequence may be regular and 
 represent a number. But this number will have no 
 special relation to the number zero, which the two 
 sequences represent. 
 
 For example, dividing the elements of the sequence 
 
 1 ' 
 
 3 ■ ■ ■ 2n-l ■ ■ ■ 
 by the corresponding elements of the sequence 
 
 ^ 2 ■ • ■ n • • • ' 
 
56 SEQUENCES 
 
 both representing zero, we have a sequence 
 
 o 
 1 ^ . . 
 
 n 
 
 3 • • • 2n-l ' ' ° ' 
 which represents ^. 
 
 Note. — The theory of irrational numbers given in 
 Chapter I is due to Richard Dedekind; or, at least, 
 most of its developments are due to Dedekind, and the 
 writings of Dedekind have done the most to bring 
 about its adoption at the present time. The reader 
 will find the theory explained in an essay, " Continuity 
 and Irrational Numbers," translated by W. W. Beman, 
 and published by the Open Court Publishing Company. 
 This theory has been adopted by Professor Fine in his 
 " CoUege Algebra." 
 
 There is another theory of irrational numbers in which 
 the numbers are defined as regular sequences and the 
 addition, multiplication, etc., of regular sequences serve 
 as definitions of these operations on irrational numbers. 
 This theory is due to Georg Cantor; some exposition of 
 it may be found in Chapter IV of Professor Fine's 
 "Number System of Algebra," and a detailed treat- 
 ment in Professor Pierpont's "Theory of Functions 
 of Real Variables." 
 
 The latter theory is more artificial than the theory 
 of Dedekind. We naturally think of an irrational 
 number as occupying a certain position among num- 
 bers, and in describing this position we mention other 
 numbers between which it lies. Thus we say that the 
 square root of 2 is greater than 1.414 and less than 
 
THE THEORY OF LIMITS 57 
 
 1.415. We know the square root of 2 by being able 
 to determine the position of any rational number with 
 reference to it. 
 
 IV. The Theory of Limits 
 
 49. We naturally think of a variable as varying con- 
 tinuously, varying through a continuous set of values 
 in a continuous interval of time; but in cases con- 
 sidered when we first study Algebra and Geometry there 
 is no continuous set of values, nor is there any question 
 of time. The values which the variable takes are dis- 
 tinct, each followed by one which comes next after it; 
 in other words, they form a sequence, and when the 
 variable in this way approaches a number a as limit its 
 values form a sequence representing a. Thus it simpli- 
 fies the subject to dismiss the idea of variable and fix 
 our attention on the sequence. 
 
 However, as the language of variable and Umit 
 permeates nearly all of our mathematics, it may be well 
 to consider a little in detail the meaning and use of 
 these terms. What we shall say applies for the most 
 part both when the variable is supposed to vary con- 
 tinuously and when it varies through a sequence of 
 values. 
 
 50. A variable approaches a niuivber a as limit if for 
 every positive number e there is a place among the 
 values of the variable beyond which they all differ 
 from o by less than e. 
 
58 SEQUENCES 
 
 51. The earliest application of the theory of hmits that 
 we meet is in connection with the ratio of incommen- 
 surable lines. 
 
 B' 
 
 Let AB be a line incommensurable with a given line 
 a. We divide a into some number of equal parts, say 
 n equal parts. If AB is shorter than a, we suppose 
 that n is taken large enough so that one of the n equal 
 parts of a is shorter than AB. 
 
 One of these parts applied to AB will be contained 
 in AB a certain number of times. Let it be contained 
 m times, reaching to a point B' and leaving a remainder 
 B'B less than one of these parts. The line AB' is 
 commensurable with o, and its ratio to a is the rational 
 
 number — . 
 n 
 
 We repeat this process, each time dividing a into 
 a larger number of parts. Each time the equal parts 
 of a are smaller than before; and we can make them 
 less than any given length by dividing a into a suffi- 
 ciently large number of parts. 
 
 The remainder B'B is always less than one of the 
 equal parts of a, and, therefore, the remainder is a 
 variable which approaches zero as limit. 
 
 The line AB' approaches AB as limit, and the ratio of 
 
 AB' to a, the rational number — , is a variable which 
 
 n 
 
THE THEORY OF LIMITS oO 
 
 approaches as limit the incommensurable ratio of yl5 
 to a. 
 
 In reality, we are simply forming a sequence of rational 
 numbers in accordance with the theorem of Art. 33 
 to represent the irrational number which is the ratio 
 of AB to a. If the incommensurable lines are the 
 side and diagonal of a square, we have the very ex- 
 ample of that article. 
 
 53. Certain things may be pointed out as not in- 
 cluded in the above definition of hmit. 
 
 (1) It is not necessary that the variable shall be 
 always less or always greater than the limit. 
 
 Those who are familiar with continued fractions 
 will recognize in the convergents a sequence of num- 
 bers alternately less and greater than the number 
 represented. We may say that the nth convergent 
 is a variable approaching a limit, its values alter- 
 nately less and greater than the limit. 
 
 We shall have a similar illustration in Art. 71. 
 
 (2) It is not necessary that the variable shall be 
 always approaching the limit, each value nearer than 
 any which precedes. 
 
 In the case of the two incommensurable lines, we 
 increase the number of parts into which a is divided, 
 so that each part is made smaller. If the part is not 
 made enough smaller to be contained one more time 
 in AB, the Une AB' will be smaller and the remainder 
 B'B larger than before. In fact, if we let the values 
 of n be the successive integers, we may find that our 
 variables move away from their limits more frequently 
 than they approach them. This is shown in the ex- 
 
60 SEQUENCES 
 
 ample of Art. 33. We can avoid this by taking n 
 each time so large that one of the n equal parts of a 
 shall be less than the last remainder, but this is not 
 necessary in order that the line AB' shall approach AB 
 and that the remainder shall approach zero as limit. 
 
 (3) It is not necessary that the variable shall never 
 equal its limit. 
 
 This does not mean that in all cases a variable 
 approaching a as limit will come finally to equal a, 
 but that we are not to exclude by the definition certain 
 cases where a is found among the values which the 
 variable takes, or even certain cases where all of its 
 values or all beyond a certain point are equal to a. 
 
 We can form a sequence representing a in which 
 the number a occurs an infinite number of times among 
 the elements of the sequence, by taking any sequence 
 that represents a and interpolating elements equal 
 to a. Such a sequence taken by itself would be in 
 no way distinguishable from a sequence in which the 
 number represented does not occur. A variable whose 
 values are the elements of such a sequence would 
 serve as a variable having a for limit just as well as if 
 the number a were not found among its values. 
 
 53. We will change a little the illustration of the 
 incommensurable lines. 
 
 Let us suppose that a hne a diminishes continuously 
 to the hmit zero, varying through all values in order 
 from some given length, each value which it takes 
 being less than any preceding but not equal to 
 zero. 
 
 If we apply o to a line AB as many times as it will 
 
THE THEORY OF LIMITS 
 
 61 
 
 go, we shall usually get a line AB' which is a multiple 
 of a and a remainder B'B less than a. 
 
 While the number of times that a is contained in 
 AB remains the same, though a grows smaller, the 
 line AB' will be continuously decreasing and the 
 line B'B will be continuously increasing. When a 
 becomes just small enough to be contained one more 
 time in AB the line AB' will become the same as AB 
 and there will be no remainder. Then, as a con- 
 tinues to decrease, AB' will again diminish continu- 
 ously, and there will be a remainder increasing con- 
 tinuously until a becomes small enough again to be 
 contained in AB one more time without a remainder. 
 
 The line AB' and the remainder B'B are in this case 
 variables which take the values AB and zero an infi- 
 nite number of times, and which at all other times are 
 constantly moving away from these values. Yet they 
 are variables approaching these values as limits; for, 
 given any length, there is a point in the process beyond 
 which they always differ from these hmits by less 
 than the given length. 
 
 The following diagram represents the way in which 
 these variables approach their limits. 
 
02 SEQUENCES 
 
 54. There are certain things which are essential to 
 an understanding of the Theory of Limits, and which 
 should be emphasized. 
 
 (1) It must be i ossible not only to make the differ- 
 ence between the variable and its limit less than any 
 assigned positive number, but also to make it remain 
 less than the assigned number. 
 
 (2) The " assigned positive number " * means every 
 positive number, and the place beyond which the vari- 
 able differs from its limit by less than the assigned 
 number is not necessarily the same for two different 
 numbers. Indeed, no one value of the variable differs 
 from its limit by less than any number whatever unless 
 it is equal to the hmit itself. 
 
 (3) We must distinguish between the question 
 whether a variable approaches some given number as 
 limit and the question whether the variable approaches 
 a limit. In one case we have to consider the differ- 
 ences between the given number and the values of 
 the variable; in the other case we must first determine 
 
 ♦Sometimes the words "however small" are added. It is not 
 correct, however, to say of any number that it is a small number 
 or that it is a large number. We can only say of two different 
 numbers that one is smaller than the other and that one is larger 
 than the other. We must not think that our system contains, 
 besides ordinary numbers, another class of numbers that are in- 
 definitely small, and that the difTerence between a variable and its 
 limit finally gets out of the region of ordinary numbers and into 
 the region of these numbers. A variable approaching zero as 
 limit is sometimes called an infinitesimal, but its values are not 
 infinitesimal numbers; they are all ordinary numbers, like 2 or .^3 
 or ^-. We have used £ to denote every positi\-e number, and when 
 we might say "every positive number however large" (e.g., in 
 Art. 19) we mean the same thingtand so we use the same symbol. 
 
THE THEORY OF LIMITS 63 
 
 a number which we can then prove is the Umit, or 
 we must prove that there is no such number. 
 
 Thus we prove, in particular, the two following 
 theorems (see Arts. 37 and 40) : 
 
 Theorem. — // a variable is constantly increasing and is 
 always less than some fixed number M, the variable ap- 
 proaches a limit, either M or some smaller number. 
 
 Theorem. — A variable approaches a limit if for every 
 e there is a place among its values beyond which they all 
 differ from one another by less than e. 
 
 55. In some of our text-books much stress is laid 
 on what is called The Theorem of Limits: If two 
 variables are constantly equal and approach Umits 
 their limits are equal. This only means that the same 
 variable cannot approach as limit two different num- 
 bers — that the variable determines the limit. In 
 fact, the variable could never differ from both of two 
 different numbers by less than a half of their difference 
 (see Art. 28). 
 
 In proving what is called the "incommensurable 
 case" of certain propositions of geometry we have two 
 incommensurable ratios which we wish to prove equal. 
 "\^"e may do this by constructing for each a sequence 
 of commensurable ratios as explained in Art. 51, every 
 ratio in one sequence being the same rational number 
 as the corresponding ratio in the other; that is, we 
 construct a single sequence of rational numbers which 
 represents both incommensurable ratios. Therefore, 
 as a sequence cannot represent two different numbers, 
 
64 SEQUENCES 
 
 the two incommensurable ratios must be the same 
 irrational number. In the language of limits we say 
 that a single variable has both of these ratios for limit, 
 and that these ratios are therefore equal, since a variable 
 cannot have two different limits. However, if we de- 
 fine the ratio of two incommensurable magnitudes as 
 the irrational number which occupies the point of 
 separation of all rational numbers into two classes, we 
 can prove the proposition directly, making no use of 
 sequences or of the theory of limits (see Art. 8). 
 
CHAPTER III 
 
 SERIES 
 I. CONVERGENCY OF SeRIES 
 
 56. A SERIES is a sequence written in a form which 
 presents the differences of successive elements as a suc- 
 cession of terms. 
 
 Suppose we have a sequence 
 
 and put 
 
 Ol a2 . . . On 
 
 u>=U2—a\ 
 
 Un==an — a„-i 
 
 then 
 
 Oi=Ml 
 
 a2 = Wl+W2 
 
 a„=Wi+W2+ . . . +Wn 
 
 and we express the sequence by writing 
 
 65 
 
66 SERIES 
 
 This is called a series. The numbers ui ua • . • Wn • • • 
 are called its terms. 
 The sequence 
 
 dl 0,2 . . . On • ' • 
 
 written in the form of a series becomes 
 
 ai + {a2—ai)+ . . . +(a„— a„_i)+ . . 4 » 
 The series 
 
 U1+U2+ . . . +Un+ . . . 
 written in the form of a sequence becomes 
 
 Ui, U1+U2, . . . , U1+U2+ . . . +u„, .... 
 
 Since the nth element of the sequence is the sum of 
 the first n terms of the series, it will often be conve- 
 nient to write this s„. Thus we shall say that the series 
 
 U1+U2+ . . . +Un+ . . . 
 
 is the same as the sequence 
 
 Si $2 . . . Sn • . > ^ 
 
 where 
 
 Sn = Ui+U2+ . . . +Un. 
 
 A series is convergent when the corresponding se- 
 quence represents a number, and this number is called 
 the value of the series. 
 
COWERGENCY OF SERIES 67 
 
 For example, the sequence 
 
 1 2 n 
 
 2 3 ■ ■ ■ n + 1 
 is the same as the series 
 
 J_ J_ 1 
 
 1.2 "^2.3"^ • • • "^n(n + l)"^ 
 
 The sequence represents 1 (p. 37) ; therefore the series 
 is convergent and its value is 1. 
 
 A series which is not convergent is divergent. 
 
 57. Theorem. — The geometrical series 
 
 a + ar+ . . . +ar'^~'^+ . . . 
 
 is convergent when r is numerically less than 1, and its 
 
 value is z . 
 
 1— r 
 
 Proof. — The sum of the first n terms is 
 a—ar'^ 
 
 That is, the series is the same as the sequence 
 
 a - ar^ a-ar" 
 « -JZ^ ■ ■ ■ l_r 
 
 The difference between the nth element of this sequence 
 
68 
 
 SERIES 
 
 and the number j^ is the numerical value of the 
 fraction 
 
 ar" 
 
 There is a place among" the powers of r beyond which 
 they are all numerically less than s', s' being any given 
 positive niunber. By taking s' so that 
 
 that is, so that 
 
 1-r 
 
 e'<e 
 
 s'<^, 
 
 1-r 
 
 we have a place in the sequence beyond which its ele- 
 ments all differ from , by less than e. 
 
 i —r 
 
 58. Theorem. — // a series is convergent, its terms them- 
 selves farm a sequence equal to zero. 
 
 Proof. — In a sequence which represents a number 
 and so is regular, there is a place for each s beyond 
 which all the elements differ from each other by less 
 than s. Now the terms of the series are the differ- 
 ences of successive elements of the corresponding 
 sequence. Therefore if the series is convergent, there 
 is for each £ a place in the series beyond which all the 
 terms are numerically less than ;. 
 
CONVERGENCY OF SERIES 69 
 
 The converse is not necessarily true. For an exam- 
 ple see Art. 61. 
 
 59. Theorem. — // a series is convergent, then for every 
 e there is a place in the series beyond which any sum of 
 successive terms is less than e. 
 
 Proof. — Any sum of successive terms is the difference 
 between some two elements of the sequence ; and if the 
 sequence represents a number and so is regular, there 
 is a place beyond which any two elements differ by 
 less than £. 
 
 Thus if the series is 
 
 U1+U2+ . . . +Un+ ...» 
 
 and the corresponding sequence 
 
 Si S2 . . . 5„ . . . , 
 
 the sum of p successive terms beginning with the 
 (n + l)th will be 
 
 60. Theorem. — Conversely, a series is convergent if for 
 every b there is a place in the series beyond which every 
 sum of successive terms is less than e. 
 
 Proof.— Every difference of elements of the corre- 
 sponding sequence is a sum of successive terms of the 
 series; and if for every e there is a place beyond which 
 
70 SERIES 
 
 these differences are all less than e, the sequence is 
 regular and the series convergent. 
 
 61. Theorem. — The series 
 
 1+77+ • • • +- + 
 2 n 
 
 is not convergent. 
 
 Proof. — The sum of p successive terms beginning with 
 
 the (n + l)th is 
 
 1 1 1 
 
 ■+-^^+ . . . +■ 
 
 n + 1 71+2 ■ ■ n+p' 
 
 The last of these being the smallest, their sum is greater 
 
 P 
 than — ; — . Therefore for a value of s less than 1 
 
 n + p 
 
 there is no place in the series beyond which every 
 sum of successive terms is less than s. For example, 
 whatever the value of n, if we take p greater than n 
 
 V 1 
 
 we shall have — ; — greater than 7:. 
 n + p * 2 
 
 If we take two or more terms of this series, and 
 
 then take as many more, we shall increase what we 
 
 had obtained before by more than •^. If we take as 
 
 many more as we have now, we shall again increase 
 
 what we have by more than -. Since there is no limit 
 
 to the number of times this may be done, there is no 
 
CONVERGENCY OF SERIES 71 
 
 limit to the number which we may obtain by taking 
 a sufficient number of terms of this series. 
 This series is called the harmonic series. 
 
 63. Theorem. — A series is convergent if the series 
 formed by taking the numerical values of its terms is 
 convergent. 
 
 Proof. — The sum of any number of successive terms 
 is numerically equal to or less than the sum of the 
 numerical values of these terms. Since the series of 
 numerical values is convergent, there is a place in 
 it beyond which any sum of successive terms is less 
 than e. The same must be true of the given series, 
 and the given series is convergent. 
 
 This theorem may also be stated as follows: 
 
 Theorem. — // a series of positive terms is convergent, 
 it will remain so after the signs of any portion of its terms 
 have been changed. 
 
 Another proof will be given on page 89. 
 
 A series which remains convergent when we replace its 
 terms by their numerical values is called an absolutely 
 convergent series. 
 
 A convergent series which becomes divergent when its 
 terms are replaced by their numerical values is called 
 a semi-convergent series. 
 
 Cor.— An absolutely convergent series is numerically 
 equal to or less than the series of numerical values 
 of its terms. 
 
 For the sum of its first n terms is numerically equal 
 
72 SERIES 
 
 to or less than the sum of their numerical values (see 
 Art. 36). . 
 
 63. Theorem. — It does not affect the convergence of a 
 series to change any finite number of its terms. 
 
 Proof. — If the ?ith term of the series is the last term 
 changed, all the elements of the sequence beyond the 
 nth, containing as they do the first n terms, are increased 
 or diminished by the same quantity, and their differ- 
 ences, which alone determine whether the sequence is 
 regular, are not changed. 
 
 Thus many of our theorems will be true when the 
 conditions do not hold throughout the series, but only 
 beyond a certain place. 
 
 64. Theorem. — A series of positive terms is convergent 
 if the elements of the corresponding sequence are all less 
 than some fixed number M. 
 
 Proof. — In a series of positive terms each element 
 of the corresponding sequence is greater than the 
 preceding. If, then, each element is also less than M, 
 the sequence represents a number equal to M or less 
 than M (Art. 37). 
 
 In other words, if the sum of the first n terms of a 
 series of positive terms is less than some number M, 
 the same for all values of n, the series is convergent. 
 
 Cor. — If a series of positive terms is not convergent, 
 then for every e there is a place in the corresponding 
 sequence beyond which all the elements are greater 
 than £. 
 
CONVERGENCY OF SERIES 73 
 
 Such a series may especially be called a divergent 
 series, but we usually call any series divergent which 
 is not convergent. 
 
 G J. Theorem. — The series 
 
 1 + 2^+ ••• +;^+ ••• 
 
 is convergent when p is any number greater than 1. 
 
 Proof. — Let s„ denote the sum of the first n terms 
 of this series ; that is, let the corresponding sequence be 
 
 Si S2 . . . S„ . . . . 
 
 The sum of m terms beginning with the mth is 
 1 1 
 
 S2m-1-Sm-1=— r+ . . . +; 
 
 TOP ■ ■ ■ (2w-l)p 
 
 The first of these terms is the largest. Therefore 
 
 . , , TO 1 
 
 their sum is less than — - or 
 
 TOP TOP"' 
 
 Putting TO = 2, 4, 8, . . . , we have 
 1 , 1 
 
 1 , 1 ( I y 
 
 S7<S3+4;rT<l+2^i + (^2^i/ , 
 Sis < 1 + 2P-1 + (o^j + (^j , 
 
 etc. 
 
74 SERIES 
 
 If ?l<2^ Sn is less than the sum of the first r terms of 
 the geometrical series 
 
 1 
 
 l + .^i+ . . . +[i^.) + 
 
 \2p-7 
 
 which is convergent, p being greater than 1. What- 
 ever the value of n, we can take r so that 2'">n 
 (Art. 19) ; and, therefore, for all values of n, s„ is less 
 than the value of this geometrical series, namely, 
 
 1 
 Sn < — J- . 
 
 1 "■ '5^1 
 
 This is a fixed number, and the given series, being 
 a series of positive terms, is convergent. 
 
 66. Theorem. — // a series of positive terms is conver- 
 gent, the sum of any terms selected from it is less than the 
 value of the series. 
 
 Proof. — I^et i<„ be the last of those selected, and $„ 
 the sum of the first n terms. The terms selected 
 are all included in s„ and their sum is equal to or lefs 
 than s„. But s„ is less than the number represented 
 by the series (see proof on page 45). Therefore the 
 sum of the terms selected is less than the value of the 
 series. 
 
 67. Theorem. — // a series of positive terms is conver- 
 gent, any series formed by takiiuj a part of its terms is 
 convergent. 
 
CONVERGENCY OF SERIES 75 
 
 Proof. — The first n terms of such a series are foiind 
 somewhere in the original series, and their sum is less 
 than the value of the original series. Therefore, as 
 the sum of the first n terms is less than a certain fixed 
 number, the series is convergent. 
 
 By virtue of Art. 62 the theorem may be stated as 
 follows : 
 
 Theorem. — If a series is absolutely convergent, any 
 series formed by taking a part of its terms is absolutely 
 convergent. 
 
 Cor, — If a series of positive terms is convergent, 
 or if a series is absolutely convergent, there is a place 
 beyond which the sum of any terms selected from 
 it, or the value of any series formed by taking a part 
 of its terms, is nimierically less than e. 
 
 Proof. — ^The spries formed by omitting the first n 
 terms of a convergent series is equal to the difference 
 between the nth element of the corresponding sequence 
 and the number which the sequence represents, and 
 there is a place beyond which this difference is less 
 than e. When a series is absolutely convergent this 
 is true of the series of numerical values of its terms, 
 and there is a place beyond which the remainder series, 
 the sum of any terms selected from it, or any series 
 formed by taking a part of its terms, will have a value 
 numerically less than s. 
 
 68. Theorem. — // the terms of a series of positive terms 
 are equal to or less than the corresponding terms of another 
 
76 SERIES 
 
 series of positive terms which is convergent, then the given 
 series is convergent. 
 
 Proof. — The sum of the first n terms of the given 
 series is equal to or less than the simi of the first n 
 terms of the series which is convergent, and there- 
 fore less than the value of the latter series. That 
 is, we have a series of positive terms, and the sum 
 of the first n terms is less than a certain fixed num- 
 ber. Hence the series is convergent. 
 
 Cor. — If the terms of a series of positive terms are 
 equal to or greater than the corresponding terms of 
 another series of positive terms which is divergent, 
 the given series is divergent. 
 
 For if the given series were convergent, the other 
 series would be convergent by the theorem. 
 
 The theorem and corollary are true whenever there 
 is a place in the series beyond which the conditions are 
 satisfied (Art. 63). 
 
 In most cases we determine the convergence of a 
 series by comparing it with other series. Two of the 
 simplest series for purposes of comparison are the 
 series 
 
 -■ 1 1 
 
 which is divergent, and the series 
 
 which is convergent. 
 
COXVERGENCY OF SERIES 77 
 
 By virtue of Art. 62 the theorem may also be stated 
 as follows : 
 
 Theorem. — // in any series there is a place beyond 
 which the terms are numerically eqval to or less than the 
 corresponding terms of another series which is absolutely 
 convergent, the given series is absolutely convergent. 
 
 69. Theorem. — The series 
 
 , 1 1 
 
 1 + 2-.+ • • • +^+ • • • 
 
 is divergent for any value of p equxd to or less than 1. 
 
 Proof. — When p = l the series is the harmonic series 
 already proved divergent, and when p<\, whether 
 positive or negative, all the terms of the series are 
 greater than the corresponding terms of the harmonic 
 series, except the first, which is the same in the two 
 series. 
 
 70. Theorem. — A series of positive terms is convergent 
 if the ratio of each term to the preceding is less than 
 some fixed positive number which is itself less than 1. 
 
 Proof. — T.et the series be 
 
 U1+U2+ . . . +Wn+ . . . , 
 
 with all of its terms positive, and suppose we have 
 — <r, - <r, . . . , <r, . . . , 
 
 7(1 M2 Ur,-\ 
 
 where r is a fixed number less than 1. 
 
78 SERIES 
 
 Multiplying together the corresponding members of 
 the first n — 1 of these inequahties (Art. 16, Cor.), we 
 have 
 
 1(2 .. • Uj, 
 
 Ml . . . W„_i 
 
 <r"-i, 
 
 or — <r"~': 
 
 Wi ' 
 
 whence, since ui is positive. 
 
 The terms of our series are, therefore, less than 
 the corresponding terms of the geometrical series 
 
 ui+uir+ . . . +Mir''~i+ . . . , 
 
 except that the first terms are the same in the two 
 series. The latter series is convergent, since r<l; 
 therefore the given series is convergent. 
 
 By virtue of Arts. 62 and 63 the theorem may be 
 stated as follows : 
 
 Theorem. — // in any series there is a place beyond 
 which the ratio of each term to the preceding is numer- 
 ically less than some fixed number which is itself less 
 than 1, the series is absolutely convergent. 
 
 Cor. — If in the series 
 
 U1+U2+ . . . +Un+ . . ~. 
 

 
 CONVERGENCY 
 
 OF 
 
 SERIES 
 
 the 
 
 sequence 
 
 of ratios 
 
 V2 "3 
 
 Ui U2 ' ' ' 
 
 w™ 
 
 "„-] 
 
 I 
 
 79 
 
 represents a number numerically less than 1, the series 
 is absolutely convergent. 
 
 Proof. — Let I be the numerical value of the number 
 which this sequence represents, and let r be a num- 
 ber between I and 1. Since r>l there is a place in 
 this sequence beyond which its elements are numer- 
 ically less than r (Art. 35). Therefore, by the preceding 
 statement of the theorem, the series is absolutely con- 
 vergent. 
 
 71. Theorem. — // the terms of a series are alternately 
 positive and negative and each is numerically less than the 
 preceding, and if they form a sequence equal to zero, the 
 series is convergent. 
 
 Proof. — Let the series be 
 
 U1—U2 + U3— . . . +U2n-1—U2n+ - • • , 
 
 and the corresponding sequence 
 
 Si S2 S3 . . . S2n-1 S2n . . . • 
 
 The elements of the sequence are alternately less 
 and greater each than the preceding, but when we sub- 
 tract a number u we then add a smaller number, and 
 
80 SERIES 
 
 when we add a number u we then subtract a smaller 
 number. Thus each element of the sequence lies be- 
 tween the preceding two, and between any two suc- 
 cessive elements of the sequence lie all that follow. 
 Now the difference between any two successive ele- 
 ments of the sequence is a term of the series, and from 
 the last part of our hypothesis it follows that there 
 are terms of the series numerically less than e. Hence 
 there is a place in the sequence beyond which the 
 elements differ from each other by less than e, and the 
 sequence is regular. 
 
 Another proof is given on page 83. 
 
 The odd-numbered elements of the sequence form by 
 themselves a sequence of elements each less than the 
 preceding and greater than any one of the even-numbered 
 elements of the sequence, and the even-numbered 
 elements of the sequence form by themselves a se- 
 quence of elements each greater than the preceding and 
 less than any one of the odd-numbered elements. 
 This would be true, and the two sequences would be 
 convergent, even if the terms of the series did not 
 form a sequence equal to zero, but the two sequences 
 would then represent different numbers. 
 
 Cor. — The value of the series of the theorem lies 
 between any two successive elements of the sequence; 
 thus the sum of any number of terms from the begin- 
 ning differs from the value of the series by less than 
 the next term. 
 
OPERATIONS UPON SERIES 81 
 
 An example is the series 
 
 ^-2+3- •■• +2^i-r.+ •••• 
 
 By 'virtue of Art. 63 the theorem and corollary may 
 be stated as follows: 
 
 Theorem. — // there is a place in a series beyond which 
 the terms are alternately positive and negative, each term 
 less than the preceding, and if the terms form a sequence 
 equal to zero, the series is convergent. 
 
 Cor. — The value of the series lies between any two 
 successive elements of the sequence beyond the place 
 where the conditions of the theorem become true. 
 
 II. Operations upon Series 
 
 72. Theorem. — // a series is convergent, its terms may 
 be grouped in parentheses in any manner without destroy- 
 ing its convergence or changing its value. 
 
 Proof. — This is the same as omitting elements from 
 the corresponding sequence (Art. 41). 
 
 73. Theorem. — When the terms of a convergent series 
 are grouped in parentheses it does not change its value to 
 remove the parentheses. 
 
 Proof. — This is but another way of stating the pre- 
 ceding theorem. 
 
82 SERIES 
 
 Removing the parentheses is the same as interpolating 
 elements in the corresponding sequence. 
 
 In particular, if a series of parentheses is convergent 
 we may remove the parentheses 
 
 (1) When all the terms in each parenthesis have 
 the same sign, or 
 
 (2) When the number of terms in each parenthesis is 
 less than some fixed number and the terms of the new 
 series form a sequence equal to zero. 
 
 Proof. — Let m be the number of terms of the series 
 of parentheses included entirely in the first n terms of 
 the new series. That is, the first n terms of the new 
 series include all of the terms of the first m parentheses 
 and perhaps some of the terms from the next parenthesis. 
 
 The sum of the first n terms of the new series will 
 differ from the sum of the first m parentheses by zero or 
 by a sum of terms from the (m + l)th parenthesis. 
 
 Now, in case (1), there is a place in the series of 
 parentheses beyond which they are numerically less 
 
 than -, and any sum of terms from a single parenthesis, 
 
 being equal to or less than the entire parenthesis, will 
 
 be less than — . 
 
 Again, in case (2), where the number of terms in 
 each parenthesis is less than a fixed number, say p, 
 since the terms form a sequence equal to zero, there is 
 a place beyond which they are numerically less than 
 
 — , and beyond Avhich, therefore, any sum of terms 
 
OPERATIONS UPON SERIES 83 
 
 from a single parenthesis will be numerically less 
 than -. 
 
 Thus, in either case, there is a place beyond which 
 we can say that the simi of the first n terms of the new 
 series differs from the sum of the first m terms of the 
 
 series of parentheses by less than -. But as the series 
 
 of parentheses is convergent there is a place beyond 
 which we can say that the sum of its first m terms 
 
 differs from the value of the series by less than -. 
 
 Beyond the farther of these two places we can say 
 that the sum of the first n terms of the new series 
 differs from the value of the series of parentheses by 
 less than £. 
 
 As an illustration we may give the following proof 
 of the theorem of Art. 71 : 
 The series 
 
 (Mi-M2)+ . . . +(M2n-l-W2n)+ . . . 
 
 is a series of positive terms. The sum of its first n 
 terms may be written 
 
 ttl-(tt2-«3)-- . .-(ti2n-2-W2n-l)-M2ni 
 
 where the expression in any parenthesis represents a 
 positive number. Hence this sum is less than the 
 fixed number u\ and the series of parentheses is con- 
 vergent. 
 
84 SERIES 
 
 Now, if the w's form a sequence equal to zero, the 
 parentheses may be removed by the second case above. 
 Therefore the given series is convergent. 
 
 74. Theorem. — The series formed by adding the corre- 
 sponding terms of two convergent series is convergent and 
 equal to the sum of their values; and the series formed by 
 subtracting the terms of one convergent series from the cor- 
 responding terms of a second convergent series is conver- 
 gent, and its value is the difference obtained by subtracting 
 the value of the first series from the value of the second. 
 
 Proof. — This is the same as adding or subtracting the 
 corresponding elements of two regular sequences; the 
 resulting sequence represents the sum or difference 
 of the numbers represented by the two sequences. 
 
 We may group the terms of two convergent series in 
 parentheses in any manner and after adding or sub- 
 tracting remove the parentheses, and the resulting series 
 will still be convergent and represent the sum or differ- 
 ence of the two given series. 
 
 For the sum of any number of terms of the final 
 series will differ from the sum of a certain number of 
 terms before the parentheses are removed by zero or by 
 a sum of terms from a single parenthesis. But the sum 
 of terms from a single parenthesis here is in any case 
 the sum of a certain number of successive terms from 
 one or the other of the original series, and there is a 
 place in each of the original series beyond which any 
 such sum of terms is numerically less than e. There- 
 fore we may remove the parentheses, as in the second 
 case considered in the last article. 
 
OPERATIONS UPON SERIES 85 
 
 Cor. — If we combine by addition or subtraction a 
 convergent and a divergent series, the resulting series 
 will be divergent. 
 
 For that one of the two series which is divergent 
 could be obtained by combining the resulting series with 
 the other given series, and if the resulting series were 
 convergent, we should have a divergent series as the 
 difference or sum of two convergent series, which is 
 contrary to the theorem. 
 
 75. Theorem. — // a series is convergent and equal to a 
 number S, the series formed by multiplying all of its terms 
 by a number m mil be convergent and equal to mS. 
 
 Proof. — This is the same as multiplying together two 
 sequences, one representing S and the other composed 
 of elements all equal to m. 
 
 Cor. — A convergent series equal to S becomes a con- 
 vergent series equal to —Sii we change the signs of 
 all of its terms. 
 
 76. Theorem. — // a series of positive terms is conver- 
 gent, the series will be convergent which we form when we 
 multiply its terms by the corresponding elements of a 
 sequence of positive numbers all less than some positive 
 number m. 
 
 .Proof. — The sum of n terms of the new series will 
 be less than the sum of n terms of a series formed by 
 multiplying the terms of the original series by ?n, and 
 so less than mS, where S is the value of the original 
 series. Therefore the new series will be convergent. 
 
86 SERIES 
 
 By virtue of Art. 62 we may state the theorem as 
 follows : 
 
 Theorem. — // a series is absolutely convergent, the series 
 will be absolutely convergent which we get when we multi- 
 ply its terms by the corresponding elements of a sequence 
 of numbers all numerically less than some fixed num- 
 ber m. 
 
 Cor. — An absolutely convergent series will remain so 
 if we multiply its terms by the corresponding elements 
 of a regular sequence. 
 
 77. Theorem. — // two series are absolutely convergent, a 
 series can be formed by multiplying every term of one 
 with every term of the other that will be absolutely con- 
 vergent and will equal the product of the valves of the 
 given series. 
 
 Proof. — Let the two series be 
 
 U1+U2+ . . . +i<„+ . . : 
 
 and V1+V2+ . . . +Vn+ . . . , 
 
 and let the corresponding sequences be 
 
 fli 02 . . . a„ . . . 
 and 61 b2 ■ • ■ bn . . . . 
 
 The ]:)rodiict of these sequences is the sequence 
 Oihi 02^2 . . . ('„h.„ .... 
 
OPERATIONS UPON SERIES 87 
 
 and this is equivalent to the series whose successive 
 terms are 
 
 aibi a2b2-aibi . . . a„?)„ - o„_ ib„_ i . . . , 
 
 that is, to the series 
 
 UiVi + {UiV2+U2Vi+U2V2)+ ■ ■ • 
 
 + (Ml?;„ + ?/2l'r,+ . . . +Wn-lVn 
 (a) +1lnVl+UnV2+ ■ ■ ■ +U„Vn-l 
 
 + Vr,T„)+ .... 
 
 Now the series of numerical values 
 
 |mi1+|w2|+ . . . +\Un\ + . . . 
 
 and |t>i|+|'y2l+ ... +|fn|+ ... 
 
 are equivalent to two sequences which we will write 
 
 ai a-z . . . an ■ . . 
 and 61' 62' . . • ?>n • • . • 
 
 The product is the sequence 
 
 ai'bi' a2'b2' ■ • • cin'bn ...» 
 
 f-quivalent to the series 
 
 |MiVil + (|wi'y2| + |w2i'i| + h*2f2|)+ • . : 
 + (lwii)„| + |u2i;„|+ . . . +|M„-l'ynI 
 
 (a') + I W„T;1 I + I MnT'2 I + • • • +|Mn1'„_i| 
 
 + |MnV„|)+ . • • • 
 
88 SERIES 
 
 In the series (a') all the terms in any parenthesis 
 have the same sign. Therefore the parentheses may 
 l>e removed without destroying its convergence or 
 changing its value. 
 
 The following series is, therefore, convergent: 
 
 |uii;i| + |wir2| + |M2ri| + |W21'2|+ . • • 
 (/?') +|WiD„l + |u2t''n|+ . . . 
 
 + |m„i)i|+ . . . +|w„?J„l+ .... 
 
 The terms of (/?') are the numerical values of the 
 terms of the series 
 
 U1V1+U1V2+U2V1+U2V2+ . . . 
 
 (jff) +UiVr, + U2Vn+ . . . 
 
 + U„Vi+ . . . +UnVn+ .... 
 
 The series (P) is, then, absolutely convergent. Its 
 value is not changed if we group the terms in any man- 
 ner, for example, in the manner of the series (a). The 
 value of the series (^) is, therefore, the same as that of 
 the series (a); it is the product of the values of the 
 two given series. 
 
 The series (a) represents the product of two given 
 series when they are convergent, even if they are not 
 both absolutely convergent. 
 
ABSOLUTE CONVERGENCE 89 
 
 III. Absolute Convergence 
 
 78. We have proved that a convergent series of 
 positive terms will remain convergent after the signs 
 of any portion of its terms have been changed (Art. C2j. 
 We may also prove this theorem as follows : 
 
 The terms which are not changed form by them- 
 selves a convergent series, and the terms which are 
 changed form by themselves a convergent series (Art. 
 67). The original series is the sum of these two series, 
 and the new series is formed by subtracting the second 
 from the first. 
 
 79. Theorem. — In a semi-convergent series the positive 
 terms taken by themselves form a divergent series and the 
 negative terms taken by themselves form a divergent series. 
 
 Proof. — If the two part-series were both convergent, 
 the series formed by taking all the terms with positive 
 sign would be convergent and would be equal to 
 the sum of the numerical values of the two series. 
 
 If one of the part-series were convergent and the 
 other divergent, the given series, formed by taking 
 them with positive and negative signs, respectively, 
 would be divergent (Art. 74, Cor.), which is contrary 
 to the hypothesis that the given series is convergent. 
 
 Therefore both part-series must be divergent. 
 
 80. Theorem. — // in each of two divergent series of 
 positive terms the terms themselves form a sequence equal 
 to zero, the two series may be put together, one with positive 
 signs and the other with negative signs, in such a manner 
 
90 SERIES 
 
 as to form a semi-convergent series equal to any given 
 number M. 
 
 Proof. — ^We may, for example, take just enough 
 positive terms from one series to make more than M, 
 then just enough negative terms from the other series 
 to make less than M, then just enough positive terms 
 from the first series again to make more than M, and 
 so on (Art. 64, Cor.). The result at each stage will 
 differ from M by an amount equal to or less than 
 the larger of the last terms taken from the two series. 
 Therefore the series so formed will equal M. 
 
 Although it might be quite impossible to write down 
 a formula for the nth term of the resulting series, the 
 term in each place is determined by the above process. 
 
 When two series contain the same terms in different 
 orders, every term of one being found somewhere in 
 the other, one series is said to be obtained from the 
 other by changing the order of its terms. This 
 does not mean a change of order like that referred 
 to on page 36. In any change in the order of the 
 terms, the terms in the new arrangement must still 
 form a sequence, each term having a definite numbered 
 position with only a finite number of terms coming 
 before it. 
 
 In the proof of the theorem the order of the terms 
 in each of the two given series is supposed to be un- 
 changed. But a divergent series of positive terms 
 can never become convergent by a change in the order 
 of its terms (see next theorem), and therefore it is not 
 
ABSOLUTE CONVERGEXCE 91 
 
 necessary to suppose the order of the terms in the 
 two given series to be unchanged. 
 
 Cor. — A semi-convergent series may be made to 
 take any value M, by properly changing the order 
 of its terms. 
 
 For examples see Art. 83. 
 
 81. Theorem. — We may change the order of terms in 
 an ahsotulely convergent series in any manner without de- 
 stroying its convergence or changing its value. 
 
 Proof. — First let all the terms of the series be posi- 
 tive, and let S be its value. If we form a new series 
 by changing the order of its terms, the first n terms 
 of the new series will be found somewhere in the origi- 
 nal series, and their sum will be less than S (Art. 66). 
 The new series is, then, convergent, and its value, 
 say S' , is equal to or less than S. 
 
 But we may obtain the original series from the new 
 series by changing the order of its terms. Thus S 
 must also be equal to or less than S' . Both of these 
 conditions can only be true if S and S' are equal. There- 
 fore any change in the order of terms of a convergent 
 series of positive terms does not change its conver- 
 gence or its value. 
 
 Now an absolutely convergent series of positive and 
 negative terms is equal to the difference of two con- 
 vergent series of positive terms. Any change in the 
 order of its terms may produce a change in the 
 order cf the terms of one or both of these two series, 
 but such a change does not affect their convergence 
 
92 SERIES 
 
 or their values. Therefore the new series is convergent 
 and its value is the same as that of the original series. 
 
 The expression "conditional convergence" is some- 
 times used with reference to the relation which the 
 order of the terms of a series has to its convergence 
 and value. A series which is convergent and has the 
 same value whatever the order of its terms is uncon- 
 ditionally convergent. A series which sometimes loses 
 its convergence, or at least changes its value when 
 the order of its terms is changed, is conditionally con- 
 vergent. 
 
 88. The product of two absolutely convergent series 
 as given in Art. 77 is an absolutely convergent series, 
 and its terms may be written in any order. One arrange- 
 ment which is often useful is obtained by putting in 
 succession those terms for which the sxmi of the subscripts 
 of It and V is the same number. 
 
 That is, the two given series being 
 
 U1+U2+ . . . +Un+ . ~. : 
 
 and Vi+r2+ . . . +Vn+ . . . , 
 
 both absolutely convergent, their product may be 
 written 
 
 U1V1+U1V2+U2V1+U1V3+U2V2+U3V1+ .... 
 
 Any number of absolutely convergent series may be 
 multiplied together in the same way. The product 
 forms an absolutely convergent series whose terms 
 may be arranged in any order. 
 
ABSOLUTE COXVERGENCE 93 
 
 83. Examples of changes in the order of terms of 
 semi-convergent series. 
 
 First we will consider some cases where there is no 
 change in the order of the positive terms among them- 
 selves or of the negative terms among themselves, but 
 only in the order in which they are put together. 
 
 Take the series 
 
 , 1 1 1 
 
 1 7=-|- . . . -I- , 7=+ .... 
 
 \/2 V2n-1 V2n 
 
 (Arts. 69 and 71). 
 
 Let s„ denote the sum of the first n terms and S the 
 value of this series, and let s„' denote the sum of the 
 first n terms when the series is arranged in some other 
 order. 
 
 Let n' denote the number of positive terms and n 
 the number of negative terms in the first n'+n terras 
 of the new arrangement. The values of n' and n are 
 determined for each value of n'+n. 
 
 In s'n'+n the positive terms are 
 
 J_ 1 
 
 and the negative terms 
 1 1 
 
 \^2 V4 ' ' " ^■27^ 
 If n' >n, S2„' will contain all of these terms and in 
 
94 SERIES 
 
 addition the negative terms 
 
 1 1 
 
 V2{n + 1) ' ' ' V2n'' 
 Hence 
 
 1 1 
 
 v2(n + l) v2n 
 
 These terms are n' -n in number. They are all 
 
 less than , — and their sum is less than — =r. The 
 
 V2n V2n 
 
 smallest of these terms is the last, and their sum is 
 greater than , — ~ . (Serret) 
 
 Suppose, for example, 
 
 n'—n^y/an and <\/an + l, 
 a being some positive number. Then 
 
 V2n \'2h ^2 V'2n 
 and 
 
 n'-ri. ~ Van la in 
 
 ^^2^'^v'2^'^^2 ' y^' 
 
 2n 
 or, as n' < n + 1 +Van, 
 
 V2n' "^2'yn + l+Vm ^2 I i 1^ 
 
 n 
 
ABSOLUTE CONVERGENCE 95 
 
 That is, 
 
 and 
 
 \a 1 
 
 „'+„-S2n'<S|g+-7=, 
 
 \/2r 
 
 la 1 
 
 Both of these expressions, for integer values of n, form 
 
 sequences representing >^-. Therefore the sequence of 
 s"s is regular and its value is 
 
 S'=S + 4 
 
 If a = 2, corresponding values of n and n' are 
 
 n =12345 6 7 8 9.. 
 n' =3 4 6 7 9 10 11 12 14 . . 
 
 1 J 1_ J L.J- 
 
 ^'°^V3"V5 V2"^V7 V4 V9 
 
 1 _ 1 
 
 Vn Ve"^ 
 
 =s+i. 
 
 Again, take the series 
 
 1 1 1, 
 
 1-2+ • ■ • +2W-1 2n^ 
 
96 SERIES 
 
 Using the same notation, if we suppose, for example, 
 
 n' — n'y.an and <an + l, 
 where o is some positive number, we shall find that 
 
 S'<5+^ and >S+ " 
 
 2 - - ' 9 
 
 (1+a)- 
 
 With this particular series we may proceed also as 
 follows : 
 
 Write 
 
 1 1 1 
 
 o:»=l+n+ . : : +-. 
 Ji n 
 
 Then y=2+4+ " ' ' +2^1' 
 
 a2n-ir = l + o+ . . . + 
 
 and 
 
 2 - '3 ' • • • ^2n-l' 
 
 S2n=a2n-an. 
 
 Suppose we change the order, taking two positive 
 terms and then one negative term; We have the series 
 
 ,11111 
 
 1+3-2 + 5+7-4+ • • • • 
 
 In s'sb there are 2n positive terms and n negative 
 
ABSOLUTE CONVERGENCE 97 
 
 terms. Therefore 
 
 S3»=(^a4n-^j— 2" 
 
 = {ain-0C2n) +;^(a2n -«„) 
 
 — S4n + (jS2n' 
 
 Here are three sequences, 
 
 S3' se' . . . s'sn . . . representing S', 
 
 S4 Sg . . . S4n ... o, 
 
 and 2^2 g** • • • 2^2n ... g*^' 
 
 and the value of the new series is 
 
 U 3, 
 
 5'. (See . 
 
 (Harkness and Morley) 
 
 <S'=5+-5=-S. (See Arts. 41 and 44.) 
 
 If we take one positive and two negative terms we 
 have the series 
 
 1_1 1_1_1 
 
 2~4'^3 6 8"^ • • • ' 
 
 and we shall find S' = :^^. In fact the series may be 
 
98 SERIES 
 
 written 
 
 1111 /T .^ 
 
 2-4+6-8+ • • • • (^^"''"*^ 
 
 One positive term taken with four negative terms 
 will give a series 
 
 1_1_1J. 
 
 2 4 6 8+ ■ • • 
 
 equal to zero. 
 
 Now, with the terms of the same series alternately 
 positive and negative, change the order of the negative 
 terms among themselves, taking first a fraction whose 
 denominator is not divisible by 4 and then two frac- 
 tions whose denominators are divisible by 4. This 
 change produces the series 
 
 ,11111 
 
 1 1 +-— — + 
 
 2^3 4 5 8 
 
 and the sum of the first 6n terms is 
 
 / a3n\ 1/ an\ 1 
 
 a2n 
 
 ■- (a'e,, — asn) +:^('a3n — «2n) — j(a2n— ««)• 
 
 The first parenthesis is equal to .sg,,. The rest may be 
 
ABSOLUTE CONVERGENCE 99 
 
 written 
 
 ir 2 2 _2_ 
 
 4L2n + l'^2n + 2"^ • ' " +3n 
 
 1 1 1 
 
 n + 1 n+2 ■ ■ ■ 2nS 
 
 Every other term in the first Une is cancelled by a 
 term in the second line. Letting n be odd, there is left 
 
 Ir 2 2 _2 
 
 4L2rH-l + 2n+3^ ' ' • ^3n 
 
 __2 2__ __2-| 
 
 371 + 1 3n+3 • • • 4n.J 
 nP 1 1 "I 
 
 ~2L(2n + l)(3n + l)"^ " " " "^3n.4nJ* 
 
 n + 1 
 Here the number of terms in the brackets is ^ . 
 
 The first is the largest and the last the smallest. There- 
 fore this expression is 
 
 n(n + 1) n + 1 
 
 ^4(2n + l)(3M + l)'^^4n' 
 
 n(n + l) 1 
 
 ^d > 4.3n.4n ^48* 
 
 It follows that 
 
 S'^S+^ and fS+^. (Art. 36) 
 
CHAPTER IV 
 
 POWER SERIES 
 
 I. The Radius of Convergence 
 
 84. A power scries is a series whose terms contain 
 as factors the successive powers of some number. 
 
 A power series is a series written in the form 
 
 U1+U2X+ . . . +u„x"'~^+ . . ~. : 
 
 An example is the geometrical series 
 
 a+ax+ . . . +ax"'~^+ .... 
 
 This is absolutely convergent if x is any number numer- 
 ically less than 1. 
 
 85. Theorem. — // the series 
 
 U1+U2X+ . . . +M„a;"-» . . : 
 
 is convergent for a particular value of x, it is absolutely 
 convergent for any value of x numerically less. 
 
 Proof. — Let Xi be a particular value of x for which 
 the series is convergent, and let x be some number 
 numerically less. 
 
 100 
 
THE RADIUS OF CONVERGENCE 101 
 
 The geometrical series 
 
 X 
 
 is absolutely convergent, since — is numerically less 
 
 than 1. This series will still be absolutely convergent 
 if we multiply its terms respectively by 
 
 Ui U2X1 . . . lt„Xi"~^ • • • > 
 
 numbers which form a sequence equal to zero, since 
 they are the terms of the given series when x has the 
 particular value for which by hypothesis the series 
 is convergent (Art. 76, Cor.)- 
 Therefore the series 
 
 U1+U2X+ . . . +'U„2;"~i+ . . ; 
 
 is absolutely convergent, x being numerically less 
 than Xi. 
 
 Cor. — If a power series is divergent for a particular 
 value of x, it is divergent for any value of x numer- 
 ically greater. 
 
 For if there were some value of x numerically greater 
 for which the given series was convergent, it would also 
 be convergent for the given value of x, for which by 
 hypothesis it is divergent. 
 
 86, Theorem. — // a power series is convergent for some 
 particular value of x and divergent for another value of x, 
 there is a positive number r such that the series is abso- 
 
102 SERIES 
 
 lutely convergent for all values of x numerically less than 
 r and divergent for all values of x numerically greater 
 than r. 
 
 Proof. — The series is always convergent or divergent, 
 and if it is convergent for a value of x, a, and diver- 
 gent for another value of x, b, then a is numerically 
 less than b unless they are positive and negative num- 
 bers numerically equal. We can separate all numbers 
 into two classes, putting into the first class all nega- 
 tive numbers and all positive numbers for which the 
 series is convergent, and into the second class all posi- 
 tive numbers for which the series is divergent. The 
 number determined by this separation is the number r. 
 
 The number r is called the radius of convergence. 
 
 If a series has a radius of convergence r it may be 
 absolutely convergent or semi-convergent or divergent 
 for x=r or x= —r. 
 
 If a series has no radius of convergence it is either 
 absolutely convergent for all values of x or divergent 
 for all values of x except a;=0. 
 
 For the geometrical series the radius of convergence 
 is 1, and the series is divergent when x = l and when 
 x=-l. 
 
 For other examples see Art. 88. 
 
 87. Theorem. — Given the series 
 
 U1+U2X+ . . . -|-M„a;"-i+ . . . , 
 if the sequence 
 
 V2 Vs u„ 
 
 Ul U2 Un-l 
 
THE RADIUS OF CONVERGENCE 103 
 
 represents a number a, then the series has a radius of con- 
 vergence equal to the reciprocal of the numerical value of a. 
 
 Proof. — The ratios of the terms of the series, each 
 to the preceding, form the sequence 
 
 11 2X U3X u„x 
 
 ■Hi U2 ' ' ' lh,_i 
 
 representing the number ax. This number is numerically 
 less than 1, and the series is absolutely convergent, if 
 
 M)n the other hand, if 
 
 the terms of the series will not form a sequence equal 
 to zero and the series will not be convergent. 
 
 88. Examples. 
 (1) The series 
 
 1+^+2+ ••• +"^+ "'• 
 
 has 1 for radius of convergence. For a; = l the series 
 '[■s divergent, being the harmonic series; for a;=-l 
 it is semi-convergent. 
 
104 SERIES 
 
 (2) The series 
 
 1+,+ + . . . ., 
 
 has no radius of convergence. It is absolutely con- 
 vergent whatever the value of x. In fact, the ratio 
 
 of the (n + l)th term to the preceding is — , and for 
 
 different values of n this fraction will form a sequence 
 equal to zero whatever the value of x. 
 
 (3) The series 
 
 H-a; + J2x2+ . . . +|nx"+ . . : 
 
 has no radius of convergence. It is divergent for all 
 values of x except zero. 
 
 (4) The series 
 
 l+C]X+C2x2+ . . . +c„x"+ . . ; , 
 where Ci = p and, in general. 
 
 yif-X) . . . (p-n+1) 
 ''"~ In 
 
 has 1 for radius of convergence except when p is zero 
 or a positive integer. The ratio of coefficients is 
 
 Cm 7)-r) + l v + 1 
 
 Cn- 1 n n 
 
THE RADIUis OP COXVEKGE.NX'E 105 
 
 and for different values of n this forms a sequence 
 equal to —1. 
 
 When x = l or —1 the convergence depends on the 
 value of p. 
 
 89. Theorem. — If each of two power aeries has a radius 
 of convergence or is convergent for all values of x, their 
 product can be written as a power series; this mil be 
 absolviely convergent for any value of x which makes both 
 of the given series absolutely convergent. 
 
 Proof. — If the series are 
 
 UI+U2X+ . . . +UnX"~'^+ . . . 
 
 and i)i + r2X+ . . . +Vr,x"~'^ i- . . . , 
 
 the product will be 
 
 UlVi + {UiV2+U2Vi)x+ . . . 
 
 + {UiVn + U2Vn-l+ • ■ • +U„Vi)x''-^ + . . . 
 
 (Art. 82) 
 
 Any number of power series may be multiplied 
 together if each has a radius of convergence or is con- 
 vergent for all values of x, and the product series will 
 be absolutely convergent for any value of x which 
 makes all the given series absolutely convergent. 
 
 90. Theorem. — // the series 
 
 U1+U2X+ . . .+i<„x"~^+ . . . 
 
 is convergent for a value of x irhose numerical value is p, 
 
106 SERIES 
 
 then there is a positive nuviber M such that for all valves 
 of n 
 
 Proof. — For the given value of x the terms of the 
 series form a sequence equal to zero and are all niuner- 
 ically less than a certain positive number M. Hence 
 
 |M„|iO"-l<M, 
 
 or 
 
 \Un\<—,. 
 
 91. Theorem. — Conversely, if there are two positive 
 numbers M and p such that for all values of n 
 
 then for any value of x numerically less than p the power 
 series is absolutely convergent and has a value which is 
 numerically less than 
 
 Mp 
 
 Proof. — The terms of the power series are numerically 
 less than the corresponding terms of the geometrical 
 series 
 
 ,, M\x\ .¥.r"-i 
 
 p p-^ 1 
 
THE RADIUS OF CONVERGENCE 107 
 
 which equals 
 
 M Mp 
 
 i— I, or r— f, 
 
 i_N p-\x\' , 
 p 
 
 when \x\<p. Therefore, when \x\<p, the given series 
 is absolutely convergent and its value is numerically less 
 
 than — rn (see Art. 68). 
 p-\x\ ' 
 
 The series 
 
 -^ Mx Mx"-! 
 
 p p^ 1 
 
 is a series of comparison for the given series. 
 
 Cor. — If u\ is zero, that is, if a power series begin ~ 
 
 with a term in x or some power of x, and if there ar 
 
 two positive numbers M and p such that for all vaUu - 
 
 of n greater than 1 the coefficient of a;""' is numerically 
 
 M 
 less than -;pij, then for values of x numerically lo- 
 
 than p the value of the series is nimierically less than 
 
 M\x\ 
 
 For the terms of the series are numerically less 
 than the corresponding terms of the series 
 
 M|zl M\xY-^ 
 
108 SERIES 
 
 which equals 
 
 M\x\ 
 
 p-\x\- 
 
 II. " Undetermined Coefficients " 
 
 93. Theorem. — A power series having a radius of con- 
 vergence or convergent for all values of x cannot equal 
 zero for values of x forming a sequence equal to zero 
 but not themselves zero. 
 
 Proof. — As we may suppose some of the coefficients 
 zero we will write the series 
 
 uixP+ . . . +w„a;P+"-i+ , . . , 
 
 where p is zero or a positive integer, and where wi 
 is not zero. 
 Write 
 
 P=Ui+U2X+ . . . +«„a;»~i+ . . . , 
 and P'=U2+V3X+ . . . +i(„+ia;"~i+ .... 
 
 Both of these series are convergent for any value of x 
 that makes the original series c^vergent, and 
 
 \ 
 
 P=ui+xP'. 
 
 P is zero for any value of x except zero, which makes 
 
 the original series zero, but P is not zero when x = 0. 
 
 Since P' has a radius of convergence or is conver- 
 
UNDETERMINED COEFFICIENTS 109 
 
 gent for all values of x, there are two positive num- 
 bers, M and p, such that for \x\<p 
 
 \xF'\<^\ 
 
 p-\x\ 
 Therefore 
 
 £ being some positive number less than |ui|, this 
 expression will be greater than £, that is, 
 
 if |^|<|(bq.l:iiL. 
 
 We will call this fraction e'. e' is less than p, since 
 |wil-£ 
 
 M+|wi|-e 
 
 <1. 
 
 Then P is numerically greater than t when x is 
 numerically less than e'. But in a sequence equal to 
 zero there is a place beyond which all the elements are 
 numerically less than e', and for none of these num- 
 bers as values of x can P be zero, nor can the original 
 series if these numbers are not themselves zero. 
 
 93. Theorem. — // iv:o series in powers of x, having radii 
 of convergence or convergent for all values of x, represent 
 
110 SERIES 
 
 the same numbers for valves of x forming a sequence equal 
 to zero and not themselves zero, then the two given, series 
 must be identically the same series, each coefficient of one 
 eqvxil to the corresponding coefficient of the other. 
 
 Proof. — If any of the corresponding coefficients were 
 different we could, by subtracting one of the given series 
 from the other, get a series equal to zero for values of 
 X forming the given sequence, which is contrary to 
 the last theorem. 
 
 Cor. — If two power series are convergent and equal 
 for all values of x numerically less than a certain num- 
 ber not zero, they are equal term by term, each coeffi- 
 cient of one equal to the corresponding coefficient of 
 the other. 
 
 For out of all values of x numerically less than the 
 number we can select values to form a sequence equal 
 to zero, and the two series, being equal for these values 
 of X, have the same coefficients, by the theorem. 
 
 This form of the theorem is given in most of our 
 Algebras as the " Theorem of undetermined coefficients. " 
 
 94. Since a polynomial in x may be regarded as a 
 power series with all the coefficients zero beyond a 
 certain place, these theorems are true of such poly- 
 nomials. 
 
 There is, however, a theorem concerning polynomials, 
 proved in our Algebras in connection with the theory 
 of equations, which is more general, and which we 
 will state here for convenience of reference: 
 
 Theorem. — A 'polynomial in x of degree n cannot equal 
 zero for more than n different values of x. 
 
UNDETERMINED COEFFICIENTS HI 
 
 Cor. — If two polynomials of degree n are equal for 
 more than n different values of x they are identically 
 the same polynomial, equal for all values of x, each 
 coefficient of one equal to the corresponding coefficient 
 of the other. 
 
CHAPTER V 
 
 THE EXPONENTIAL, BINOMIAL, AND LOGARITHMIC 
 
 SERIES 
 
 I. The Exponential Series 
 
 95. We shall assume in this chapter that the bi- 
 nomial theorem has been proved for positive integer 
 exponents. 
 
 In particular, r being any positive integer less than n, 
 the (r + l)th term in the expansion of (a+a;)" may be 
 written 
 
 In 
 
 n—r r 
 
 96. Theorem. — x being any rational number, 
 
 x^ a;" 
 
 = l+x+|2+ . . . +^+ . . . . 
 
 Proof.— Write 
 
 5,=l+x+^+ . . . +^+ . . 
 
 112 
 
THE EXPONENTIAL SERIES 113 
 
 SO that, y being any other niunber, 
 
 'S„ = l+2/+^+ . . . +P+ .... 
 
 The product of these two series is the series 
 
 l + (a;+2/) + (|+a;2/+^)+ . . . 
 
 + l|n + [^^T|i+ • • • +j^;+ • •• 
 
 , . . {x+yY (x + y)" 
 = l + (x + y) + '^^+ . . . +'—^+ 
 
 That is, we may write it Sx+y and say 
 
 These series are convergent and this relation holds 
 true for all values of x and y. 
 If we' put y=x, 2a;, ... ra;, ... we shall have 
 
 Sx^=S2x, 
 
 and by induction S^=Srx, 
 
 r being any positive integer. 
 
 TX 
 
 In this formula put — for x and s for r: 
 
114 SERIES 
 
 or, taJdng the sth root, 
 
 « 
 
 That is, for positive fractional values of r 
 
 If in the product series we put ^= — x, it will reduce 
 to its first term, 1 ; that is, 
 
 or S-x=g-' 
 
 Now S^'^st'sZ' 
 
 and from what we have just shown this is equal to 5_„. 
 Hence, for negative rational values of r and, there- 
 fore, for all rational values of r. 
 
 If we put x = l the series becomes 
 
 /S = l + l + |-7^+ . . . +]-+ . , . . 
 
 This series represents a number which is denoted by 
 the letter e. It is a positive number greater than 2. 
 
THE EXPONENTIAL SERIES 115 
 
 Putting also x in place of r, we have for all rational 
 values of x 
 
 e- = l+x+^+ . . . +f-+ 
 
 2 \n 
 
 97. Theorem. — The same relation holds true for irra- 
 tional values of x. 
 
 Proof. — If X and y are any two numbers, rational 
 or irrational, x<y, then Sx<Sy. 
 
 For, in the first place, if x and y are both positive, 
 the terms in both series are positive and the terms in 
 Sx are less than the corresponding terms in Sy, except 
 the first, which is the same in both series. Again, 
 if x and y are both negative, say x= —x' and y= —y', 
 x'>y', then 
 
 Sx=-^ and Sy=-^—. 
 
 Ox' ^y' 
 
 Sx'>Sy' and, therefore, Sx<Sy. 
 Finally, if a; is negative, say x=—x', and y positive, 
 
 <Sx = -5-<l and Sy>l. 
 
 Ox' 
 
 Let a be an irrational number separating all rational 
 numbers into two classes, and for the moment let x stand 
 for any rational number in the first class and y any 
 rational number in the second class. The number e° occu- 
 pies the point of separation of the numbers e" and e" 
 into two classes and is determined by this separation 
 
116 SERIES 
 
 (Art. 24). The same numbers written in the form Sx 
 and Sy determine the same point of separation, which 
 is, therefore, occupied by the number Sa. 
 
 That is, writing now x in place of a, we have proved 
 for irrational values of x, also, 
 
 |2 \n 
 
 II. The Binomial Theorem for any Rational 
 
 Exponent 
 
 98. Theorem. — p being any rational number and x any 
 number numerically less than 1, 
 
 il+x)P = l+pCiX+ . . . +pCnX''+ . . . , 
 
 where pCi = p, and, in general, 
 
 p(p-l) . . . (p-n + 1) 
 '^"~ \n 
 
 Proof. — We assume that the theorem has already- 
 been proved for positive integer values of p. 
 Write 
 
 5p = l + pCiX+ . . . +pCnX"+ . . . , 
 
 so that, q being some other number, 
 
 Sg = l + qCix+ . . . +gCna;"+ .... 
 
THE BINOMIAL SERIES 117 
 
 The product of these two series is the series 
 
 l + (pCi+gCi)a;+ . . . 
 
 + (pc„ + pc„_i,ci+ . . . +5C„)a;"+ . . . ; 
 
 When p and q are positive integers we know that this 
 is the same as the series 
 
 l + p+<2Ci2:+ . . . +p+^„a;"+ .... 
 
 If p is a positive integer less than n, pCn=0; and if q 
 is a positive integer less than n, ^„ = Q. But even 
 for such values of p and q, as well as for all integer 
 values greater than n, we have the relation 
 
 pCn + pCn_ ijCi + . . . -\- (fn^^ p+^n, 
 
 n being any positive integer. 
 
 The two members of this equation are polynomials 
 of degree n in p and q. For any positive integer values 
 of p and q they are equal. If we put for q any posi- 
 tive integer they become polynomials of degree n in 
 p, equal for all positive integer values of p; that is, 
 equal for more than n values of p, and, therefore, 
 for all values of p. 
 
 The two members of this equation are, then, equal 
 for any positive integer value of q combined with 
 any value whatever of p. If we put for p any value 
 whatever they become polynomials of degree n in q, 
 equal for all positive integer values of q; that is, equal 
 for more than n values of q, and, therefore, for all 
 values of q. 
 
118 SERIES 
 
 That is, for any value whatever of p and any value 
 whatever of q we have 
 
 Putting q=p, 2p, etc., and proceeding as in the case 
 of the exponential series, we prove for every positive 
 or negative rational number r the relation 
 
 For p=l the relation becomes 
 
 Si = l+x. 
 
 Therefore, putting also p in place of r, we have for all 
 rational values of p, x being numerically less than 1, 
 
 (l+x)P = l + pCia;+ . . . +pC„a;'»+ .... 
 
 This proof and the proof of the last section for the 
 series representing e* are due in part to Euler. 
 
 III. The Binomial Theoeem for an Irrational 
 Exponent and the Logarithmic Series 
 
 99. In what follows we shall take e for the base of 
 
 all logarithms. 
 If l+x=e'', so that /(=log (l+x), then the formula 
 
 e'"'=l + fiy+ . . . +^+ . . : 
 n 
 
EXPONENTIAL AND BINOMIAL SERIES COMPARED 119 
 becomes 
 
 (1) (l+i)''=i+ilog(i+x)!2/+ . . : 
 
 + ll0g(l+T)|''|'^ + 
 
 But if X is numerically less than 1, and ?/ is a rational 
 number, we have also 
 
 {\+x)y = \+yx+ '^%r^x^+ . . 
 
 y(y-l) . . . (y-n + 1) 
 + "~ i — -I" + 
 
 = l+?y.r + (^]|-g)+ ... 
 
 /7/"T" Ui. — 1 \ 
 
 + (^-...+(-1)-^./.")+.... 
 
 This series is convergent when x is numerically 
 less than 1, whatever the value of y. Now in each 
 parenthesis the terms of even degree in x and y taken 
 together have plus signs and the terms of odd degree 
 have minus signs. If we put for x and y the negative 
 numbers which have, respectively, the same numerical 
 values, we shall be putting for these terms their numer- 
 ical values. X being numerically less than 1, the series 
 will be convergent and the parentheses may be removed 
 (Art. 73 (1)). 
 
120 SERIES 
 
 That is, whatever the values of x and y, | a; | < 1, the 
 parentheses in the above series may be removed and 
 the series as then written will be absolutely convergent, 
 namely, 
 
 l+2/x + ^-Y+ • • • 
 (a) 
 
 +n • • • +(-1)"-'^—+ . . : , 
 
 \n ' n ' 
 
 representing {1+x)^ when y is also rational. 
 
 If from the series (a) we select all the terms that 
 
 contain y", we have a series from which we can take 
 
 the factor y", say u„y", Un being a power series in x 
 
 convergent for values of x numerically less than 1 (Art. 
 
 i" 
 67). The first term of w„ is-j — , and its coefficients are 
 
 alternately positive and negative. 
 In particular, 
 
 "1=^-2+3- ••• +(-l)""'n+ •••• 
 
 Consider the series 
 
 (i8) l+i'iy+ . . . +w„y"+ ■ . . ■ 
 
 The terms of this series are numbers represented by 
 part-series taken from the series (a). If in the series 
 (a) we replace the terms by their numerical values, we 
 
EXPONENTIAL AND BINOMIAL SERIES COMPARED 121 
 have the convergent series of positive terms 
 
 («') 
 
 +—, '+ . . . 4-'-^ — ^ + 
 
 and corresponding to the series (/?) the series 
 (/?') l+ui'\y\+ . . . +w„'|2/"|+ . . . , 
 
 where w„' is the series of numerical values of the terms 
 
 of Un. 
 
 The sum of the first n terms of (/?') is the sum of a 
 certain number of part-series from («')> ^.nd as a whole 
 may be regarded as a part-series taken from (a'). 
 Its value is less than the value of {a'), and therefore 
 the series (/?') is convergent and its value is equal to 
 or less than the value of the series (a'). 
 
 But all the terms of (a') are found in the different 
 series which represent the terms of (^'), and therefore 
 the svun of any number of terms of (a') is less than 
 the sum of a certain number of terms of (/?')• That is, 
 the value of the series (a') is equal to or less than the 
 value of the series (/?')• 
 
 It follows that the two series must have the same 
 value. 
 
 Since \un\^Un (Art. 62, Cor.), the series (j9) is 
 absolutely convergent whenever the series (/?') is conver- 
 gent. 
 
 If we take from the series (a') the part-series corre- 
 sponding to the sum of the first n terms of the series 
 
122 SERIES 
 
 (/?') we shall have a remainder series fl„' of positive 
 terms whose value will be less than e when n is taken 
 sufficiently large (Art. 67, Cor.). 
 
 If in the same way we take from the series (a) the 
 part-series corresponding to the sum of the first n 
 terms of the series (/?) we shall have a remainder series 
 R„ whose terms have for numerical values the cor- 
 responding terms of the series fl„'. Therefore the 
 numerical value of Rn will also be less than e, and the 
 sum of the first n terms of the series (/?) will differ from 
 the value of the series (a) by less than e. 
 
 This proves that the series (/9) has the same value as 
 the series (a). 
 
 Now for rational values of y, x being numerically 
 less than 1, the series (a) represents (l+x)*, which is 
 also represented by the series (1). That is, for any 
 value of X numerically less than 1 the series (/?) and the 
 series (1) are convergent power series in y, equal for 
 all rational values of y. 
 
 But from all rational values of y we can select a se- 
 quence equal to zero. Therefore, by the theorem of 
 Art. 93, the two series (/?) and (1) are identical, each 
 coefficient of one equal to the corresponding coefficient 
 of the other. 
 
 100. Since the series (a) has the same value as the 
 series (/3) for all values of ?/, I^:] <1, and since the scries 
 (ifl) is identical with the series (1), which for all values 
 of y represents (1+x)'', it follows that the series (a) 
 represents (1+a;)^ for all values of j/, |2:|<1. 
 
 Thus we prove the binomial theorem for irrational 
 values of the exponent; namely, writing p for y, 
 
THE LOGARITHMIC SERIES 123 
 
 Theorem. — For irrational values of p, x being numer- 
 ically less than 1, 
 
 {l+x)P = l + fCiX+ . . . +^c„x"+ .... 
 
 101. The coefficient of y in (1) is log (l+x) ; therefore 
 we have the following theorem: 
 
 Theorem. — For all values of x numerically less than 1 , 
 \og{l+x)=x-^+ . . . +(-l)"-i^+ .... 
 
SHORT-TITLE CATALOGUE 
 
 OF THB 
 
 PUBLICATIONS 
 
 OF 
 
 JOHN WILEY & SONS, 
 
 New York. 
 Lokdon: chapman & HALL, Limtebd. 
 
 ARRANGED UNDER SUBJECTS. 
 
 Descriptive circulars sent on application. Books marked with an asterisk are sold 
 at net prices only, a double asterisk <.**) books sold under the rules of the American 
 Publishers* Association at net prices subject to an extra charge for postage. All books 
 are bound in cloth unless otherwise stated. 
 
 AGRICULTURE. 
 Aimsby's Hanual of Cattle-feeditig i2mo. Si 75 
 
 Principles of Animal Nutrition 8vo 4 00 
 
 Budd and Hansen's American Horticultural Manual: 
 
 Part I. — ^Propagation, Culture, and Improvement izmo, i 50 
 
 Part n. — Systematic Pomology i2mo, i 50 
 
 Downing's Fruits and Fruit-trees of America 8vo, s 00 
 
 Elliotfs Engineering for Land Drainage i2mo, i 50 
 
 Practical Farm Drainage i2mo, i 00 
 
 Green's Principles of American Forestry i2mo, i 50 
 
 Grotenfelt's Principles of Modem Dairy Practice. (Woll.) i2mo, a 00 
 
 Kemp's Landscape Gardening izmo, 2 50 
 
 Maynard's Landscape Gardening as Applied to Home Decoration i2mo, t 50 
 
 Sanderson's Insects Injurious to Staple Crops iimo, i 50 
 
 Insects Injurious to Garden Crops, (/n preporaiiiwi.) 
 
 Insects Injuring Fruits, (/n ■preyaratian.') 
 
 Stockbridge's Rocks and Soils 8vo, 3 so 
 
 Woll s Handbook for Farmers and Dairymen i6mo, i 50 
 
 ARCHITECTURE. 
 
 Baldwin's Steam Heating for Buildings i2mo, 2 50 
 
 Berg's Buildings and Structures of American Railroads 4to, 5 00 
 
 Birkmire^ Planning and Construction of American Theatres 8vo, 3 00 
 
 Architectural bon and SteeL 8vo, 3 50 
 
 Compotmd Riveted Girders as Applied in Buildings 8vo, 2 00 
 
 Planning and Construction of High Office Buildings 8vo, 3 50 
 
 Skeleton Construction in Buildings 8vo, 3 00 
 
 Briggs's Modem American School Buildings 8vo, 4 00 
 
 Carpenter's Heating and Ventilating of Buildings 8vo, 4 00 
 
 Freitag's Architectural Engineering. 2d Edition, Rewritten 8vo, 350 
 
 Fireprooflng of Steel Buildings 8vo, 3 50 
 
 French and Ives's Stereotomy 8vo, 2 50 
 
 Gerhard's Guide to Sanitary House-inspection x6mo, t 00 
 
 Theatre Fires and Panics i2mo, i 50 
 
 Holly's Carpenters' and Joiners' Handbook i8mo, 75 
 
 Johnson's Statics by Algebraic and Graphic Methods 8vo, 2 00 
 
 1 
 
s 
 
 00 
 
 5 
 
 oo 
 
 4 
 
 oo 
 
 4 
 
 oo 
 
 5 
 
 oo 
 
 7 
 
 50 
 
 3 
 
 00 
 
 z 
 
 so 
 
 3 
 
 so 
 
 2 
 
 oo 
 
 3 
 
 oo 
 
 6 
 
 oo 
 
 b 
 
 so 
 
 S 
 
 oo 
 
 5 
 
 so 
 
 3 
 
 oo 
 
 4 
 
 00 
 
 2 
 
 so 
 
 I 
 
 2S 
 
 I 
 
 oo 
 
 Kidder's Architect's and Bnilder's Pocket-book. Rewritten Edition. l6mo,iliOf.( 
 
 Merrill's Stones for Building and Decoration 8vo, 
 
 Ron-metallic Minerals: Their Occurrence and Uses s . .8to, 
 
 Uonckton's Stair-building 4to, 
 
 Patton's Practical Treatise on Foundations 8vo, 
 
 Peabody's Naval Architecture 8vo, 
 
 Richey's Handbook {or Superintendents of Construction. (7n preu.) 
 
 Sabin's Industrial and Artistic Technology of Paints and Varnish 8to, 
 
 Siebert and Biggin's Modem Stone-cutting and Masonry 8to, 
 
 Snow's Principal Species of Wood 8vo, 
 
 Sondericker's Graphic Statics with Applications to Trusses, Beams, and Arches. 
 
 Sto, 
 
 Towne's Locks and Builders' Hardware i8mo, morocco. 
 
 Wait's Engineering and Architectural Jurisprudence 8vo, 
 
 Sheep, 
 Law of Operations Preliminary to Construction in Engineering and Archi- 
 tecture Sto, 
 
 Sheep. 
 
 Law of Contracts 8to, 
 
 Wood's Rustless Coatings : Corrosion and Electrolysis of Iron and Steel . . . 8to, 
 
 Woodbury's Fire Protection of Mills 8n>, 
 
 Worcester and Atkinson's Small Hospitals, Establishment and Maintenance, 
 Suggestions for Hospital Architecture, with Plans for a Small Hospital. 
 
 lamo. 
 The World's Columbian Exposition of 1893 ,. Large 4to. 
 
 ARMY AND NAVY. 
 Bemadou's Smokeless Powder, Nitro-cellulose, and the Theory of the Cellulose 
 
 Molecule lamo, 2 50 
 
 * BruS's Text-book Ordnance and Gunnery 8to. 6 00 
 
 Chase's Screw Propellers and Marine Propulsion 8to. 3 00 
 
 Craig's Azimuth 4to, 3 50 
 
 Cnhore and Squire's Polarizing Photo-chronograph 8to, 3 00 
 
 Cronkhite's Gunnery for non-commissioned Officers 34mo. morocco, 2 00 
 
 * Davis's Elements of Law 8vo, 2 50 
 
 * Treatise on the Military Law of United States 8to, 7 00 
 
 Sheep, 7 SO 
 
 De Brack's Cavalry Outpost Duties. (Carr.) 34mo morocco, 200 
 
 Dietz's Soldier's First Aid Handbook i6mo, morocco, i 25 
 
 * Dredge's Modem French Artillery 4to, half morocco, 15 00 
 
 Durand's Resistance and Propulsion of Ships 8vo, 5 00 
 
 * Dyer's Handbook of Light Artillery lamo, 3 00 
 
 Eissler's Modem High Explosives 8vo, 4 00 
 
 * Fiebeger's Text-book on Field Fortification Small 8vo, 2 00 
 
 Hamilton's The Gunner's Catechism i8mo, i oo 
 
 * Hofi's Elementary Naval Tactics 8vo, i so 
 
 Ingalls's Handbook of Problems in Direct Fire Svo. 4 00 
 
 * Ballistic Tables. 8vo, i so 
 
 * Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and II.. 8vo. each. 600 
 
 * Mohan's Permanent Fortifications. (Mercur.) 8vo, half morocco, 7 so 
 
 Manual for Courts-martial x6mo, morocco, z 50 
 
 * Mercur's Attack of Fortified Places z2mo, 2 00 
 
 * Elements of the Art of War 8vo, 4 00 
 
 Hetcalf 's Cost of Manufactures — And the Administration of Workshops, Public 
 
 and Private 8vo, s 00 
 
 * Ordnance and Gunnery, 2 vols. X2mo, 5 00 
 
 Murray's Infantry Drill Regulations i8mo, paper, zo 
 
 Peabody's Naval Architecture Sto 7 so 
 
 2 
 
* Phelps's Practical Marine Surreying 8to, 2 50 
 
 Powell's Army Officer's Examiner x2mo, 4 00 
 
 Sharpe's Art of Subsisting Armies in War i8mo, morocco, i 50 
 
 * Walke's Lectures on Explosives 8vo, 4 00 
 
 * Wheeler's Siege Operations and Military Mining 8vo, 2 00 
 
 Winthrop's Abridgment of Military Law i2mo, 250 
 
 Woodhull's Notes on Military Hygiene i6mo, i so 
 
 Young's Simple Elements of Navigation i6mo morocco, i 00 
 
 Second Edition, Enlarged and Revised i6mo, morocco, 2 00 
 
 ASSAYING. 
 
 Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe. 
 
 lamo, morocco, i 50 
 
 Funnan's Manual of Practical Assaying 8vo, 3 00 
 
 Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. . . .8vo, 3 00 
 
 Miller's Manual of Assaying i2mo, i 00 
 
 O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 00 
 
 Ricketts and Miller's Notes on Assaying 8vo, 3 00 
 
 Hike's Modem Electrolytic Copper Refining 8vo, 3 00 
 
 Wilson's Cyanide Processes lamo, i so 
 
 Cblorination Process lamo, i so 
 
 ASTRONOMY. 
 
 Comstock's Field Astronomy for Engineers 8vo, 3 so 
 
 Craig's Azimuth 4to, 3 50 
 
 Doolittle'a Treatise on Practical Astronomy 8vo, 4 00 
 
 Gore's Elements of Geodesy 8vo, 2 50 
 
 Hayford's Text-book of Geodetic Astronomy 8vo, 3 00 
 
 Merriman's Elements of Precise Surveying and Geodesy Svo, 2 so 
 
 * Michie and Harlow's Practical Astronomy 8vo> 3 00 
 
 * White's Elements of Theoretical and Descriptive Astronomy ismo, 2 00 
 
 BOTANY. 
 
 Davenport's Statistical Methods, with Special Reference to Biological Variation. 
 
 i6mo, morocco, t 25 
 
 Thom^ and Bennett's Structural and Physiological Botany. i6mo, 2 2S 
 
 Westermaier's Compendium of General Botany. (Schneider.) 8vo, 2 00 
 
 CHEMISTRY. 
 
 kdriance's Laboratory Calculations and Specific Gravity Table* lamo, i 25 
 
 Allen's Tables for Iron Analysis. 8vo, 3 00 
 
 Arnold's Compendium of Chemistry. (MandeL) Small 8vo, 3 so 
 
 Austen's Notes for Chemical Students lamo, i 50 
 
 * Austen and Langworthy. The Occurrence of Aluminium in Vegetable 
 
 Products, Animal Products, and Natural Waters 8vo. 2 00 
 
 Beraadou's Smokeless Powder. — Hitro-eellulose, and Theory of the Cellulose 
 
 Molecule "«no. 2 50 
 
 Bolton'a Quantitative Analysis 8". ' so 
 
 * Browning's Introduction to the Rarer Elements 8*0, i 50 
 
 Brush and Penfield's Manual of Determinative Mineralogy 8vo. 4 00 
 
 Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.) . . . .8vo. 3 00 
 
 Cohn's Indicators and Test-papers Ijmo, 2 00 
 
 Tests and Reagents 8»<>« 3 00 
 
 Craft's Short Course in Qualitative Chemical Analysis. (SchaeSer.) lamo, i so 
 
 Dolezalek'B Theory of the Lead Accumulator (Storage Battery). (Von 
 
 Drechsel's Chemical Reactions. (MerriU.) = i2mo, 1 2S 
 
 Dohem'a Thermodynamics and Chemistry. (Burgess.) 8vo, 4 00 
 
 Eissler's Modem High Explosives ?"• 4 00 
 
 BBront's Enzymes and their Applications. (Prescott) 8vo, 3 00 
 
 3 
 
Brdmann's Introduction to Chemical Preparations. (Dunlap.)., i2mo, i 25 
 
 Fletcher's Practical Instructions in Quantitative Assaying with the Blowpipe 
 
 zamo* morocco, x 50 
 
 Fowler's Sewage Works Analyses ismo, 2 00 
 
 Fretenius's Manual of Qualitative Chemical Analysis. (Wells.) 8vo, 5 oo 
 
 Manual of Qualitative Chemical Analysis. Parti. Descriptive. (Wells.) 8vo, 3 oa 
 System of Instruction in Quantitative Chemical Analysts. (Cohn.) 
 
 a vols. 8vo, 12 50 
 
 Fnertes's Water and Public Health lamo. i so 
 
 Furman's Manual of Practical Assaying 8vo, 3 oa 
 
 Oetman't Exercises in Physical Chemistry lamo. 
 
 Gill's Gas and Fuel Analysis for Eneineers lamo, i 25 
 
 Orotenfelt's Principles of Modem Dairy Practice. (Woll.) zamo, 2 00 
 
 Hammarsten's Text-book of Physiological Chemistry. (MandeL) 8vo, 400 
 
 Helm's Principles of Mathematical Chemistry. (Morgan.) lamoi i 50 
 
 Berisg's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 50 
 
 Hinds's Inorganic Chemistry 8vo. 3 00 
 
 * Laboratory Manual for Students i2mo, 75 
 
 Holleman'B Text-book of Inorganic Chemistry. (Cooper.) 8vo, 2 so 
 
 Text-book of Organic Chemistry. (Walker and Mott.) 8vo. 2 50 
 
 ** Laboratory Manual of Organic Chemistry. (Walker.) lamo, i oa 
 
 Hopkins's Oil-chemists' Handbook 8vo, 3 oo 
 
 Jackson's Directians for Laboratory Work in Physiological Chemistry. .8va, i 25 
 
 Keep's Cast Iron 8vo, 2 50 
 
 Ladd'* Manual of Quantitative Chemical Analysis i2mo, i oo 
 
 Landauer's Spectrum Analysis. (Tingle.) 8vo, 3 00 
 
 Lassar-Cohn's Practical Urinary Analysis. (Lorenz.) i2mo. i 00 
 
 Application of Some General Reactions to Investigations in Organic 
 
 Chemistry. (Tingle.) i2mo, i 00 
 
 Leach's The Inspection and Analysis of Food with Special Reference to State 
 
 Control 8vo, 7 50 
 
 LSb's Electrolysis and Electrosynthesis of Organic Compounds. (Lorenz.) lamo, i oo 
 
 Lodge's Notes on Assaying and Metallurgical Laboratory Experiments. . . . 8vo, 3 00 
 
 Lunge's Techno-chemical Analysis. (Cohn.) lamo, i 00 
 
 Mandel's Handbook for Bio-chemical Laboratory lamo, z so 
 
 • Martln't Laboratory Guide to Qualitative Analysis with the Blowpipe . . zamo, 60 
 Mason's Water-supply. (Considered Principally from a Sanitary Standpoint.) 
 
 3d Edition, Rewritten 8vo, 4 00 
 
 Examination of Water. (Chemical and BacteriologicaL) zamo, z 25 
 
 Matthews's The Textile Fibres gvo, 
 
 Meyer's Determination of Radicles in Carbon Compounds. (Tingle.). . zamo, i 00 
 
 Miller's Manual of Assaying z2mo, i 00 
 
 Hixtei*s Elementary Text-book of Chemistry z2mo. i so 
 
 Morgan's Outline of Theory of Solution and its Results i2mo, z 00 
 
 Elements of Physical Cheiziistry zamo. 
 
 Hone's Calcnlationa used in Cane-sugar Factories z6mo, morocco. 
 
 Hulliken's General Method for the Identification of Pure Organic Compounds. 
 
 VoL I Large 8vo, 
 
 O'Brine's Laboratory Guide in Chemical Analysis 8vo, 
 
 O'Driscoll's Notes on the Treatment of Gold Ores gvo, 
 
 Ostwald's Conversations on Chemistry. Part One. (Ramsey.) (In press.) 
 * Penfield's Notes on Deterzninative Mineralogy and Record of Mineral Tests. 
 
 8vo, paper, 50 
 
 Pictet's The Alkaloids and their Chemical Constitution. (Biddle.) 8vo. 5 00 
 
 Pinner's Introduction to Organic Chemistry. (Austen.) Z2mo, z so 
 
 Poole's Calorific Power of Fuels gyo 3 00 
 
 Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- 
 ence to Sanitary Water Analysis ' ijmo. z 25 
 
 4 
 
 50 
 
 00 
 50 
 
 00 
 00 
 00 
 
• Reisis's Guide to Piece-dyeing 8vo, 25 00 
 
 RldurdsondWoodman'BAiriWater.andFoodframaSanitaryStandpoint.STO, 2 00 
 Kicbards'i Cost of Living as Modified by Sanitary Science i2mo i 00 
 
 Cost of Food a Study in Dietaries i2mo, i 00 
 
 • Richards and Williams's The Dietary Computer 8vo, i so 
 
 Ricketts and Russell's Skeleton Notes upon Inorganic Chemistrr. rPart I. — 
 
 Ron-metallic Elements.) 8to, morocco. 75 
 
 Ricketts and Miller's Rotes on Assaying 8vo, 3 00 
 
 Rldeal's Sewage and the Bacterial Purification of Sewage 8vo, 3 50 
 
 Disinfection and the Preservation of Food Svo. 4 00 
 
 Riggs's Elementary Manual for the Chemical Laboratory 8vd. i 25 
 
 Rostoski's Serum l)iagnosis. (Bolduaa.) i2mo, i 00 
 
 Ruddiman's Incompatibilities in Prescriptions. Svo, 2 00 
 
 Sabin's Industrial and Artistic Technology of faints and Varnish Svo, 3 00 
 
 Salkowski's Physiological and Pathological Chemistry. (OmdorS.).. ..Svo. 2 so 
 Schimpf's Text-book of Volumetric Analysis i2mo, 2 50 
 
 Essentials of Volumetric Analysis lamo, i 25 
 
 Spencer's Hanabook for Chemists of Beet-sugar Houses i6mo« morocco, ^ 00 
 
 Handbook for Sugar Manufacturers and their Chemists.. j6mo> morocco, 2 00 
 Stockbridge'i Rocka and Soils Svo, 2 50 
 
 • Tillman's Elementary Lessons in Heat Svo, i 50 
 
 • Descriptive General Chemistry Svo, 3 oo 
 
 Treadwell's Qualitative Analysis. (Hall.) a Svo, 3 00 
 
 Quantitative Analysis. (HaH.) Svo, 4 00 
 
 Tomeaure and Russell's Public Water-supplies Svo, s 00 
 
 Tan Deventer's Physical Chemistry for Beginners. (Boltwood.) ismo, i 50 
 
 • Walke's Lectures on Explosives Svo, 4 00 
 
 Washington's Manual of the Chemical Analysis of Rocks Svo, 2 00 
 
 Wassermann's Immune Sera: Hsemolysins, Cytotoxins, and Precipitins. (Bol- 
 
 duan.) i3mo, 1 00 
 
 Wells'i Laboratory Guide in Qualitative Chemical Analysis Svo, i so 
 
 Short Course in Inorganic Qualitative Chemical Analysis for Engineering 
 
 Students i2mo, i 50 
 
 WUpple'i Microscopy of Drinking-water Svo, 3 so 
 
 Wiechmann's Sugar Analysis Small Svo. 2 50 
 
 Wilson's Cyanide Processes. i2mo, i 50 
 
 Chlorioation Process. ., i2mo, i so 
 
 Wulling's Elementary Course in Inorganic Pharmaceutical and Medical Chem- 
 istry i2mo, 2 00 
 
 CIVIL ENGIIfEERIH6> 
 BRIDGES AlID ROOFS. HYDRAULICS. MATERIALS OF EKGINEERIHO 
 
 RAILWAY ENGIHEERinG. 
 
 taker's Engineers' Surveying Instruments i2mo, 3 00 
 
 Eixby's Graphical Computing Table Paper 19^X24^ inches. 25 
 
 *• Burr's Ancient and Modem Engineering and the Isthmian CanaL (Postage, 
 
 37 cents additionaL) Svo, net, 3 so 
 
 Comitock's Field Astronomy for Engineers. Svo, 2 50 
 
 Davis's Elevation and Stadia Tables Svo, i 00 
 
 EUiotfs Engineering for Land Drainage i2mo, 1 so 
 
 Practical Farm Drainage "™o, i 00 
 
 Folwell's Sewerage. (Designing and Maintenance.) Svo, 3 00 
 
 Freitag'a Architectural Engineering. 2d Edition Rewrinen Svo 3 50 
 
 French and Ives's Stereotomy 8vo, 2 so 
 
 Goodhue's Municipal Improvements "™0' ' 7S 
 
 Goodrich's Economic DisposBi of Towns' Refuse Svo, 3 50 
 
 (Sore's Elemente of Geodesy ?™' ^ so 
 
 Hayford's Text-book of (Seodetic Astronomy Svo, 3 00 
 
 Bniag'a Ready Reference Table* (Conversion Factors) i6mo, morocco, a 50 
 
 6 
 
Howe's Retaininc Walls for Earth i2nio, z 25 
 
 Johnson's (J. B.) Theory and Practice 01 Surveying Small 8vo, 4 00 
 
 Johnson's (L. J.) Statics by Algebraic and Graphic Methods 8vo, 2 00 
 
 Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) ismo. 2 00 
 
 Uahan's Treatise on Civil Engineering. (1873.) (Wood.) 8vo. s 00 
 
 * Descriptive Geometry Svo, 1 50 
 
 Herriman's Elements of Precise Surveying and Geodesy JSvo, 2 50 
 
 Elements of Sanitary Engineering 8v0) 2 00 
 
 Uerriman and Brooks's Handbook for Surveyors i6mo> morocco, 2 co 
 
 Hngenfs Plane Surveying Svo 3 5° 
 
 Ogden'f Sewer Design. i2mo, 2 00 
 
 Patton's Treatise on Civil Engineering Svo half leather, 7 50 
 
 Reed's Topographical Drawing and Sketching 4tOi 5 00 
 
 Rideal's Sewage and the Bacterial Purification of Sewage Svo, 3 SO 
 
 Siebett and Biggin's Modem Stone-cutting and Masonry Svo, i so 
 
 Smith's Manual of Topographical Drawing. (McMillan.) Svo, 2 So 
 
 Sondericker's Graphic Statics, with Applications to Trusses. Beams, and 
 
 Arches .,8va, 2 00 
 
 Taylor and Thompson's Treatise on Concrete,^ lau and Reinforced. (In prat.) 
 
 * Trantwine's Civil Engineer's Pocket-book i6mo, morocco, 5 00 
 
 Walt's Engineering and Architectural Jurisprudence 8v6, 6 00 
 
 Sheep, 6 50 
 Law of Operations Preliminary to Construction in Engineering and Archi- 
 tecture Svo, 5 00 
 
 Sheep, 5 50 
 
 Law of Contracts Svo, 3 00 
 
 Warren's Stereotomy — Problems in Stone-cutting Svo, 2 50 
 
 Webb's Problems in the Use and Adjustment of Engineering Instruments. 
 
 i6mo, morocco, i 25 
 
 * Wheeler's Elementary Course of Civil Engineering Svo, 4 00 
 
 Wilson's Topographic Surveying Svo, 3 50 
 
 BRIDGES KS-D ROOFS. 
 BoUar's Practical Treatise on the Construction of Iron Highway Bridges, .Svo, 2 00 
 
 * Thames River Bridge 4to, paper, 5 00 
 
 Burr's Course on the Stresses in Bridges and Roof Trusses. Arched Ribs, and 
 
 Suspension Bridges Svo, 3 50 
 
 Du Bols's Mechanics of Engineering. Vol. U Small 4to, i o 00 
 
 Foster's Treatise on Wooden Trestle Bridges 4to, 5 00 
 
 Fowler's Cofier-dam Process for Piers : Svo, 2 so 
 
 Ordinary Foundations Svo, 3 50 
 
 Oreene's Roof Trusses Svo, i 25 
 
 Bridge Trusses Svo, 2 so 
 
 Arches in Wood, Iron, and Stone Svo, 2 50 
 
 Howe's Treatise on Arches Svo, 4 00 
 
 Design of Simple Roof -trusses in Wood and Steel Svo, 2 00 
 
 Johnson^Bryan, and Tumeaure's Theory and Practice in the Designing of 
 
 Modern Framed Structures Smfil 4to, 10 00 
 
 Herriman and Jacoby's Text-book on Roofs and Bridges : 
 
 Partl. — Stresses in Simple Trusses Svo, 2 50 
 
 Part n. — Graphic Statics Svo, 2 so 
 
 Part ni, — Bridge Design. 4th Edition, Rewritten Svo, 2 so 
 
 Part IV. — Higher Structures Svo, 2 So 
 
 Horison's Memphis Bridge 4to, 10 00 
 
 Waddell's De Pontibus, a Pocket-book for Bridge Engineers. . . i6mo. morocco^ 3 00 
 
 Specifications for Steel Bridges lamo, i 25 
 
 Wood's Treatise on the Theory of the Construction of Bridges and Roofs. Svo, 2 00 
 Wright's Designing of Draw-spans: 
 
 Part L — Plate-girder Draws '.Svo, 2 50 
 
 Part n.— Riveted-tniss and Pin-connected Long-span Draws Svo, 2 50 
 
 Two parts in one volume 8vO| 3 50 
 
 6 
 
HYDRAULICS. 
 Bazlii's Ezperiments upon the Contraction of the Liquid Vein Issuing from an 
 
 Orifice. (Trautwine.) 8vo, 2 00 
 
 BoTey*! Treatise on Hydraulics 8vo, 5 00 
 
 Cbnrch's Mechanics of Engineering 8vo, 6 00 
 
 Diagrams of Mean Velocity of Water in Open Channels paper, i so 
 
 Coffin's Graphical Solution of Hydraulic Problems idmo. morocco, 2 50 
 
 Vlather's Dynamometers, and the Measurement of Power zamo, 3 00 
 
 Folwell's Water-supply Engineering 8vo, 4 00 
 
 Viizell's Water-power 8vo, 5 00 
 
 Vuertes's Water and Public Health i2mo, i 50 
 
 Water-filtration Works i2mo, 2 so 
 
 Oangnillet and Kutter's General Formula for the Uniform Flow of Water in 
 
 Rivera and Other Channels. (Hering and Trautwine.) 8vo, 400 
 
 Hazen's Filtration of Public Water-supply 8yo, 3 00 
 
 Hazlehorst's Towers and Tanla for Water-works 8to, 2 50 
 
 Herschel's 1 15 Ezperiments on the Carrying Capacity of Large, Riveted, Metal 
 
 Conduits 8vo, 2 00 
 
 Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
 point) 3d Edition, Rewritten 8vo, 4 00 
 
 Mertiman's Treatise on Hydraulics. 9th Edition, Rewritten 8vo, s 00 
 
 * Michie's Elements of Analytical Mechanics 8vo, 4 00 
 
 Schuyler's Reservoirs for Irrigation, Water-power, and Domestic Water- 
 supply Large 8vo, s 00 
 
 •* Thomas and Watt's Improvement of Riyen. (Post., 44 c. additional), 4to, 6 00 
 
 Tnmeaure and Russell's Public Water-supplies 8vo, 5 00 
 
 Wegmann's Desisn and Construction of Dams 4to, 5 00 
 
 Water-supplyof the City of New York from 1658 to 1895 4to, 10 00 
 
 Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.) 8vo, 5 00 
 
 Wilson's Manual of Irrigation Engineering Small 8vo, 4 00 
 
 Wolff's Windmill as a Prime Mover 8vo, 3 00 
 
 Wood's Turbines 8vo, 2 50 
 
 Elements of Analytical Mechanics 8vo, 3 00 
 
 MATERIALS OP ERGIKEERIIIG. 
 
 Baker's Treatise on Masonry Construction 8vo, 5 00 
 
 Roads and Pavements. 8vo, s 00 
 
 Black's United States Public Works Oblong 4to, s 00 
 
 Bovey's Strength of Materials and Theory of Structures 8vo, 750 
 
 Bnrr's Elastic!^ and Resistance of the Materials of Engineering. 6th Edi- 
 tion, Rewritten 8vo, 7 50 
 
 Byrne's Highway Construction Svo, s 00 
 
 Inspection of the Materials and Workmanship Employed in Construction. 
 
 x6mo, 3 00 
 
 Church's Mechanics of Engineering Svo, 6 00 
 
 Du Bois's Mechanics of Engineering. VoL I Small 4to, 7 so 
 
 Johnson's Materials of Construction Large Svo, 6 00 
 
 Fowler's Ordinary Foundations 8vo, 3 so 
 
 Keep's Cast Iron 8vo, 2 50 
 
 Lanza's Applied Mechanics 8vo, 7 so 
 
 Marteni's Handbook on Testing Materials. (Henning.) a vols. Svo, 7 so 
 
 Merrill's Stones for Building and Decoration 8vo, s 00 
 
 Merriman's Teit-book on the Mechanics of Materials 8vo, 4 00 
 
 Strength of Materials "mo, i 00 
 
 Metcalf's Steel. A Manual for Steel-users "mo. 2 00 
 
 Patton's Practical Treatise on Foundations 8vo, s 00 
 
 Richey's Handbook for Building Superintendents of Construction. (.In press. ) 
 
 Rockwell's Roads and Pavements in France i2mo, i 25 
 
 1 
 
Sabin*s Industrial and Artistic TechnologT of Paints and Varnish 8to, 3 00 
 
 Smith's Materials of Machines i2mo» x 00 
 
 Snow's Principal Species of Wood 8vo, 3 50 
 
 Spalding's Hydraulic Cement i2mo, 2 00 
 
 Text-book on Roads and Pavements x2mo, 2 00 
 
 Taylor and Thompson's Treatise on Concrete* Plain and Reinforced. (In 
 
 press. ) 
 
 Thtu'ston's Materials of Engineering:. 3 Parts 8vo, 8 00 
 
 Part 1. — Non-metallic Materials of Engineering and Metallurgy 8to» a 00 
 
 Part II. — Iron and SteeL 8vo, 3 50 
 
 Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their 
 
 Constituents 8vo, 2 50 
 
 Thurston's Text-book of the Materials of Construction 8to» 5 00 
 
 Tillson's Street Pavements and Paving Materials 8vo, 4 00 
 
 Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.) . . x6mo, mor., 3 00 
 
 Specifications for Steel Bridges X2mo, i 25 
 
 Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on 
 
 the Preservation of Timber 8vo, 2 00 
 
 Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00 
 
 Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and 
 
 SteeL 8vo, 4 00 
 
 RAILWAY ENGINEERING. 
 
 Andrews's Handbook for Street Railway Engineers 3x5 inches* morocco, i 25 
 
 Berg's Buildings and Structures of American Railroads 4to, 5 00 
 
 Brooks's Handbook of Street Railroad Location idmo* morocco, i 50 
 
 Butts's Civil Engineer's Field-book z6mo, morocco, 2 50 
 
 Crandall's Transition Curve i6mo, morocco, i 50 
 
 Railway and Other Earthwork Tables 8vo, i 50 
 
 Dawson's "Engineering" and Electric Traction Pocket-book. z6mo, morocco, 5 00 
 
 Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 on 
 
 * Drinker's TunneHng, Explosive Compounds, and Rock Drills, 4to, half mor., 25 00 
 
 Fisher's Table of Cubic Yards Cardboard, 25 
 
 Godwin's Railroad Engineers* Field-book and Explorers' Guide .... i6mo, mor., z 50 
 
 Howard's Transition Curve Field-book i6mo, morocco, i 50 
 
 Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- 
 bankments 8vo, I 00 
 
 Molitor and Beard's Manual for Resident Engineers i6mo, x 00 
 
 Nagle's Field Manual for Railroad Engineers x6mo, morocco, 3 00 
 
 Philbrick's Field Manual for Engineers i6mo, morocco, 3 00 
 
 Searles's Field Engineering x6mo, morocco, 3 00 
 
 Railroad Spiral x6mo, morocco, x 50 
 
 Taylor's Prismoidal Formulee and Earthwork 8vo, z 50 
 
 * Trautwine's Method ot Calculating the Cubic Contents of Excavations and 
 
 Embankments by the Aid of Diagrams 8vo, 2 00 
 
 The Field Practice of Laying Out Circular Curves for Railroads'. 
 
 x2mo, morocco, 2 50 
 
 Cross-section Sheet Paper, 25 
 
 Webb's Railroad Construction. 2d Edition, Rewritten x6mo, morocco, 5 00 
 
 WeUington's Economic Theory of the Location of Railways Small 8vo, $ 00 
 
 DRAWING. 
 
 Barr's Kinematics of Machinery 8vo, 2 50 
 
 * Bartlett's Mechanical Drawing 8vo, 3 00 
 
 * " Abridged Ed 8vo, x 50 
 
 Coolidge's Manual of Drawing 8vo, paper, i 00 
 
 Coolidge and Freeman's Elements of General Drafting for Mechanical Engi- 
 neers Oblong 4to. 2 50 
 
 Diirley's Kinematics of Machines 8vo, 4 00 
 
 8 
 
2 
 
 00 
 
 2 
 
 SO 
 
 1 
 
 so 
 
 3 
 
 oo 
 
 3 
 
 oo 
 
 S 
 
 oo 
 
 4 
 
 oo 
 
 1 
 
 so 
 
 1 
 
 so 
 
 3 
 
 SO 
 
 3 
 
 OO 
 
 3 
 
 OO 
 
 3 
 
 OO 
 
 2 
 
 so 
 
 I 
 
 OO 
 
 I 
 
 2S 
 
 I 
 
 so 
 
 I 
 
 oo 
 
 J 
 
 2S 
 
 
 75 
 
 3 
 
 SO 
 
 3 
 
 oo 
 
 7 
 
 SO 
 
 2 
 
 SO 
 
 Hill's Text*book on Shades and Shadows, and Perspective Svo. 
 
 Tamison's Elements of Mechanical Drawing Svo, 
 
 Jones's Machine Design : 
 
 Part I. — Kinematicanf Machinery Svo, 
 
 Part n. — Form, Strength, and Proportions of Parts Svo, 
 
 HacCord's Elements of Descriptive Geometry Svo, 
 
 Kinematics; or. Practical Mechanism Svo, 
 
 Mechanical Drawing 4to, 
 
 Velocity Diagrams Svo, 
 
 Mahan's Descriptive Geometry and Stone-cutting Svo, 
 
 Industrial Drawing. (Thompson.) Svo, 
 
 Moyer's Descriptive Geomet-.y. (In preaa.) 
 
 Reed's Topographical Drawing and Sketching 4to, 
 
 Reid's Course in Mechanical Drawing Svo, 
 
 Text-book of Mechanical Drawing and Elementary Machine Design . . Svo, 
 
 Robinson's Principles of Mechanism Svo, 
 
 Schwamb and Merrill's Elements of Mechanism Svo, 
 
 Smith's Manual of Topographical Drawing. (McMillan.) Svo, 
 
 Warren's Elements of Plane and Solid Free-hand Geometrical Drawing . . i2mo. 
 
 Drafting Instruments and Operations i2mo. 
 
 Manual of Elementary Projection Drawing i2mo. 
 
 Manual of Elementary Problems in the Linear Perspective of Form and 
 
 Shadow i2mo. 
 
 Plane Problems in Elementary Geometry. i2mo. 
 
 Primary Geometry i2mo, 
 
 Elements of Descriptive Geometry, Shadows, and Perspective 8vo, 
 
 General Problems of Shades and Shadows Svo 
 
 Elements of Machine Construction and Drawing Svo. 
 
 Problems, Theorems, and Examples in Descriptive Geometry Svo, 
 
 Weisbach's Kinematics and the Power of Transmission. (Hermann and 
 
 Klein.) Svo, s oo 
 
 Whelpley's Practical Instruction in the Art of Letter Engraving i2mo, 2 oo 
 
 Wilson's (H. M.) Topographic Surveying Svo, 3 so 
 
 Wilson's (V. T.) Free-hand Perspective Svo, 2 50 
 
 Wilson's (V. T.) Free-hand Lettering 8vo, i 00 
 
 Woolf's Elementary Course in Descriptive Geometry Large Svo, 3 00 
 
 ELECTRICITY AND PHYSICS. 
 
 Anthony and Brackett's Text-book of Physics. (Magie.) Small Svo, 3 00 
 
 Anthony's Lecture-notes on the Theory of Electrical Measurements. . . . i2mo, 
 Benjamin's History of Electricity Svo, 
 
 Voltaic Cell 8vo, 
 
 Classen's Quantitative Chemical Analysis by Electrolysis. (Boltwood.). Svo, 
 
 Crehore and Squier's Polarizing Photo-chronograph Svo, 
 
 Dawson's "Engineering" and Electric Traction Pocket-book. . i6mo, morocco, 
 Dolezalek's Theory of the Lead Accumulator (Storage Battery). (Von 
 
 Ende.) "™°' 
 
 Duhem's Thermodynamics and Chemistry. (Burgess.) Svo, 
 
 Flather's Dynamometers, and the Measurement of Power i2mo, 
 
 Gilbert's De Magnete. (Mottelay.) 8vo, 
 
 Hancbett's Alternating Currents Explained i2mo, 
 
 Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 
 
 Hohnan's Precision of Measurements 8vo, 2 
 
 Telescopic Mirror-scale Method, Adjustments, and Tests Large Svo, 
 
 L«nd«uer's Spectrum Analysis. (Tingle.) 8vo, 3 
 
 Lc ChateUer'i High-temperature Measurements. (Boudouard— Burgess. )i 2mo 3 
 Ub'tBlwtrelyii<aadStoetrei]rBtfaMi««fOrruucCempeiudf. (Lereai.) lams. i 
 
 
 
 3 
 
 00 
 
 3 
 
 00 
 
 3 
 
 00 
 
 3 
 
 00 
 
 5 
 
 00 
 
 2 
 
 50 
 
 4 
 
 00 
 
 3 
 
 00 
 
• Lyons's Treatise on Electromagnetic Phenomena. Vols. I. and IL 8vo, each, 600 
 
 • Hicbie. Elements of Wave Motion Relating to Sound and Light. 8vo, 4 00 
 
 Niaudet's Elementary Treatise on Electric Batteries. (Fishoack.) izmo, 2 50 
 
 • Rosenberg^ Electrical Engineering. (Haldane Gee — Kiszbrunner.) 8to, i 50 
 
 Ryan, Ilorris, and Hozie's Electrical Machinery. VoL L 8to, 2 50 
 
 Thurston's Stationary Steam-engines 8to, 2 50 
 
 • Tillman's Elementary Lessons in Heat. 8to, i 50 
 
 Tory and Pitcher's Manual of Laboratory Physics Small Svo, 2 00 
 
 Ulke',^ Modern Electrolytic Copper Refining Svo, 3 00 
 
 LAW. 
 
 • Davis's Elements of Law 8vo, 2 50 
 
 • Treatise on the Military Law ol United States Svo, 7 00 
 
 • Sheep, 7 50 
 
 Manual for Courts-martial i6mo, morocco, i 50 
 
 Waifs Engineering and Architectural Jurisprudence Svo, 6 00 
 
 Sheep, 6 50 
 
 Law of Operations Preliminary to Construction in Engineering and Archi- 
 tecture Svo, 50a 
 
 Sheep, 5 so 
 
 Law of Contracts Svo, 3 00 
 
 Winthrop's Abridgment of Military Law tamo, 2 so 
 
 MAHUFACTURES. 
 
 Bemadou's Smokeless Powder — Nitro-cellulose and Theory of the Cellulose 
 
 Molecule i2mo, 2 50 
 
 Holland's Iron Founder iimo, 2 so 
 
 "The Iron Founder," Supplement i3mo, 2 50 
 
 Encyclopedia of Founding and Dictionary of Foundry Terms Used in the 
 
 Practice of Moulding ismo, 3 00 
 
 Bissler's Modem High Explosives Svo, 4 00 
 
 BSront's Enzymes and their Applications. (Prescott.) Svo 300 
 
 Fitzgerald's Boston Machinist iSmo, i 00 
 
 Ford's Boiler Making for Boiler Makers iSmo, x 00 
 
 Hopkins's Oil-chemists' Handbook Svo, 3 00 
 
 Keep's Cast Iron Svo, 2 so 
 
 Leach's The Inspection and Analysis of Food with Special Reference to State 
 
 Control. (In preparation.) 
 
 Matthews's The Textile Fibres Svo, 3 50 
 
 Metcatf's Steel. A Manual for Steel-users i2mo, 2 00 
 
 Metcalfe's Cost of Manufactures — And the Administration of Workshops, 
 
 Public and Private Svo, 5 00 
 
 Meyer's Modern Locomotive Construction 4to, 10 00 
 
 Morse's Calculations used in Cane-sugar Factories. i6mo, morocco, i 50 
 
 • Reisig's Guide to Piece-dyeing Svo, 25 00 
 
 Sabin's Industrial and Artistic Technology of Paints and Varnish Svo, 3 00 
 
 Smith's Press-working of Metals Svo, 3 00 
 
 Spalding's Hydraulic Cement i2mo, 2 00 
 
 Spencer's Handbook for Chemists of Beet-sugar Houses i6mo,morocco, 3 00 
 
 Handbook for Sugar Manufacturers and their Chemists.. .x6mo morocco, 2 00 
 Taylor and Thompson's Treatise on Concrete, Plain and Reinforced, (/n 
 
 press.) 
 Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- 
 tion Sto, 5 00 
 
 • Walke's Lectures on Explosives Svo, 4 00 
 
 West's American Foundry Practice xamo, 2 so 
 
 Moulder's Tezt-book izmo, 2 50 
 
 10 
 
Wolff*! Windmill as a Prime Hover 8vo, 3 oo 
 
 Woodbury's Fire Protection of Mills 8vo, 2 so 
 
 Wood's Rustless Coatings: Corrosion and ElectrolTsis of Iron and Steel. . .Svo, 4 00 
 
 MATHEUATICS. 
 
 Baker's Elliptic Functions 8vo, i 50 
 
 * Bass's Elements of Differential Calculus I2ma, 4 00 
 
 Briggs's Elements of Plane Analytic Geometry i2mo, i 00 
 
 Compton's Manual of logarithmic Computations i2mo, i so 
 
 DaTis's Introduction to the Logic of Algebra Svo, i 50 
 
 * Dickson's College Algebra Large i2mo, i so 
 
 * Answers to Dickson's College Algebra Svo, paper, 25 
 
 * Introduction to the Theory of Algebraic Equations Large izmo, i 25 
 
 Halsted's Elements of Geometry Svo, i 75 
 
 Elementary Synthetic Geometry 8vo> i 50 
 
 Rational Geometry i2mo, 
 
 * Johnson's (J. B.) Three-place Logarithmic Tables; Vest-pocket size, .paper, 15 
 
 100 copies for 5 00 
 
 * Mounted on heavy cardboard, S X 10 inches, 25 
 
 10 copies for 2 00 
 
 Johnson's (W. W.) Elementary Treatise on Differential Calculus. . .Small Svo, 3 00 
 
 Johnson's (W. W.) Elementary Treatise on the Integral Calculus. .Small Svo, i so 
 
 Johnson's (W. W.) Curve Tracing in Cartesian Co-ordinates i2mo, i 00 
 
 Johnson's (W. W.) Treatise on Ordinary and Partial Differential Equations. 
 
 Small Svo, 3 50 
 
 Johnson's (W. W.) Theory of Errors and the Method of Least Squares. . i2mo, 1 50 
 
 * Johnson's (W. W.) Theoretical Mechanics i2mo, 3 00 
 
 Laplace's Philosophical Essay on Probabilities. (Truscott and Emory.) ijmo, 2 00 
 
 * Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other 
 
 Tables Svo, 3 00 
 
 Trigonometry and Tables published separately Each, 2 00 
 
 * Ludlow's Logarithmic and Trigonometric Tables Svo, i 00 
 
 Haurer's Technical Mechanics Svo, 4 00 
 
 Herriman and Woodward's Higher Mathematics Svo, 5 00 
 
 Herrimon's Method of Least Squares Svo, 2 00 
 
 Rice and Johnson's Elementary Treatise on the Differential Calculus. Sm., Svo, 3 00 
 
 Differential and Integral Calculus. 3 vols, in one Small Svo, 2 50 
 
 Wood's Elements of Co-ordinate Geometry Svo, 2 00 
 
 Trigonometry: Analytical, Plane, and Spherical i2mo, i 00 
 
 MECHAHICAL EKGIITEERIIIG. 
 MATERIALS OF ENGESTEERIHG, STEAM-ENGINES AND BOILERS. 
 
 Bacon's Forge Practice i2mo, .1 so 
 
 Baldwin's Steam Heating for Buildings i2mo, 2 50 
 
 Borr's Kinematics of Machinery Svo, 2 so 
 
 * Bartletf s Mechanical Drawing Svo, 3 00 
 
 * " " " Abridged Ed Svo, i 50 
 
 Benjamin's Wrinkles and Recipes i2mo, 2 00 
 
 Carpenter's Experimental Engineering Svo, 6 00 
 
 Heating and Ventilating Buildings Svo, 4 00 
 
 Cory's Smoke Suppression in Plants using Bituminous CoaL (7n prep- 
 anUioTu) 
 
 Clerk's Gas and Oil Engine SmaU Svo, 4 00 
 
 Coolidge's Manual of Drawing 8vo, paper, i 00 
 
 Coolidge and Freeman's Elements of General Drafting for Mechanical En- 
 gineers Oblong 4to, 2 50 
 
 11 
 
Cromwell's Treatise on Toothed Gearing X2mo i 50 
 
 Treatise on Belts and Pulleys i2mo. i so 
 
 Durley's Kinematics of Machines 8to, 4 00 
 
 Flather's Dynamometers and the Ueasnrement of Power lamo, i oo 
 
 Rope Driving i2mo, 2 00 
 
 Sill's Gas and Fuel Analysis for Engineers i2mo, i 25 
 
 Hall's Car Lubrication i2mo, i 00 
 
 Bering's Ready Reference Tables (Conversion Factors) i6mo, morocco, 2 so 
 
 Huiton's The Gas Engine 8vo, 
 
 Jamison's Mechanical Drawing .8vo, 
 
 Jones's Machine Design: 
 
 Part I. — Kinematics of Machinery 8vo, 
 
 Part IL — Form, Strength, and Proportions of Parts 8vo, 
 
 Kent's Mechanical Engineer's Pocket-book i6mo, morocco, 
 
 Ken's Power and Power Transmission 8vo, 
 
 Leonard's Machine Shops. Tools, and Methods. (In press.) 
 
 MacCord's Kinematics; or. Practical Mechanism 8vo, 
 
 Mechanical Drawing 4to. 
 
 Velocity Diagrams 8vo, 
 
 Hahan's Industrial Drawing. (Thompson.) 8vo, 
 
 Poole's Calorific Power of Fuels 8vo. 
 
 Reid's Course in Mechanical Drawing 8vo. 
 
 Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 
 
 Richards's Compressed Air X2mo, 
 
 Robinson's Principles of Mechanism 8vo, 
 
 Schwamb and Merrill's Elements of Mechanfsm 8vo, 
 
 Smith's Press-working of Metals 8vo, 
 
 Thurston's Treatise on Friction and Lost Work in Machinery and Hill 
 Work 8vo, 
 
 Animal as a Machine and Prime Motor, and the Laws of Energetic*. i2mo, 
 
 Warren's Elements of Machine Constructior and Drawing 8ro, 
 
 Weisbach's Kinematics and the Power of Transmission. Herrmann — 
 
 Klein.) 8vo, 
 
 Machinery of Transmission and Governors. (Herrmann — Klein.). .8vo 
 
 Hydraulics and Hydraulic Motors. (Du Bois.) Svo, 
 
 Wolff's Windmill as a Prime Mover gvo. 
 
 Wood's Turbines , . . . ,8vo. 2 50 
 
 MATERIALS OF ENGINEERING. 
 
 Bovey's Strength of Materials and Theory of Structures Svo 7 so 
 
 Burr's Elasticity and Resistance of the Materials of Engineering. 6th Edition 
 
 Reset Svo 
 
 Church's Mechanics of Engineering Svo. 
 
 Johnson's Materials of Construction Large Svo, 
 
 Keep's Cast Iron Svo, 
 
 Lanza's Applied Mechanics Svo. 
 
 Hartens's Handbook on Testing Materials. (Henning.) Svo, 
 
 Uerriman's Tezi-book on the Mechanics of Materials Svo, 
 
 Strength of Materials i2mo, 
 
 Metcalf s SteeL A Manual for Steel-users i2mo 2 00 
 
 Sabin's Industrial and Artistic Technology of Paints and Varnish Svo, 3 00 
 
 Smith's Materials of Machines i2mo, i 00 
 
 Thurston's Materials of Engineering 3 vols , Svo. 8 00 
 
 Part n. — Iron and Steel Svo, 3 50 
 
 Part III. — A Treatise on Brasses, Bronzes, and Other Alloys and their 
 Constituents Svo 
 
 TW-bsok of the MatarUla o( Cooatructlon. gve. 
 
 s 
 
 00 
 
 2 
 
 50 
 
 I 
 
 SO 
 
 3 
 
 00 
 
 5 
 
 00 
 
 2 
 
 00 
 
 S 
 
 00 
 
 4 
 
 00 
 
 I 
 
 so 
 
 3 
 
 SO 
 
 3 
 
 00 
 
 2 
 
 00 
 
 3 
 
 00 
 
 X 
 
 so 
 
 3 
 
 00 
 
 3 
 
 00 
 
 3 
 
 00 
 
 3 
 
 00 
 
 I 
 
 00 
 
 7 
 
 50 
 
 S 
 
 00 
 
 5 
 
 00 
 
 S 
 
 00 
 
 3 
 
 00 
 
 7 
 
 SO 
 
 6 
 
 00 
 
 6 
 
 00 
 
 2 
 
 SO 
 
 7 
 
 SO 
 
 7 
 
 SO 
 
 4 
 
 00 
 
 I 
 
 00 
 
 2 50 
 
 g 00 
 
Wood's (De V.) Treatise on the Resistance of Materials and an Appt..dijt on 
 
 the Preservation of Timber 8vo, i oo 
 
 Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00 
 
 Wood's (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. 
 
 8vo, 4 00 
 
 STEAU-ERGmES AND BOILERS. 
 
 Cunot's Reflections on the Motive Power of Heft. (Thiimton.) 12110, i 50 
 
 Dawson's "Engineering" and Electric Traction Pocket-book. .i6moi mor., 5 00 
 
 Ford's Boiler Making for Boiler Makers iSmo, i 00 
 
 Boss's Locomotive SpKiss 8vo, 2 00 
 
 Hemenway's Indicator Practice and Steam-encine Economy lamo, 2 00 
 
 Button's Mechanical Engioseriiie of Power Plants 8vo, s 00 
 
 Heat and Heat-engines 8vo. s 00 
 
 Kent's Steam-boiler Economy 8vo, 4 00 
 
 Kneais's Practice and Theory of the Injector 8vo, i 50 
 
 HacCord's Slide-valves 8vo, 2 00 
 
 Meyer's Modem Locomotive Construction 4to. 10 00 
 
 Peabody's Manual of the Steam-engine Indicator i2mo, i 50 
 
 Tables of the Properties of Saturated Steam and Other Vapors Svo, i 00 
 
 Thermodynamics of the Steam-engine and Other Heat-engines Svo, 5 00 
 
 Valve-gears for Steam-engines Svo, 2 50 
 
 Peabody and Miller's Steam-boilers Svo, 4 00 
 
 Pray's Twenty Tears with the Indicator Large Svo, 2 50 
 
 Puplu's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. 
 
 (Osterberg.) lamo, i 25 
 
 Reagan's Locomotives : Simplei Compound, and Electric i2mo, ^ So 
 
 Rontgen's Principles of Thermodynamics. (Du Bois.) Svo, 5 00 
 
 Sinclair's Locomotive Engine Running and Management 1 2mo, 200 
 
 Smart's Handbook of Engineering Laboratory Practice i2mo, 2 50 
 
 Snow's Steam-boiler Practice Svo, 3 00 
 
 Spangler's Valve-gears Svo, 2 50 
 
 Notes on Thermodynamics i2mo, i 00 
 
 Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 00 
 
 Thurston's Handy Tables Svo, i so 
 
 Maniini of the Steam-engine 2 vols. Svo, 10 00 
 
 Part L — History. Structuce, and Theory Svo, 6 00 
 
 Part H. — Design, Construction, and Operation Svo, 6 00 
 
 Handbook of Engine and Boiler Trials, and the Use of the Indicator and 
 
 the Prony Brake Svo, s 00 
 
 Stationary Steam-engines Svo, 2 s" 
 
 Steam-boiler Explosions in Theory and in Practice i2mo, i 50 
 
 Manualof Steam-boilers, Their Designs, Construction.and Operation. Svo, 5 00 
 
 Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) Svo, s 00 
 
 nrhitham's Steam-engine Design Svo, s 00 
 
 Wilson's Treatise on Steam-boilers. (Flather,) i6mo, 2 50 
 
 Wood's Thermodynamics Heat Motors, and Refrigerating Machines Svo, 4 00 
 
 MECHAHICS AND MACHINERY. 
 
 Barr's Kinematics of Machinery Svo, 2 so 
 
 Bovey's Strength of Materials and Theory of Structures Svo, 7 50 
 
 Chase's The Art of Pattern-making i2mo, 2 50 
 
 ChordaL— Extracts from Letters "mo, 2 00 
 
 Church's Mechanics of Engineering Svo, 6 00 
 
 13 
 
Church's Rote* and Examples in Mechanics Sro, 2 oo 
 
 Compton's First Lessons in Metal-worlcini; i2mo, i so 
 
 Compton and De Groodf s The Speed Lathe izmo, i so 
 
 Cromwell's Tieatise on Toothed Gearing i2mo, i 50 
 
 Treatise on Belts and Pulleys i2mo, i 50 
 
 Dana's Text-book of Elementary Mechanics for the Use of Colleges and 
 
 Schools i2mo, I so 
 
 Dingey's Machinery Pattern Making i2mo, 2 00 
 
 Dredge's Record of the Transportation Exhibits Building of the World's 
 
 Columbian Exposition of 1893 4to half morocco, 5 oo 
 
 Du Boil's Elementary Principles of Mechanics: 
 
 VoL L — Kinematics 8vo, 3 So 
 
 Vol. n. — Statics 8to, 4 00 
 
 Vol. m. — Kinetics 8vo, 3 so 
 
 Mechanics of Engineering. Vol. I Small 4to, 7 50 
 
 VoL n. Small 4to, 10 00 
 
 Durley's Kinematics of Machines 8vo, 4 00 
 
 Fitzgerald's Boston Machinist i6mo, i 00 
 
 Flather's Dynamometers, and the Measurement of Power. lamo, 3 00 
 
 Rope Driving i2mo, 2 00 
 
 Goss's Locomotive Sparks 8vo, 2 00 
 
 Hall's Car Lubrication i2mo, i 00 
 
 Holly's Art of Saw Filing i8mo, 75 
 
 * Johnson's ("W. W.) Theoretical Mechanics izmo, 3 00 
 
 Johnson's (L. J.) Statics by Graphic and Algebraic Methods 8to, 2 00 
 
 Jones's Machine Design : 
 
 Part I. — Kinematics of Machinery 8to, i so 
 
 Part n. — Form, Strength, and Proportions of Parts 8vo, 3 00 
 
 Kerr's Power and Power Transmission 8vo, 2 00 
 
 Lanza's AppUed Mechanics 8to, 7 50 
 
 Leonard s Machine Shops, Tools, and Methods, (/n press.) 
 
 MacCord's Kinematics; or. Practical Mechanism 8vo, 5 00 
 
 Velocity Diagrams 8vo, i so 
 
 Maorer's Technical Mechanics Bvo, 4 00 
 
 Merriman's Text-book on the Mechanics of Materials 8vo, 4 00 
 
 * Mlchie'i Elements of Analytical Mechanics Sn>, 4 00 
 
 Reagan's Locomotives: Simple, Compound, and Electric lamo, 2 so 
 
 Reid's Course in Mechanical Drawing 8vo, 2 00 
 
 Text-book of Mechanical Drawing and Elementary Machine Design. .8vo, 3 00 
 
 Richards's Compressed Air izmo, i 50 
 
 Robinson's Principles of Mechanism 8vo, 3 00 
 
 Ryan, Rorris, and Hoxie's Electrical Machinery. VoL 1 8vo, 2 50 
 
 Schwamb and Merrill's Elements of Mechanism : 8vo, 3 00 
 
 Sinclair's Locomotive-engine Running and Management i2mo, 2 00 
 
 Smith's Press-working of Metals 8vo, 3 00 
 
 Materials of Machines xamo, x 00 
 
 Spangler, Greene, and Marshall's Elements of Steam-engineeiing 8vo, 3 00 
 
 Thurston's Treatise on Friction and Lost Work in Machinery and Mill 
 
 Work 8vo, 3 00 
 
 Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, i 00 
 
 Warren's Elements of Machine Construction and Drawing 8vo, 7 so 
 
 Weisbach's Kinematics and the Power of Transmission. (Herrmann — 
 
 Klein.) 8vo, s 00 
 
 Machinery of Transmission and Governors. (Herrmann— Klein.). 8vo, 5 00 
 
 Wood's Elements of Analytical Mechanics Svo, 3 00 
 
 Principles of Elementary Mechanics izmo, i 25 
 
 Turbines Svo, 2 50 
 
 The World's Columbian Exposition of i8g3 > 4to, i 00 
 
 14 
 
3 SO 
 2 so 
 
 METALLURGY. 
 Egleston's MetallurEy of SilTer, Gold, and Menurj: 
 
 VoL 1.— Silver 8vo, 7 50 
 
 Vol. n. — Gold and Mercury 8vo, 7 so 
 
 •• Iles's Lead-smelting. (Postage cents additionaL) i2mo, 2 so 
 
 Keep's Cast Iron 8vo, 2 so 
 
 Kunhardt's Practice of Ore Dressing in Europe 8vo, i 50 
 
 Le Chatelier's High-temperature Measurements. (Boudouard — Burgess.). lamo, 3 00 
 
 Hetcalf's Steel. A Manual for Steel-users i2mo, 2 00 
 
 Smith's Materials of Machines lamo, i 00 
 
 Thurston's Materials of Engineering. In Three Parts 8vo, 8 00 
 
 Part n. — Iron and Steel 8vo, 
 
 Part HL — A Treatise on Brasses, Bronzes, and Other Alloys and their 
 
 Constituents 8vo, 
 
 Olke'i Modem Electrolytic Copper Refining 8vo, 
 
 MIKERAL0G7. 
 Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 
 
 Boyd's Resources of Southwest Virginia 8vo. 
 
 Map of Southwest Virginia Pocket-book form, 
 
 Bmsh'f Manual of DeterminatiTe Mineralogy. (Penfield.) 8vo, 
 
 Cbester'g Catalogue of Minerals 8vo, paper. 
 
 Cloth, 
 
 Dictionary of the Names of Minerals 8vo, 
 
 Dana's System of Mineralogy. Large 8to, half leather. 
 
 First Appendix to Dana's Bew "System of Mineralogy.'!.... Large 8to, 
 
 Text-book of Mineralogy 8to, 
 
 Minerals and How to Study Them lamo. 
 
 Catalogue of American Localities of Minerals Large 8to, 
 
 Mannal of Mineralogy and Petrography lamo, 
 
 Douglas's Untechnical Addresses on Technical Subjects i2mo, 
 
 Eakk's Mineral Tables. 8to, 
 
 Egleston's Catalogue of Minerals and Synonyms 8vo, 
 
 Hussak's The Determination of Rock-forming Minerals. (Smith.) Small 8vo, 2 00 
 Merrill's Bon-metallic Minerals; Their Occurrence and Uses. Svo, 4 00 
 
 * Penfield's Botes on Determinative Mineralogy and Record of Mineral Tests. 
 
 Svo, paper, o 50 
 Rosenbnsch's Microscopical Physiography of the Rock-making Minerals. 
 
 (Iddings.) Svo, 5 00 
 
 • Tillman's Text-book of Important Minerals and Docks Svo, 2 00 
 
 Williams's Manual of Lithology Svo, 3 00 
 
 Mmrno. 
 
 Beard's Ventilation of Mines lamo, 2 50 
 
 Boyd's Resources of Southwest Virginia Svo, 3 00 
 
 Map of Southwest Virginia Pocket-book form, 2 00 
 
 Douglas's Untechnical Addresses on Technical Subjects i2mo, 1 00 
 
 • Drinker's Tunneling, Explosive Compounds, and Rock Drills. 
 
 4to, half morocco, 25 00 
 
 Eissler's Modem High Explosives Svo, 4 oo 
 
 Fowler's Sewage Works Analyses ismo, 2 00 
 
 Goodyear's Coal-mines of the Western (^ast of the United States i2mo, 2 50 
 
 Ihlseng's Manniii of Mining Svo, 4 00 
 
 ** Ues's Lead-smelting. (Postage pc. additional.) izmo, 2 50 
 
 Kunhardt's Practice of Ore Dressing in Europe Svo, i 50 
 
 O'Driscoll's Botes on the Treatment of Gold Ores Svo, 2 00 
 
 * Walke's Lectures on Explosives Svo, 4 00 
 
 Wilson's Cyanide Processes zsmo, i so 
 
 Chlorination Process ismo, i 5c 
 
 15 
 
 2 
 
 SO 
 
 3 
 
 00 
 
 2 
 
 00 
 
 4 
 
 00 
 
 I 
 
 00 
 
 I 
 
 2S 
 
 3 
 
 SO 
 
 12 
 
 so 
 
 Z 
 
 00 
 
 4 
 
 00 
 
 1 
 
 so 
 
 I 
 
 00 
 
 2 
 
 00 
 
 I 
 
 00 
 
 I 
 
 2S 
 
 2 
 
 so 
 
Wilson's Hydraulic and Placer Mining l2mo, 2 oo 
 
 Treatise on Practical and Theoretical Mine Ventilation i2mo, i 25 
 
 SANITARY SCIENCE. 
 
 FolwelPs Sewerage. (Designing, Construction, and Maintenance.) Svo, 3 00 
 
 Water-supply Engineering 8vo, 4 00 
 
 Fuertes's Water and Public Health izmo, i 50 
 
 Water-filtration Works i2mo, 2 50 
 
 Gerhard's Guide to Sanitary House-inspection i6mo, x 00 
 
 Goodrich's Economical Disposal of Town's Refuse Demy 8to, 3 50 
 
 Hazen's Filtiation of Public Water-supplies 8vo» 3 00 
 
 Leach's The Inspection and Analysis of Food with Special Reference to State 
 
 ControL 8vo, 7 50 
 
 Mason's Water-supply. (Considered Principally from a Sanitary Stand- 
 point.) 3d Edition, Rewritten 8vo, 4 00 
 
 Examination of Water. (Chemical and BacteriologicaL) i2mo, z 25 
 
 Merriman's Elements of Sanitary Engineering 8vo, 2 00 
 
 Ogden's Sewer Design x2mo, 2 00 
 
 Prescott and Winslow's Elements of Water Bacteriology, with Special Reference 
 
 to Sanitary Water Analysis i2mo, z 25 
 
 * Price's Handbook on Sanitation Z2mo, i 50 
 
 Richards's Cost of Food. A Study in Dietaries X2mo, x 00 
 
 Cost of Living as Modified by Sanitary Science Z2mo, z 00 
 
 Richards and Woodman's Air, Water, and Food from a Sanitary Stand- 
 point 8vo, 2 00 
 
 * Richards and Williams's The Dietary Computer 8vo, z 50 
 
 Rideal's Sewage and Bacterial Purification of Sewage 8to, 3 50 
 
 Turneaure and Russell's Public Water-supplies 8vo, 5 00 
 
 Von Behring's Suppression of Tuberculosis. (Bolduan.) x2mo, z 00 
 
 Whipple's Microscopy of Drinking-water 8to, 3 50 
 
 Woodhull's Notes and Military Hygiene z6mo, x 50 
 
 MISCELLANEOUS. 
 
 Emmons's Geological Guide-book of the Rocky Mountain Excursion of the 
 
 International Congress of Geologists Large 8to, 
 
 Ferrel's Popular Treatise on the Winds 8vo, 
 
 Haines's American Railway Management Z2mo 
 
 Mott's Composition, Digestibility, and Nutritive Value of Food. Mounted chart. 
 
 Fallacy of the Present Theory of Sound z6mo, 
 
 Ricketts's History of Rensselaer Polytechnic Institute, Z824-2894. Sznail 8vo, 
 
 Rostoski's Serum Diagnosis. (Bolduan.) i2mo, 
 
 Rotherham's Emphasized New Testament Large 8vo, 
 
 Steel's Treatise on the Diseases of the Dog 8vo, 
 
 Totten's Important Question in Metrology 8vo, 
 
 The World's Columbian Exposition of 1893 4to, z 00 
 
 Von Behring's Suppression of Tuberculosis. (Bolduan.) i2mo, x 00 
 
 Worcester and Atkinson. Small Hospitals, Establishment and Maintenance- 
 and Suggestions for Hospital Architecture, with Plans for a Small 
 Hospital i2mo, z 25 
 
 HEBREW AND CHALDEE TEXT-BOOKS. 
 
 Green's Grammar of the Hebrew Language 8vo, 3 00 
 
 Elementary Hebrew Grammar i2mo. 1 25 
 
 Hebrew Chrestomathy 8vo, 2 00 
 
 Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. 
 
 f Tregelles.) Small 4to, half morocco, s 00 
 
 Letteris'^ Hebrew Bible Svo, 2 25 
 
 16 
 
 I 
 
 SO 
 
 4 
 
 00 
 
 2 
 
 SO 
 
 I 
 
 SS 
 
 I 
 
 00 
 
 3 
 
 00 
 
 I 
 
 00 
 
 2 
 
 00 
 
 3 
 
 SO 
 
 2 
 
 50 
 
borneii University Libraries 
 APR 1 8 1991 
 
 MATHEMATICS UBRAW