SOURCE BOOKS IN THE HISTORY OF THE SCIENCES
Gregory D. Walcott • General Editor
A SOURCE BOOK IN MATHEMATICS
SOURCE BOOKS IN THE HISTORY OF THE SCIENCES
Gregory D. Walcott General Editor
7^0 w ready A SOURCE BOOK IN ASTRONOMY
Harlow Shapley • Dnector, Harvard Observatory Harvard University
AND
Helen E. Howarth • Harvard Observatory
A SOURCE BOOK IN MATHEMATICS
David Eugene Smith • Columbia University
OTHER volumes TO BE
announced later
Endorted bj the American Philoiophicjl Aisocuition, tht Am^rxan Auoc\Atum for Ac Advancement of Science, and the Htsicny of Science Society. Alto by the Ameucan Anihropological Association, th« Maihc matical Assoaatum of America, the American Miiihan<jl:c4 1 Society and the American Astronomical Society in their respective fields.
A SOURCE BOOK
in
MATHEMATICS
By
DAVID EUGENE SMITH, ph.d., ll.d.
Professor Emeritus m Teachers College, Columbia University, ?-{ew Tor\ City
FIRST EDITION
McGRAW'HILL BOOK COMPANY, Inc. NEW YORK: 370 SEVENTH AVENUE
LONDON: 6 &> 8 BOUVERIE ST.: E. C. 4 1929
3
CkiPYRIGHT, 1929, BY THE
McGraW'Hill Book Company, Inc.
Printed in the United States of America
THB MAFLE PRESS CXJMPANY, YORK, PA.
SOURCE BOOKS IN THE HISTORY OF THE SCIENCES
General Editor's Preface
THIS series of Source Books aims to present the most significant passages from the works of the most important contributors to the major sciences during the last three or four centuries. So much material has accumulated that a demand for selected sources has arisen in several fields. Source books in philosophy have been in use for nearly a quarter of a century, and history, economics, ethics, and sociology utilize carefully selected source material. Recently, too, such works have appeared in the fields of psychology and eugenics. It is the purpose of this series, there- fore, to deal in a similar way with the leading physical and biologi- cal sciences.
The general plan is for each volume to present a treatment of a particular science with as much finality of scholarship as possible from the Renaissance to the end of the nineteenth century. In all, it is expected that the series will consist of eight or ten vol- umes, which will appear as rapidly as may be consistent with sound scholarship.
In June, 1924, the General Editor began to organize the follow- ing Advisory Board:
Pbilosopby
Philosophy
Philosophy
Philosophy
Philosophy
Philosophy
Philosophy
Physics
Chemistry
Geology
Zoology
Astronomy
Mathematics
Anthropology
* No longer chairman of a committee, because of the pressure of other duties, but remains on the Board in an advisory capacity.
Harold C. Brown Morris R. Cohen Arthur O. Lovejoy George H. Mead William P. Montague WiLMON H. Sheldon Edward G. Spaulding Joseph S. Ames* Frederick Barry R. T. Chamberlin* Edwin G. Conklin Harlow Shapley David Eugene Smith Alfred M. Tozzer
Stanford University College of the City of N. Y. Johns Hopkins University University of Chicago Columbia University Yale University Princeton University Johns Hopkins University Columbia University University of Chicago Princeton University Harvard University Columbia University Harvard University
4-S»\S
VIII EDITOR'S PREFACE
Each of the scientists on this board, in addition to acting in a general advisory capacity, is chairman of a committee of four or five men, whose business it is to make a survey of their special field and to determine the number of volumes required and the contents of eacli volume.
In December, 1925, the General Editor presented the project to the Eastern Division of the American Philosophical Association. After some discussion by the Executive Committee, it was approved and the philosophers of the board, with the General Editor as chairman, were appointed a committee to have charge of it. In November, 1927, the Carnegie Corporation of New York granted $10,000 to the American Philosophical Association as a revolving fund to help finance the series. In December, 1927, the American Association for the Advancement of Science approved the project, and appointed the General Editor and Professors Edwin G. Conklin and Harlow Shapley a committee to represent that Association in cooperation with the Advisory Board. In February, 1928, the History of Science Society officially endorsed the enterprise. Endorsements have also been given by the Ameri- can Anthropological Association, the Mathematical Association of America, the American Mathematical Society, and the American Astronomical Society w'ithin their respective fields.
The General Editor wishes to thank the members of the Advisory Board for their assistance in launching this undertaking; Dr. J. McKeen Cattell for helpful advice in the early days of the project and later; Dr. William S. Learned for many valuable suggestions; the several societies and associations that have given their endorse- ments; and the Carnegie Corporation for the necessary initial financial assistance.
Gregory D. Walcott.
Long Island University,
Brooklyn, N. Y.
December, 1928.
A SOURCE BOOK IN MATHEMATICS
Author^ s Preface
The purpose of a source book is to supply teachers and students with a selection of excerpts from the works of the makers of the subject considered. The purpose of supplying such excerpts is to stimulate the study of the various branches of this subject — in the present case, the subject of mathematics. By knowing the beginnings of these branches, the reader is encouraged to follow the growth of the science, to see how it has developed, to appre- ciate more clearly its present status, and thus to see its future possibilities.
It need hardly be said that the preparation of a source book has many difficulties. In this particular case, one of these lies in the fact that the general plan allows for no sources before the advent of printing or after the close of the nineteenth century. On the one hand, this eliminates most of mathematics before the invention of the calculus and modern geometry; while on the other hand, it excludes all recent activities in this field. The latter fact is not of great consequence for the large majority of readers, but the former is more serious for all who seek the sources of elementary mathematics. It is to be hoped that the success of the series will permit of a volume devoted to this important phase of the development of the science.
In the selection of material in the four and a half centuries closing with the year 1900, it is desirable to touch upon a wide range of interests. In no other way can any source book be made to meet the needs, the interests, and the tastes of a wide range of readers. To make selections from the field, however, is to neglect many more sources than can possibly be selected. It would be an easy thing for anyone to name a hundred excerpts that he would wish to see, and to eliminate selections in which he has no
X AUTHOR'S PREFACE
special interest. Some may naturally seek for more light on our symbols, but Professor Cajori's recent work furnishes this with a Siitisfactory approach to completeness. Others may wish for a worthy treatment of algebraic equations, but Matthiessen's Grundziige contains such a wealth of material as to render the undertaking unnecessary. The extensive field of number theory will appeal to many readers, but the monumental work of Professor Dickson, while not a source book in the ordinary sense of the term, satisfies most of the needs in this respect. Consideration must always be given to the demands of readers, and naturally these demands change as the literature of the history of mathe- matics becomes more extensive. Furthermore, the possibility of finding source material that is stated succinctly enough for purposes of quotation has to be considered, and also that of finding material that is not so ultra-technical as to serve no useful purpose for any considerable number of readers. Such are a few of the many difficulties which will naturally occur to everyone and which will explain some of the reasons which compel all source books to be matters of legitimate compromise.
Although no single department of "the science venerable" can or should be distinct from any other, and although the general trend is strongly in the direction of unity of both purpose and method, it will still serve to assist the reader if his attention is called to the rough classification set forth in the Contents.
The selections in the field of Number vary in content from the first steps in printed arithmetic, through the development of a few selected number systems, to the early phases of number theory. It seems proper, also, to consider the mechanics of com- putation in the early stages of the subject, extending the topic to include even as late a theory as nomography. There remains, of course, a large field that is untouched, but this is a necessary condition in each branch.
The field of Algebra is arbitrarily bounded. Part of the articles classified under Number might have been included here, but such questions of classification are of little moment in a work of this nature. In general the articles relate to equations, symbolism, and series, and include such topics as imaginary roots, the early methods of solving the cubic and biquadratic algebraic equations and numerical equations of higher degree, and the Fundamental Theorem of Algebra. Trigonometry, which is partly algebraic, has been considered briefly under Geometry. Probability, which
AUTHOR'S PREFACE xi
is even more algebraic, is treated by itself, and is given somewhat more space than would have been allowed were it not for the present interest in the subject in connection with statistics.
The field of Geometry is naturally concerned chiefly with the rise of the modern branches. The amount of available material is such that in some cases merely a single important theorem or statement of purpose has been all that could be included. The topics range from the contributions of such sixteenth-century writers as Fermat, Desargues, Pascal, and Descartes, to a few of those who, in the nineteenth century, revived the study of the subject and developed various forms of modern geometry.
The majority of the selections thus far mentioned have been as non-technical as possible. In the field of Probability, however, it has been found necessary to take a step beyond the elementary bounds if the selections are to serve the purposes of those who have a special interest in the subject.
The fields of the Calculus, Function Theory, Quaternions, and the general range of Mathematics belong to a region so extensive as to permit of relatively limited attention. It is essential that certain early sources of the Calculus should be considered, and that some attention should be given to such important advances as relate to the commutative law in Quaternions and Ausdehnungs- lehre, but most readers in such special branches as are now the subject of research in our universities will have at hand the material relating to the origins of their particular subjects. The limits of this work would not, in any case, permit of an extensive offering of extracts from such sources.
It should be stated that all the translations in this work have been contributed without other reward than the satisfaction of assisting students and teachers in knowing the sources of certain phases of mathematics. Like the editor and the advisory com- mittee, those who have prepared the articles have given their services gratuitously. Special mention should, however, be made of the unusual interest taken by a few who have devoted much time to assisting the editor and committee in the somewhat difficult labor of securing and assembling the material. Those to whom they are particularly indebted for assistance beyond the preparation of special articles are Professor Lao G. Simons, head of the department of mathematics in Hunter College, Professor Jekuthiel Ginsburg, of the Yeshiva College, Professor Vera Sanford of Western Reserve University, and Professor Helen M.
xii AUTHOR'S PREFACE
Walker, of Teachers College, Columbia University. To Professor Sanford special thanks are due for her generous sacrifice of time and effort in the reading of the proofs during the editor's prolonged absence abroad.
The advisory committee, consisting of Professors Raymond Clare Archibald of Brown University, Professor Florian Cajori of the University of California, and Professor Leonard Eugene Dickson of the University of Chicago, have all contributed of their time and knowledge in the selection of topics and in the securing of competent translators. Without their aid the labor of preparing this work would have been too great a burden to have been assumed by the editor.
In the text and the accompanying notes, the remarks of the translators, elucidating the text or supplying historical notes of value to the reader, are inclosed in brackets [ ]. To these con- tributors, also, are due slight variations in symbolism and in the spelling of proper names, it being felt that they should give the final decision in such relatively unimportant matters.
David Eugene Smith.
New York, September, 1929.
Contents
Page General Editor's Preface vii
Author's Preface ix
I. THE FIELD OF NUMBER
The First Printed Arithmetic. Treviso, 1478 1
Selection translaced from the Italian by David Eugene Smith Robert Recorde on "The Declaration of the Profit of Arith-
meticke" 13
Selected from The Ground of Artes, by David Eugene Smith Stevin on Decimal Fractions 20
Translated from the French by Vera Sanford Dedekind on Irrational Numbers 35
Translated from the German by Wooster Woodruff Beman. Selec- tion made and edited by Vera Sanford John Wallis on Imaginary Numbers 46
Selected and edited by David Eugene Smith Wessel on Complex Numbers 55
Translated from the Danish by Martin A. Nordgaard Pascal on the Arithmetic Triangle 67
Translated from the French by Anna Savitsky Bombelli and Cataldi on Continued Fractions 80
Translated from the Italian by Vera Sanford Bernoulli on "Bernoulli Numbers" 85
Translated from the Latin by Jekuthiel Ginsburg EuLER ON Every Integer as a Sum of Four Squares 91
Translated from the Latin by E. T. Bell
EuLER ON THE UsE OF € TO REPRESENT 2.718- • • 95
Selections translated from the Latin by Florian Cajori
HeRMITE on THE TRANSCENDENCE OF C 99
Translated from the French by Laura Guggenbiihl Gauss on the Congruence of Numbers 107
Translated from the Latin by Ralph G. Archibald Gauss on the Third Proof of the Law of Quadratic Reciprocity 112
Translated from the Latin by D. H. Lehmer KuMMER ON Ideal Numbers 119
Translated from the German by Thomas Freeman Cope Chebyshev (Tchebycheff) on the Totality of Primes 127
Translated from the French by J. D. Tamarkin Napier on the Table of Logarithms 149
Selected and edited by W. D. Cairns
xiii
XIV CONTENTS
Page
Delamain on the Slide Rule 156
Edited by Florian Cajori
OUCHTRED ON THE SlIDE RuLE 160
Edited by Florian Cajori Pascal on His Calculating Machine 165
Translated from the French by L. Leiand Locke Leibniz on His Calculating Machine 173
Translated from the Latin by Mark Kormes Napier on the Napier Rods 182
Translated from the Latin by Jekuthiel Ginsburg Galileo Galilei on the Proportional or Sector Compasses ... 186
Translated from the Italian by David Eugene Smith D'Oc.^GNE on No.mogr-^phy 192
Translated from the French by Ne\Tn C. Fisk
n. THE FIELD OF ALGEBRA
Carda.n on I.maginary Roots 201
Translated from the Latin by Vera Sanford Cardan on the Cubic Equation 203
Translated from the Latin by R. B. McCIenon Ferrari-Cardan on the Biquadr.\tic Equatio.n 207
Translated from the Latin by R. B. McCIenon, with additional
notes by Jekuthiel Ginsburg Fermat on the Equation a:" + y" = z" 213
Translated from the French by Vera Sanford Fermat on the So-called Pell Equation 214
Translated from the Latin by Edward E. Whitford John Wallis on General Expo-nen-ts 217
Translated from the Latin by Eva M. Sanford Wallis ant) Newton on the Bino.mial Theorem for Fractional and
Negative Exponen-ts 219
Selection from Wallis's Algebra, by David Eugene Smith Newton on the Binomial Theorem for Fractional and Negative
Exponents 224
Translated from the Latin by Eva M. Sanford Leibniz ant> the Bernoullis o.n the Polynomial Theorem .... 229
Translated from the Latin by Jekuthiel Ginsburg FIoRNER ON Numerical Higher Equations 232
Selected and edited by Margaret iMcGuire RoLLE ON the Location of Roots 253
Translated from the French by Florian Cajori Abel on the Quintic Equation 261
Translated from the French by W. H. Langdon, with notes by
O^'stein Ore Leibniz on Deter.mina.nts 267
Translated from the Latin by Thomas Freeman Cope Bernoulli. Verses on Intintte Series 271
Translated from the Latin bv Helen M. Walker
CONTENTS XV
Paob Bernoulli on the Theory of Combinations 272
Translated from the Latin by Mary M. Taylor Galois on Groups and Equations 278
Translated from the French by Louis Weisner Abel's Theorem on the Continuity of Functions Defined by Power
Series 286
Translated from the German by Albert A, Bennett Gauss on the Fundamental Theorem of Algebra 292
Translated from the Latin by C. Raymond Adams
IIL THE FIELD OF GEOMETRY
Desargues om Perspective Triangles 307
Translated from the French by Lao G. Simons Desargues on the 4-rayed Pencil 311
Translated from the French by Vera Sanford Poncelet on Projective Geometry 315
Translated from the French by Vera Sanford Peaucellier's Cell 324
Translated from the French by Jekuthiel Ginsburg Pascal, "Essay Pour Les Coniques" 326
Translated from the French by Frances Marguerite Clarke Brianchon's Theorem 331
Translated from the French by Nathan Altshiller-Court Brianchon and Poncelet on the Nine-point Circle Theorem . . 337
Translated from the French by Morris Miller Slotnick Feuerbach on the Theorem Which Bears His Name 339
Translated from the German by Roger A. Johnson The First Use of tt for the Circle Ratio 346
Selection made by David Eugene Smith from the original work Gauss on the Division of a Circle into n Equal Parts 348
Translated from the Latin by J. S. Turner Saccheri on Non-Euclidean Geometry 351
Translated from the Latin by Henry P. Manning
LoBACHEVSKY ON NoN-EuCLlDEAN GEOMETRY 360
Translated from the French by Henry P. Manning Bolyai ON NoN-EuCLiDEAN Geometry 371
Translated from the Latin by Henry P. Manning Fermat on Anal\tic Geometry 389
Translated from the French by Joseph Scidlin Descartes on Analytic Geometry 397
Translated from the French by David Eugene Smith and Marcia L.
Latham Pohlke's Theorem 403
Translated from the German by Arnold Emch Riemann on Surfaces ant) Analysis Situs 404
Translated from the German by James Singer Riemann on the Hypotheses Which Lie at the Foundations of
Geometry 411
Translated from the German by Henry S. White
xvi CONTENTS
Page MONGE ON THE PuRPOSE OF DESCRIPTIVE GeOMETRY 426
Translated from the French by Arnold Emch Regiomontanus on the Law of Sines for Spherical Triangles . . 427
Translated from the Latin by Eva M. Sanford Regiomontanus on the Relation of the Parts of a Triangle. , 432
Translated from the Latin by Vera Sanford PiTiscus ON the Laws of Sines and Cosines 434
Translated from the Latin by Jekuthiel Ginsburg PiTiscus on Burgi's Method of Trisecting an Arc 436
Translated from the Latin by Jekuthiel Ginsburg De Moivre's Formula 440
Translated from the Latin and from the French by Raymond Clare
Archibald Clavius on Prosthaphaeresis as Applied to Trigonometry. . . . 455
Translated from the Latin by Jekuthiel Ginsburg Clavius on Prosthaphaeresis 459
Translated from the Latin by Jekuthiel Ginsburg Gauss on Conformal Representation 463
Translated from the German by Herbert P. Evans Steiner on Quadratic Transformation between Two Spaces . . . 476
Translated from the German by Arnold Emch Cremona on Geometric Transformations of Plane Figures . . . 477
Translated from the Italian by E. Amelotti Lie's Memoir on a Class of Geometric Transformations .... 485
Translated from the Norwegian by Martin A. Nordgaard MoBius, Cayley, Cauchy, Sylvester, and Clifford on Geometry of
Four or More Dimensions 524
Note by Henry P. Manning MoBius ON Higher Space 525
Translated from the German by Henry P. Manning Cayley on Higher Space 527
Selected by Henry P. Manning Cauchy on Higher Space 530
Translated from the French by Henry P. Manning Sylvester on Higher Space 532
Selected by Henry P. Manning Cufford on Higher Space 540
Selected by Henry P. Manning
IV. THE FIELD OF PROBABILITY
Fermat and Pascal on Probability 546
Translated from the French by Vera Sanford De Moivre on the Law of Normal Probability 566
Selected and edited by Helen M. Walker Legendre on Least Squares 576
Translated from the French by Henry A. Ruger and Helen M.
Walker Chebyshev (Tchebycheff) on Mean Values 580
Translated from the French by Helen M. Walker
CONTENTS xvii
Page Laplace on the Probability of Errors in the Mean Results of a
Great Number OF Observations, Etc 588
Translated from the French by Julian L. C. A. Gys
V. FIELD OF THE CALCULUS, FUNCTIONS, QUATERNIONS
Cavalieri on an Approach to the Calculus 605
Translated from the Latin by Evelyn Walker Fermat ON Maxima AND Minima 610
Translated from the French by Vera Sanford Newton on Fluxions 613
Translated from the Latin by Evelyn Walker Leibniz on THE Calculus 619^
Translated from the Latin by Evelyn Walker ^
Berkeley's "Analyst" 627
Selected and edited by Florian Cajori Cauchy on Derivatives and Differentials 635
Translated from the French by Evelyn Walker EuLER ON Differential Equations OF THE Second Order 638
Translated from the Latin by Florian Cajori Bernoulli on the Brachistochrone Problem 644
Translated from the Latin by Lincoln La Paz Abel ON Integral Equations 656
Translated from the German by J. D. Tamarkin Bessel ON His Functions 663
Translated from the German by H. Bateman M6BIUS ON THE Barycentric Calculus 670
Translated from the German by J. P. Kormes Hamilton on Quaternions 677
Selected edited by Marguerite D. Darkow Grassmann on Ausdehnungslehre 684
Translated from the German by Mark Kormes
Index 697
SOURCE BOOK IN MATHEMATICS
I. FIELD OF NUMBER
The First Printed Arithmetic
Treviso, Italy, 1478
(Translated from the Italian by Professor David Eugene Smith, Teachers College, Columbia University, New York City.)
Although it may justly be said that mere computation and its simple appli- cations in the lives of most people are not a part of the science of mathematics, it seems proper that, in a source book of this kind, some little attention should be given to its status in the early days of printing. For this reason, these extracts are selected from the first book on arithmetic to appear from the newly established presses of the Renaissance period. ^ The author of the work is unknown, and there is even some question as to the publisher, although he seems to have been one Manzolo or Manzolino. It is a source in the chronological rather than the material sense, since the matter which it con- tains had apparently but little influence up>on the other early writers on arithmetic. The work is in the Venetian dialect and is exceedingly rare.^ The copy from which this translation was made is in the library of George A. Plimpton of New York City. As with many other incunabula, the book has no title. It simply begins with the words, Incommincia vna practica molto bona et vtilez a ciascbaduno cbi vuole vxare larte dela mercbadantia. chiamata vulgarmente larte de labbacbo. It was published at Treviso, a city not far to the north of Venice, and the colophon has the words "At Treviso, on the 10th day of December, 1478."
Here beginneth a Practica, very helpful to all who have to do with that commercial art commonly known as the abacus.
I have often been asked by certain youths in whom I have much interest, and who look forward to mercantile pursuits, to put into writing the fundamental principles of arithmetic, commonly
1 For the most part, these selections are taken from an article by this translator which appeared in Isis, Vol. VI (3), pp. 311-331, 1924, and are here published by permission of the editor. For a more extended account of the book, the reader is referred to this periodical.
- A critical study of it from the bibliographical standpoint was made by Prince Boncom- pagni in the Alti delC Accademia Pontificia de' Nuovi Lined, tomo XVI, 1862-1863.
1
SOURCE BOOK IN MATHEMATICS
called the abacus. Therefore, being impelled by my affection for them, and by the value of the subject, I have to the best of my small ability undertaken to satisfy them in some slight degree, to the end that their laudable desires may bear useful fruit. There- fore in the name of God I take for my subject this work in algorism, and proceed as follows:
All things which have existed since the beginning of time have owed their origin to number. Furthermore, such as now exist
are subject to its laws, and therefore in all domains of knowledge this Practica is necessary. To enter into the subject, the reader must first know the basis of our science. Number is a multitude brought together or assembled from sev- eral units, and always from two at least, as in the case of 2, which is the first and the smallest number. Unity is that by virtue of which any- thing is said to be one. Fur- thermore be it known that there are three kinds of num- bers, of which the first is called a simple number, the second an article, and the third a com- posite or mixed number. A simple number is one that con- tains no tens, and it is repre- sented by a single figure, like i, 2, 3, etc. An article is a number that is exactlydivisibleby ten, like iO, 20, 30 and similar numbers. A mixed number is one that exceeds ten but that cannot be divided by ten without a remainder, such as ii, i2, i3, etc. Furthermore be it known that there are five fundamental operations which must be understood in the Practica, viz., numeration, addition, subtraction, multiplication, and division. Of these we shall first treat of numeration, and then of the others in order.
Numeration is the representation of numbers by figures. This is done by means of ten letters or figures, as here shown, .i., .2.,
Jnremmincta wia ptacfia moltobo«a ttttiet a dafcbaduno cbi vuole v^are larrc «f la nirrcba^' dqniujbianiaM vul<;amiriite lane ice Ubbacfco*
1 "RfSoto ptu e p'u rolte ta a!cl:ur< ^ouani e mi tnolto ciUca/Timi : li I qiuii p:rtmdf uano a vourr rokr Uare la mercbadanriatcbe per loio Munoiemcfiaafff aflTadigarme o 'no ptioc}:0:'oe «arg[i in Tcntco qualcbc roiidairicta cerca lam vr arifmetrura:* bianiata rulgarmc nte" labbacbo.Unde to ronftrrtto prr amo: vi I020: rt eoadic^ad vnljtati cutt cbi prmndiiiio a qiiella:ri2 gondola picola intrlligcntia vcl injegno mio:^ drLberato fe non in tu(o:in parte tame fanffjre a lo:o.ario cbe I020 rirtiiofi cffidf m rale frutto re. ceuere pofTeano. Jn nome w uio adoncba : tojjLo per {Qinpto mio el cirto wr algoJifm* coft vuedo. t ' Vte quelle core:cbe va la p^ima o:igtne baiio babuto p:oduninfto:per ratone «e nuniero fono fta formade.^ co(T come fo/ nottanoM fircognofaidr.pcrone la cogiiinonc cc fuce le coff -.cjucfta p:actica e nrcrffatia . £ ^ cr tntrarnel^poficomioiprmio fapi lccto:c*.f be qn/ to fa al pjopofito noftro:Outnf ro e t na moltitu. ^ine congrrgata cncro inftmbiada tsa moltc xnU tade.eta! menova 70 vnitadr.come e.i.ti quale c lo p:jmo c me no:c nuniero:cbe fe truoiia.La v* ruffde e que lla cofa : t)a la qiK le Cf m cofa fj ntta tma.Sesodano fapr.cbe fe truona numcn vc tre nianicre.^1 prime fe cbiama numr.ro fimpbce.Ul (TO namno srdculo • •£! terjo fe cbuma mmcn
THE TREVISO ARITHMETIC 3
.3., A., .5., .6., .7., .8., .9., .0.. Of these the first figure, i, is not called a number but the source of number. The tenth figure, 0, is called cipher or "nulla," i. e., the figure of nothing, since by itself it has no value, although when joined with others it increases their value. Furthermore you should note that when you find a figure by itself its value cannot exceed nine, t. e., 9; and from that figure on, if you wish to express a number you must use at least two figures, thus: ten is expressed by iO, eleven by ii, and so on. And this can be understood from the following figures.^
'6
1 o -o
c
3
X
c
1
<:» O
CO
C
3 O
O
-o
"^
3 X
■o
c
03 (0
3 O
o
tn
C
CO
C
cS
(0
3 O
h i
en "V
c
3
i
2
CO
C
i
2 3
CO
'5 D
i
2 3 4 5 6 7 8 9 0 0 0
i
2
3
4
0
i
2
3
4
S
0
i
2
3
4
5
6
0
i
2
3
4
5
6
7
0
i
2
3
4
5
6
7
8
0
i
2
3
4
5
6
7
8
9
0
2
3
4
5
6
7
8
9
0
0
3
4
5
6
7
8
9
0
0
0
4
5
6
7
8
9
0
0
0
0
5
6
7
8
9
0
0
0
0
0
6
7
8
9
0
0
0
0
0
0
7
8
9
0
0
0
0
0
0
0
8
9
0
0
0
0
0
0
0
0
9000000000
* The figure 1 was not always in the early fonts of type, the letter "i" being then used in its stead.
4 SOURCE BOOK IN MATHEMATICS
To understand the figures it is necessary to have well in mind the following table :^
i times i makes i i times iO makes iO
i times 2 makes 2 2 times iO makes 20
i times 3 makes 3 3 times iO makes 30
i times 4 makes 4 4 times iO makes 40
i times 5 makes 5 5 times iO makes SO
i times 6 makes 6 6 times iO makes 60
i times 7 makes 7 7 times iO makes 70
i times 8 makes 8 8 times iO makes 80
i times 9 makes 9 9 times iO makes 90
i times 0 makes 0 0 times iO makes 0
And to understand the preceding table it is necessary to observe that the words written at the top^ give the names of the places occupied by the figures beneath. For example, below 'units' are the figures designating units, below 'tens' are the tens, below 'hundreds' are the hundreds, and so on. Hence if we take each figure by its own name, and multiply this by its place value, we shall have its true value. For instance, if we multiply i, which is beneath the word 'units,' by its place, — that is, by units, — we shall have *i time i gives i,' meaning that we have one unit. Again, if we take the 2 which is found in the same column, and multiply by its place, we shall have 'i time 2 gives 2,' meaning that we have two units, . . . and so on for the other figures found in this column . . . This rule applies to the various other figures, each of which is to be multiplied by its place value.
And this suffices for a statement concerning the 'act*^ of numeration.
Having now considered the first operation, viz. numeration, let us proceed to the other four, which are addition, subtraction, multiplication, and division. To differentiate between these operations it is well to note that each has a characteristic word, as follows :
^ [The tabic continues from "i times iOO makes iOO" to "0 times iOO makes 0."]
* [That is, the numeration table shown on p. 3.]
3 [The fundamental operations, which the author calls "acts" (atti) went by various names. The medieval Latin writers called them "species," a word that appears in The Crajte oj Nombryng, the oldest English manuscript on arithmetic, where the author speaks of " 7 spices or partes of this craft." This word, in one form or another, is also found in various languages. The Italians used both 'atti' and 'passioni.'J
THE TREVISO ARITHMETIC 5
Addition has the word andt
Subtraction has the word from. Multiplication has the word times, Division has the word in.
It should also be noticed that in taking two numbers, since at least two are necessary in each operation, there may be determined by these numbers any one of the above named operations. Furthermore each operation gives rise to a different number, with the exception that 2 times 2 gives the same result as 2 and 2, since each is 4. Taking, then, 3 and 9 we have:
Addition: 3 and 9 make i2
Subtraction : 3 from 9 leaves 6
MuItipHcation: 3 times 9 makes 27
Division: 3 in 9 gives 3
We thus see how the different operations with their distinctive words lead to different results.
In order to understand the second operation, addition, it is necessary to know that this is the union of several numbers, at least of two, in a single one, to the end that we may know the sum arising from this increase. It is also to be understood that, in the operation of adding, two numbers at least are necessary, namely the number to which we add the other, which should be the larger, and the number which is to be added, which should be the smaller. Thus we always add the smaller number to the larger, a more convenient plan than to follow the contrary order, although the latter is possible, the result being the same in either case. For example, if we add 2 to 8 the sum is iO, and the same result is obtained by adding 8 to 2. Therefore if we wish to add one number to another we write the larger one above and the smaller one below, placing the figures in convenient order, i. e., the units under units, tens under tens, hundreds under hundreds, etc. We always begin to add with the lowest order, which is of least value. Therefore if we wish to add 38 and 59 we write the numbers thus:
59
38
Sum 97
We then say, '8 and 9 make i7,* writing 7 in the column which
was added, and carrying the i (for when there are two figures in
one place we always write the one of the lower order and carry
the other to the next higher place). This i we now add to 3,
6 SOURCE BOOK IN MATHEMATICS
making 4, and this to the 5, making 9, which is written in the column from which it is derived. The two together make 97.
The proof of this work consists in subtracting either addend from the sum, the remainder being the other. Since subtraction proves addition, and addition proves subtraction, I leave the method of proof until the latter topic is studied, when the proof of each operation by the other will be understood.
Besides this proof there is another. If you wish to check the sum by casting out nines, add the units, paying no attention to 9 or 0, but always considering each as nothing. And whenever the sum exceeds 9, subtract 9, and consider the remainder as the sum. Then the number arising from the sum will equal the sum of the numbers arising from the addends. For example, suppose that you wish to prove the following sum:
.59.
.38. Sum .97. I 7
The excess of nines in 59 Is 5; 5 and 3 are 8; 8 and 8 are i6; subtract 9 and 7 remains. Write this after the sum, separated by a bar. The excess of nines in 97 is 7, and the excess of nines in 7 equals 7, since neither contains 9. In this way it is possible to prove the result of any addition of abstract numbers or of those having no reference to money, measure, or weight. I shall show you another plan of proof according to the nature of the case. If you have to add 816 and 1916,^ arrange the numbers as follows:
1916
816
Sum 2732
Since the sum of 6 and 6 is 12, write the 2 and carry the 1. Then add this 1 to that which follows to the left, saying, *1 and 1 are 2, and the other 1 makes 3.' Write this 3 In the proper place, and add 8 and 9. The sum of this 8 and 9 is 17, the 7 being written and the 1 carried to the other 1, making 2, which Is written in the proper place, the sum being now complete. If you wish to prove by 9 arrange the work thus:
1916 816 The sum 2732 | 5
> [From now on, the figure 1 will be used in the translation instead of the letter *i' which appears always in the original.]
THE TREVISO ARITHMETIC 7
You may now effect the proof by beginning with the upper number, saying '1 and 1 are 2, and 6 are 8, and 8 are 16. Subtract 9, and 7 remains. The 7 and 1 are 8, and 6 are 14. Subtract 9, and 5 remains,' which should be written after the sum, separated by a bar. Look now for the excess of nines in the sum : 2 and 7 are 9, the excess being 0; 3 and 2 are 5, so that the result is correct.^
Having now considered the second operation of the Practica of arithmetic, namely the operation of addition, the reader should give attention to the third, namely the operation of subtraction. Therefore I say that the operation of subtraction is nothing else than this: that of two numbers we are to find how much difference there is from the less to the greater, to the end that we may know this difference. For example, take 3 from 9 and there remains 6. It is necessary that there should be two numbers in subtraction, the number from which we subtract and the number which is subtracted from it.
The number from which the other is subtracted is written above, and the number which is subtracted below, in convenient order, viz., units under units and tens under tens, and so on. If we then wish to subtract one number of any order from another we shall find that the number from which we are to subtract is equal to it, or greater, or less. If it is equal, as in the case of 8 and 8, the remainder is 0, which 0 we write underneath in the proper column. If the number from which we subtract is greater, then take away the number of units in the smaller number, writing the remainder below, as in the case of 3 from 9, where the remainder is 6. If, however, the number is less, since we cannot take a greater number from a less one, take the complement of the larger number with respect to 10, and to this add the other, but with this condition: that you add one to the next left-hand figure. And be very careful that whenever you take a larger number from a smaller, using the complement, you remember the condition above mentioned. Take now an example: Subtract 348 from 452, arranging the work thus:
452 348
Remainder 104
First we have to take a greater number from a less, and then an equal from an equal, and third, a less from a greater. We proceed
^ (The addition of larger numbers and of the compound numbers like 916 lire 14 soldi plus 1945 lire 15 soldi are now considered.]
8 SOURCE BOOK IN MATHEMATICS
as follows: We cannot take 8 from 2, but 2 is the complement of 8 with respect to 10, and this we add to the other 2 which is above the 8, thus: 2 and 2 make 4, which we write beneath the 8 for the remainder. There is, however, this condition, that to the figure following the 8 (viz., to 4), we add 1, making it 5. Then 5 from 5, which is an equal, leaves 0, which 0 we write beneath.
Then 3 from 4, which is a less from a greater, is 1, which 1 we write under the 3, so that the remainder is 104.
If we wish to prove this result, add the number subtracted to the remainder, and the result will be the number from which we subtracted. We may arrange the work as follows:
452
348
104
452
Now add, 4 and 8 are 12; write 2 under the 4 and carry 1; then 1 and 4 are 5 ; write this 5 under the 0; then add 1 and 3, making 4, and write this 4 under the 1, and the work checks. Thus is found that which was promised you, as you can see^. . .
Having now explained the third operation, namely that of sub- traction, the reader should give attention to the fourth, namely that of multiplication. To understand this it is necessary to know that to multiply one number by itself or by another is to fmd from two given numbers a third number which contains one of these numbers as many times as there are units in the other. For example, 2 times 4 are 8, and 8 contains 4 as many times as there are units in 2, so that 8 contains 4 in itself twice. Also the 8 contains 2 as many times as there are units in 4, and 4 has in itself four units, so that 8 contains 2 four times. It should be well understood that in multiplication two numbers are necessary, namely the multiplying number and the number multiplied, and also that the multiplying number may itself be the number multiplied, and vice versa, the result being the same in both cases. Nevertheless usage and practice demand that the smaller number shall be taken as the multiplying number, and not the larger. Thus we should say, 2 times 4 makes 8, and not 4 times 2 makes 8,
' [The author now gives a further proof of subtraction by tlie casting out of nines, after which he devotes about six or seven pages to checks on subtraction and to the subtraction of lire, soldi, grossi, pizoli, and the like.]
THE TREVISO ARITHMETIC
although the results are the same. Now not to speak at too great length I say in brief, but sufficiently for the purposes of a Practica, that there are three methods of multiplication, viz., by the tables, cross multiplication, and the chess-board plan. These three methods I will explain to you as briefly as I am able. But before I give you a rule or any method, it is necessary that you commit to memory the following state-
Vtos\iopttOchetamKniichtTonoi\trin]Odi xiej
molnpbcare pn fcacbierorli qiuli faffa'ro nl ftiidi
o tiw.tnr ttendo U eftmpU fot fohmente m fojma*
come pojsi wdere qui fbtto
O: togfi w firelopiedntofcachiero.ioe.j 14.
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ments, without which no one can understand all of this oper- ation of multiplication^ . .
I have now given you to learn by heart all the statements needed in the Practica of arith- metic, without which no one is able to master the Art. We should not complain, however, at having to learn these things by heart in order to acquire readiness; for I assure you that these things which I have set forth are necessary to any one who would be proficient in this art, and no one can get along with less. Those facts which are to be learned besides these are valuable, but they are not necessary.
Having learned by heart all of the above facts, the pupil
may with zeal begin to multiply by the table. This operation arises when the multiplier is a simple number, and the number multiplied has at least two figures, but as many more as we wish. And that we may more easily understand this operation we shall call the first figure toward the right, units; the second toward the left, tens, and the third shall be called hundreds. This being under- stood, attend to the rule of working by the table, which is as follows: First multiply together the units of the multiplier and
' [The author now gives the multiplication table, omitting all duplications like 3X2 after 2X3 has been given, but extending for "those who are of scholarly tastes" the table to include multiples of 12, 20, 24, 32 and 36, as needed in the monetary systems used by merchants of the time.]
Bomma*
l.TTToZ
\o/\o /\0/\ 9\/<*\/s\/4r\i
I'j /VTA I A
y ? 4
UVtNl i\l4U IoNJo\|o\fi(i
|t\lo\[t\|Wt * 9 5
10 SOURCE BOOK IN MATHEMATICS
the number multiplied. If from this multiplication you get a simple number, write it under its proper place; if an article, write a 0 and reserve the tens to add to the product of the tens; but if a mixed number is found, write its units in the proper place, and save the tens to add to the product of the tens, proceeding in the same way with all the other orders. Then multiply together the units of the muItipHer with the tens; then with the hundreds, and so on in regular order ^ . .
In order to understand the fourth operation, viz., division, three things are to be observed, viz., what is meant by division; second, how many numbers are necessary in division; third, which of these numbers is the greater. As to the first I say that division is the operation of finding, from two given numbers, a third number, which is contained as many times in the greater number as unity is contained in the less number. You will find this number when you see how many times the less number is contained in the greater. Suppose, for example, that we have to divide 8 by 2; here 2 is contained 4 times in 8, so we say that 4 is the quotient demanded. Also, divide 8 by 4. Here the 4 is contained 2 times in 8, so that 2 is the quotient demanded.
Second, it is to be noticed that three numbers are necessary in division, — the number to be divided, the divisor, and the quotient, as you have understood from the example above given, where 2 is the divisor, 8 the number to be divided, and 4 the quotient. From this is derived the knowledge of the third thing which is to be noted, that the number which is to be divided is always greater than, or at least is equal to, the divisor. When the numbers are equal the quotient is always 1.
Now to speak briefly, it is sufficient in practice to say that there are two ways of dividing, — by the table and the galley method. In this operation you should begin with the figure of highest value, that is by the one which is found at the left, proceeding thence to the right. If you can divide by the table you will be able to divide by the galley method, and it is well, for brevity, to avoid the latter when you can. Therefore this is the method of dividing by the table: See how many times your divisor is found in the first
^ [The author now gives an example in multiplying by a one-figure number, proving the work by casting out nines. He then gives a proof by casting out sevens, after which he sets forth various methods of multiplication, such as that of the chessboard, that of the quadrilateral, or the one known by the name of gelosia, all of which were in common use at the time.]
THE TREVISO ARITHMETIC 11
left-hand figure, if it is contained in it, and write the quotient beneath it. If it is not so contained, consider this figure as tens and take together with it the following figure; then, finding the quotient write it beneath the smaller of the two figures. If there is any remainder, consider this as tens, and add it to the next number to the right, and see how many times your divisor is found in these two figures, writing the quotient under the units. In this same way proceed with the rest of the figures to the right. And when you have exhausted them all, having set down the quotient, write the remainder at the right, separated by a bar; and if the remainder is 0, place it where I have said. In the name of God I propose the first example, so attend well.
Divide 7624 ducats into two parts, viz. by 2, arranging your work as follows:
The divisor .2. 7624 The quotient 3812
0 the remainder*
The operations which I have set forth above being understood, it is necessary to take up the method and the rules of using them. The rule you must now study is the rule of the three things. Therefore that you may have occasion to sharpen your under- standing in the four operations above mentioned, — addition, subtraction, muItipHcation, and division, — I shall compare them. As a carpenter (wishing to do well in his profession) needs to have his tools very sharp, and to know what tools to use first, and what next to use, &c., to the end that he may have honor from his work, so it is in the work of this Practica. Before you take the rule of the three things it is necessary that you should be very skilled in the operations which have been set forth in addition, subtraction, multiplication, and division, so that you may enter enthusiastically into your work. Furthermore, that the rule of the three things, which is of utmost importance in this art, may be at your com- mand, you must have at hand this tool of the operations, so that you can begin your labors without spoiling your instruments and without failing. Thus will your labors command high praise.
* [The author now devotes twelve quarto pages to completing the explana- tion of division, which shows the degree of difficulty which the subject then offered. The rest of the text is devoted largely to the solution of mercantile problems by the Rule of Three. The preliminary statement and four problems will suffice to show the nature of the work. The subject is treated much more fully in Isis, Vol. VI (3), pp. 311-331, 1924.]
12 SOURCE BOOK IN MATHEMATICS
The rule of the three things is this: that you should multiply the thing which you wish to know, by that which is not like it, and divide by the other. And the quotient which arises will be of the nature of the thing which has no term like it. And the divisor will always be dissimilar (in weight, in measure, or in other differ- ence) to the thing which we wish to know.
In setting forth this rule, note first that in every case which comes under it there are only two things of different nature, of which one is named twice, — by two different numbers, — and the other thing is named once, by one number alone. For example:
If 1 lira of saffron is worth 7 lire of pizoli, what will 25 lire of this same saffron be worth? Here are not mentioned together both saf- fron and money, but the saffron is mentioned twice by two different numbers, 1 and 25; and the money is mentioned once, b}'^ the one number 7. So this is not called the rule of three things because there are three things of different nature, for one thing is mentioned twice.
Three merchants have invested their money in a partnership, whom to make the problem clearer I will mention by name. The first was called Piero, the second Polo, and the third Zuanne. Piero put in 112 ducats. Polo 200 ducats, and Zuanne 142 ducats. At the end of a certain period they found that they had gained 563 ducats. Required to know how much falls to each man so that no one shall be cheated.
There are two merchants of whom the one has cloth worth 22 soldi a yard, but who holds it in barter at 27 soldi. The other has wool which is worth in the country 19 lire per hundredweight. Required to know how much he must ask per hundredweight in barter so that he may not be cheated.
The Holy Father sent a courier from Rome to Venice, command- ing him that he should reach Venice in 7 days. And the most illustrious Signoria of Venice also sent another courier to Rome, who should reach Rome in 9 days. And from Rome to Venice is 250 miles. It happened that by order of these lords the couriers started on their journeys at the same time. It is required to find in how many days they will meet.
What availeth virtue to him who does not labor? Nothing. At Treviso, on the 10th day of December, 1478.
RECORDE On "The Declaration of the Profit of Arithmeticke"
(Selected by Professor David Eugene Smith, Teachers College, Columbia University, New York City.)
Robert Recorde (c. 1510-1558), a student and later a private teacher at both Oxford and Cambridge, wrote several works on mathematics. His arithmetic, The Grovnd oj Artes, was not the first one published in England, but it was by far the most influential of the early books upon the subject as far as the English-speaking peoples are concerned. This is not because of its catechetic style, although it doubtless influenced other writers to adopt this form of textbook instruction, but rather because through its subject matter and style of problems it set a standard that has been followed until com- paratively recent times. On the principle that a source book should touch at least lightly upon the elementary branches, "The declaration of the profit of Arithmeticke" is here set forth. The exact date of the first edition is uncer- tain, but it was about 1540 to 1542. Although a number of the early editions are available in the library of George A. Plimpton of New York City, it has been thought best to select one which represents the results of Recorde's influence for a full century, — that of 1646. As the title page says, this was "afterward augmented by M. John Dee," the promoter of the first English edition of Euclid; "enlarged — By John Mellis," and "diligently perused, cor- rected, illustrated and enlarged by R. C", and its tables "diligently calculated by Rv: Hartwell, Philomathemat." It therefore represents the best efforts of the teaching profession for a hundred years.
The following is an extract from Recorde's preface:
TO THE LOVING Readers, The Preface of Mr. Robert Record
Sore oft times have I lamented with my self the unfortunate condition of England, seeing so many great Clerks to arise in sundry other parts of the world, and so few to appear in this our Nation: whereas for pregnancy of naturall wit (I think) few Nations do excell Englishmen: But I cannot impute the cause to any other thing, then to be contempt, or misregard of learning. For as Englishmen are inferiour to no men in mother wit, so they passe all men in vain pleasures, to which they may attain with great pain and labour: and are as slack to any never so great
13
14 SOURCE BOOK IN MATHEMATICS
commodity; if there hang of it any painfull study or travelsome labour.
Howbeit, yet all men are not of that sort, though the most part be, the more pity it is: but of them that are so glad, not onely with painfull study, and studious pain to attain learning, but also with as great study and pain to communicate their learning to other, and make all England (if it might be) partakers of the same; the most part are such, that unneath they can support their own necessary charges, so that they are not able to bear any charges in doing of that good, that else they desire to do.
But a greater cause of lamentation is this, that when learned men have taken pains to do things for the aid of the unlearned, scarce they shall be allowed for their wel-doing, but derided and scorned, and so utterly discouraged to take in hand any like enterprise again.
The following is "The declaration of the profit of Arithmeticke " and con- stitutes the first ten pages of the text. It may be said to represent the influ- ence of this text upon establishing for a long period what educators at present sjjeak of as "the objectrves" of elementary arithmetic.
A Dialogue between the Master and the Scholar: teaching the Art
and use of Aritbmetick u-itb Pen.
The Scholar speaketh.
SIR, such is your authority in mine estimation, that I am content to consent to your saying, and to receive it as truth, though I see none other reason that doth lead me thereunto: uhereas else in mine own conceit it appearetb but vain, to bestow ariy time privately in learning oj that thing, that every childe may, and doth learn at all times and hours, when he doth any thing himself alone, and much more when he talketh or reasoneth with others.
Master. Lo, this is the fashion and chance of all them that seek to defend their blinde ignorance, that when they think they have made strong reason for themselves, then have they proved quite contrary. For if numbring be so common (as you grant it to be) that no man can do anything alone, and much lesse talk or bargain with other, but he shall still have to do with number: this proveth not number to be contemptible and vile, but rather right excellent and of high reputation, sith it is the ground of all mens affairs, in that without it no tale can be told, no communication without it can be continued, no bargaining without it can duely be ended, or no businesse that man hath, justly completed. These commodi-
RECORDS 15
ties, if there were none other, are sufficient to approve the worthi- nesse of number. But there are other innumerable, farre passing all these, which declare number to exceed all praise. Wherefore in all great works are Clerks so much desired? Wherefore are Auditors so richly fed? What causeth Geometricians so highly to be enhaunsed? W^hy are Astronomers so greatly advanced? Because that by number such things they finde, which else would farre excell mans minde.
Scholar. Verily, sir, if it bee so, that these men by numbring, their cunning do attain, at whose great works most men do wonder, then I see well I was much deceived, and numbring is a more cunning thing then I took it to be.
Master. If number were so vile a thing as you did esteem it, then need it not to be used so much in mens communication. Exclude number, and answer to this question: How many years old are you?
Scholar. Mum.
Master. How many dayes in a weeke? How many weeks in a year? What lands hath your Father? How many men doth hee keep? How long is it since you came from him to me?
Scholar. Mum.
Master. So that if number want, you answer all by Mummes: How many miles to London?
Scholar. A poak full of plums.
Master. Why, thus you may see, what rule number beareth, and that if number bee lacking it maketh men dumb, so that to most questions they must answer Mum.
Scholar. This is the cause, sir, that I judged it so vile, because it is so common in talking every w^hile: Nor plenty is not dainty, as the common saying is.
Master. No, nor store is no sore, perceive you this? The more common that the thing is, being needfully required, the better is the thing, and the more to be desired. But in numbring, as some of it is light and plain, so the most part is difficult, and not easie to attain. The easier part serv^eth all men in common, and the other requireth some learning. Wherefore as without num- bring a man can do almost nothing, so with the help of it, you may attain to all things.
Scholar. Yes, sir, why then it were best to learn the Art of numbring, first of all other learning, and then a man need learn no more, if all other come with it.
16 SOURCE BOOK IN MATHEMATICS
Master. Nay not so: but if it be first learned, then shall a man be able (I mean) to learn, perceive, and attain to other Sciences; which without it he could never get.
Scholar. I perceive by your former words, that Astronomy and Geometry depend much on the help of numbring: but that other Sciences, as Musick, Physick, Law, Grammer, and such like, have any help of Arithmetick, I p>erceive not.
Master. I may perceive your great Clerk-Iinesse by the ordering of your Sciences: but I will let that passe now, because it toucheth not the matter that I intend, and I will shew you how Arithmetick doth profit in all these somewhat grosly, according to your small understanding, omitting other reasons more substantial!.
First (as you reckon them) Musick hath not onely great help of Arithmetick, but is made, and hath his perfectnesse of it: for all Musick standeth by number and proportion: And in Physick, beside the calculation of critical! dayes, with other things, which I omit, how can any man judge the pulse rightly, that is ignorant of the proportion of numbers?
And so for the Law, it is plain, that the man that is ignorant of Arithmetick, is neither meet to be a Judge, neither an Advocate, nor yet a Proctor. For how can hee well understand another mans cause, appertaining to distribution of goods, or other debts, or of summes of money, if he be ignorant of Arithmetick? This oftentimes causeth right to bee hindered, when the Judge either delighteth not to hear of a matter that hee perceiveth not, or cannot judge for lack of understanding: this commeth by ignorance of Arithmetick.
Now, as for Grammer, me thinketh you would not doubt in what it needeth number, sith you have learned that Nouns of all sorts. Pronouns, Verbs, and Participles are distinct diversly by numbers: besides the variety of Nouns of Numbers, and Adverbs. And if you take away number from Grammer, then is all the quantity of Syllables lost. And many other ways doth number help Grammer. Whereby were all kindes of Meeters found and made? was it not by number?
But how needfull Arithmetick is to all parts of Philosophy, they may soon see, that do read either Aristotle, Plato, or any other Philosophers writings. For all their examples almost, and their probations, depend of Arithmetick. It is the saying of Aristotle, that hee that is ignorant of Arithmetick, is meet for no Science.
RECORDS 1 7
And Plato his Master wrote a little sentence over his Schoolhouse door, Let none enter in hither (quoth he) that is ignorant of Geometry. Seeing hee would have all his Scholars expert in Geometry, much rather he would the same in Arithmetick, with- out which Geometry cannot stand.
And how needfull Arithmetick is to Divinity, it appeareth, seeing so many Doctors gather so great mysteries out of number, and so much do write of it. And if I should go about to write all the commodities of Arithmetick in civill acts, as in governance of Common-weales in time of peace, and in due provision & order of Armies, in time of war, for numbering of the Host, summing of their wages, provision of victuals, viewing of Artillery, with other Armour; beside the cunningest point of all, for casting of ground, for encamping of men, with such other like: And how many wayes also Arithmetick is conducible for all private Weales, of Lords and all Possessioners, of Merchants, and ail other occupiers, and generally for all estates of men, besides Auditors, Treasurers, Receivers, Stewards, Bailiffes, and such like, whose Offices without Arithmetick are nothing: If I should (I say) particularly repeat all such commodities of the noble Science of Arithmetick, it were enough to make a very great book.
Scholar. No, no sir, you shall not need: For I doubt not, but this, that you have said, were enough to perswade any man to think this Art to be right excellent and good, and so necessary for man, that (as I think now) so much as a man lacketh of it, so much hee lacketh of his sense and wit.
Master. What, are you so farre changed since, by hearing these few commodities in generall: by likelihood you would be farre changed if you knew all the particular Commodities.
Scholar. I beseech you Sir, reserve those Commodities that rest yet behinde unto their place more convenient: and if yee will bee so good as to utter at this time this excellent treasure, so that I may be somewhat inriched thereby, if ever I shall be able, I will requite your pain.
Master. I am very glad of your request, and will do it speedily, sith that to learn it you bee so ready.
Scholar. And I to your authority my wit do subdue; whatso- ever you say, I take it for true.
Master. That is too much; and meet for no man to bee beleeved in all things, without shewing of reason. Though I might of my Scholar some credence require, yet except I shew reason, I do it
18 SOURCE BOOK IN MATHEMATICS
not desire. But now sith you are so earnestly set this Art to attaine, best it is to omit no time, lest some other passion coole this great heat, and then you leave off before you see the end.
Scholar. Though many there bee so unconstant of mind, that flitter and turn with every winde, which often begin, and never come to the end, I am none of this sort, as I trust you partly know. For by my good will what I once begin, till I have it fully ended, I would never blin.
Master. So have I found you hitherto indeed, and I trust you will increase rather then go back. For, better it were never to assay, then to shrink and flie in the mid way: But I trust 30U will not do so; therefore tell mee briefly: What call you the Science that you desire so greatly.
Scholar. Why sir, you know.
Master. That maketh no matter, I would hear whether you know, and therefore I ask you. For great rebuke it were to have studied a Science, and yet cannot tell how it is named.
Scholar. Some call it Arsemetrick, and some Augrime.
Master. And what do these names betoken?
Scholar. That, if it please you, of you would I learn.
Master. Both names are corruptly written: Arsemetrick for Arithmetick, as the Greeks call it, and Augrime for Algorisme, as the Arabians found it: which both betoken the Science of Numbring: for Arithmos in Greek is called Number: and of it commeth Arithmetick, the Art of Numbring. So that Arithmetick is a Science or Art teaching the manner and use of Numbring: This Art may be wrought diversly, with Pen or with Counters. But I will first shew you the working with the Pen, and then the other in order.
Scholar. This I will remember. But how many things are to bee learned to attain this Art fully?
Master. There are reckoned commonly seven parts or works of it.
Numeration, Addition, Subtraction, MultipHcation, Division, Progression, and Extraction of roots: to these some men adde Duphcation, Triplation, and Mediation. But as for these three last they are contained under the other seven. For Duplication, and Triplation are contained under Multiplication; as it shall appear in their place: And Mediation is contained under Division, as I will declare in his place also.
Scholar. Yet then there remain the first seven kinds of Numbring.
RECORDE 19
Master. So there doth: Howbeit if I shall speak exactly of the parts of Numbrlng, I must make but five of them: for Progression is a compound operation of Addition, Multiplication and Division. And so is the Extractions of roots. But it is no harme to name them as kindes severall, seeing they appear to have some several! working. For it forceth not so much to contend for the number of them, as for the due knowledge and practising of them.
Scholar. Then you will that I shall name them as seven kindes distinct. But now I desire you to instruct mee in the use of each of them.
Master. So I will, but it must be done in order: for you may not learn the last so soon as the first, but you must learn them in that order, as I did rehearse them, if you will learn them speedily, and well.
Scholar. Even as you please. Then to begin; Numeration is the first in order: what shall I do with it?
Master. First, you must know what the thing is, and then after learn the use of the same.
STEVIN On Decimal Fractions
(Translated from the French by Professor Vera Sanford, Western Reserve University, Qeveland, Ohio.)
The invention of the decimal fraction cannot be assigned to any single individual. Pellos (1492) used a decimal point to set off one, two, or three places in the dividend when the divisor was a multiple of 10, 100, or 1000. Adam Reise (1522) printed a table of square roots in which values to three places were computed for the irrationals. Most imjxtrtant of all, Rudolfl (1530) used the symbol | as a decimal point in a compound interest table.^
The first person to discuss the theory of decimal fractions and their arithme- tic was Simon Stevin (c.l548-c.l620), a native of Bruges and a firmsupp>orterof William the Silent in the struggle of the Low Countries against Spain. Stevin was tutor to Maurice of Nassau, served as quartermaster general in the Dutch army, and acted as commissioner of certain public works, especially of the dikes. He is reported to have been the first to adapt the principles of commercial bookkeeping to national accounts, and his studies in hydraulics resulted in theorems which foreshadowed the integral calculus.
Stevin's work on decimal fractions was published in 1585, two editions appearing in that year — one in Flemish with the title La Tbiende, the other in French with the title La LHsme,
The translation that follows was made from Les Oeuvres Matbematiques de Simon Stevin, edited by Girard and published in Leyden in 1634.*
La Disme
Teaching how all Computations that are met in Business may be performed
by Integers alone without the aid of Fractions
Written first in Flemish and now done into French
by
Simon Stevin of Bruges
To Astrologers, Surveyors, Measurers of Tapestry, Gaugers, Stereometers in General, Mint-masters, and to All Merchants Simon Stevin Sends Greeting
A person who contrasts the small size of this book with your greatness, my most honorable sirs to whom it is dedicated, will
'These instances are discussed with facsimiles of the cases in point in "The Invention of the Decimal Fraction," by David Eugene Smith, Ttacbcrs College Bulletin, First Series. No. 5.
- A facsimile of the original edition with an introduction by the late Father Bosmans was printed by the Soci6t6 des Bibliophiles Anversois in Antwerp, in 1924, with the title La "Tbiende" de Simon Stevin.
20
STEVIN 21
think my idea absurd, especially if he imagines that the size of this volume bears the same ratio to human ignorance that its usefulness has to men of your outstanding ability; but, in so doing, he will have compared the extreme terms of the proportion which may not be done. Let him rather compare the third term with the fourth.
What is it that is here propounded? Some wonderful inven- tion? Hardly that, but a thing so simple that it scarce deserves the name invention; for it is as if some stupid country lout chanced upon great treasure without using any skill in the finding. If anyone thinks that, in expounding the usefulness of decimal numbers, I am boasting of my cleverness in devising them, he shows without doubt that he has neither the judgment nor the intelligence to distinguish simple things from difficult, or else that he is jealous of a thing that is for the common good. However this may be, I shall not fail to mention the usefulness of these numbers even in the face of this man's empty calumny. But, just as the mariner who has found by chance an unknown isle, may declare all its riches to the king, as, for instance, its having beautiful fruits, pleasant plains, precious minerals, etc., without its being imputed to him as conceit; so may I speak freely of the great usefulness of this invention, a usefulness greater than I think any of you anticipates, without constantly priding myself on my achievements.
As your daily experience. Messieurs, makes you sufficiently aware of the usefulness of number, which is the subject of La Disme, it will not be necessary to say many words with reference to this. The astrologer^ knows that, by computation, using tables of declinations, the pilot may describe the true latitude and longi- tude of a place and that by such means every point upon the earth's surface may be located. But as the sweet is never without the bitter, the labor of such computations cannot be disguised, for they involve tedious multipfications and divisions of sexagesimal fractions, 2 degrees, minutes, seconds, thirds, etc. The surveyor
* [This is used for "astrologer" and for "astronomer" as well.]
* [Fractions whose denominators were the powers of sixty. They were not restricted to the measurement of time or angles but were used by scientists and mathematicians in all sorts of computations. They afforded a convenient way of expressing the approximate root of an equation, — in one case, for instance a root is given to the tenth sexagesimal, — but although their use facilitated the comparing of one number with another and although they were well suited to addition and subtraction, multiplication, division, and square root were difficult and were frequently performed by tables.]
22 SOURCE BOOK IN MATHEMATICS
knows the great benefit which the world receives from his science by which it avoids many disputes concerning the unknown areas of land. And he who deals in large matters, cannot be ignorant of the tiresome multiplications of rods, feet, and inches^ the one by the other, which often give rise to error tending to the injury of one of the parties, and to the ruin of the reputation of the sur- veyor. So too, with mint-masters, merchants, etc., each in his own business. The more important these calculations are, and the more laborious their execution, so much the greater is this dis- covery of decimal numbers which does away with all these diffi- culties. To speak briefly. La Disme teaches how all computations of the type of the four principles of arithmetic — addition, sub- traction, multiplication and division — may be performed by whole numbers with as much ease as in counter-reckoning. ^
If by these means, time may be saved which would otherwise be lost, if work may be avoided, as well as disputes, mistakes, lawsuits, and other mischances commonly joined thereto, I willingly submit La Disme to your consideration. Someone may raise the point that many inventions which seem good at first sight are of no eff"ect when one wishes to use them, and as often happens, new methods good in a few minor cases are worthless in more important ones. No such doubt exists in this instance, for we have shown this method to expert surveyors in Holland and they have abandoned the devices which they have invented to lighten the work of their computations and now use this one to their great satisfaction. The same satisfaction will come to each of you, my most honorable sirs, who will do as they have done.
Argument
La Disme consists of two parts, — definitions and operations. In the first part, the first definition explains what decimal numbers' are, the second, third, and fourth explain the meaning of the terms unit,^ prime, second, etc., and the other decimal numbers.
1 [Here Stevin uses the units verge, pied, doigt.]
^ [That is, reckoning with jetons or counters, a method of reckoning that was still in vogue in Stevin's time.]
' [In this translation, the words "decimal numbers" are used where the literal translation would be "the numbers of La Disme."]
* [In the Flemish version, Stevin uses the word Begbin and in the French one. Commencement.]
STEVIN
23
In the operations, four propositions show the addition, subtrac- tion, multiplication, and division of decimal numbers. The order of these topics may be succinctly represented in a table.
Decimals j Unit
Definitions \ pj-ime. Second, etc.
Decimal Numbers
La Disme has two divisions.
Operations.
Addition Subtraction Multiplication Division
At the end of this discussion, there will be added an appendix setting forth the use of decimal numbers in real problems.
The First Division of La Disme
Of Definitions
Definition I
Decimal numbers^ are a kind of arithmetic based on the idea
of the progression by tens, making use of the ordinary Arabic
numerals, in which any number may be written and by which all
computations that are met in business may be performed by
integers alone without the aid of fractions.
Explanation Let the number one thousand one hundred eleven be written in Arabic numerals 1111, in which form it appears that each 1 is the tenth part of the next higher figure. Similarly, in the number 2378, each unit of the 8 is the tenth part of each unit of the 7, and so for all the others. But since it is convenient that the things which we study have names, and since this type of computation is based solely upon the idea of the progression by tens^ as will be seen in our later discussion, we may properly speak of this treatise as La Disme and we shall see that by it we may perform all the computations we meet in business without the aid of fractions.
^ [Disme est une espece d'arithmetique.]
* [Disme: "tithe," later the word was contracted into dime. Earlier forms in English use are dyme and dessime. Disme came into the language when Stevin's work was translated in 1608. It was used as a noun meaning a tenth and as a synonym for decimal arithmetic; also as a verb, to divide into tenths.]
24 SOURCE BOOK IN MATHEMATICS
Definition II Any given number is called the unit and has the sign (5).
Explanation In the number three hundred sixty four, for example, we call the three hundred sixty four units and write the number 364(0). Similarly for other cases.
Definition III The tenth part of a unit is called a Prime, and has the sign ®, and the tenth of a prime is called a Second, and has the sign 0. Similarly for each tenth part of the unit of the next higher figure.
Explanation Thus 3(T)7@509® is 3 primes, 7 seconds, 3 thirds, 9 fourths, and we might continue this indefinitely. It is evident from the defini- tion that the latter numbers are ^o» l^io6> /-fooo. ^fo»ooo. and that this number is 375^^q,qqq. Likewise 8(0)9®3®7@ has the value 8^^o. Koo. Kooo. or S^^Kooo- And so for other numbers. We must also realize that in these numbers we use no fractions and that the number under each sign except the "unit" never exceeds the 9. For instance, we do not write 701 2@ but 8020 instead, for it has the same value.
Definition IV The numbers of the 2nd and 3d definitions are called Decimal
Numbers.
The End of the Definitions^
1 [These same names and symbols are given a more general application in Stevin's other works. The following discussion is from his work on the subject in V Arilbmetique where geometric progressions, or geometric numbers play a prominent part.
He says, " When the ancients realized the value of progressions of the sort where the first term multiplied by itself gives the second term. . ., they saw that it would be necessary to choose meaningful names for these numbers so that they might the more readily distinguish them. Thus they called the first term Prime which we denote by ©, the next Second which wc write as (2) etc. "For example,
®2®4{D 8®16... ®3@9®27®81...
"We intend that the Unit of a quantity shall mean something distinct from the first quantity or prime. Any arithmetic number or radical which one uses in algebraic computation as 6 or \/3 or 2 -f V^I. . ., wc will call
STEVIN 25
The Second Division of La Disme Of Operations
Proposition I. — To add decimal numbers. Given three decimal numbers, 27@8®4@7®, 37(o)6®7®5®, 875(o)7®8©2®.
Required to find their sum.
Construction. — Arrange the numbers as in the accompanying figure, adding them in the usual manner of adding (Q)®@® integers. This (by the first problem of rArithme- 2 7 8 4 7 tique^) gives the sum 941304,^ which, as the signs 3 7 6 7 5 above the numbers show^ is 941(o)3®0@4®. And 8 7 5 7 8 2 this is the sum required. 9 4 13 0 4
Proof. — By the third definition of this book, the given number 27@8®4©7® is 27^1o. ^loo, Kooo, or 27S4y^ooo- Similarly, the 37(o)6®7®5® is 3767^^ooo. and the 875(o)7®8@3® is 87578^000- These three numbers 27847^^^^^ 37675,^^^^^ g7578^^QQQ added, according to the 10th problem of rArithmetique, make 94130^^qqq, but 941(o)3®0®4® has this same value, and is therefore the true sum which was to be shown.
Conclusion. — Having been given decimal numbers to add, we have found their sum which was to be done.
Note. — If, in the numbers in question, some figure of the natural order be lacking, fill its place with a zero. For example, in the numbers 8(0)5®6® and 5@7@ where the second lacks a (Q)®@ figure of order prime, insert 0® and take 5 (5)0® 7® as 8 5 6 the given number and add as before. This note applies 5 0 7 to the three following propositions also. ~T'3~6~3
the Unit and we will give it the symbol @: but this symbol shall be used only when the arithmetic number or radical is not denominate, (quand les nombres Arithmetiques ou radicaux ne sont pas absoluement descripts)."
Stevin writes denominate numbers as 1 hour 3® 5©, 5 degrees 4® 18®, 2790 verges 5®9@. He later notes (rAritbmetique, p. 8) that Bombelli has used this symbolism also except for the (o). In Bombelli's Algebra (1572) the symbols vL', k2j, ksj. . . are used for the p>owers of the unknown quantity just as Stevin used his ®, @, ®, . . . , i. e., a specialized form of the geometric progression.
The names for these quantities except that of the unit, are easily traced to the pars minuta prima, etc., of the sexagesimals.]
' [La Pratiqve D' Aritbmktiqve De Simon Stevin De Brvges, Leyden, 1585.]
^ [Stevin has three ways of writing these numbers, depending upon the
exigencies of the c^: 27@8®4@7@. ®®®®; g^^^'j
26 SOURCE BOOK IN MATHEMATICS
Proposition II. — To subtract decimal numbers. Given the number 237(o)5©7@8® from which the number 59(0)7®4@9@ is to be subtracted.
Required to find the remainder. (Q)®@@
Construction. — Place the numbers in order as m ^ the adjoining figure, subtracting after the usual manner of subtracting integers (by the 2nd problem 1 7 7 8 2 9 of I' Arithmkique) . There remains 177829 which, as indicated by the signs above the numbers, is 1 77(0)8® 2@90; and this is the remainder required.
Proof. — By the third definition of la Disme, the 237 @ 5®7®8(D is 237^^0, Koo. ^looo or 23757^fooo. And, by the same reasoning, the 59@7®4(2)9@ is 59'^4^^qqq; subtracting this from 2375'7^-{qqq, according to the tenth problem of l' Aritbmetique, leaves 177^^^000- But the aforesaid 1 77(0)8® 2®9® has this same value and is, therefore, the true remainder, which was to be proved.
Conclusion. — Having been given a decimal number and a similar number which is to be subtracted from it, we have found the remainder which was to be done.
Proposition III. — To multiply decimal numbers.
Given the number 32(o)5®7@ and the multiplier 89®4®6®.
Required to find their product.
Construction. — Place the numbers in order and multiply in the ordinary way of multiplying whole numbers (by (OXD®
the third problem of I'Arithmetique). This gives 3 2 5 7
the product 29137122. To find what this is, add 8 9 4 6
the last two signs of the given numbers, the one 1~9~5~4~2~
@ and the other ® also, which together are ®. 13 0 2 8 We say, then, that the sign of the last figure of 2 9 3 13 the product will be ®. Once this is estabhshed, 2 6 0 5 6
ail the signs are known on account of their con-
tinuous order. Therefore, 2913(0)7®1@2®2® ^ ^ ll^ll^ IS the required product.
Proof. — As appears by the third definition of La Disme, the given number 32(0)5®7@ is 32^^o» Koo. or 325J.foo. andfikewise the multipfier 89@4®6® is 894^foo- Multiplying the aforesaid 32^}ioo by this number gives the product 2913 "^12^^^^^^^ (by the twelfth problem of I'Arithmetique). But the aforesaid product 291 3(0)7® 1®2®2® has this value and is, therefore, the true
STEVIN 27
product which we were required to prove. We will now explain why second multiplied by second gives the product fourths, which is the sum of their signs, and why fourth by fifth gives the product ninth and why unit by third gives the product third and so forth.
Let us take for example %o ^^^ Moo which, by the definitions of La Disme, are the values of 2(i) and 30. Their product is Mooo> w'hich by the third defintion given above is 6®. Hence, multiplying prime by second gives a product in thirds, that is, a number whose sign is the sum of the given signs.
Conclusion. — Having been given a decimal number to multiply and the multiplier we have found the product, which was to be done.
Note. — If the last sign of the multiplicand is not equal to the last sign of the multiplier, for example, 3®706® 0®®
and 5040, proceed as above. The placing of the 3 7 8
figures will appear as here shown. 5 40
15 12 18 9 0
2 0 4 12
®0®®0
Proposition IV. — To divide decimal numbers.
Given 3(0)4040305020 to be divided by 9060.
Required to find their quotient.
Construction. — Omitting their signs, divide ). the given numbers in the ordinary way of J?) dividing whole numbers by the fourth problem p/^^ of r Aritbmkique. This gives the quotient 7^^J (0)000 3587. To determine the signs, subtract the ^^^^^ (3587 last sign of the divisor 0 from the last sign of P^^0f5 the dividend 0, leaving 0 as the sign of the 9^^ last digit of the quotient. Once this is determined, all the other signs are known because of their continuous order. 3(0)508070 is, therefore, the quotient required.
Proof. — By the third definition of La Disme the dividend 3(0)4040305020 is 3 fio, Moo. Mooo, Mo.ooo Moo.ooo or 344.35^^00,000. The divisor 9©60_is Ho, Moo or Q^foo- By the thirteenth problem of I' Arithmetique, the quotient of these numbers is 3^^}iooo. The aforesaid 3(0)508070 is therefore the true quotient which was to be shown.
Conclusion. — Having been given a decimal number to be divided and the divisor, we have found the quotient, which was to be done.
28 SOURCE BOOK IN MATHEMATICS
Note I. — If the signs of the divisor be higher than the signs of tlie dividend, add to the dividend as many zeros as may be neces- sary. For example, in dividing 7@ by 40, I place zeros after the 7 and divide as above getting the quotient 1750.
It sometimes happens that the quotient cannot ^^ @
be expressed by whole numbers as in the case of 4® 7PPP (175 0 divided by 30. Here, it appears that the quotient ^4^ will be infinitely many threes with always one third in addition. In such a case, we may approach as near to Xl)^ (§)®©
the real quotient as the problem requires and ^PPPOOO (13 3 3 omit the remainder. It is true indeed that ^>?i.3^ 13(0)3030 or 13(o)30303>^0 is the exact result, but in this work we propose to use whole numbers only, and, moreover, we notice that in business one does not take account of the thou- sandth part of a maille^ or of a grain. Omissions such as these are made by the principal Geometricians and Arithmeticians even in computations of great consequence. Ptolemy and Jehan de Montroyal, for instance, did not make up their tables with the utmost accuracy that could be reached with mixed numbers, for in view of the purpose of these tables, approximation is more useful than perfection.
Note II. — Decimal numbers may be used in the extraction of roots. For example, to find the square root of 502090, work according to the ordinary method of extracting square root and the root will be 2030. The last sign of the root is always J. one half the last sign of the given number. If, however, the ^^^ last sign is odd, add (a zero in place of) the next sign and 2, 3 extract the root of the resulting number as above. By a Aqn similar method with cube root, the third of the last sign of the given number will be the sign of ther oot — similarly for all other roots.
The End of La Disme Appendix Decimal numbers have been described above. We now come to their applications and in the following six articles we shall show how all computations which arise in business may be performed by them. We shall begin with the computations of surveying as this subject was the first one mentioned in the introduction.
' [H ounce.)
STEVIN 29
Article One Of the Computations of Surveying When decimal numbers are used in surveying, the verge^ is called the unit, and it is divided into ten equal parts or primes, each prime is divided into seconds and, if smaller units are required, the seconds into thirds and so on so far as may be necessary. For the purposes of surveying the divisions into seconds are sufficiently small but in matters that require greater accuracy as in the measur- ing of lead roofs, thicknesses etc., one may need to use the thirds. Many surveyors, however, do not use the verge, but a chain three, four, or five verges long, and a cross-staffs with its shaft marked in five or six feet divided into inches. These men may follow the same practise here substituting five or six primes with their seconds. They should use these markings of the cross-staff without regard to the number of feet and inches that the verge contains in that locality, and add, subtract, multiply, and divide the resulting numbers as in the preceding examples. Suppose, for instance, four areas' are to be added: 34S(o)708@, and 872(0)- 5®3(2), 615(0)408®, and 956(g)8®6®. Add these as in the first proposition of La Disme. This gives the sum (Q)®@
2790 verges 5 primes, 9 seconds. This number, 3 4 5 7 2 divided by the number of verges in an arpent^ gives 8 7 2 5 3 the number of arpents required. To find the number 6 15 4 8 of small divisions in 5 primes 9 seconds, look on the 9 5 6 8 6 other side of the verge to see how many feet and inches 2 7 9 0 5 0 match with them; but this is a thing which the sur- veyor must do but once, i. e., at the end of the account which he
* [The word verge was used for a measuring rod of that length.]
2 [The cross-staff was a piece of wood mounted at its mid-point p>erpendicular to a shaft and free to move along this shaft to positions parallel to the first one. To use this instrument, the observer adjusts it so the lines of sight from the ends of the staff to the tip of the cross-piece coincide with the end-points of the line to be measured, and the distances are computed from one measurment and similar triangles. When neither point of the required line is accessible, the distance is computed from two observations at known distances from each other.]
' [Stevin's area units evidently proceed directly from his decimal scheme. Thus the prime of the area unit is one tenth of the unit itself, not a square whose side is the prime of the linear unit.]
* fThe arpent was the common unit of area. It varied from about 3000 to 5100 square meters, according to the locality.]
30 SOURCE BOOK IN MATHEMATICS
gives to the proprietaries, and often not then, as the majority think it useless to mention the smaller units.
Secondly, to subtract 32(O)5®70 from 57(0)3©2©, work according to the second proposition of La Disme, and (0)®@ there remains 24 verges, 7 primes, 5 seconds. 5 7 3 2
3 2 5 7
24 75 Thirdly, to multiply 8(0)703(2) by 7(o)5®40 (these might be the sides of a rectangle or quadrangle), proceed as (0)00
above according to the third proposition of La Disme, 8 7 3
getting the product, or area, 65 verges 8 primes etc. 7 5 4
3 4 9 2 4 3 6 5 6 111
65824 2 ©000 0 Fourthly, suppose that the rectangle ABCD has the side AD 26(0)30. From what point on AB should a 1
line be drawn parallel to AD to cut off a ^^ rectangle of area 367(0)60? Divide 367(0)60 7^ by 26(0)30 according to the fourth proposition ^,508 of La Disme getting the quotient 13(5)9070 ^^^Ji which is the required distance AE. If greater 70^7^9 @(J)® accuracy is desired, this division may be carried ^^7^PP (13 9 7 further, although such accuracy does not seem necessary. The proofs of these problems are given above in the propositions to which we have ^^ referred.
Article Two
0/ the Computations oj the Measuring of Tapestry
The aiine^ is the unit of the measurer of tapestry. The blank side of the aune should be divided into ten equal parts each of which is 1 prime, just as is done in the case of the verge of the surveyor. Then each prime is divided into ten equal parts, each 1 second etc. It is not necessary to discuss the use of this measure as examples of it would be similar in every respect to those given in the article on surveying.
1 [About 46 inches.]
STEVIN 31
Article Three Of Computations Used in Gauging and in the Measuring of all Casks
Article Four 0/ Computations of Volume Measurement in General
It is true, indeed, that gauging is stereometry that is the science of measuring volumes, but all stereometry is not gauging. We therefore distinguish them in this treatise. The stereometer who uses the method of La Disme, should mark the customary measure of his town, whether this be the verge or the aune, with the decimal divisions described in the first and second articles above.
Let us suppose that he is to find the volume of a rectangular column whose length is 3®2@, breadth 2©4(2), 3 2
2@3®5@. He should multiply the length by the 2 4
breadth (according to the 4th proposition of La 12 8
Disme) and this product by the height, getting as 6 4
his result 108(2)408©. 7 68
2 35
3 8 40 2 3 0 4 15 3 6
18 0 4 80
©00000 Note. — Someone who is ignorant of the fundamentals of stereo- metry— for it is such a man that we are addressing now — may wonder why we say that the volume of the above column is but 1© etc., for it contains more than 180 cubes of side 1 prime. He should realize that a cubic verge is not 10 but 1000 cubes of side 1 prime. Similarly, 1 prime of the volume unit is 100 cubes each of side 1 prime. The like is well known to surveyors for when one says 2 verges 3 feet of earth, he does not mean 2 verges 3 feet square, but 2 verges and (counting 12 feet to the verge) 36 feet square.
If however the question had been how many cubes of side 1 prime are in the above column, the result would have been changed to conform to this requirement, bearing in mind that each prime of volume units is 100 cubes of side 1 prime, and each second is 10 such cubes. If the tenth part of the verge is the greatest measure that the stereometer intends using, he should call it the unit and proceed as above.
32 SOURCE BOOK IN MATHEMATICS
Article Five Of Astronomical Computations The ancient astronomers who divided the circle into 360 degrees, saw that computations with these and their fractional parts were too laborious. They divided the degree, therefore, into sub- multiples and these again into the same number of equal parts, and in order to always work with integers, they chose for this division the sexagesimal progression for sixty is a number com- mensurate with many whole numbers, to wit with 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. If we may trust to experience, however, and we say this with all reverence for the past, the decimal and not the sexagesimal is the most convenient of all the progressions that exist potentially in nature. Thus, we would call the 360 degrees the unit and we would divide the degree into ten equal parts, or primes, and the prime in turn into ten parts and so forth, as has been done several times above. Having once agreed upon this division, we might describe the easy methods of adding, sub- tracting, multiplying, and dividing these numbers, but as this does not differ from the preceding propositions such a recital would be but a waste of time. We therefore let those examples illustrate this article. Moreover we would use this division of the degree in all astronomical tables and we hope to publish one such^ in our own Flemish language which is the richest, the most ornate, and the most perfect of all languages. Of its exquisite uniqueness, we contemplate a fuller proof than the brief one which Pierre and Jehan made in the Beivysconst or Dialectique^ which was recently published.
1 [Stevin did not make this promise good, however, for in his work on astronomy, the Wiscontige Gedacbtenissen (1608), he keeps to the old partition of the degree. Father Bosmans is of the opinion that this was due to the tremendous labor involved, but he also points out that the errors incident to converting readings to the decimal system for computation and then shifting back again would be greater than those involved in the mere computation with sexagesimals.]
2 [Dialectike Ojte Beivysconst. Leerende van alien saeken recbt ende constelick Oirdeelen; Oock openende den wecb tot de alderiepste verborgentbeden der Nature- ren. Bescbreven int Neerdytacb door Simon Stevin van Brugghe. Tot Ley den by Cbristoffel Plantijn M.D.LXXXV.
In this book, which was an imitation of Gcero's Tusculan Disputations, Pierre and Jehan discuss the beauties of the Flemish tongue. (First edition, pp. 141-166.)
The more complete proof comes as a digression at the end of the preface of Stevin's Begbinselen der Weegbconst, 1596.)
STEVIN 33
Article Six Of the Computations of Mint-masters, Merchants, and in General
of All States
To summarize this article, we might say that all measures — linear, liquid, dry, and monetary — may be divided decimally, and that each large unit may be called the unit. Thus the marc is the unit of weight for gold and silver; the Hvre, for other common weights; and the livre gros in Flanders, the Hvre esterlain in England, and the ducat in Spain, are the units of money in those countries. In the case of the money, the highest sj^mbol (and the lowest denomination) of the marc would be the fourth, for the prime weighs half the Es of Antwerp. The third suffices for the highest symbol of the livre gros, for the third is less than the fourth of a penny.
Instead of the demi-Iivre, once, demi-once, esterlain, grain etc. the subdivisions of the weights should be the 5, 3, 2, 1 of each sign, that is to say, after a livre would follow a weight of 5 primes (or 3^ lb.) then 3 primes, then 2, then 1. Similar parts of other weights would have the 5 and the other multiples of the division following.
We think it essential that each subdivision should be named prime, second, third etc., whatever sort of measure it may be, for it is evident to us that second multiplied by third gives the product fifth (2 & 3 make 5 as was said above), and third divided by second gives the quotient prime, facts which cannot be shown so neatly by other names. But when one wishes to name them so as to distinguish the systems of measure, as we say demi-aune, demi- livre, demi-pinte, etc., we may call them prime of marc, second of marc, second of livre, second of aune, etc.
As examples of this, let us suppose that 1 marc of gold is worth 36 lb. 5®3@, how much is 8 marcs 3®5(2)40 worth? Multiply 3653 by 8354 getting the product 305 lb. 1®7®10 which is the required solution.^ As for the 602®, these are of no account here.
Again, take the case of 2 aunes 3® (of cloth) which cost 3 lb. 2®5@, what would be the cost of 7 aunes 5®3@? To find this, multiply the last of the given numbers by the second and divide this product by the third according to the usual custom, that is to
^ [Stevin is not consistent in his approximation of results. When he com- putes his interest tables, he says he considers ^°9^oi an extra unit 'for it is more than one half.']
34 SOURCE BOOK IN MATHEMATICS
say 753 by 325 which gives 244725. This divided by 23 gives the quotient and solution 10 lb. 6®4(D.
We might give examples of all the common rules of arithmetic that pertain to business, as the rules of partnership, interest, exchange etc., and show how they may be carried out by integers alone and also how they may be performed by easy operations with counters, but as these may be deduced from the preceding, we shall not elaborate them here. We might also show by com- parison with vexing problems with fractions the great difference in ease between working with ordinary numbers and with decimal numbers, but we omit this in the interest of brevity.
Finally, we must speak of one difference between the sixth article and the five preceding articles, namely that any individual may make the divisions set forth in the five articles, but this is not the case in the last article where the results must be accepted by every one as being good and lawful. In view of the great useful- ness of the decimal division, it would be a praiseworthy thing if the people would urge having this put into effect so that in addition to the common divisions of measures, weights, and money that now exist, the state would declare the decimal division of the large units legitimate to the end that he who wished might use them. It would further this cause also, if all new money should be based on this system of primes, seconds, thirds, etc. If this is not put into operation so soon as we might wish, we have the consolation that it will be of use to posterity, for it is certain that if men of the future are like men of the past, they will not always be neglect- ful of a thing of such great value.
Secondly, it is not the most discouraging thing to know that men may free themselves from great labor at any hour they wish.
Lastly, though the sixth article may not go into effect for some time, individuals may always use the five preceding articles indeed it is evident that some are already in operation.
The End of the Appendix
DEDEKIND On Irrational Numbers
(Translated from the German by the Late Professor Wooster WoodruflF
Beman, University of Michigan, Ann Arbor, Michigan. Selection
made and edited by Professor Vera Sanford, Western
Reserve University, Cleveland, Ohio)
Julius Wilhelm Richard Dedekind (1831-1916) studied at Gi)ttingen and later taught in Zurich and Braunschweig. His essay Stetigkeit und irra- tionale Zablen published in 1872 was the outcome of researches begun in Zurich in 1858 when Dedekind, teaching differential calculus for the first time, became increasingly conscious of the need for a scientific discussion of the concept ot continuity.
This work is included in Essays on the Theory of Numbers, by Richard Dedekind, translated by the late Professor Wooster Woodruff Beman (Open Court Publishing Company, Chicago, 1901). The extract here given is on pages 6 to 24 of this translation and is reproduced by the consent of the publishers.
The author begins with a statement of three properties of rational numbers and of the three corresponding properties of the points on a straight line. These properties are as follows:
For Numbers
I. If a > b, and b > c, then a > c.
II. If a, c are two different numbers, there are infinitely many different numbers lying between a, c.
III. If a is any definite number, then all numbers of the system R fall into two classes, Ai and A2, each of which contains infinitely many individuals; the first class Ai comprises all numbers ai that are < a, the second class A2 comprises all numbers 02 that are > a; the number a itself may be assigned at pleasure to the first or second class, being respectively the greatest number of the first class or the least of the second.
For the Points on a Line
I. If p lies to the right of q, and q to the right of r, then p lies to the right of r ; and we say that q lies between the points p and r.
II. If p, r are two different points, then there always exist infinitely many points that lie between p and r.
III. If p is a definite point in L, then all points in L fall into two classes. Pi and P2 each of which contains infinitely many individuals; the first class Pi contains all the points pi that lie to the left of p, and the second class Pt contains all the points pj that lie to the right of p; the point p itself may be assigned at pleasure to the first or second class. In every case, the separation
35
36 SOURCE BOOK IN MATHEMATICS
of the straight line L into the two classes or portions Pi, Pj is of such a charac- ter that every point of the first class Pi lies to the left of every point of the second class P2.
III.
CONTINUITY OF THE STRAIGHT LINE.
Of the greatest importance, however, is the fact that in the straight line L there are infinitely many points which correspond to no rational number. If the point p corresponds to the rational number a, then, as is well known, the length op is commensurable with the invariable unit of measure used in the construction, i. e., there exists a third length, a so-called common measure, of which these two lengths are integral multiples. But the ancient Greeks already knew and had demonstrated that there are lengths incommensurable with a given unit of length, e. g., the diagonal of the square whose side is the unit of length. If we lay off such a length from the point 0 upon the hne we obtain an end-point which corresponds to no rational number. Since further it can be easily shown that there are infinitely many lengths which are incommensurable with the unit of length, we may affirm: The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number individuals.
If now, as is our desire, we try to follow up arithmetically all phenomena in the straight line, the domain of rational numbers is insufficient and it becomes absolutely necessary that the instru- ment R constructed by the creation of the rational numbers be essentially improved by the creation of new numbers such that the domain of numbers shall gain the same completeness, or as we may say at once, the same continuity, as the straight fine.
The previous considerations are so familiar and well known to all that many will regard their repetition quite superfluous. Still I regarded this recapitulation as necessary to prepare properly for the main question. For, the way in which the irrational numbers are usually introduced is based directly upon the con- ception of extensive magnitudes — which itself is nowhere carefully defined — and explains number as the result of measuring such a magnitude by another of the same kind.^ Instead of this I demand that arithmetic shall be developed out of itself.
^The apparent advantage of the generality of this definition of number disappears as soon as we consider complex numbers. According to my view, on the other hand, the notion of the ratio between two numbers of the same kind can be clearly develojjed only after the introduction of irrational numbers.
DEDEKIND 3 7
That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number- concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.
The above comparison of the domain R of rational numbers with a straight hne hf's led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. In what then does this continuity consist? Every- thing must depend on the answer to this question, and only through it shall we obtain a scientific basis for the investigation of all continuous domains. By vague remarks upon the unbroken connection in the smallest parts obviously nothing is gained; the problem is to indicate a precise characteristic of continuity that can serve as the basis for vahd deductions. For a long time I pondered over this in vain, but finally I found what I was seeking. This discovery will, perhaps, be differently estimated by different people; the majority may find its substance very commonplace. It consists of the following. In the preceding section attention was called to the fact that every point p of the straight line pro- duces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i. e., in the follow- ing principle:
"If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which pro- duces this division of all points into two classes, this severing of the straight line into two portions."
As already said I think I shall not err in assuming that every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed. To this I may say that I am glad if every one finds the above
38 SOURCE BOOK IN MATHEMATICS
principle so obvious and so in harmony with his own ideas of a line; for I am utterly unable to adduce any proof of its correctness, nor has any one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. If space has at all a real existence it is not necessary for it to be con- tinuous; many of its properties would remain the same even were it discontinuous. And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it continuous; this filling up would consist in a creation of new point- individuals and would have to be effected in accordance with the above principle.
IV.
CREATION OF IRRATIONAL NUMBERS.
From the last remarks it is sufficiently obvious how the discon- tinuous domain R of rational numbers may be rendered complete so as to form a continuous domain. In Section I it was pointed out that every rational number o effects a separation of the system R into two classes such that every number oi of the first class Ai is less than every number a2 of the second class A2; the number a is either the greatest number of the class Ai or the least number of the class A 2. If now any separation of the system R into two classes Ai, A2, is given which possesses only this characteristic property that every number oi in Ai is less than every number 02 in A 2, then for brevity we shall call such a separation a cut [Schnitt] and designate it by (Ai, A2). We can then say that every rational number a produces one cut or, strictly speaking, two cuts, which, however, we shall not look upon as essentially different; this cut possesses, besides, the property that either among the numbers of the first class there exists a greatest or among the numbers of the second class a least number. And conversely, if a cut possesses this property, then it is produced by this greatest or least rational number.
But it is easy to show that there exist infinitely many cuts not produced by rational numbers. The following example suggests itself most readily.
Let D be a positive integer but not the square of an integer, then there exists a positive integer X such that X2<D<(X+ 1)2.
DEDEKIND 39
If we assign to the second class A 2, every positive rational number 02 whose square is >D, to the first class Ai all other rational numbers ai, this separation forms a cut (Ai, A2), i. e., every number ai is less than every number 02. For if ai = 0, or is negative, then on that ground Oi is less than any number 02, because, by definition, this last is positive; if ai is positive, then is its square ^D, and hence ai is less than any positive number 02 whose square is >D.
But this cut is produced by no rational number. To demon- strate this it must be shown first of all that there exists no rational number whose square =D. Although this is known from the first elements of the theory of numbers, still the following indirect proof may find place here. If there exist a rational number whose square = D, then there exist two positive integers t, u, that satisfy the equation
f2-Du2 = 0,
and we may assume that u is the least positive integer possessing the property that its square, by multiphcation by D, may be converted into the square of an integer t. Since evidently
\u<t<{\+l)u,
the number u' = t—\u is a positive integer certainly less than u. If further we put
t' = Du-\t,
t' is likewise a positive Integer, and we have
f'2_Du'2=(X2-D)(f2-Du2) = 0,
which is contrary to the assumption respecting u.
Hence the square of every rational number x is either <D or >D. From this it easily follows that there is neither in the class A I a greatest, nor in the class A2 a least number. For if we put
x(x2+3D)
y =
we have
3x^+D
^ 2x{D-x^) ^ ^ 3x2+D and
^ (3x2+D)2
If In this we assume x to be a positive number from the class Ai, then x^<D, and hence y>x and y^<D. Therefore y likewise belongs to the class Ai. But if we assume x to be a number from
40 SOURCE BOOK IN MATHEMATICS
the class As, then x^>D, and hence y<x, y>0, and y^>D. Therefore y likewise belongs to the class A 2. This cut is therefore produced by no rational number.
In this property that not all cuts are produced by rational numbers consists the incompleteness ordiscontinuity of the domain R of all rational numbers.
Whenever, then, we have to do with a cut (Ai, A 2) produced by no rational number, we create a new, an irrational number a, which we regard as completely defined by this cut (Ai, A2); we shall say that the number a corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as different or unequal always and only when they correspond to essentially different cuts.
In order to obtain a basis for the orderly arrangement of all real, i. e., of all rational and irrational numbers we must investigate the relation between any two cuts (Ai, A 2) and (Bi, B2) produced by any two numbers a and j8. Obviously a cut (Ai, A 2) is given completely when one of the two classes, e. g., the first Ai is known, because the second A 2 consists of all rational numbers not con- tained in Ai, and the characteristic property of such a first class lies in this that if the number Oi is contained in it, it also contains all numbers less than ai. If now we compare two such first classes Ai, JBi with each other, it may happen
1. That they are perfectly identical, i. e., that every number contained in Ai is also contained in Bi, and that every number contained in fii is also contained in Ai. In this case A 2 is neces- sarily identical with J52, and the two cuts are perfectly identical, which we denote in symbols by a = /3 or /3 = a.
But if the two classes Ai, Bi are not identical, then there exists in the one, e. g., in Ai, a number a'i=«6'2 not contained in the other Bi and consequently found in B2; hence all numbers 61 contained in Bi are certainly less than this number a'i = 6'2 and therefore all numbers 61 are contained in Au
2. If now this number a'l is the only one in Ai that is not contained in Bi, then is every other number ai contained in Ai also contained in Bi and is consequently <a'i, i. e., a\ is the greatest among all the numbers oi, hence the cut (Ai, A 2) is produced by the rational number a = a'i = 6'2. Concerning the other cut (Bi, B2) we know already that all numbers 61 in Bi are also contained in Ax and are less than the number a'i = h'i which is
DEDEKIND 41
contained in B%', every other number 62 contained in Bz must, however, be greater than h\, for otherwise it would be less than a'l, therefore contained in Ai and hence in Bx\ hence ^'2 is the least among all numbers contained in B2, and consequently the cut (Bi, B2) is produced by the same rational number /3 = 6'2 = a'i = a. The two cuts are then only unessentially different.
3. If, however, there exist in A 1 at least two different numbers a'i = 6'2 and a"i = h"-2, which are not contained in Bi, then there exist infinitely many of them, because all the infinitely many numbers lying between a\ and a" i are obviously contained in Ai (Section I, 11) but not in Bi. In this case we say that the numbers a and j3 corresponding to these two essentially different cuts (Ai, A2) and (Bi, Bi) are different, and further that a is greater than j3, that /3 is less than a, which we express in symbols by a>/3 as well as /3<a. It is to be noticed that this definition coin- cides completely with the one given earlier, when a, /3 are rational.
The remaining possible cases are these:
4. If there exists in Bi one and only one number b\ = a'2, that is not contained in Ai then the two cuts (Ai, A2) and (Bi, B2) are only unessentially different and they are produced by one and the same rational number a = a'2 = 6'i = /3.
5. But if there are in Bi at least two numbers which are not contained in Ai, then /3>a, a</3.
As this exhausts the possible cases, it follows that of two different numbers one is necessarily the greater, the other the less, which gives two possibiHties. A third case is impossible. This was indeed involved in the use of the comparative (greater, less) to designate the relation between a, /3; but this use has only now been justified. In just such investigations one needs to exercise the greatest care so that even with the best intention to be honest he shall not, through a hasty choice of expressions borrowed from other notions already developed, allow himself to be led into the use of inadmissible transfers from one domain to the other.
If now we consider again somewhat carefully the case a>/3 it is obvious that the less number /3, if rational, certainly belongs to the class A 1; for since there is in Ai a number a'i = b'2 which belongs to the class B2, it follows that the number /3, whether the greatest number in Bi or the least in B2 is certainly ^a'l and hence con- tained in Ai. Likewise it is obvious from a>/3 that the greater number a, if rational, certainly belongs to the class B2, because a^a'i. Combining these two considerations we get the following
42 SOURCE BOOK IN MATHEMATICS
result: If a cut is produced by the number a then any rational number belongs to the class A i or to the class A 2 according as it is less or greater than a; if the number a is itself rational it may belong to either class.
From this we obtain finally the following: If a>^, i. e., if there are infinitely many numbers in Ai not contained in Bi then there are infinitely many such numbers that at the same time are differ- ent from a and from /S; every such rational number c is <a, because it is contained in Ai and at the same time it is >/3 because contained in ^2.
V.
CONTINUITY OF THE DOMAIN OF REAL NUMBERS.
In consequence of the distinctions just established the system $R of all real numbers forms a well-arranged domain of one dimen- sion; this is to mean merely that the following laws prevail:
I. If a>/3, and IS>y, then is also a>y. We shall say that the number /3 lies between a and 7.
II. If a, 7 are any two different numbers, then there exist infinitely many diff^erent numbers /3 lying between a, 7.
III. If a is any definite number then all numbers of the system 9? fall into two classes Ui and U2 each of which contains infinitely many individuals; the first class Ui comprises all the numbers ai that are less than a, the second U2 comprises all the numbers 02 that are greater than a; the number a itself may be assigned at pleasure to the first class or to the second, and it is respectively the greatest of the first or the least of the second class. In each case the separation of the system 9i into the two classes Ui, U2 is such that every number of the first class Ui is smaller than every number of the second class U2 and we say that this separation is produced by the number a.
For brevity and in order not to weary the reader I suppress the proofs of these theorems which follow immediately from the defini- tions of the previous section.
Beside these properties, however, the domain 9? possesses also continuity; 1. e., the following theorem is true:
IV. If the system 9? of all real numbers breaks up into two classes Ui, U2 such that every number ai of the class Ui is less than every number 02 of the class U2 then there exists one and only one number a by which this separation is produced.
DEDEKIND 43
ProoJ. By the separation or the cut of $R into Ui and U2 we obtain at the same time a cut (Ai, A 2) of the system R of all rational numbers which is defined by this that Ai contains all rational numbers of the class Ui and A 2 all other rational numbers, i. e., all rational numbers of the class U2. Let a be the perfectly definite number which produces this cut (Ai, A2). If ^3 is any number different from a, there are always infinitely many rational numbers c lying between a and ^. If /3<a, then c<a; hence c belongs to the class Ai and consequently also to the class Ui, and since at the same time ^<c then /? also belongs to the same class Ui, because every number in U2 is greater than every number c in Ui. But if )3>a, then is c>a; hence c belongs to the class A 2 and consequently also to the class U2, and since at the same time /3>c, then j3 also belongs to the same class U2, because every number in Ui is less than every number c in U2. Hence every number /3 different from a belongs to the class Ui or to the class U2 according as /3<a or fi>a\ consequently a itself is either the greatest number in Ui or the least number in Us, i. e., a is one and obviously the only number by which the separation of 9? into the classes Ui, U2 is produced. Which was to be proved.
VI.
OPERATIONS WITH REAL NUMBERS.
To reduce any operation with two real numbers a, /3 to operations with rational numbers, it is only necessary from the cuts (Ai, A 2), (Bi, B2) produced by the numbers a and /3 in the system R to define the cut (Ci, C2) which is to correspond to the result of the operation, 7. I confine myself here to the discussion of the simplest case, that of addition.
If c is any rational number, we put it into the class Ci, provided there are two numbers one ai in Ai and one 61 in Bi such that their sum ai+6i^c; all other rational numbers shall be put into the class C2. This separation of all rational numbers into the two classes Ci, C2 evidently forms a cut, since every number c\ inCi is less than every number c% in C2. If both a and jS are rational, then every number ci contained in Ci is ^a+/3, because Oi^a, fei^/3, and therefore ai+6i^a+i3; further, if there were contained in Cz a number C2<a4-j3, hence a+i3 = C2+p, where p is a positive rational number, then we should have
C2=(a-ip) + (i5-ip),
44 SOURCE BOOK IN MATHEMATICS
which contradicts the definition of the number C2, because a — ^p is a number in Ai, and jS — ^p a number in Br, consequently every number cz contained in Cz is ^a+jS. Therefore in this case the cut (Ci, C2) is produced by the sum a+^S. Thus we shall not violate the definition which holds in the arithmetic of rational numbers if in all cases we understand by the sum a+j3 of any two real numbers a, /S that number y by which the cut (Ci, €■>) is produced. Further, if only one of the two numbers a, ^ is rational, e, g., a, it is easy to see that it makes no difference with the sum Y = a+i3 whether the number a is put into the class Ai or into the class A 2.
Just as addition is defined, so can the other operations of the so-called elementary arithmetic be defined, viz., the formation of differences, products, quotients, powers, roots, logarithms, and in this way we arrive at real proofs of theorems (as, e. g., \/T'\/3 = \/6)> which to the best of my knowledge have never been established before. The excessive length that is to be feared in the definitions of the more complicated operations is partly inher- ent in the nature of the subject but can for the most part be avoided. Very useful in this connection is the notion of an interval, i. e., a system A of rational numbers possessing the following characteristic property: if a and a' are numbers of the system A, then are all rational numbers lying between a and a' contained in A. The system R of all rational numbers, and also the two classes of any cut are intervals. If there exist a rational number ai which is less and a rational number 02 which is greater than every number of the interval A, then A is called a finite interval; there then exist infinitely many numbers in the same condition as Oi and infinitely many in the same condition as oo; the whole domain R breaks up into three parts Ai, A, A2 and there enter two perfectly definite rational or irrational numbers ai, a-i which may be called respectively the lower and upper (or the less and greater) limits of the interval; the lower limit ai is determined by the cut for which the system Ai forms the first class and the upper 02 by the cut for which the system A 2 forms the second class. Of every rational or irrational number a lying between ai and aa it may be said that it lies within the interval A. If all numbers of an interval A are also numbers of an interval B, then A is called a portion of B.
Still lengthier considerations seem to loom up when we attempt to adapt the numerous theorems of the arithmetic of rational
DEDEKIND 45
numbers (as, e. g., the theorem {a-\-h)c = ac-\-hc) to any real numbers. This, however, is not the case. It is easy to see that it all reduces to showing that the arithmetic operations possess a certain continuity. What I mean by this statement may be expressed in the form of a general theorem :
"If the number X is the result of an operation performed on the numbers a, /3, 7, . . . and X lies within the interval L, then inter- vals A, B, Cy . . , can be taken within which He the numbers a, jS, 7, . . . such that the result of the same operation in which the numbers a, /3, 7, . . . are replaced by arbitrary numbers of the intervals A, B, C, . . . is always a number lying within the interval L." The forbidding clumsiness, however, which marks the statement of su^h a theorem convinces us that something must be brought in as an aid to expression; this is, in fact, attained in the most satisfactory way by introducing the ideas of variable magnitudes, Junctions, limiting values, and it would be best to base the definitions of even the simplest arithmetic operations upon these ideas, a matter which, however, cannot be carried further here.
WALLIS On Lmaginary Numbers
(Selected from the English Version, by Professor David Eugene Smith, Teachers College, Columbia University, New York City)
John Wallis (1616-1703), Savilian professor of geometry at Oxford (1649- 1705), contemporary of Newton (see also p. 217), was the first to make any considerable contribution to the geometric treatment of imaginary numbers. This appeared in his Algebra (1673), cap. LXVI (Vol. II, p. 286), of the Latin edition. The following extract is from his English translation:
CHAP. LXVI.i 0/ Negative Squares, mid their Imaginary Roots in Algebra.
We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary' Roots, (as contradistinguished to what they call Real Roots, whether Affirmative or Negative:) But referred the fuller consideration of them to this place.
These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square, (when they happen,) are reputed to imply that the Case proposed is Impossible.
And so indeed it is, as to the first and strict notion of what is proposed. For it is not possible, that any Number (Negative or Affirmative) Multiplied into itself, can produce (for instance) —4. Since that Like Signs (whether + or — ) will produce +; and therefore not —4.
But it is also Impossible, that any Quantity (though not a Supposed Square) can be Negative. Since that it is not possible that any Magnitude can be Less than Nothing, or any Number Fewer than None.
Yet^ is not that Supposition (of Negative Quantities,) either Unuseful or Absurd; when rightly understood. And though, as to the bare Algebraick Notation, it import a Quantity less than nothing: Yet, when it comes to a Physical Application, it denotes as Real a Quantity as if the Sign were + ; but to be interpreted in a contrary sense.
1 [Page 264.] « [Page 265.]
46
WALLIS 47
As for instance: Supposing a man to have advanced or moved forward, (from A to B,) 5 Yards; and then to retreat (from B to C) 2 Yards: If it be asked, how much he had Advanced (upon the whole march) when at C? or how many Yards he is now Forwarder than when he was at A? I find (by Subducting 2 from 5,) that he is Advanced 3 Yards. (Because +5 — 2 = +3.)
D A C B
\-V-\-\ I I I I I
But if, having Advanced 5 Yards to B, he thence Retreat 8 Yards to D; and it be then asked. How much he is Advanced when at D, or how much Forwarder than when he was at A: I say —3 Yards. (Because +£ -8 = —3.) That is to say, he is advanced 3 Yards less than nothing.
Which in propriety of Speech, cannot be, (since there cannot be less than nothing.) And therefore as to the Line AB Forward, the case is Impossible.
But if (contrary to the Supposition,) the Line from A, be con- tinued Backward, we shall find D, 3 Yards Behind A. (Which was presumed to be Before it.)
And thus to say, he is Advanced — 3 Yards; is but what we should say (in ordinary form of Speech), he is Retreated 3 Yards; or he wants 3 Yards of being so Forward as he was at A.
Which doth not only answer Negatively to the Question asked. That he is not (as was supposed,) Advanced at all: But tells moreover, he is so far from being advanced, (as was supposed) that he is Retreated 3 Yards; or that he is at D, more Backward by 3 Yards, than he was at A.
And consequently — 3, doth as truly design the Point D; as + 3 designed the Point C. Not Forward, as was supposed; but Backward, from A.
So that + 3, signifies 3 Yards Forward; and — 3, signifies 3 Yards Backward: But still in the same Streight Line. And each designs (at least in the same Infinite Line,) one Single Point: And but one. And thus it is in all Lateral Equations; as having but one Single Root.
Now what is admitted in Lines, must on the same Reason, be allowed in Plains also.
As for instance: Supposing that in one Place, we Gain from the Sea, 30 Acres, but Lose in another Place, 20 Acres: If it be now asked. How many Acres we have gained upon the whole: The
48 SOURCE BOOK IN MATHEMATICS
Answer is, 10 Acres, or + 10. (Because of 30-20 = 10.) Or, which is all one 1600 Square Perches. (For the Etiglish Acre being Equal to a Plain of 40 Perches in length, and 4 in breadth, whose Area is 160; 10 Acres will be 1600 Square Perches.) Which if it lye in a Square Form, the Side of that Square will be 40 Perches in length; or (admitting of a Negative Root,) — 40.
But if then in a Third place, we lose 20 Acres more; and the same Question be again asked. How much we have gained in the whole; the Answer must be — 10 Acres. (Because 30 — 20 — 20= —10.) That is to say. The Gain is 10 Acres less than nothing. Which is the same as to say, there is a Loss of 10 Acres: or of 1600 Square Perches.
And hitherto, there is no new Difficulty arising, nor any other Impossibihty than what we met with before, (in supposing a Negative Quantity, or somewhat Less than nothing:) Save only that ■\/l600 is ambiguous; and may be + 40, or — 40. And from such Ambiguity it is, that Quadratick Equations admit of Two Roots,
But now (supposing this Negative Plain, — 1600 Perches, to be in the form of a Square;) must not this Supposed Square be supposed to have a Side? And if so, What shall this Side be?
We^ cannot say it is 40, nor that it is —40. (Because either of these Multiplyed into itself, will make 4- 1600; not —1600).
But thus rather, that it is V~1600, (the Supposed Root of a Negative Square;) or (which is Equivalent thereunto) 10 V~16, or20 V-4, or 40 V-l-
Where V implies a Mean Proportional between a Positive and a Negative Quantity. For like sls \/ b c signifies a Mean Propor- tional between -{-b and +c; or between —6, and — c; (either of which, by Multiplication, makes +6c:) So doth \/ — bc signify a Mean Proportional between + 6 and — c, or between — b and + c; either of which being Multiphed, makes —be. And this as to Algebraick consideration, is the true notion of such Imaginary Root, y/ — be.
CHAP. LXVII. The same Exemplified in Geometry.
What hath been already said of \/~ ^c in Algebra, (as a Mean Proportional between a Positive and a Negative Quantity:) may be thus ExempHfied in Geometry.
1 [Page 266].
WALLIS
49
If (for Instance,) Forward from A, I take A B = +6; and Forward from thence, BC =+c; (making AC = +AB4-BC= +6+c, the Diameter of a Circle:) Then is the Sine, or Mean Proportional BP=V+6c.
But if Backward from A, I take AB =—b; and then Forward from that B, BC=+c; (making AC = — AB+BC= — 6-Hc, the Diameter of the Circle:) Then is the Tangent or Mean Propor- tional BP = \/-bc.
So that where ^/-{-b c signifies a Sine; yZ—b c shall signify a Tangent, to the same Arch (of the same Circle,) AP, from the same Point P, to the same Diameter AC.
Suppose now (for further Illustration,) A Triangle standing on the Line AC (of indefinite length;) whose one Leg AP = 20 is given; together with (the Angle PAB, and consequently) the Height PC =12; and the length of the other Leg PB = 15: By which we are to find the length of the Base AB.
'Tis manifest that the Square of AP being 400; and of PC, 144; their Difference 256 (= 400-144) is the Square of AC.
And therefore AC(= \/2S6) = +16, or —16; Forward or Back- ward according as we please to take the Affirmative or Negative Root. But we will here take the Affirmative.
Then, because the Square of PB is 225; and of PC, 144; their Difference 81, is the Square of CB. And th-erefore CB = \/81; which is indifferently, +9 or —9: And may therefore be taken Forward or Backward from C. Which gives a Double value for the length of AB; to wit, AB = 16+9 = 25, or AB = 16-9 = 7. Both Affirmative. (But if we should take. Backward from A, AC=-16; AB=-16+9=-7, and AB = -16-9 = -25. Both Negative.)
Suppose^ again, AP=15, PC = 12, (and therefore AC = \/: 225 -144 : =V81=9,) PB = 20 (and therefore BC = V:400-144:
MPart 267.]
50
SOURCE BOOK IN MATHEMATICS
= V2S6 = +16, or -16:) Then is AB = 9+16 = 25, or AB = 9— 16= — 7. The one Affirmative, the other Negative. (The same values would be, but with contrary Signs, if we take AC = V81 = -9: That is, AB = -9+16 = +7, AB= -9-16= -25.)
In all which cases, the Point B is found, (if not Forward, at least Backward,) in the Line AC, as the Question supposeth.
And of this nature, are those Quadratick Equations, whose Roots are Real, (whether Affirmative or Negative, or partly the one, partly the other;) without any other Impossibihty than (what is incident also to Lateral Equations,) that the Roots (one or both) may be Negative Quantities.
3.
But if we shall Suppose, AP = 20, PB = 12, PC=15, (and there- fore AC = \/175:) When we come to Subtract as before, the Square of PC (225,) out of the Square PB (144,) to find the Square of BC, we find that cannot be done without a Negative Remainder, 144-225 = -81.
So that the Square of BC is (indeed) the Difference of the Squares of PB, PC; but a defective Deference; (that of PC proving the greater, which was supposed the Lesser; and the Triangle PBC, Rectangled, not as was supposed at C, but at B:) And therefore BC=V-81.
Which gives indeed (as before) a double value of AB, \/175, -f-V— 81, and \/l7S, — V — 81: But such as requires a new Impossibihty in Algebra, (which in Lateral Equations doth not happen;) not that of a Negative Root, or a Quantity less than nothing; (as before,) but the Root of a Negative Square. Which in strictness of speech, cannot be: since that no Real Root (Affirma- tive or Negative,) being MuItipHed into itself, will make a Nega- tive Square.
This Impossibility in Algebra, argues an Impossibihty of the case proposed in Geometry; and that the Point B cannot be had, (as was supposed,) in the Line AC, however produced (forward or backward,) from A.
WALLIS 51
Yet are there Two Points designed (out of that Line, but) in the same Plain; to either of which, if we draw the Lines AB, BP, we have a Triangle; whose Sides AP, PB, are such as were required: And the Angle PAC, and Altitude PC, (above AC, though not above AB,) such as was proposed; And the Difference of Squares of PB, PC, is that of CB.
And Hke as in the first case, the Two values of AB (which are both Affirmative,) make the double of AC, (16+9, +16 — 9, = 16+16 = 32:) So here, V175+V-81, +V175-V'-81, = 2V175.
And (in the Figure,) though not the Two Lines themselves, AB, AB, (as in the First case, where they lay in the Line AC;) yet the Ground-hnes on which they stand, A/3, A/S, are Equal to the Double of AC: That is, if to either of those AB, we join Ba, equal to the other of them, and with the fame Declivity; ACa (the Distance of Aa) will be a Streight Line equal to the double of AC; as is AC a in the First case.
The greatest difference is this; That in the first Case, the Points B, B, lying in the Line AC, the Lines AB, AB, are the fame with their Ground-Lines, but not so in this last case, where BB are so raised above j3 13 (the respective Points in their Ground-Lines, over which they stand,) as to make the case feasible; (that is, so much as is the versed Sine of CB to the Diameter PC:) But in both ACa (the Ground-Line of ABa) is Equal to the Double of AC.
So that, whereas in case of Negative Roots, we are to say. The Point B cannot be found, so as is supposed in AC Forward, but Backward from A it may in the same Line: We must here say, in case of a Negative Square, the Point B cannot be found so as was supposed, in the Line AC; but Above that Line it may in the same Plain.
This I have the more largely insisted on, because the Notion (I think) is new; and this, the plainest Declaration that at present I can think of, to exphcate what we commonly call the Imaginary Roots of Quadratick Equations. For such are these.
For instance; The Two Roots of this Equation, aa — 2a\/l7S +256 = 0;area = V175+V-81,anda = -v/175-V-81. (Which are the values of AB in the last case.) For if from 175 (the Square of half the Coefficient,) we Subduct the Absolute Quantity 256, the Remainder is —81; the Root of which. Added to, and Sub- ducted from, the half Coefficient,) makes \/l75±\/ — 8l: Which are therefore the Two Roots of that Equation. In the same man-
52 SOURCE BOOK IN MATHEMATICS
ner as in the Equation 0^—32 a+17S=0; if from 256 (the Square of Half 32,) we Subduct 175, the Remainder is +81; whose Root V81 =9, Added to and Subducted from, 16 (the half Coefficient,) makes 16 + 9; which are the values of A B in the First case.
CHAP. Lxvni.
The Geometrical Construction accommodated hereunto.
In the former Chapter, we have shewed what in Geometry answers to the Root of a Negative Square in Algebra.
I shall now shew some Geometrical EfFections, answering to the Resolution of such Quadratick Equations whose Roots may have (what we call) Imaginary values, arising from such Negative Squares.
The natural Construction of this Equation aa + ba-\-a=0; is this. The Coefficient b being the Sum of Two Quantities, whose Rectangle is a, the Absolute Quantity: This cannot be more
naturally expressed, in Magnitudes, than
jf by making b { = Aa) the Diameter of a
^^^^^-J/>""~'^>\^ Circle, and \/a ( = BS) a Right Sine or
/ \ i ; \ Ordinate thereunto. (For it is one of the
/ \\ \ \ most known Properties of a Circle, that
A B C B ^ ^^^ ^^^^ ^^ Ordinate is a mean Propor- tional between the Two Segments of the Diameter.) And because BS (of the same length,) may be taken indifferently on either side of CT, we have therefore, in the Diameter, two Points B, B, (answering to SS in the Semi- circumference,) either of which divide the Diameter into AB, Ba, the Two Roots desired. (Both Affirmative, or both Negative, according as in the Equation we have —6a, or +6a.) And as BS increaseth, so B approacheth (on either Side) to C; and CB (the Co-Sine, or Semi-difference of Roots,) decreaseth.
But because the Sine BS can never be greater than CT the Semidiameter: Therefore, whenever y/a. is greater than \h\ the Case according to this construction is Impossible.
1. The Geometrical Effection, therefore answering to this Equation, a a + b a-\-a = 0, (so as to take in both cases at once. Possible and Impossible; that is, whether |6 6 be or be not less than a;) may be this.
On^ ACa = 6, bisected in C, erect a Perpendicular CP=\/a- And taking PB = |6, make (on whether Side you please of CP,)
* [Page 269 on this p. 52, a has been used for the as of Wallis.]
WALL/5 53
PBC, a Rectangled Triangle. Whose Right Angle will therefore be at C or B, according as PB or PC is bigger; and accordingly, BC a Sine or a Tangent, (to the Radius PB,) terminated in PC.
The Streight Lines AB, B a, are the two values of a. Both Affirmative if (in the Equation,) it be — 6a: Both Negative, if -\-b a. Which values be (what we call) Real, if the Right-Angle be at C: But Imaginary if at B.
P
A-^^
P
p C ^
In both cases (whether the Right Angle be at C or B,) the Point B may indifferently be taken on either side of PC, in a like Posi- tion. And the Two Points B, B, are those which the Equation designs.
In the former case; ABa is a Streight Line, and the same with ACa.
In the latter; ABa makes at B, such an Angle, as that ACa is the distance of Aa; and is the Ground-hne, on which if ABa be Ichnographically projected, B falls on /3, the point just under it.
And therefore, if (in the Problem which produceth this Equa- tion) ABa were supposed to be a Streight Line; or the Point B, in the Line ACa; or the same with /3; or that ACa be Equal to the Aggregate of AB + Ba; or any thing which doth imply any of these: This Construction shews that Case (so understood) to be Impossible; but how it may be qualified, so as to become possible.
The difi"erence between this ImpossibiHty, and that incident to a Lateral Equation, is this. When in a Lateral Equation, we are reduced to a Negative value; it is as much as to say the Point B demanded, cannot be had (in the Line AC proposed,) Forward from A, as is presumed: But backward from A it may, at such a distance Behind it. But when in a Quadratick Equation, we be reduced, (not to a Negative value; wherein it communicates with the Lateral; but) to (what is wont to be called) an Imaginary value; it is as much as to say, The Point B cannot be had in the Line AC, as was presumed; but, out of that Line it may (in the same Plain;) at such a distance Above it.
The other form of Quadratick Equations, aa^ha — x = 0\ is naturally thus Eff"ected. Taking CA, or CP, = \h\ and PB
54
SOURCE BOOK IN MATHEMATICS
= \/3e; containing a Right Angle at P. The Hypothenuse, BC
continued, will cut the Circle PAa, in Aa. And the two Roots
desired, are AB, Ba; between which the
Tangent PB is a mean Proportional, and Aa
their Difference. But one of them is to be
understood Affirmative, the other Negative.
(Because if AB be Forward, Ba is Backward;
if that be Backward, this Forward.) To wit,
+AB, — Ba, if we have (in the Equation)
4-6a; or — AB, +Ba, if —6a.
But this Construction belongs not properly to this place: Because
in this form of Equation, we are never reduced to these Imaginary
values. For PB, of whatever length, may be a Tangent to that
Circle.
WESSEL On Complex Numbers
(Translated from the Danish by Professor Martin A. Nordgaard, St. Olaf College, Northfield, Minnesota.)
Caspar Wessel (1745-1818) was a Norwegian surveyor. In 1797, he read a paper upon the graphic representation of complex numbers. The paper was printed in 1798 and appeared in the memoirs of the Royal Academy of Denmark in 1799. This paper may be said to have been the first noteworthy attempt at the modern method. Within a few years thereafter, numerous other attempts were made, all leading to similar results (see Smith, History of Matbemalics, Vol. II, pp. 263-267). Wessel's work attracted little attention at the time and was almost unknown until the French translation appeared in 1897. The present translation of certain essential passages is made from the original Danish.
On the Analytical Representation of Direction;
AN Attempt,^
Applied Chiefly to the Solution of Plane and Spherical Polygons (By Caspar Wessel, Surveyor.)
This present attempt deals with the question, how may we represent direction analytically; that is, how shall we express right lines so that in a single equation involving one unknown line and others known, both the length and the direction of the unknown hne may be expressed.
To help answer this question I base my work on two propositions which to me seem undeniable. The first one is: changes in direc- tion which can be effected by algebraic operations shall be indi- cated by their signs. And the second: direction is not a subject for algebra except in so far as it can be changed by algebraic operations. But since these cannot change direction (at least, as
^ [In recent histories of mathematics, there have come about very misleading translations into English of Wessel's title word "forsog" as "essay on, etc." This possibility comes from the word "essai" used in the French translation of Wessel's memoir, the French word meaning both an attempt or endeavor, and a treatise (essay.) Wessel's word "forsog" can only mean attempt or endeavor.]
SS
56 SOURCE BOOK IN MATHEMATICS
commonly explained) except to its opp>osite, that is, from positive to negative, or vice versa, these two are the only directions it should be possible to designate, by present methods; for the other directions the problem should be unsolvable. And I suppose this is the reason no one has taken up the matter.^ It has undoubtedly been considered impermissible to change anything in the accepted explanation of these operations.
And to this we do not object so long as the explanation deals only with quantities in general. But when in certain cases the nature of the quantities dealt with seems to call for more precise definitions of these operations and these can be used to advantage, it ought not to be considered impermissible to offer modifications. For as we pass from arithmetic to geometric analysis, or from operations with abstract numbers to those with right lines, we meet with quantities that have the same relations to one another as numbers, surely; but they also have many more. If we now give these operations a wider meaning, and do not as hitherto limit their use to right lines of the same or opposite direction; but if we extend somewhat our hitherto narrow concept of them so that it becomes applicable not only to the same cases as before, but also to infinitely many more; I say, if we take this liberty, but do not violate the accepted rules of operations, we shall not con- travene the first law of numbers. We only extend it, adapt it to the nature of the quantities considered, and observe the rule of method which demands that we by degrees make a diflficult principle intelhgible.
It is not an unreasonable demand that operations used in geometry be taken in a wider meaning than that given to them in arithmetic. And one will readily admit that in this way it should be possible to produce an infinite number of variations in the directions of lines. Doing this we shall accomplish, as will be proved later, not only that all impossible operations can be avoided — and we shall have light on the paradoxical statement that at times the possible must be tried by impossible means — , but also that the direction of all lines in the same plane can be expressed as analytically as their lengths without burdening the mind with new signs or new rules. There is no question that the general validity of geometric propositions is frequently seen with greater ease if direction can be indicated analytically and governed by alge-
1 Unless it be Magister Gilbert, in Halle, whose prize memoir on Calculus Situs possibly contains an explanation of this subject.
W ESS EL 57
braic rules than when ft is represented by a figure, and that only in certain cases. Therefore it seems not only permissible, but actually profitable, to make use of operations that apply to other lines than the equal (those of the same direction) and the opposite. On that account my aim in the following chapters will be:
I. First, to define the rules for such operations; II. Next, to demonstrate their application when the lines are in the same plane, by two examples;
III. To define the direction of lines lying in diff'erent planes by a new method of operation, which is not algebraic;
IV. By means of this method to solve plane and spherical poly- gons;
V. Finally, to derive in the same manner the ordinary formulas of spherical trigonometry.
These will be the chief topics of this treatise. The occasion for its being was my seeking a method whereby I could avoid the impossible operations; and when I had found this, I appHed it to convince myself of the universality of certain well-known formulas. The Honorable Mr. Tetens, Councillor-of-state, was kind enough to read through these first investigations. It is due to the encour- agement, counsel, and guidance of this distinguished savant that this paper is minus some of its first imperfections and that it has been deemed worthy to be included among the publications of the Royal Academy.
A Method Whereby from Given Right Lines to Form Other
Right Lines by Algebraic Operations; and How to
Designate Their Directions and Signs
Certain homogeneous quantities have the property that if they are placed together, they increase or diminish one another only as increments or decrements.
There are others which in the same situation effect changes in one another in innumerable other ways. To this class belong right lines.
Thus the distance of a point from a plane may be changed in innumerable ways by the point describing a more or less inclined right line outside the plane.
For, if this line is perpendicular to the axis of the plane, that is, if the path of the point makes a right angle with the axis, the
58 SOURCE BOOK IN MATHEMATICS
point remains in a plane parallel to the given plane, and its path has no effect on its distance from the plane.
If the described line is indirect, that is, if it makes an oblique angle with the axis of the plane, it will add to or subtract from the distance by a length less than its own; it can increase or diminish the distance in innumerable ways.
If it is direct, that is, in line with the distance, it will increase or diminish the same by its whole length; in the first case it is positive, in the second, negative.
Thus, all the right lines which can be described by a point are, in respect to their effects upon the distance of a given point from a plane outside the point, either direct or indirect or perpendicular^ according as they add to or subtract from the distance the whole, a part, or nothing, of their own lengths.
Since a quantity is called absolute if its value is given as immedi- ate and not in relation to another quantity, we may in the preced- ing definitions call the distance the absolute line; and the share of the relative line in lengthening or shortening the absolute line may be called the "effect" of the relative line.
There are other quantities besides right lines among which such relations exist. It would therefore not be a valueless task to explain these relations in general, and to incorporate their general concept in an explanation on operations. But I have accepted the advice of men of judgment, that in this paper both the nature of the contents and plainness of exposition demand that the reader be not burdened here with concepts so abstract. I shall conse- quently make use of geometric explanation only. These follow.
§1
Two right lines are added if we unite them in such a way that the second line begins where the first one ends, and then pass a right line from the first to the last point of the united lines. This line is the sum of the united lines.
For example, if a point moves forward three feet and backward two feet, the sum of these two paths is not the first three and the last two feet combined; the sum is one foot forward. For this path, described by the same point, gives the same effect as both the other paths.
' "Indifferent" would be a more fitting name were it not so unfamiliar to our ears.
W ESS EL 59
Similarly, if one side of a triangle extends from a to 6 and the other from 6 to c, the third one from a to c shall be called the sum. We shall represent it by ab + 6c, so that ac and ab + 6c have the same meaning; or ac = a6 + 6c = — 6a + 6c, if 6a is the opposite of a6. If the added lines are direct, this definition is in complete agreement with the one ordinarily given. If they are indirect, we do not contravene the analogy by calHng a right line the sum of two other right lines united, as it gives the same effect as these. Nor is the meaning I have attached to the symbol + so very
unusual; for in the expression a6 + ^ = :^a6 it is seen that -y is
not a part of the sum. We may therefore set a6 + 6c = ac without, on that account, thinking of 6c as a part of ac; a6 + 6c is only the symbol representing ac.
§2
If we wish to add more than two right lines we follow the same procedure. They are united by attaching the terminal point of the first to the initial point of the second and the terminal point of this one to the initial point of the third, etc. Then we pass a right line from the point where the first one begins to the point where the last one ends; and this we call their sum.
The order in which these lines are taken is immaterial; for no matter where a point describes a right line within three planes at right angles to one another, this line has the same effect on the distances of the point from each of the planes. Consequently any one of the added lines contributes equally much to the deter- mination of the position of the last point of the sum whether it have first, last, or any other place in the sequence. Consequently, too, the order in the addition of right lines is immaterial. The sum will always be the same; for the first point is supposed to be given and the last point always assumes the same position.
So that in this case, too, the sum may be represented by the added lines connected with one another by the symbol +. In a quadrilateral, for example, if the first side is drawn from a to 6, the second from 6 to c, the third from c to d, but the fourth from a to d, then we may write: ac? = a6 + 6c + cd.
§3 If the sum of several lengths, breadths and heights is equal to zero, then is the sum of the lengths, the sum of the breadths, and the sum of the heights each equal to zero.
60 SOURCE BOOK IN MATHEMATICS
§4
It shall be possible in every case to form the product of two right lines from one of its factors in the same manner as the other factor is formed from the positive or absolute line set equal to unity. That is:
Firstly, the factors shall have such a direction that they both can be placed in the same plane with the positive unit.
Secondly, as regards length, the product shall be to one factor as the other factor is to the unit. And,
Finally, if we give the positive unit, the factors, and the product a common origin, the product shall, as regards its direction, lie in the plane of the unit and the factors and diverge from the one factor as many degrees, and on the same side, as the other factor diverges from the unit, so that the direction angle of the product, or its divergence from the positive unit, becomes equal to the sum of the direction angles of the factors.
§5
Let +1 designate the positive rectilinear unit and +e a certain other unit perpendicular to the positive unit and having the same origin; then the direction angle of +1 will be equal to 0", that of -1 to 180^ that of +€ to 90°, and that of -e to -90° or 270°. By the rule that the direction angle of the product shall equal the sum of the angles of the factors, we have: ( + 1)(+1) = +1; ( + 1)(-1) = -1; (-1)(-1) = +1; (+l)(+6) = +6; (+l)(-6) = -.; (-!)(+,) = -,;(-l)(-e) = +e;(+.)(+0 = -l;(+6) (-0 = +l;(-e)(-6) = -1.
From this it is seen that t is equal to \/— 1; and the divergence of the product is determined such that not any of the common rules of operation are contravened.
§6
The cosine of a circle arc beginning at the terminal point of the radius +1 is that part of the radius, or of its opposite, which begins at the center and ends in the perpendicular dropped from the terminal point of the arc. The sine of the arc is drawn perpendicu- lar to the cosine from its end point to the end point of the arc.
Thus, according to §5, the sine of a right angle is equal to V— 1. Set \/— 1 = €. Let V be any angle, and let sin v represent a right line of the same length as the sine of the angle r, positive, if the measure of the angle terminates in the first semi-circumference.
WESSEL 61
but negative, if in the second. Then it follows from §§4 and 5 that € sin v expresses the sine of the angle v in respect to both direction and extent. . . .
§7 In agreement with §§1 and 6, the radius which begins at the center and diverges from the absolute or positive unit by angle v is equal to cos t; + « sin v. But, according to §4, the product of the two factors, of which one diverges from the unit by angle v and the other by angle u, shall diverge from the unit by angle V -{• u. So that if the right line cos v + e sin v is muItipHed by the right hne cos u + e sin u, the product is a right line whose direction angle is r + u. Therefore, by §§1 and 6, we may represent the product by cos (v -\- u) -\- e sin (v -\- u),
§8 The product (cos r + c sin v)(cos u + € sin u), or cos (r + u) + € sin {v + u), can be expressed in still another way, namely, by adding into one sum the partial products that result when each of the added lines whose sum constitutes one factor is multiplied by each of those whose sum constitutes the other. Thus, if we use the known trigonometric formulas
cos (v -\- u) = cos V cos u — sin v sin u, sin {v -\- u) = cos V sin u + cos u sin v,
we shall have this form:
(cos f + c sin r)(cos u + c sin u) = cos r cos u — sin u
+ c(cos r sin u -f- cos u sin v).
For the above two formulas can be shown, without great difficulty, to hold good for all cases, — be one or both of the angles acute or obtuse, positive or negative. In consequence, the propositions derived from these two formulas also possess universality.
§9 By §7 cos r + € sin r is the radius of a circle whose length is equal to unity and whose divergence from cos 0° is the angle v. It follows that r cos v + re sin v represents a right line whose length is r and whose direction angle is v. For if the sides of a right angled triangle increase in length r times, the hypotenuse increases r times; but the angle remains the same. However, by §1, the sum of the sides is equal to the hypotenuse; hence,
r cos V -\- re s'mv = r(cos r + c sin v).
62 SOURCE BOOK IN MATHEMATICS
This Is therefore a general expression for every right line which lies in the same plane with the lines cos 0° and e sin 90®, has the length r, and diverges from cos 0° by r degrees.
§10
If a, b, c denote direct lines of an}^ length, positive or negative, and the two indirect lines a -\- eb and c + ed lie in the same plane with the absolute unit, their product can be found, even when their divergences from the absolute unit are unknown. For we need only to multiply each of the added lines that constitute one sum by each of the lines of the other and add these products; this sum is the required product both in respect to extent and direction: so that (a + e6)(c + ed) = ac - bd + e{ad + be).
Proof. — Let the length of the line a -\- ebbe A, and its divergence from the absolute unit be v degrees; also let the length oi c -}- ed be C, and its divergence be u. Then, by §9, a + et = A cos v + Re sin r, and c -\- ed = C cos u -\- Ce sin u. Thus a = A cos v, b = A sin V, c = C cos u, d = C sin u (§3). But, by §4, (a + (a + c6)(c + €c/) = AC[cos (r + u) + e sin (r + u)] = Ac[cosvcosu — sin V sin u + 6(cos r sin u + cos u sin v)] (§8). Consequently, if instead of A C cos v cos u we write ac, and for A C sin v sin u write bd, etc., we shall derive the relation we set out to prove.
It follows that, although the added lines of the sum are not all direct, we need make no exception in the known rule on which the theory of equations and the theory of integral functions and their simple divisors are based, namely, that if two sums are to be multi- plied, then must each of the added quantities in one be multiplied by each of the added quantities in the other. It is, therefore, certain that if an equation deals with right lines and its root has the form a + ^b, then an indirect line is represented. Now, if we should want to multiply together right lines which do not both lie in the same plane with the absolute unit, this rule would have to be put aside. That is the reason why the multiplication of such lines is omitted here. Another way of representing changes of direction is taken up later, in §§24—35.
The quotient multiplied by the divisor shall equal the dividend. We need no proof that these lines must lie in the same plane with the absolute unit, as that follows directly from the definition in §4. It is easily seen also that the quotient must diverge from the absolute unit by angle v — u, ii the dividend diverges from the same unit by angle v and the divisor by angle u.
T^ewtoii
Fiontispiece
WESSEL 63
Suppose, for example, that we are to divide A (cos v -{- e sin r by B(cos u + € sin u). The quotient is
A
■d[cos (r — u) + € sin (v — u)] since
A
-^[cos (f — u) + e sin (v — u)] X B(cos u + c sin u)
= A (cos v + € sin v),
A by §7. That is, since -^Icos (r — u) + c sin (u — u)] multiplied
by the divisor J5(cos u + e sin u) equals the dividend A (cos v +
A € sin v), then -o[cos {v — u) -\- e sin {v — u)] must be that
required quotient. . . .
§12 If a, b, c, and d are direct lines, and the indirect lines a -\- eb and c -\- ed are in the same plane with the absolute unit: then
1 _ c — ed J ^, _ ^. ^ a -\- eb
c -\- ed c
c — ed
— ed , T . a -\- eb / , iv 1
- — r,; and the quotient — ; , = (a + eb). — ; ;
+ a^ ^ c -{- ed ^ c -\- ed
= (a + eb). '^f^^, = [ac + 6c/ + e{bc - ad)]:{c' + c/^).
For by §9 we may set a -{- eb = A (cos v -{- e sin v), and
c -{- ed = C(cos w + 6 sin u).
so that
Since
then
c — ed = C(cos u — € sin u), by §3. (c + €£/)(c - ed) = c2 + J2 = O, by §10,
^2-qrj2 = ^^^^^ w - € sin u), by §10; or
^H^V^ " C^^^^ (-^0 + € sin (-u)] = ^-:p^' by §11. Multiplying by a + e6 = A (cos r + e sin i'), gives (^ + ^^)-c2 _[_^j2 = -^^cos (r - w) + csin (r - u)] = " _^ J^ by §11.
Indirect quantities of this class have also this in common with direct, that if the dividend is a sum of several quantities, then each of these, divided by the divisor, gives a quotient, and the sum of these constitute the required quotient.
64 SOURCE BOOK IN MATHEMATICS
§13
If m is an integer, then cos \- e sin — multiplied by itself m
times gives the power cos v -{- e sin v (§7); therefore we have:
(cos t; + e sm v)'" = cos H e sm — .
m 771
But, according to §11,
COS ( I + e sm ( l = =
V , . V
COS f- e sm —
771 777
1 , I
J- = (cos r 4- € sm V) ">.
(cos r + e sin v)"* Consequently, whether 77i is positive or negative, it is always true that
cos f- € sm — = (cos r + € sm r)"*.
771 771
Therefore, if both 77i and 77 are integers, we have;
n , , • N - 77 , . 77
(cos f + 6 sm I')'" = cos — r + e sm —v.
m m
In this way we find the value of such expressions as \6 + c-\/— 1
or \a \6 + c\/— 1. For example, \4\/3 +4\/— 1 denotes a right fine whose length is 2 and whose angle with the absolute unit is 10°.
§14 If two angles have equal sines and equal cosines their difference is 0, or +4 right angles, or a multiple of ±4 right angles; and con- versely, if the difference between two angles is 0 or +4 right angles taken once or several times, then their sines as well as their cosines are equal.
§15
1
If 777 is an integer and w is equal to 360°, then (cos i; + e sin v)"» has only the following 777 different values:
7r-\-V , . T-\-V 2t-\-v , . 2v-\-v cos r+€ sm v, cos \-t sm , cos 1-€ sm ...,
771 771 771 771
(777 — l)7r -\- V , . (m — l)7r + v cos -=^ 1- c sm -^ ;
777 777
for the numbers by which tt is muItipHed in the preceding series are in the arithmetical progression 1, 2, 3, 4,.. .771 — 1. Conse-
WESSEL 65
quently the sum of every two of them is m, if the one is as far from 1 as the other is from m — 1 ; and if their number is not even, then
the middle one taken two times equals m. Therefore if ^^
m
is added to ^^ , and the latter is as far from
m
TT + f . ^i • (m — n)7r + r . - (m — l)7r -\- v ,
, m the series, as ^ — is irom -^ ■^— , then
m mm
, . , ^ 2m — u — n , 2v , 2y „ , ,.
the sum is equal to — t -\ = t -\ . But adding
m m m
(m — n)7r . • i ^ ^ u* *• (m — n)( — tt) , .
— IS equivalent to subtracting ^ — -; and since
m m
the dilterence is t, has the same cosine and sine
m
(m — n)T -[- V jj /N- I -T,
as -^^ . Hence ( — r) gives no values not given by +7r.
m
However, none of these values are equal; for the difference between any two angles of the series is always less than tt and never equal to 0. Nor will any more values result if the series is con- tinued; for then the new angles will be tt H ,t -\ , r -\ ,
m m m
etc., and according to §14 the values of the sines and cosines of these will be the same as in the angles we already have. There can be no angle outside of the series; for then t would not be multi- plied in the numerator by an integer, and the angles muItipHed by m would not produce any angle which subtracted from v gives 0, or ±7r, or a multiple of +7r; consequently the mth power of the cosine and sine of such angles could not equal cos r + e sin v.
§16 Without knowing the angle which the indirect hne I -{- x makes with the absolute, we may find, if the length of x is less than 1,
the power (1 -|- x) "* = 1 + -^ -j- y. — ^ — x^ + etc. If this series
is arranged according to the powers of m, it has the same value and is changed into the form
i , ml , mH'^ , mH^ ,
where
, x^ , x^ x^ ,
/ = x-2--|-y-j + etc..
66 SOURCE BOOK IN MATHEMATICS
and is a sum of a direct and a perpendicular line. If we call the direct line o and the perpendicular by/~\, then 6 is the smallest measure of the angle which 1 + x makes with +1. If we set
1,1.1, 1 ,
^ + 1 + U + L2J + "''• = ''
then
/I . \ 1 , fnl , mH^ , m^l^ ,
(1 + x)-, or 1 + y- + -yj- + j-y^ + etc.,
may be represented by e'^'' + "'^^"^-^ ; that is, (1 + x)"* has the length e^^ and a direction angle whose measure is mb, assuming m to be either positive or negative. Lines lying in the same plane may thus have their direction expressed in still another way, namely, by the aid of the natural logarithms. I shall produce complete proofs for these statements at another time, if privileged to do so. Now, that I have rendered an account of my plan for finding the sums, products, quotients, and powers of right lines, I shall next give a couple of examples illustrating the use of this method.
PASCAL
On the Arithmetic Triangle
(Translated from the French by Anna Savitsky, A. M., Columbia University,
New York City.)
Although Pascal (see p. 165) was not the originator of the arithmetic tri- angle, such an arrangement of numbers having been anticipated, his name has been linked v/ith the triangle by his development of its properties, and by the applications which he made of these properties. The historical interest of the work is to be found, perhaps, in its bearing on probability discussions and on the early developments of the binomial theorem. Since Pascal's contribu- tions to the theory of probability are considered elsewhere in this Source Book, passages pertinent to that theory are omitted in the present translation. Other omissions, also, are necessarily made with great freedom. The original article is found in the works of Pascal, the latest edition of which was edited by Leon Brunschvicg and Pierre Boutroux (Paris, 1908).
Treatise on the Arithmetic Triangle Definitions
I designate as an arithmetic triayigle a figure whose construction is as follows:
I draw from any point, G, two lines perpendicular to each other, GV, Gf,* in each of which I take as many equal and^ continuous parts as I please, beginning at G, which I name 1, 2, 3, 4, etc.; and these numbers are the indices^ of the divisions of the lines.
Then I join the points of the first division in each of the two lines by another line that forms a triangle of which it is the base.
I join in this manner the two points of the second division by another line that forms a second triangle of which it is the base.
And joining in this manner all the points of division which have the same index I form with them as many triangles and bases.
I draw through each of the points of division lines parallel to the sides, which by their intersections form small squares that I call cells.
1 [The editor of the French edition uses ^ instead of f by mistake.]
* [Pascal employs the words "continues" and "contigiies" interchangeably. In the translation, they have been rendered literally.]
2 [The term used is "exposans."]
67
68
SOURCE BOOK IN MATHEMATICS
The cells which lie between two parallels going from left to right are called cells of the same parallel rank, like the cells G, a, w, etc.,
or 4), ypy 6, etc.
And those which he between two lines going from the top down- ward are called cells of the same perpendicular rank, like the cells G, <i>. A, D, etc., and also a, \l/, B, etc.
Those which are crossed diagonally by the same base are called cells of the same base, hke the following: D, B, 6, X, or A, \{/, r.
2
1
I
3
4.
5
6
7
8j
9j
lOl
1
\G/
It / /\
/I
>5/ /I
/I
y
/l
/I
/
Z
y
/1\
/3
R/ /*
S/
A
/6
y
/8
/9
3
A/
B/ A
A\
A
As
/
/e8
/36
4
D/
E/
F/
/lO
A^
y/
As
z
A\.
Parallel Ranks
5
7
M/
A
k/
/is
As
/ra\
Alb
6
P/
0/
A
/2I
/S6
/f26
7
/
/
/28
/M
u e
.3
ARITHMETIC TRIANSLE
6
/^
/e
/36
9
/
/9
10
/
The cells of the same base equally distant from its ends are called reciprocals, as E, R, and B, 6, because the index of the paral- lel rank of the one is the same as the index of the perpendicular rank of the other, as is apparent in the example where E is in the second perpendicular rank and in the fourth parallel, and its reciprocal R is in the second parallel rank and reciprocally in the fourth perpendicular; and it is quite easy to show that those cells which have their indices reciprocally equal are in the same base and equally distant from its extremities.
It is also quite easy to show that the index of the perpendicular rank of any cell whatsoever, added to the index of its parallel rank, exceeds by unity the index of its base.
For example, cell F is in the third perpendicular rank, and in the fourth parallel rank, and in the sixth base; and the two indices of the ranks 3+4 exceed by unity the index of the base 6, which
PASCAL 69
arises from the fact that the two sides of the triangle are divided into an equal number of parts; but this is rather understood than demonstrated.
The above statement is equivalent to saying that each base contains one cell more than the preceding base, and each as many as the number of units in its index; thus, the second (/)cr has two cells, the third Axpir has three of them, etc.
Now the numbers which are placed in each cell are found by this method:
The number of the first cell, which is in the right angle, is arbitrary; but when that has been decided upon, all the others necessarily follow; and for this reason, it is called the generator of the triangle. Each of the others is determined by this one rule:
The number of each cell is equal to that of the cell which precedes it in its perpendicular rank, added to that of the cell which precedes it in its parallel rank. Thus, the cell F, that is, the number of the cell F, is equal to the cell C, plus the cell E; and likewise for the others.
From these facts there arise several consequences. Below are the principal ones, in which I consider those triangles whose generator is unity; but what is said of them will apply to all others.
Corollary 1. — In every arithmetic triangle, all the cells of the first parallel rank and of the first perpendicular rank are equal to the generator.
For, by the construction of the triangle, each cell is equal to that of the cell which precedes it in its perpendicular rank, added to that which precedes it in its parallel rank. Now the cells of the first parallel rank have no cells which precede them in their per- pendicular ranks, nor those of the first perpendicular rank in their parallel ranks; consequently they are all equal to each other and thus equal to the generating first number.
Thus <}) equals G + zero, that is, <^ equals G.
Likewise A equals 0 + zero, that is, </>.
Likewise c equals G + zero, and r equals <r + zero.
And likewise for the others.
Corollary 2. — In every arithmetic triangle, each cell is equal to the sum of all those of the preceding parallel rank, comprising the cells from its perpendicular rank to the first, inclusively.
Consider any cell w: I assert that it is equal to R -\- d -\- \l/ + <t>, which are cells of the parallel rank above, from the perpendicular rank of co to the first perpendicular rank.
70 SOURCE BOOK IN MATHEMATICS
This is evident by defining the cells, merely, in terms of the cells from which they are formed. For u) equals R -\- C.
e-^ B
0, for A and 4> are equal to each other by the preceding.
Hence co equals R -\- d -{- \l/ -\- <j).
Corollary 3. — In every arithmetic triangle, each cell is equal to the sum of all those of the preceding perpendicular rank, com- prising the cells from its parallel rank to the first, inclusively.
Consider any cell C: I assert that it is equal to B -{- \p -^ cr, which are the cells of the preceding perpendicular rank, from the parallel rank of the cell C to the first parallel rank.
This appears likewise by the very definition of the cells.
For C equals B -{- 6.
(T, for T equals a by the first (corollary). Hence C equals B -{- \l/ -\- a.
Corollary 4. — In every arithmetic triangle, each cell diminished by ttftity is equal to the sum of all those which are included between its perpendicular rank and its parallel rank, exclusively.
Consider any cell ^: I assert that ^ — g equals K + ^ + r^ + <^ + X4-7r-f-(T + C/, which are all the numbers included between the rank ^coCBA and the rank ^Sn, exclusively.
This appears in like manner from the definition.
For ^ equals X + K + co.
T -h e -\- C
e-\-^p + B
C+CA + A
G. Hence ? equals \-\-R-\-r + d + <T + yp + G+<}> + G.
Note. — I have said in the statement: each cell dimmished by unity, because unity is the generator; but if it were another
PASCAL 71
number, it would be necessary to say: each cell diminished by the generating number.
Corollary 5. — In every arithmetic triangle, each cell is equal to its reciprocal.
For in the second base (^o-, it is evident that the two reciprocal cells 4>, a, are equal to each other and to G,
In the third Ai^tt, it is hkewise seen that the reciprocals tt, A, are equal to each other and to G.
In the fourth, it is seen that the extremes D, X, are again equal to each other and to G.
And those between the two are evidently equal, since B equals A + i/', and 6 equals ^ + tt; now ir -\- \}/ are equal to A + 'A, as has been shown; hence, etc.
Likewise it can be shown in all the other bases that the recipro- cals are equal, because the extremes are always equal to G, and the rest can always be defined by their equals in the preceding base which are reciprocal to each other.
Corollary 6. — In every arithmetic triangle, a parallel rank and a perpendicular one which have the same index are composed of cells which are respectively equal to each other.
For they are composed of reciprocal cells.
Thus, the second perpendicular rank axf/BEMQ is exactly equal to the second parallel rank 4>\}/dRSN.
'^ Corollary 7. — In every arithmetic triangle, the sum of the cells of each base is twice those of the preceding base.
Consider any base DBdX. I assert that the sum of its cells is double the sum of the cells of the preceding base Ai/'tt.
For extremes D, X,
are equal to the extremes A, tt,
and each of the others B, 6,
is equal to two of the other base A -\- \{/, ^ + tt.
Hence D + X + B + 0 equal 2A + 2^p + 27r.
The same thing may be demonstrated for all the others.
Corollary 8. — In every arithmetic triangle, the sum of the cells of each base is a number of the^ geometric progression which begins with unity, and whose order is the same as the index of the base.
For the first base is unity.
The second is twice the first, hence it is 2.
^ [The term used, "double progression," refers to a geometric progression.]
72 SOURCE BOOK IN MATHEMATICS
The third is twice the second, hence it is 4.
And so on to infinity.
Note. — If the generator were not unity, but another number like 3, the same thing would be true; however, one should not take the numbers of the geometric progression beginning with unity, that is, 1, 2, 4, 8, 16, etc., but those of another geometric progression beginning with the generator 3, as, 3, 6, 12, 24, 48, etc. k;^ Corollary 9.— In every arithmetic triangle, each base dimin- ished by -unity is equal to the sum of all the preceding ones.
For this is a property of the double (geometric) progression.
Note. — If the generator were other than unity, it would be necessary to say: each base diminished by the generator.
Corollary 10. — In every arithmetic triangle, the sum of ,a many continuous cells as desired of a base, beginning at one end, is equal to as many cells of the preceding base, taking as many again less one.
Let the sum of as many cells as desired of the base DX be taken: for example, the first three D -\- B -\- 6.
1 assert that it is equal to the sum of the first three cells of the preceding base A + ^ + tt, adding the first two of the same base
For D. B. e.
equals A. A -\- ^. \}/ -\- t.
Hence D -\- B + d equals 2A + l^p + tt.
Definition. — I designate as cells oj the dividend those which are crossed diagonally by the line which bisects the right angle, as G, \l/, C, p, etc.
Corollary 11. — Every cell of the dividend is twice that which precedes it in its parallel or perpendicular rank.
Consider a cell of the dividend C. I assert that it is twice d, and also twice B.
For C equals d ■{■ B, and 6 equals B, by Corollary 5.
Note. — All these corollaries are on the subject of the equalities which are encountered in the arithmetic triangle. Now we shall consider those relating to proportions; and for these, the following proposition is fundamental.
Corollary 12. — In every arithmetic triangle, if two cells are contiguous in the same base, the upper is to the lower as the number of cells from the upper to the top of the base is to the number of those from the lower to the bottom, inclusive.
PASCAL 73
Consider any two contiguous cells of the same base, E, C: I assert that:
£ is to C as 2 is to 3
lower, upper, because there are because there are
two cells from E to three cells from C to
the bottom, that is, the top, that is, C,
E, H; R, fx.
Although this proposition has an infinite number of cases, I will give a rather short demonstration, assuming two lemmas.
Lemma 1 : which is self-evident, that this proportion is met with in the second base; for it is apparent that 0 is to o- as 1 is to 1.
Lemma 2: that if this proportion is found in any base, it will necessarily be found in the following base.
From which it will be seen that this proportion is necessarily in all the bases: for it is in the second base by the first lemma; hence by the second, it is in the third base, hence in the fourth, and so on to infinity.
It is then necessary only to prove the second lemma in this way. If this proportion is met with in any base, as in the fourth DX, that is, if D is to B as 1 is to 3, and B is to ^ as 2 is to 2, and 0 is to X as 3 is to 1, etc., I say that the same proportion will be found in the following base Hfx, and that, for example, E is to C as 2 is to 3.
For D is to B as 1 is to 3, by the hypothesis.
Hence
D + B is to B as 1 + 3 is to 3.
E is to B as 4 is to 3. In the same way B is to ^ as 2 is to 2, by the hypothesis. Hence
B + e is to B as 2 + 2 is to 4.
But
Hence by the^ mixed proportion, C is to £" as 3 is to 2: Which was to be proved.
The same may be demonstrated in all the rest, since this proof is based only on the assumption that the proportion occurs in the
^ [The term used is "proportion troublee."]
c
is to B as
4
is to 2.
B
is to E as
3
is to 4,
74 SOURCE BOOK IN MATHEMATICS
preceding base, and that each cell is equal to its preceding plus the one above it, which is true in all cases.
Corollary 13. — In every arithmetic triangle, if two cells are continuous in the same perpendicular rank, the lower is to the upper as the index of the base of the upper is to the index of its parallel rank.
Consider any two cells in the same perpendicular rank, F, C.
I assert that F is to C as 5 is to 3
the lower, the upper, index of the index of the parallel
base of C, rank of C.
For £" is to C as 2 is to 3. Hence
£■ + C is to C as 2 + 3 is to 3.
F is to C as 5 is to 3. Corollary 14. — In every arithmetic triangle, if two cells are continuous in the same parallel rank, the greater is to the preced- ing one as the index of the base of the preceding is to the index of its perpendicular rank.
Consider two cells in the same parallel rank, F, E. I assert that F is to F as 5 is to 2
the greater, the preceding, index of the index of the per-
base of F, pendicular rank of F. For F is to C as 2 is to 3.
Hence
F + C is to F as 2 + 3 is to 2.
F is to F as 5 is to 2.
Corollary 15. — In every arithmetic triangle, the sum of the cells of any parallel rank is to the last cell of the rank as the index of the triangle is to the index of the rank.
Consider any triangle, for example, the fourth GD\: I assert that for any rank which one takes in it, like the second parallel rank, the sum of its cells, that is </> + i/' + 0, is to ^ as 4 is to 2. For (j) -\- \p -\- 6 equals C, and C is to ^ as 4 is to 2, by Corollary 13.
Corollary 16. — In every arithmetic triangle, any parallel rank is to the rank below as the index of the rank below is to the number of its cells.
Consider any triangle, for example the fifth uGH: I assert that, whatever rank one may choose in it, for example the third, the sum of its cells is to the sum of those of the fourth, that is
PASCAL 75
A + 5 + C is to D + £■ as 4, the index of the fourth rank, is to 2, which is the index of the number of its cells, for it contains 2 of them.
For A + B + C equals F, and D + E equals M.
Now F is to M as 4 is to 2, by Corollary 12.
Note. — It may also be stated in this manner: Every parallel rank is to the rank below as the index of the rank below is to the index of the triangle minus the index of the rank above.
For the index of a triangle, minus the index of one of its ranks, is always equal to the number of cells contained in the rank below.
Corollary 17. — In every arithmetic triangle, any cell whatever added to all those of its perpendicular rank is to the same cell added to all those of its parallel rank as the number of cells taken in each rank.
Consider any cell B: I assert that B + i^ + <r is to B + A as 3 is to 2.
I say 3, because there are three cells added in the antecedent, and 2, because there are two of them in the consequent.
For B -\- \p -\- (X equals C, by Corollary 3, and B + A equals E, by Corollary 2.
Now C is to F as 3 is to 2, by Corollary 12.
Corollary 18. — In every arithmetic triangle, two parallel ranks equally distant from the ends are to each other as the number of their cells.
Consider any triangle GVf, and two of its ranks equally distant from the ends, as the sixth P -{- Q, and the second (t> -\- ^ + 6 -\- R -\- S -{- N: I assert that the sum of the cells of the one is to the sum of the cells of the other as the number of cells of the first is to the number of cells of the second.
For, by Corollary 6, the second parallel rank (j)\f/dRSN is the same as the second perpendicular rank aypBEMQ, for which we have demonstrated this proportion.
Note. — It may also be stated: In every arithmetic triangle, two parallel ranks, whose indices added together exceed by unity the index of the triangle, are to each other inversely as their indices.
For it is the same thing as that which has just been stated.
Final Corollary. — In every arithmetic triangle, if two cells in the dividend are continuous, the lower is to the upper taken four times as the index of the base of the upper is to a number greater (than the base) by unity.
76 SOURCE BOOK IN MATHEMATICS
Consider two cells of the dividend p, C: I assert that p is to 4C as 5, the index of the base of C, is to 6.
For p is twice w, and C twice 6; hence 40 equal 2C.
Hence 4d is to C as 2 is to 1.
Now p is to 4C as w is to 4d, or by a ratio composed of co to C + C to 40
5 to 6
Hence p is to 4C as 5 is to 6. Which was to be proved. Note. — Thence many other proportions may be drawn that I have passed over, because they may be easily deduced, and those who would like to apply themselves to it will perhaps find some, more elegant than these which I could present.^
Application of the Arithmetic Triangle To Find the Powers of Binomials and^ Apotomes
If it is proposed to find a certain power, like the fourth degree, of a binomial whose first term is A and the other unity, that is to say, if it is required to find the fourth power of A + 1, take the fifth base of the arithmetic triangle, namely, the one whose index 5 is greater by unity than 4, the exponent of the proposed order. The cells of this fifth base are 1, 4, 6, 4, 1; the first number, 1, is to be taken as the coefficient of A to the proposed degree, that is» of A*; then take the second number of the base, which is 4, as the coefficient of A to the next lower degree, that is to say, of A^ and take the following number of the base, namely 6, as the coefficient of A to the lower degree, namely, of A^ and the next number of the base, namely 4, as the coefficient of A to the lower degree,
' [At this point, Pascal establishes a theorem which would be stated in modern notations as follows: The cell in the n-th parallel and r-th perpendicular ranks contains the number
n(n + l).--(n + r - 2)
(r- 1)! He then indicates applications of the arithmetic triangle in the theory of com- binations, and in the elementary analysis of questions of mathematical proba- bility suggested by games of chance. All of this material is omitted in the present translation.]
* [By "apotome," Pascal means a binomial which is the difference between two terms.]
PASCAL 77
namely, of the root A, and take the last number of the base, 1, as the absolute number; thus we obtain: lA'* + 4A^ + 6A'^ + 4A + 1, which is the fourth (square-square) power of the bino- mial A + 1- So that if A (which represents any number) is unity, and thus the binomial A + 1 becomes 2, this power lA^ + 4A3 + eA"" + 4A + 1, now becomes 1.1* + 4.P + 6.V + 4.1 + 1. That is, one times the fourth power of A, which is unity 1
Four times the cube of 1, that is 4
Six times the square of 1, that is 6
Four times unity, that is 4
Plus unity 1
Which added together make 16
And indeed, the fourth power of 2 is 16.
If A is another number, like 4, and thus the binomial A + 1 is 5, then its fourth power will always be, in accordance with this method,
lA' + 4A3 + 6A2 + 4A + 1 which now means,
1.4* + 4.4^ + 6.42 + 4.4 + 1. That is to say, one times the fourth power of 4, namely 256
Four times the cube of 4, namely 256
Six times the square of 4 96
Four times the root 4 16
Plus unity 1
whose sum 625
produces the fourth power of 5 : and indeed, the fourth power of 5 is 625.
Likewise for other examples.
If it is desired to find the same degree of the binomial A + 2, take the same expression lA* + 4A^ + 6A'^ + 4A + 1, and then write the four numbers, 2, 4, 8, 16, which are the first four degrees of 2, under each of the numbers of the base, omitting the first, in this way
lA* + 4A3 + 6A2 + 4A1 + 1 2 4 8 16.
and multiply the numbers which correspond to each other 1A4 + 4A3+ 6A^ + 4A1 + 1 2 4 8 16
in this way lA* + %A^ + 24A2 -f 32Ai + 16
78 SOURCE BOOK IM MATHEMATICS
Thus the fourth power of the binomial A + 2 is obtained; if A is unity, the fourth power will be as follows:
One times the fourth power of A, which is unity 1
Eight times the cube of unity 8
24, 12 24
32, 1 32
Plus the fourth power of i 16
Whose sum 81
is the fourth power of 3. And indeed, 81 is the fourth power of 3. If A is 2, then A + 2 is 4, and its fourth power will be One times the fourth power of A, or of 2,
namely 16
8,23 64
24, 22 96
32, 2 64
Plus the fourth power of 2 16
whose sum 256
is the fourth power of 4.
In the same way, the fourth power of A + 3 can be found, by writing likewise
A' + 4A3 + 6A2 + 4A + 1 and below, the numbers,
3 9 27 81
lA* + 12A3 + 54A2 + 108A + 81
which are the first four degrees of 3; and by multiplying the corresponding numbers, we obtain the fourth power of A + 3.
And so on to infinity. If in place of the fourth power, the square-cube, or the fifth degree, is desired, take the sixth base and apply it as I have described in the case of the fifth; and likewise for all the other degrees.
In the same way, the powers of the apotomes A — 1, A — 2, etc., may be found. The method is wholly similar, and difi'ers only in the matter of signs, for the signs + and — always alter- nate, and the sign + is always first.
Thus the fourth power of A — 1 may be found in this way. The fourth power of A + 1 is, according to the preceding rule, lA* -f 4A3 + 6A2 + 4A + 1- Hence, by changing the signs in the way described, we obtain lA — 4:A^ -\- 6A^ — 4 A + 1. Thus the cube of A — 2 is likewise found. For the cube of A -f 2, by the preceding rule, is A^ + ^A- -\- 12A -f 8. Hence
PASCAL 79
the cube of A — 2 is found by changing the signs, A^ — 6A^ + 12A — 8. And so on to infinity.
I am not giving a demonstration of all this, because others have already treated it, like Herigogne; besides, the matter is self-evident.
BOMBELLI AND CATALDI On Continued Fractions
(Translated from the Italian by Professor Vera Sanford, Western Reserve University, Cleveland, Ohio.)
The study of continued fractions seems to have arisen in connection with the problem of finding the approximate values of the square roots of numbers that are not perfect squares. Various methods of finding such roots had been advanced^ at an earlier period, but, in general, their operation was difficult and clumsy.
The first mathematician to make use of the concept of continued fractions was Rafael Bombelli (born c. 1530). Little is known of his career, but his contribution to mathematics was the writing of a work which has been charac- terized as "the most teachable and the most systematic treatment of algebra that had appeared in Italy up to that time."^ The title wsiS L' Algebra parte maggiore dell' arimetica divisa in tre libri and the work was published in Bologna in 1572 and brought out in a second edition in that same city in 1579 under the title L' Algebra Opera, the editions being identical except for the title pages and the dedicatory letter. This algebra was noteworthy for its treatment of the cubic and biquadratic equations. The selection here given appears on pages 35 to 37 of the edition of 1579.
Method of Forming Fractions in the Extraction of Roots
Many methods of forming fractions have been given in the works of other authors; the one attacking and accusing another without due cause (in my opinion) for they are all looking to the same end. It is indeed true that one method may be briefer than another, but it is enough that all are at hand and the one that is the most easy will without doubt be accepted by men and be put in use without casting aspersions on another method. Thus it may happen that today I may teach a rule which may be more acceptable than those given in the past, but if another should be discovered later and if one of them should be found to be more vague and if another should be found to be more easy, this [latter] would then be accepted at once and mine would be discarded; for as the saying goes, experience is our master and
1 See Smith, D. E., History oj Mathematics, Vol. II, pp. 144, and 2S3, Boston, Massa- chusetts, 1925.
2 Ibid., Vol. I. p. 301.
80
BOMBELLI AND CATALDI 81
the result praises the workman. In short, I shall set forth the
method which is the most pleasing to me today and it will rest
in men's judgment to appraise what they see: mean while I shall
continue my discourse going now to the discussion itself.
Let us first assume that if we wish to find the approximate
root^ of 13 that this will be 3 with 4 left over. This remainder
2 should be divided by 6 (double the 3 given above) which gives :^-
This is the first fraction which is to be added to the 3, making
2 . . 3:: which is the approximate root of 13. Since the square of
4 . . 4 this number is 13q> it is 5 too large, and if one wishes a closer
approximation, the 6 which is the double of the 3 should be added
2 . . 2 to the fraction :^> givmg 6:^ > and this number should be divided
into the 4 which is the difference between 13 and 9. The result
. 3 . 3 . .
is ^ which, added to the 3 makes 3 ^- This is a closer approxi-
24 mation to the root of 13, for its square is 12^} which is closer
2 . .
than that of the 3^-^ But if I wish a closer approximation,
3 I add this fraction to the 6 making 6^> divide 4 by this, obtaining
20
x:=' This should be added to the 3 as was done above, making
3:^- This is a closer approximation for its square is 13:jyrgQ>
4 which is TTjnq too large. If I wish a closer approximation, I
20 . . 109
divide 4 by 6jx> obtaining j^? [and] add this to 3, obtain-
109 mg 3.-QX- This is much closer than before for its square is loU
' [Bombelli's term latus was a popular one based on the concept of a square
root as the side of a square of given area. In this translation, however, the
term root will be used because of its greater significance.]
4 * [In modem notation, this would, of course be written as: 3 + 7*
Bombelli gives no hint as to the reasons for the success of this method, nor does he tell how he discovered it.]
82 SOURCE BOOK IN MATHEMATICS
13 jfTr^' which is ^^^^ too large. If I wish to continue this
109 . . 729
even further, I divide 4 by 6^^ obtaining tjkq' which
A . . 4
is the root of 13 TTVjfyr' which is . .. ^--. too large, and this
process may be carried to within an imperceptable difference. Care should be taken, however, in the formation of these frac- tions in the many cases when the number whose root is to be found falls just short of being a perfect square (as 8, for example). In this case, since 4 is the largest square number, and since 4 is
4 also the remainder, the fraction becomes ^ which is equal to 1.
Adding this to 2 gives 3, whose square is 9. Subtracting the number 8 whose root is required from this number, 1 remains.
This should be divided by 6, the double of the 3 giving ^• Subtracting this from the 3 gives 2p as the approximate root of 8. The square of this number is 8:r^^ which is ^ too large. If a closer approximation is desired, add the 2^ to the 3 getting
Spj and divide 1 by this as was done above, giving yf' which
29 should be subtracted from 3 leaving l^r- This will be a
nearer root. If a still closer approximation is desired, divide 1
29 by Sj^' Proceed (as was done above) as close as any one may
desire.
Pietro Antonio Cataldi^ (1548-1626) was professor of mathematics and astronomy at Florence, Perugia, and Bologna. He was the author of works on arithmetic, theory of numbers, and geometry and also wrote treatises on topics in algebra. He seems to have been the first to develop a symbolism for continued fractions, and this appears in an essay with the title Trattato del modo brevissimo Di trouare la Radice quadra delli numeri, Et Regole da approssimarsi di continuo al vero nelle Radici de'numeri non quadrati, con le cause et inuentioni loro, Et anco il modo di pigliarne la Radice cuba, appli-
' (Here Bombelli gives jt^> evidently a misprint.)
5 Sometimes given as Cattaldi.
BOMBELLI AND CATALDI 83
\
cando il tutto alle Operationi Militari &" aUre. Bologna, 1613. The selection here given app>ears on page 70.
Let US now proceed to the consideration of another method of finding roots continuing by adding row on row {di mano in mano) to the denominator of the fraction, which finally yields a fraction equal to the fraction of the preceding rule. But for greater con- venience, I shall assume a number whose root may be easily taken and I shall assume that the first part of the root is an integer. Then let 18 be the proposed number, and if I assume that the first
2 . 1 . . . 1
root is 4. & o' that is 4t' this will be in excess by ^^ which is 8 4 -^16
the square of the fraction ^- The second root will be found by
2 1 8
the above mentioned method to be 4. & q- & - which is 4. & ^^'
o 4 33
2 which is T7^oQ too small. This arises from multiplying the entire
8 1 . .
fraction ^ by y^ in which the whole fraction is less than the
7 which is the added fraction.^ 4
Let the root of 18. be
2 2 2
4 &•&•&-
8 8 8
The total fraction added is
makes
11 1 136 33
2^ 1089 1088
*^4 136
A 1 V 33 X 1
33 136 X 33 ^ 136
1
2 8 That is, 4. & Q-& :j^ o 33
18,496
-^
272
^ [The work which follows appears in a column at the side of the page, and is rearranged in this translation.]
84 SOURCE BOOK IN MATHEMATICS
which is
c . 1089 Squaring, ^g^^-
136
17^1 1088
The square is 18 toTqz' which is too large by yqaqZ'
Be it noted that in the printing when proceeding hurriedly, it is not possible to form fractions and fractions of fractions con- veniently in this form, as for instance in the case of
4.&2.
as we are forcing ourselves to do in this example, but we may denote
2 2 2 all of them by adopting this device: 4. & ^ & ^ & ^ letting a
o. o. o. period by the 8 in the denominator of each fraction mean that the following fraction is a fraction of the denominator.
I shall find the third fraction by the above mentioned method to
2 2 2 2 2 1
be 4. & ^ & p & 5 ; or as I might say 4. & 5 & ^ & j' o. o. o. o. o. 4
0 Q '2 '2
which is 4, & 5 & ^^' or 4. & . > which is 4. & ^n^?-'
8 33 A , ^ 136
4+33
33 which will be in excess since y^- the whole fraction is greater
g than j-^. the added fraction. The excess of the square over 18 is
1 . . . . 33
Tn7n2 which arises from multiplying y^» the whole fraction by
1 . . 8
..^^ -,-, in which the whole fraction is greater than the -^
which is the added fraction."'
^ (Cataldi continues this work until he reaches the fifteenth fraction.)
JACQUES (I) BERNOULLI On the "Bernoulli Numbers"
(Translated from the Latin by Professor Jekuthiel Ginsburg, Yeshiva G)IIege,
New York City.)
Of the various special kinds of numbers used in analysis, there is hardly a species that is so important and so generally applicable as the Bernoulli Num- bers. Their numerous properties and applications have caused the creation of an extensive literature on the subject which still continues to attract the attention of scholars. The first statement of the properties of these numbers was given to the world uy their inventor Jacques (1) Bernoulli (1654-1705) in his posthumously printed work, Ars Conjeclandi (Basel, 1713), pages 95 to 98. These pages are here translated.
The excerpt is interesting from more than one point of view. First, we witness in it the first stroke of genius that caused ripples in human thought that have not died out even to the present day. Second, the memoir is as fresh and vigorous today as when it was written; in fact, it could be used even now as a p>opular exposition of the simpler properties of the Bernoulli Num- bers. Third, the text reveals the personal touch, the unbounded enthusiasm of the author over the power of the numbers later called by his name. His Cfemark that the results of Bullialdus's enormous treatise could, by means of his numbers, be compressed in less than one page, is both striking and illumi- nating. Nor is the element of puzzle and mystery lacking. Regardless of the fact that the discovery is more than 200 years old, mathematicians have not been able as yet to find by what process Bernoulli derived the properties of his numbers which he gives in these pages. They can readily be derived by various modern methods, but how did he derive them with the means at his disposal? It is also interesting to compare his criticism of Wallis's use of incomplete induction with his own use of the same imperfect tool. In short, in the compass of three printed pages we get not only information about the invention but also glimpses of the person of the great master.
We will observe here in passing that, many [scholars] engaged in the contemplation of figurate numbers (among them Fauihaber'
^ [Johann Faulhaber, a successful teacher of mathematics in Ulm, was born there on May 5, 1580, and died therein 1635 (D. E. Smith, History of Mathe- matics, Vol. I, p. 418). With the help of his friend and protector Johann Remmelin he published a number of mathematical works. In his Mysterium Aritbmeticumf 1615, he discussed the properties of figurate numbers. Ber« noulli possibly refers to this work of his. Faulhaber also developed formulas for Sn' from c = ltoc=17 (Tropfke, Gescbicbte der Elemenlar Matbema- tik. Vol. VI, p. 22).]
85
86 SOURCE BOOK IN MATHEMATICS
and Remmelin of Ulm, Wallls, Mercator.^ In his Logaritbmo- tecbnia and others) but I do not know of one who gave a general and scientific proof of this property.^
Wailis in his Aritbmetica Infinitorum investigated by means of induction the ratios that the series of squares, cubes, and other powers of natural numbers have to the series of terms each equal to the greatest term. This he put in the foundation of his method. His next step was to establish 1 76 properties of trigonal, pyramidal, and other figurate numbers, but it would have been better and more fitting to the nature of the subject if the process would have been reversed and he would have first given a discussion of figurate numbers, demonstrated In a general and accurate way, and only then have proceeded with the investigation of the sums of powers of the natural numbers. Even disregarding the fact that the method of Induction is not sufficiently scientific and, moreover, requires special work for every new series; it is a method of com- mon judgment that the simpler and more primitive things should precede others. Such are the figurate numbers as related to the powers, since they are formed by addition, while the others are formed by multiphcatlon; chiefly, however, because the series of figurate numbers, supplied with the corresponding zeros^ have a submuitlple ratio to the series of equals.'* In case of powers (when
^ [Nicolaus Mercator was born near Cismar in Holstein, c. 1620, and died in Paris in February, 1687. His Aritbmotecbnia sive metbodus construendi logaritbmos nova accurate et facilis... was published in London in 1678 (Smith, /. c, I, 434). Bernoulli fails to mention Oughtred who pointed out the correspondence between the binominal coefficients and the figurate num- bers, as did also Nicolo Tartaglia, Pascal, and others.]
2 [The property refers to the method of finding the nth term and the sum of n terms in a series of figurate numbers.]
^ [The number of zeros to put in the triangle of figurate numbers to make it look like a square
0
0
0
0
0
0
1
0
0
0
0
0
2
1
0
0
0
0
3
3
1
0
0
0
4
6
4
I
0
0
S
10
10
5
1
0
6
IS
20
15
6
1
7
21
35
35
21
7
Bernoulli counts each zero as a term. Thus the sum of the terms of the third column is 0 + 0+1+3+6+ 10 + 15 + 21.]
* [That is taking for example column three in the preceding footnote, the ratio of the sum of any number of terms beginning with two zeros to the sum of
JACQUES (/) BERNOULLI 87
the number of terms is finite) this does not hold without some excess or defect no matter how many zeros be added. With the known sums of the figurate numbers it is not difficult to derive the sums of the powers. I will show briefly how it is done.
Let the series of natural numbers 1, 2, 3, 4, 5, etc. up to n be given, and let it be required to find their sum, the sum of the squares, cubes, etc. Since in the table of combinations the general term of the second column is n — 1 and the sum of all terms, that is, all n — \, or Jn — \ in consequence of above is^ n.n — 1 _ nn — n
The sum /n — 1 or Therefore :
Jji - Jl = — — -•
r nn — n , ri J n = — 2 ^ J ^•
But Jl (the sum of all units ) = n. Therefore the sum of all n or
J
nn - n ^ ^
n = 2 — + n = fnn + fn.
A term of the third column is generally taken to be n — l.n — 2 _ nn — 3n + 2 1:2 2~~~
a series of terms each equal to the last term of the first series will be ^- Thus,
0 + 0 + 1 + 3 ^ A ^ 1 0 + 0 + 1+3+6 + 10 ^ 1
3 + 3 + 3+3 12 3' 10 + 10 + 10 + 10 + 10 + 10 3'
0 + 0 + 1 + 3+6 + 10 + 15 1
= ^' etc.
15 + 15 + 15 + 15 + 15 + 15 + 15 3 In the fourth column we get }i, and in the fifth we have J^. In every case the first series is a submultiple of the second, which he calls the series of equal terms.l
* [/. e., the sum ofO + l+2+... + n — 1 is as was stated above ^ of (n — 1) + (n — 1) + (n — 1) . . . (n times) since the ratio 0 + 1 +2 +...+ (n - 1) _1 Hence ^ -^ .-_"("-!).
(n- 1) + (n- 1)...+ (n - 1) 2 n(n - 1) 2' 1.2
Throughout the work Bernoulli uses the old form of 5, our present integral sign (/) where we would now use 2. His usage has been followed in the translation. He also writes n.n — 1 where we would write n(n — 1) and he expresses equality by the sign tx but in this translation the sign = will be used. His use of nn instead of n' should also be noted.]
88 SOURCE BOOK IN MATHEMATICS
and the sum of all terms ( that is, of all ^^ ~ ^^ ) is
n.n - l.n - 2 _ n^ - 3nn + 2n 1.2.3 6
P j1 R S S EC U Nl> A.
97
30 7^- J eriuunque J j-3Lll _£, hoc eft,
nn^-Cnn+rin-'fiOO "'"'"'^ "'"'"'% indequc /-i^J C»
Yi rfnn — f j»-tfu Etquoniam peimodo in-
vcttti fnn X }'»'+i':"+.'«-, necnon f'^'n five l'/Jj X Ti'"»+}I*» 8c /"i X »> hinc faila horum fubftitutione cmergct f^n* CO
M^ •— 6n1 ■^ 1 1 nn — St . , . ,
=^ + [«' +i'"' + ^«-}l""— ji»+«X
ri'''*+ il«'+2;^»> ejusque proin /extuplum /«» ((umiracubo- rum ) X 4»* + i'«'+^"«. Atquc fie porr6 ad altiores gradarini potelUcespergere, levique negotio fequentem adornaie Ucccculvm Lcsc:
Summs PoteJlMum. fn X i»« +f ».
/n' X ^»' +i"^ +Tr»* *— ^n^jjc+Ti""'
/»« X ^n9 4-i»» + f»^*— tV»' * + I«' *— t^".
/fl9 XT'o^'^+i"' + i"* *— /o»* *+i"* *— t'z""*
/l.'<'XT'T»"+i'»"'+ |"9 5jc— l»^5^C+ l»^ *— i»' ^-f-^"*
Quin imb qui legem progreflfionis inibi attentius infpexcrit » eitndem etiatn continuare poterit abfij; his ratiociniorum ambagibus : Sumtl enim t pro poteftatis cujuslib«t cxponente, fit fumma omniumn' icu
/n* X r:p,"*+' + !»'+ iA»'- + ^T?H r ^"'""'^
< .c~t.t».i.t-l .c-4 ^ f — J . t.c — t.« — i.c— }.t-4.t- ;.f--« '».}.4.J.6 •" l.j-4-5-6;7.3
Dn'"7 . . , . & ita deinceps , exponcntcm poteftatis ipfius » con- tinue minaendobinario, quoufque pervcniatur ad « vel n« . Literae capitales A, B,C) O &c. ordine denotant cocfficientes ultimo- lum teiminorum pro /9i«, /»"♦,/»*> /»' &c, ncmpc A X i^> B
N »-j'o
We will have then that
- 3n + 2
Cn^ - 3n
or
?i^ — 3nn + 2n
r§2„„_ r|„+ fi =
JACQUES (!) BERNOULLI 89
and
r, n^ - 3nn + 2n , C r
but
/l^ = f /" = Inn 4- f n and
/I = n.
Substituting, we have
J n^ — 3nn + 2n 3nn + 3n 13,1 , ,
i„„ = _ 1 ^ n = In^ -\- \nn + -^n,
of which the double ^nn (the sum of the squares of all n) = 3^n' + ^i'nn + 3^n. A term of the fourth column is generally
n — l.n — l.n — 3 _ n* — 6nn + 11^ — 6 1X3 6 '
and the sum of all terms is
n.n — l.n — l.n — 3 _ n* — 671^ + linn — 6n 03:4 ~ "24 •
It must certainly be that
'n^ — 6nn + 11^ — 6
/■
that is
Ji .' - Jnn + Jv n - Jl = "^ - 6n' + Hnn - 6n
Hence
fi 3 n^ - 6n3 + linn - 6n , f f, , . f ,
And before it was found that ^nn = \n^ + \nn + ^n, ^^n or
VJ" = H"^ + ii"» and /l = n.
When all substitutions are made, the following results:
/
.3 n^ — 6n=' + linn - (^n ..... . . .
1 n3 = 24 H ^ n3 + i nn + i n - \\ nn
- -\\n-\-n
= H4"^ + K2^' + K4wn; or, multiplying by 6,
Jn3 = >^n* + Hn^ + 3^nn.
Thus we can step by step reach higher and higher powers and with slight effort form the following table :^
^ [Bernoulli uses ^ to mean what we now designate by . . . ]
90 SOURCE BOOK IN MATHEMATICS
Sum oj Powers jn = }4nn + >^n,
Jn^ = }in^ + }W + >^n3 ^ -Hon.
Jn^ = ^^n^ + }in' + ^^2"' X -K2nn,
/nB = 3..fn^ + i.^n« + Hn^ X ->^n' X +3^2",
Jn^ = 3^n« + Kn^ + K2n« X -K4n* X +K2nn,
jn' = Kon^° + Mn« + Hn' X -Kon« X +Mn^ X -M2nn,
Whoever will examine the series as to their regularity may be able to continue the table. Taking c to be the power of any exponent, the sum of all n" or
/
+ c.c — l.c — 2.e — 3.C — 4 ^
Cn"
2.3.4.5.6
-u c-c - 1-c - 2.C - 3.C - 4.C - 5.C - 6 ^ ^ 2.3.4.5.6.7.8 '
and so on, the exponents of n continually decreasing by 2 until n or 7m is reached. The capital letters A, B, C, D denote in order the coefficients of the last terms in the expressions for jnn, jn*.jn^ namely A, is equal to 3^^, J5 is equal to — 3^^o» ^ is equal to 3-^2» ^ is equal to —}io-
These coefficients are such that each one completes the others in the same expression to unity. Thus D must have the value -3^0 because 3.^ + i.^ + ^ - K5 + ^^ + (+£>) - Vso = 1-
With the help of this table it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum
91,409,924,241.424,243,424,241,924,242,500
From this it will become clear how useless was the work of Ismael Bullialdus^ spent on the compilation of his voluminous Aritbmetica Infinitorum in which he did nothing more than compute with immense labor the sums of the first six powers, which is only a part of what we have accomplished in the space of a single page.
' [The title of BuIIialdus's (160S-1694) work is Opus novum ad aritbmeticum infinitorum. It was published in Paris 1682 and consists of six parts.]
EULER Proof that Every Integer is a Sum of Four Squares
(Translated from the Latin by Professor E. T. Bell, California Institute of Technology, Pasadena, California.)
Leonard (Leonhard) Euler (1707-1783), a pupil of Jean (I) Bernoulli, was not only one of the greatest mathematicians and astronomers of his century, but he was also versed in theology, medicine, botany, physics, mechanics, chemistry, and the Oriental as well as the modern languages. He was a voluminous writer, and there was hardly a branch of mathematics to which he did not contribute. The selection here translated serves to illustrate his method of attacking a problem in the theory of numbers. It is taken from his Commenlationes Aritbmeticae Collectx, Petropoli, 1849, edited by P. H. Fuss and N. Fuss (Vol. I, pp. 543-546) but appeared earlier in the Acta Eruditorum (p. 193, Leipzig, 1773) and the Acta Petrop., (p. 48, I. II., 1775. Exhib. Sept. 21, 1772). In preparing the article, the effort has been made to give a free translation that shall clearly convey Euler's meaning, in preference to following too closely the rather poor Latin of the day. Of his two proofs for the exceptional case of n = 2, only the simpler one has been given. From the modern point of view, the proof of the theorem is not very satisfactory, but it serves to illustrate the theory of numbers of the eighteenth century.
Lemma. — The product of two numbers, each oj which is a sum of Jour squares, may always be expressed as a sum of Jour squares. Let such a product be
Write
(a2 -I- 62 + c2 + d^){a^ + /32 + 72 4- 52).
A = aa + 6/3 + C7 + dd, B = 0/3 — 60; — c5 -f dy, C = ay -{- bd — ca — c//3, D = a5 — 67 + c/3 — da. Then
A2 + jB2 + C2 + D2 = (a2 -f 6'' + c2 + d')(a^ + ^2 + 72 _}. 8^),
since obviously the cross products in A^, B^, O, D^ cancel.
Theorem 1. — // N is a divisor oJ a sum oJ Jour squares, say oJ p2 -|- g2 _|_ ^2 _j_ ^2^ „Q Q„g qJ |^6{c^ is divisibk by N, then N is the sum oJ Jour squares.
91
92 SOURCE BOOK IN MATHEMATICS
It will first be shown that each of the four roots p, q, r, s may be chosen less than j-iN.''-
I. Let 71 be the quotient on dividing the sum of four squares by
N, so that Nn = p^ + g^ + r^ + 5^. Then we may write p = a -\- na, q = b -\- n^, r = c -\- ny, s = d + n8,
where each remainder a, h, c, d does not exceed 3^n in absolute value. ^ Hence
a2 + 62 + C2 + (/2 < „2^
II. By substituting the above values of p, q, r, s in
iVn = p2 + g2 ^ r2 + s\ we get yVn = a2 + 62 + c^ + (/2 + 2n(aa + 6)3 + ct + d8)
+ n2(«' + /5' + 7' + 5');
whence it follows that n must be a divisor of a^ -{- 6^ + c^ + d"^.
Put
a2 + 6=* + c^ + (/2 = nn\
Then n > n', or n' < n. By division we get
N = n' + 2A + ?i(a2 + |32 + 7' + 52).
III. Multiply now by n'. Then, since
nn' = a2 + 62 + c2 + d\ we have, by the Lemma,
nn'(a2 + ^2 ^ ^2 ^ 52) = ^2 ^ 52 _|_ (72 + D\
Combining this with the preceding equation we find
Nn' = n'2 + 2n'A -\- A^ + B^ -\- O + D\ and therefore
in' + Ay 4- B' + O + £>' = Nn'.
IV. By repeating the foregoing argument we obtain a decreasing sequence of integers Nn', Nn", etc., and hence finally we reach NA and its expression as a sum of four squares.
^ [It is to be observed in the following proof that n is different from 2; this case is tacitly ignored until the so-called corollary, following the proof, which disposes of the exceptional case implicit in the argument as presented.]
2 [But see the preceding footnote. The condition as to absolute values safe- guards the assertion above, but it does not take care of all possibilities in the proof which immediately follows, unless, as with Euler, and as indicated in the preceding footnote, we attend to the corollary.]
(I-ucing ijau,t i)2.)
EULER 93
Corollary. — To dispose of the apparent exception, let p, 5, r, 5 be odd numbers and n an even number. Then, since
Nn = p2 + g2 _j- r2 + s\ we have
and the four squares on the right are integers. A like reduction may be performed so long as the roots of all the squares are odd. Thus the exception when n = 2 disappears.
Theorem 2. — // N is prime, not only 4 squares not divisible by N, can be found in an infinity oj ways, whose sum is divisible by N, but also 3 squares.
For, with respect to N, all numbers are of one or other of the N forms
\N,\N-i-l,\N + 2,\N + 3,...,\N + N- 1.
Disregard the first form, \N, which contains all the multiples of N. There remain N — I forms, and we observe that the square of a number of the form \N + 1, likewise the square of a number of the form \N -\- N — 1, belongs to the same form \N + 1. Similarly the square of a number of either form \N + 2, \N + N — 2 is of the form \N + 4; and so on. Thus the squares of all numbers not of the form XiV are comprised in the 3^(iV — 1)
forms
XiV +1, \N + 4, \N + 9, etc.,
which will be called forms of the first class, and will be denoted by
m + a, \N-\-b, \N + c, \N-\- d, etc., so that a, 6, c, d,.. .denote the squares 1, 4, 9, 16,. . .or, if these exceed N, their residues on division by N. The remaining 3*2 (N — 1) forms will be denoted by
\N + a,\N + p,\N + y, etc., which will be called forms of the second class. It is easy to prove the following three properties concerning these classes.^
I. The product of two numbers of the first class is again con- tained in the first class, since evidently XN + ab is in the first class. If ab > N, the residue of ab on division by N is to be under- stood.
^ [These merely are the well known elementary properties of quadratic residues, which, since the time of Gauss, are phrased more briefly in modern terminology. The like applies to the proof given presently.]
94 SOURCE BOOK IN MATHEMATICS
II. Numbers of the first class a, h, c, d, etc., multiplied into any numbers of the second class a, /3, 7, 5, etc., give products in the second class.
III. A product of two numbers in the second class, say afi, falls into the first class.
We shall now proceed to the proof of Theorem 2, by means of a contradiction.
Suppose then that there are no three squares, not all divisible by N, whose sum is divisible by N. Then, so much the more, there are no two such squares. Hence it follows at once that the form \N — a, or what amounts to the same, \N + (N — a), cannot occur in the first class. For, if there were a square of the form \N — a, the sum of this and \N + a would be divisible by N, contrary to hypothesis. Hence the form XN — a is necessarily in the second class; the numbers — 1, —4, —9, etc., are among those of the set a, /3, 7, 5, etc. Let / by any number of the first class, so that there exist squares of the form \N + /. If to one of these be added a square of the form \N + 1, the sum of the two will have the form XN+/+ !• Now if there were squares of the form \N — / — 1, there would exist a sum of three squares divisible by N. Since this is denied, the form \N — / — 1 is not contained in the first class, and hence it is in the second. But in the second class there appear the numbers — 1 and — / — 1, and hence, by III above, their product/ + 1 is in the first class. In the same way it may be shown that the numbers
/+2, /+3, / + 4, etc.,
must occur in the first class. Hence, taking/ = 1, we see that all the numbers
\N +1, XN + 2, \N + 3, etc.,
occur in the first class, and therefore that there are none left for the second class. But, by the same reasoning, we see that the numbers —1, — / — 1, — / — 2, etc., occur in the second class, and hence all forms are in the second class. This obviously is a con- tradiction. It follows therefore that it is false that there are not three squares whose sum is divisible by N. Hence there are indeed three squares, and much more therefore four squares, of the prescribed kind whose sum is divisible by N.
Corollary. — From this theorem, combined with the preceding, it follows obviously that every number is a sum of four or fewer squares.
EULER Use of the Letter e to Represent 2.718...
(Selections Translated by Professor Florian Cajori, University of California, Berkeley, California.)
Prominent among the mathematicians who have contributed notations which have met with general adoption is the Swiss Leonhard Euler (1707- 1783). One of his suggestions, made when he was a young man of twenty or twenty-one, at the court in St. Petersburg, was the use of the letter e to stand for 2.718. . ., the base of the natural system of logarithms. It occurs in a manuscript of Euler entitled " Meditation upon Experiments made recently on the firing of Canon" (Meditatio in Experimenta explosione tormentorum nuper instituta). The manuscript was first printed in 1862 in Euler's Opera pos- tuma matbematica et pbysica, Petropoli, 1862, edited by P. H. Fuss and N. Fuss (Vol. II, p. 800-804). In this article, seven experiments are cited, which were performed between Aug. 21 and Sept. 2, 1727. These dates, and the word "recently" (nuper) in the title, would indicate that the article was written in 1727 or 1728. In it the letter e occurs sixteen times to represent 2.718. . . From page 800, we translate the following:
Let c designate the diameter of a globe [spherical projectile], in scruples of Rhenish feet,^ m:n the ratio of the specific gravity of the globe to the specific gravity of the air or the medium in which the globe moves, let t seconds be the length of time of the globe in air, let also the required height to which the body rises be X. For the number whose logarithm is unity, let e be written, which is 2,7182817. . .whose logarithm^ according to Vlacq is 0, 4342944. Also let N indicate the number of degrees of an arc, whose tangent is:
Ve*"*' - 1, the sinus totus [or radius] = 1. The required altitude x may be obtained from the following equation:
m\/c / / / i^ I _n_x \\
' = 447650V3n(m - n) O^W- 7162 log. (V.«""-V.'--0)-
/ 3nx
That the analysis may proceed more easily, let us call \e*'^ — 1
1 [Rhenish foot = 1000 scruples.] * (That is, logarithm to the base 10.)
95
96 SOURCE BOOK IN MATHEMATICS
= y, then N will be the number of degrees of the arc whose tangent
is y, . . .
In a letter of Nov. 25, 1731, addressed to Goldbach' (first published in 1843), Euler solves the differential equation
dz — Izdv -\ = — ; thus:
V V
This multiplied by e'''-^^ or what is the same, by e-'^'v (e denotes that number, whose hyperbolic logarithm is = 1), becomes
e-^'vdz — 2e-^''zvdv + e-'^'zdv = e-'^^'dv,
which, integrated, gives
e-^^vz = Const. — ^e"^''
or
2vz + I = ae^" . . .
The earliest occurrence in print of the letter e for 2.718. . . is in Euler's Mecbanica, 1736. It is found in Vol. I, page 68, and in other places, as well as in Vol. II, page 251, and on many of the 200 pages following. We quote, in translation, from Vol. I, page 68, where c means the velocity of the point under consideration:
Corollary II 171. Although in the foregoing equation the force p does not occur, its direction still remains, which depends upon the ratio of the elements dx and dy. Given therefore the direction of the force which moves the point and the curve along which the point moves, one can, from these data alone, derive the velocity of the
J .1 Cdyds
point at any place. For there will be — = ^-r- ot c = eJ "^j where e denotes the number whose hyperbolic logarithm is 1.
The use of the letter e, affected by imaginary exponents, in analytica expressions that were new to mathematics, occurs in a dissertation of Euler's entitled "On the sums of reciprocal series arising from the Powers of the natural Numbers" (De summis serierum reciprocarum ex potestatibus numerorum naturalium ortarum).* He lets 5 denote a circular arc and develops sin x into the now familiar infinite series. On page 177 he gives without explanation the exponential expression for sin s, and the fundamental limit for e* in the follow- ing passage:
^ Correspondance malbimatiquc tt physique de quelques celebres giomitres du XVIll^ sxicle. Par P. H. Fuss, St. Pttersbourg. 1843. Tome I, p. S8. ^ Miscellanea BiTolinensia, p. 172, Vol. VIl. Berlin, 1743.
EULER 97
Hence I am now able to write down all the roots or factors of the following infinite expression
C •'"^^ 4. ^' -. ^' 4. _J?!__ _ it
•^ 1.2.3 "^ 1.2.3.4.5 1.2.3.. .7"^ 1.2.3.9 ^'-
pt-<l/—l p—s-^—l
Indeed that expression is equivalent to this — 7^ — -.- — ^ — » e denoting the number whose logarithm is = 1, and, since e' = ( 1 H — ) , when n emerges an infinite number, the given in- finite expression is reduced to this:
{^-'-^y-i^-'-^y
^ 2V-1
More systematic development is found in Euler's Introductio in analysin infinitorum. Vol. I, Lausannae, 1748. We quote from § 138, in which the letter i is an infinitely great number:
. . . Substituting gives
COS. V
and
sin. V
(.H-^y-0-^^y
2V-1 In the preceding chapter we saw that
(-;)■-
e'
e denoting the base of hyperbolic logarithms; writing for z, first +r\/ — 1, then — rV — 1, there will be
g+vV— l-l- g— fV-l cos. V = ^
and
p+V-\/—l __ p—v-\/—l
''"■ " = 2V-1 From these it is perceived how imaginary exponential quantities are reduced to the sine and cosine of real arcs. For, there is
g+t-v-i _ ^.Qg_ J, _|_ y' _ i_ sin. V
g-BV-i = COS. V — V — !• sin. v.
98 SOURCE BOOK IN MATHEMATICS
If in the formula for e"*^"^"* one substitutes tt and V, there results the famous formula e^~^ = — i, indicating the strange interrelation of tt and e. Euler states this relation in the logarithmic form and generaHzed, in his paper, "De la Controverse entre Mrs. Leibnitz & Bernoulli sur les logarithmes des nombres negatifs et imaginaires," Histoire de Vacademie royale des sciences et belles lettres, annee 1749, Berlin, 1751, where on page 168 he refers to:
. . .this formula cos <p -\- \/ — I . sin <p, all logarithms of which are included in this general formula
/(cos <p -\- \/ — I. sin (f) = { <p -{- pTr)\/ — 1,
p indicating any even integer, either affirmative, or negative, or even zero. From this we derive. . .
/-! = (!+ p)W -I =qW -h
taking q to mark any odd integer. One has therefore:
/ _ 1 = +7rV - 1; ±3tV - 1; ±57rv/ - 1; ± /ttV - 1; &c.
HERMITE On the Transcendence of e
(Translated from the French by Dr. Laura Guggenbiihl, Hunter College, New
York City.)
Charles Hermite (1822-1901) was one of the best-known writers upon the function theory in the second half of the nineteenth century. He was a professor in the Ecole Polytechnique, an honorary professor in the University of Paris, and a member of the Academic des Sciences. His memoir on the transcendence of e was published in 1873. As is well known, ^ the character of the number tt was a source of disturbance in ancient times because of its connection with the classic problem of the quadrature of the circle. From the Greek period, names of famous mathematicians have been connected with transcendental numbers, but it was not until 1844 that a definite step forward was made in the general investigation of the subject. At this time, Liouville proved the existence of these numbers, thus justifying the classification of algebraic and transcendental. Liouville had already proved that e could not be a root of quadratic equation with rational coefficients. Finally, in 1873, Hermite's proof of the transcendence of e appeared. A few years later (1882), Lindemann, modeling his proof upon that of Hermite, proved the tran- scendence of TT.
The memoir is somewhat over thirty pages long and can be divided roughly into three parts. In the first two parts, two distinct proofs of the transcend- ence of e are given — but as Hermite says, the second is the more rigorous of the two. In the third part, Hermite obtains, applying the method suggested in the second proof, the following approximations:^
58291 , 158452
The translation here given includes, with indicated omissions, the portion referred to above as the second part of the memoir. Since the time when this paper first appeared, many simplifications have been made, so that now one rarely, if ever, sees more than an acknowledgement of the existence and impor- tance of this proof. The name "Hermite's Theorem" is still, however, given to the statement that e is a transcendental number.
' Monographs on Topics oj Modern Mathematics, edited by J. W. A. Young, Monograph IX, "The History and Transcendence of ir," by D. E. Smith. Additional references are there given.
2 Correct to six decimal places, e = 2.718282. This fraction gives e ■= 2.718289.
The correction of a numerical mistake pointed out by Picard, in his edition of Hermite's work, increases the accuracy of this approximation.
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100 SOURCE BOOK IN MATHEMATICS
. . . But, as a more general case, take
F(z) = (Z — ZoYoiz - Zi)"'- • .(Z - Zn)'^
for any Integral values whatever of the exponents, upon integrating both members of the identity
d[e-^F{z)] dz one obtains
= e-^[F'(z) - F(z)].
e
from which it follows that »z
F{7) = j e-'F'{z)dz - I e-'F{z)dz, Dws that r e-'F{z)dz = r e-'F{z)dz.'
Now the formula
F'{z) _ fio , Ml I I Atn
F(z) z— Zo z — Zi ' ' ' z — Zt,
yields the following decomposition,
■r-
+...+,. I -cQ?)rf!,.
Z - Zn
. . .We shall prove that it is always possible to determine two integral polynomials of degree n, 9(2) and 6i(z), such that, upon representing one of the roots 2!o> Si, . . . z^ by f , one has the following relation:
. . .And further, upon writing 0(2, f) in place of 9(z), to emphasize the presence of f , we have
e(z, f ) = Z» + 0i(f)z"-2 + 02a)z"-3 + . . . + 0n(r).'
' [Where Z represents any one of the roots Zo, Zi, . . .z„.] ^ /(z) = (2 — Zo)(z — Zi) . . . (z — z„). The proof of this statement, which is given in detail in the text, is here omitted.
* [It is shown in the text that e,(f ) is a polynomial of degree i in f, having for coefficients integral functions, with integral coefficients, of the roots Zo, I Zi, .... z„. ■
Gi(f) for t = 1, is not to be confused with 61(2), mentioned above in con- nection with 0(z).]
HERMITE
101
From this there follows, for the polynomial 6i(z), the formula
61(2) ^ tioQizo, r) ^ Mie(zi. r) _j_ _,_ Mne(zn, o_
/(z) Z — Zo Z — Zi ■ ' ■ 1 — Zn
... It is sufficient to take the integrals between the limits Zo and Z in the relation
and thus we obtain the equation
*Z t:/j\ f/-\ nz
Ji»
Jz„ 2 — 2i
(/2
+Mne(z„, r) r^^^^ c/2.
Jz, Z — Zn
We use this equation, in particular, in the case Mo = Ml = . . . =)u„ = m; in this case, if one writes
7719(2;, Zk) = (ik)
and if one takes f successively equal to Zo, 21, ... , 2„, the above relations evidently become
c/z« 2 Zi J20 2 2(,
re-i-{z)
Ji. 2 - 2i
(fe
+ ...'
for i = 0, 1, 2, . . . , 77. But for the general case, we must still prove the following theorem. Let A and 5 be the determinants
G(2o, Zo) e(2i, Zo).. .e(2n, Zo)
0(2o, zi) e(2i, 2i)...e(2„, 2i) e{Zo, Zn) e(2l, Zn) ... 6(2n, 2n)
102 and
SOURCE BOOK IN MATHEMATICS
1
1 ..
. 1
Zo
Zi . .
. Zn
Zo'
Zl'.
.z„'
Zo''
Zl"
Zn"
then A = 8^^ Now, let
1.2 ... m - 1 Jz. z — Zi
the relation proved above
•z /»z„_
I e-'j^{z)dz = m — ^— ^ (/z + m — ^— ^ cfz
c/«» ,/Zo 2 ^O »/Z(, Z Z\
Jzo Z - Zn
becomes simply
and the relation •z
+ . . . -\-m i '' ^~' dz
^m "I Cm "i • • • I Cm >
•z„_.
t/zo ^ i »/zo 2 Zo
f/2
+ ...
+me(z„, f)
upon taking f successively equal to Zo, Zi,..., z„, gives us the following substitution, which we shall represent by Sm, namely
C^m+l = 6(Zo, Zo)€j' 4- 6(2l, 2o)€n,^ +• . . + 0(2n> Zo)im'', i^m+\ = Q{Zo, Zi)€m° + Q{ZU Zl)e^^ +. . . + ©(Zn, ^Ocm",
e"m+l = G(Zo, Z„)fm'' + 0(2l, 2»)€m^ + • • • + 6(z„, z„)e„".
If now, one builds up in turn Si, Sz,..., Sm-i, one concludes from these, expressions for em", Cm^.., €m" in terms of ei", €i', . . . , £"1", which we shall write as follows.
* [A short and simple proof for this statement is given in the text.j
HERMITE
103
€„»' = Boil" + Bici» +...+ B„ex'»,
and the determinant of this new substitution, being equal to the product of the determinants of the partial substitutions, will be g2(m-i)^ \l remains for us to replace ei°, ci^..., ei", by their values so that we shall have expressions for the quantities e^* in form suitable for our purpose. These values are easily obtained, as will be seen.
For this purpose, we apply the general formula
taking
that is
J
€~'F{z)dz = —e~'y{z),
F(z) = -=i-i-^
F{z) = 2" + r
+
+pir
H-P2
It is easily seen that y{z) will be an expression integral in z and f, entirely similar to Q{z, f), such that if one represents it by $(s, f), one has
Hz, f) = 2" + v5i(f)2"-i + v'2(f)2'*-2 + . . . + ^rXt),
where <p,(f ) is a polynomial in f of degree i, in which the coefficient of f ' is unity . . . and the analogy of the form with Q{z, f ) shows that the determinant
HZo, Zo) *(2i, Zo) . . .4)(2„, Zq) $(2o, 2l) *(2l, 2l). . .*(Zn, 2l)
4>(2o, 2„) $(2i, Zn) . . .^(2n, 2„)
is also equal to 5^. Next, we conclude from the relation '^-iiz)
I
taking f = Zi, the desired value
ci^ = e-'-Hzo, Zi)
dz = e-^-^iZo, f) - e-^<J>(Z, r),
'(Z, 2,).
Consequently we have the expressions given below for e™*.
104 SOURCE BOOK IN MATHEMATICS
Let
S3 = BoHZ, Zo) + BxHZ, zi) + . . . + BnHZ, Zn),
S = LMZ, Zo) + LMZ, 2i) + . . . + U^{Z, Zn), and let 2l<„ 58<„. . ., So be the values obtained for Z = Zo; one has
ern" = e-^°2lo - e-^" 21 €m' = e-'-S&o - e-^33
In these formulas, Z represents any one whatever of the quantities Zo, Zi,. . ., Zn, now if we wish to state the result for Z = Zk, we shall agree at the outset, to represent on the one hand, by 21^,58 a, . . . ?;■, and on the other, by 77^;°, rjk,. . ., Vk"', the values which the coeffi- cients 21, S3, . . ., S, and the quantities tm", ej,. . ., em"> take on in this case. Thus one obtains the equations
•nk" = e-^2(o - e-nik Vk' = e-^»S3o - e-'>'kS8
17^ = e-'-2o - e-'''2k, which will lead us to the second proof, we have mentioned, of the impossibihty of a relation of the form
where the exponents Zo, 2i, . . . , z„, as also the coefficients No, Ni, . . . , N„, are assumed to be integers.
Note in the first place, that €m* can become smaller than any given quantity for a sufficiently large value of m. For, the exponential e~^ being always positive, one has, as is known,
I e-'Fiz)dz = F(^) r e-'dz = F(^)(e-^<' - e"^),
F{z) being any function whatever, and ^ a quantity taken between Zo and Z, the limits of the integral. Now, upon taking
one obtains the expression
'" 1.2...m- l^-zr ^'
HERMITE
105
which demonstrates the property quoted above. Now, we obtain from the equations
7/2° = e-">%o - e-''%2.
the following relation,
e''77iWi + e'^aWa +. . . + e'"77nW„
If the condition
e'oNo + e''Ni + . . . + e'Wn = 0 is introduced, this relation becomes
e''77iWi + e'^zWa + . . . + e'''r]r,'>Nn
= - (SToiVo + 2tiiVx + . . . + 2I„iV„). However, under the assumption that Zo, Zi,..., Zn are integers, the quantities 9(2,-, Zk), ^(Zi, Zk) and consequently %„, 5(i, . - ., 21^ are also integers. Then we have a whole number
2IoiV<, + 5liNi+...+ 2In/V„, which decreases indefinitely with tji", 171^. . ., 771", as m increases; it follows that for a certain value of m and for all larger values,
510^0 + 5liNi+...+ 5I„N„ = 0, and, since one obtains similarly the relations
^oNo + S3i/Vi + . . . + 33„M. = 0,
the relation
. + ?nNn
= 0.
e'-No + e'^Ni + .
. + e'W„
= 0
demands that the determinant
2I0 9Ii .
.. 2ln
A =
be equal to zero. But, because of the expressions for 2to, 53o, . . . , ?o, it follows that A is the product of these two other determinants
Ao Ai . . . An
Bo Bi ... B„
Lo Li
Ln
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SOURCE BOOK IN MATHEMATICS
^{Zo, Zo) *(Zl, Zo). . .^Zn, Zo) *(3o, 2l) ^{Zi, Zi) . . .^{Zn, Zi)
^{Zo, Zn) $(2, Zr).. .^{Zn, Zn)
of which the first has for its value 82("*~^), and the second 6^ One has then A = S-*", and it is easily shown in an entirely rigorous manner, that the assumed relation is impossible,^ and that there- fore, the number e is not among the irrational algebraic numbers.
1 [It can be shown that 1 1 ... 1
Zo Zi . . . Zn
Zo^ Zi^ ... Z„2 = + (Z„ — Zn-l) (Z„ — Zn-2) . . . (Z„ — Zo)
(Z„_i - Z„_2) . . . (Zi - Zo)
Zo^ Zi"
and therefore that 5 is not zero, assuming, as is of course assumed throughout, that the exponents, Zo, zi, . . . , z„, are distinct.]
GAUSS On the Congruence of Numbers
(Translated from the Latin by Professor Ralph G. Archibald, Columbia University, New York City.)
Carl Friedrich Gauss (1777-1855), the son of a day laborer, was the founder of the modern school of mathematics in Germany and was, perhaps, equally well known in the fields of physics and astronomy. Kronecker (1823-1891) said of him that "almost everything which the mathematics of our century has brought forth in the way of original scientific ideas is connected with the name of Gauss." His work in the theory of numbers began when he was a student at Gottingen, and much of it appeared in his Disquisitiones Aritbmeticae, published in 1801, when he was only twenty-four years old. In this is found his treatment of the congruence of numbers, a translation of portions of which is here given. It also appears in the first volume of his Werke (Gottingen, 1870).
First Section
Concerning Congruence of Numbers in General
Congruent Numbers, Moduli, Residues, and Non-residues
1 If a number a divides the difference of the numbers 6 and c, b and c are said to be congruent with respect to a; but if not, incon- gruent. We call a the modulus. In the former case, each of the numbers b and c is called a residue of the other, but in the latter case, a non-residue.
These notions apply to all integral numbers both positive and negative,^ but not to fractions. For example, —9 and +16 are congruent with respect to the modulus 5; —7 is a residue of +15 with respect to the modulus 11, but a non-residue with respect to the modulus 3. Now, since every number divides zero, every number must be regarded as congruent to itself with respect to all moduli.
2 U k denotes an indeterminate integral number, all residues of a given number a with respect to the modulus m are contained in
* Obviously, the modulus is always to be taken absolutely, — that is, without any sign.
107
108 SOURCE BOOK IN MATHEMATICS
the formula a + km. The easier of the propositions which we shall give can be readily demonstrated from this standpoint; but anyone will just as easily perceive their truth at sight.
We shall denote in future the congruence of two numbers by this sign, =, and adjoin the modulus in parentheses when neces- sary. For example, — 16 = 9 (mod 5), — 7 = 15 (mod 11).^
3
Theorem. — // there be given the m consecutive integral numbers
a, a + 1, a + 2,. . ., a + m — 1,
and another integral number A, then some one of the former will be
congruent to this number A with respect to the modulus m; and, in
fact, there will be only one such number.
a — A . If, for Instance, Is an Integer, we shall have a = A; but
771
if it is fractional, let k be the Integer Immediately greater (or, when it is negative, immediately smaller if no regard is paid to sign). Then A + km will fall between a and a -{- m, and will therefore be the number desired. Now, it is evident that all the quotients
^ > ? , etc., are situated between k — I
mm m
and fe + 1 . Therefore not more than one can be integral.
Least Residues 4 Every number, then, will have a residue not only in the sequence 0, 1, 2,..., 771—1, but also in the sequence 0, —1, —2,..., — (77t — 1). We shall call these least residues. Now, it Is evident that, unless 0 is a residue, there will always be two: one positive, the other negative. If they are of different magnitudes, one of
them will be less than y; but If they are of the same magnitude,
each will equal y when no regard is paid to sign. From this it is
evident that any number has a residue not exceeding half the modulus. This residue Is called the absolute minimum.
^ We have adopted this sign on account of the great analogy which exists between an equality and a congruence. For the same reason Legendre, in memoirs which will later be frequently quoted, retained the sign of equality itself for a congruence. We hesitated to follow this notation lest it introduce an ambiguity.
GAUSS 109
For example, with respect to the modulus 5,-13 has the posi- tive least residue 2, which at the same time is the absolute mini- mum, and has —3 as the negative least residue. With respect to the modulus 7, +5 is its own positive least residue, —2 is the negative least residue and at the same time the absolute minimum.
Elementary Propositions Concerning Congruences
5
From the notions just established we may derive the following obvious properties of congruent numbers.
The numbers which are congruent with respect to a composite modulus, will certainly be congruent with respect to a7xy one of its divisors.
If several numbers are congruent to the same number with respect to the same modulus, they will be congruent among themselves (with respect to the same modulus).
The same identity of moduli is to be understood in what follows.
Congruent numbers have the same least residues, incongruent numbers different least residues.
6 // the numbers A, B, C, etc. and the numbers a, b, c, etc. are congruent each to each with respect to any modulus, that is, if
A = a, B = b, etc., then we shall have
A+ B -\- C + etc. = a + 6 + c + etc.
If A = a and B ^ b, we shall have A — B = a — b.
7
IJ A ^ a, we shall also have kA = ka.
If fe is a positive number, this is merely a particular case of the proposition of the preceding article when we place A = B = C etc. and a = 6 = c etc. If /s is negative, —k will be positive. Then — feA = —ka, and consequently kA = ka.
IJ A = aandB = 6, we shall have A B = ab. For, AJ5 = Ab = ba.
8
// the numbers A, B, C, etc. and the numbers a, b, c, etc. are con- gruent each to each, that is, ij A ^ a, B = b, etc., the products of the numbers of each set will be congruent; that is, ABC etc. = abc etc.
From the preceding article, AB = ab, and for the same reason ABC = abc; in a like manner we can consider as many factors as desired.
110 SOURCE BOOK IN MATHEMATICS
If we take all the numbers A, B, C, etc. equal, and also the corresponding numbers a, b, c, etc., we obtain this theorem: IJ A ^ a and ij k is a positive integer, we shall have A* s a''.
9
Let X be a Junction of the indeterminate x, of the form Ax" + Bx* + Cx' + etc., where A, B, C, etc., denote any integral numbers, and a, b, c, etc., non-negative integral numbers. IJ, now, to the indeterminate x there be assigned values which are congruent with respect to any stated modulus, the resulting values oj the Junction X will then be congruent.
Let/ and g be two congruent values of x. Then by the preceding articles /" = g" and A/" = Ag°^; in the same way B/* = JBg*, etc. Hence
A/<^ + BJ^ + C/'^ + etc. = A^ + Bg^ + Cg" + etc. Q. E. D.
It is easily seen, too, how this theorem can be extended to func- tions of several indeterminates.
10 If, therefore, all consecutive integral numbers are substituted for X, and if the values of the function X are reduced to least residues, these residues will constitute a sequence in which the same terms repeat after an interval of m terms (m denoting the modulus) ; or, in other words, this sequence will be formed by a period oJ m terms repeated indefinitely. Let, for example, -Y = x' — 8x + 6 and m = 5. Then for x = 0, 1, 2, 3, etc., the values of X give the positive least residues, 1, 4, 3, 4, 3, 1, 4, etc., where the first five, namely, 1, 4, 3, 4, 3, are repeated without end. And furthermore, if the sequence is continued backwards, that is, if negative values are assigned to x, the same period occurs in the inverse order. It is therefore evident that terms different from those constituting the period cannot occur in the sequence.
11
In this example, then, X can be neither =0 nor =2 (mod 5), and can still less be =0 or =2. Whence it follows that the equa- tions x' — 8x -^ 6 = 0 and x' — 8x + 4 = 0 cannot be solved in integral numbers, and therefore, as we know, cannot be solved in rational numbers. It is obviously true in general that, if it is impossible to satisfy the congruence X = 0 with respect to some particular modulus, then the equation X = 0 has no rational root when AT is a function of the unknown x, of the form X" + Ax"-i 4- Bx"-2 + etc. + N,
GAUSS 111
where A, B, C, etc. are integers and n is a positive integer. (It is well known that all algebraic equations can be brought to this form.) This criterion, though presented here in a natural manner, will be treated at greater length in Section VIII. From this brief indication, some idea, no doubt, can be formed regarding the utility of these researches.
Some Applications
12
Many of the theorems commonly taught in arithmetic depend upon theorems given in this section; for example, the rules for testing the divisibility of a given number by 9, 11, or other num- bers. With respect to the modulus 9, all powers of 10 are congruent to unity. Hence, if the given number is of the form a + 106 + 100c + etc., it will have, with respect to the modulus 9, the same least residue as a + 6 + c + etc. From this it is evident that, if the individual figures of the number, expressed in the denary scale, are added without regard to their position, this sum and the given number will exhibit the same least residues; and further- more, the latter can be divided by 9 if the former be divisible by 9, and conversely. The same thing also holds true for the divisor 3. Since with respect to the modulus 11, 100 = 1, we shall have generally 10^^ = 1 and 10^^+^ = 10 = — 1. Then a number of the form a + 106 + 100c + etc. will have, with respect to the modulus 11, the same least residue as a — 6 -j- c etc.; whence the known rule is immediately derived. On the same principle all similar rules are easily deduced.
The preceding observations also bring out the principle under- lying the rules commonlv relied upon for the verification of arith- metical operations. These remarks, of course, are applicable when from given numbers we have to deduce others by addition, sub- traction, multiplication, or raising to powers: in place of the given numbers, we merely substitute their least residues with respect to an arbitrary modulus (generally 9 or 11; since, as we have just now observed, in our decimal system residues with respect to these moduli can be so very easily found). The numbers thus obtained should be congruent to those which have been deduced from the given numbers. If, on the other hand, this is not the case, we infer that an error has crept into the calculation.
Now as these results and others of a similar nature are so very well known, it would serve no purpose to dwell on them further.
GAUSS Third Proof of the Law of Quadratic Reciprocity
(Translated from the Latin by D. H. Lehmer, M.Sc, Brown University, Providence, Rhode Island.)
The theorem with which the following pages are concerned and to which Gauss gave the name of Fundamental Theorem is better known today as Legendre's Law of Quadratic Reciprocity. Although a statement of a theorem equivalent to this law is found in the works of Euler^ without proof, the first enunciation of the law itself is attributed to Legendre,^ whose proof, however, is invalid. It tacitly assumes that there exist infinitely many primes in certain arithmetical progressions, a fact which was first established by Dirichlet half a century later. The first proof of this theorem was given by Gauss^ in 1801 and was followed by seven others in an interval of 17 years. The proof given below is the third one published,* although it is really his fifth proof. It is considered by Gauss and many others to be the most direct and elegant of his eight demonstrations.
In fact, in the first two paragraphs of the present proof Gauss expresses himself as follows:
§L The questions of higher arithmetic often present a remark- able characteristic which seldom appears in more general analysis, and increases the beauty of the former subject. While analytic investigations lead to the discovery of new truths only after the fundamental principles of the subject (which to a certain degree open the way to these truths) have been completely mastered; on the contrary in arithmetic the most elegant theorems frequently arise experimentally as the result of a more or less unexpected stroke of good fortune, while their proofs lie so deeply embedded in the darkness that they elude all attempts and defeat the sharpest inquiries. Further, the connection between arithmetical truths which at first glance seem of widely different nature, is so close that one not infrequently has the good fortune to find a proof (in an entirely unexpected way and by means of quite another
> EuLER, Opuscula, Vol. 1, p. 64. 1783.
-Legendre, Histoire de P Acadkmie des Sciences, pp. S16-S17, 1785; Tbiorie des Nombres, Ed. 1, pp. 214-226. 1798; Ed. 2, pp. 198-207. 1808.
'Gauss, Disquisitiones Aritbmeticae, Sect. 4, Leipzig, 1801; Werke, Gottingen, 1870, Bd. 1, pp. 73-111.
* Gauss, Comment ationes Societatis Regise Scientiarum Collingensis, Vol. 16, Gottingen, 1808; Werke, Gottingen, 1876. Bd. 2, pp. 1-8.
112
GAUSS 113
inquiry) of a truth which one greatly desired and sought in vain in spite of much effort. These truths are frequently of such a nature that they may be arrived at by many distinct paths and that the first paths to be discovered are not always the shortest. It is therefore a great pleasure after one has fruitlessly pondered over a truth and has later been able to prove it in a round-about way to find at last the simplest and most natural way to its proof. §2. The theorem which we have called in sec. 4 of the Disquisi- tiones Arithmeticae, the Fundamental Theorem because it contains in itself all the theory of quadratic residues, holds a prominent position among the questions of which we have spoken in the preceding paragraph. We must consider Legendre as the dis- coverer of this very elegant theorem, although special cases of it had previously been discovered by the celebrated geometers Euler and Lagrange. I will not pause here to enumerate the attempts of these men to furnish a proof; those who are interested may read the above mentioned work. An account of my own trials will suffice to confirm the assertions of the preceeding paragraph. I discovered this theorem independently in 1795 at a time when I was totally ignorant of what had been achieved in higher arithmetic, and consequently had not the shghtest aid from the Hterature on the subject. For a whole year this theorem tormented me and absorbed my greatest efTorts until at last I obtained a proof given in the fourth section of the above-mentioned work. Later I ran across three other proofs which were built on entirely diff"erent principles. One of these I have already given in the fifth section, the others, which do not compare with it in elegance, I have reserved for future publication. Although these proofs leave nothing to be desired as regards rigor, they are derived from sources much too remote, except perhaps the first, which however proceeds with laborious arguments and is overloaded with extended operations. I do not hesitate to say that till now a natural proof has not been produced. I leave it to the authorities to judge whether the following proof which I have recently been fortunate enough to discover deserves this discription.
Inasmuch as Gauss does not give any mathematical background in the introduction to his third proof or even a formal statement of the theorem itself (these having been given in his first proof), we shall attempt to supply in a few sentences the information necessary for the proper understanding of the theorem.
The integer p is said to be a quadratic residue or non-residue of an integer q relatively prime to p according as there exist or not solutions x of the congru-
114 SOURCE BOOK IN MATHEMATICS
ence x^ s p (mod q). These two cases may be written symbolically as pRq and pNq, respectively. If p and r are both residues or both non-residues of g, then they are said to have the same quadratic character with respect to q. With this understanding, the fundamental theorem may be stated in words as follows: // p and q are any distinct odd primes, then the quadratic character of p with respect to q is the same as that oj q with respect to p except when both p and q are of the form 4n — 1, in which case the characters are opposite.
The quadratic character of p with respect to q may be expressed by the symbol of Legendre
CO'
which has the value +1 or —1 according as pRq or pNq. The use of this symbol enables us to state the theorem analytically as follows / \ / \ (P - 1) (g - 1)
(0(l);<-»""^~"
We proceed with the translation of Gauss's proof in full:
§3. Theorem.^ — Let p be a positive prime number and k be any number not divisible by p. Further let A be the set oj numbers
(p-1)
and B the set
1,2,3,
(p + 1) (p + 3)
. . , p 1.
2 2
We determine the smallest positive residue modulo p oJ the product oj k by each oj the numbers in the set A. These will be distinct arid will belong partly to A and partly to B. Ij we let p, be the number oj these residues belonging to B, then k is a quadratic residue oj p or a non-residue oj p according as p, is odd or even.
Prooj. — Let a, a', a", ... be the residues belonging to the class A and 6, b', b", ... be those belonging to B. Then it is clear that the complements of these latter: p — b, p — b' , p — b",. . .are not equal to any of the numbers a, a', a", . . ., and together with them make up the class A. Consequently we have
1.2.3...^^ = a.a'.a"...{p - b){p - b')ip - b") . . .
The right-hand product evidently becomes, modulo p:
= {-\Yaa'a"...bb'b"... = {-iyk.2k.3k. . .k^-^
= {-irk^^n.2.3...^-^
' [This theorem is known to-day as Gauss's Lemma and the number m 's called the characteristic number.)
GAUSS
US
2
)
Hence
1 = {-iyk\
p-i that is fe 2 = + 1 according as n is even or odd. Hence our theorem follows at once.^
§4. We can shorten the following discussion considerably by introducing certain convenient notations. Let the symbol (k, p)^ represent the number of products among
fCy ^rCy D f\y • • ■ /v
2
whose smallest positive residues modulo p exceed p/1. Further if X is a non-integral quantity we will express by the symbol [x] the greatest integer less than x so that x — [x] is always a positive quantity between 0 and 1. We can readily establish the following relations:
I. M + [-.v] = -1.
\\. [x] + h = [x -{- b], whenever h is an integer. HI. [x]-^[h- x\ = h - 1. IV. If X — [x] is a fraction less than ]/2, then [2.v] — 2[x] = 0.
If on the other hand x — [x] is greater than 3^, then
[2x] - 2[x] = 1. V. If the smallest positive residue of h{mod p) is less than p/2,
then [2h/p] - 2[h/p] = 0.
If however it is larger than p/2, then [2h/p] — 2[b/p] = 1. VI. From this it follows that:
(k p) =
- 2
2k LP
+ 2k
4k LP
- 2
-+...+
(P - 1)^
- 2
Hp - l)/2l P J
VII. From VI and I we obtain without difficulty: (/e.p) + (-fe,p) =^^
From this it follows that the quadratic character of — /e with respect to p is the same as or opposite to the quadratic character
' [This follows from the famous Euler's criterion : k ^ = + 1 according as k is or is not a quadratic residue of p.]
^ [The symbol {k, p) replaces the characteristic number ^ of the preceding theorem.]
116
SOURCE BOOK IN MATHEMATICS
of k with respect to p, according as p is of the form 4n + 1 or 4n + 3. It is evident that in the first case —1 is a residue and in the second a non-residue of p.
VIII. We transform the formula given in VI as follows: From III we have
r(p-5)
L p
When we apply these substitutions to the last .
above series we have first, when p is of the form 4n + x,
{k - l)(p - 1)
-'— [f]
P + 1
terms of the
iK P) =
4 - 2
-1 +
.Pj
LP.
[?1
+
+ - +
5k P . 3k
l3l LP
+ ...+
+ ...+
fe(p - 3)/2"
P ^(P - l)/2'
second, when p is of the form 4n + 3 - 2
ir^"
ru"
\'^k^
+
+
I LP.
IP .
LP J
r^i
'7.k'\
'U]
+
+
lpJ
.p J
-P J
+ ...+
+ ...+
■fe(p - l)/2
■/e(p - l)/2
IX. In the special case k = +2 it follows from the above formu- las^ that (2, p) = (p T l)/4, where we take the upper or lower sign according as p is of the form 4n + 1 or 4n + 3. Therefore (2, p) is even and hence 2jRp in case p is of the form 8?! + 1 or 8?t + 7; on the other hand (2, p) is odd and hence 2A/p when p is of the form 8n + 3 or 8n + 5.
§5. Theorem. — // x is a positive non-integral quantity among whose multiples x, 2x, 3x,. . ., nx there exist no integers; putting
1 [Each term in the braces is zero in this case, since the quantities in the square brackets are less than unity.]
GAUSS
117
[fix] = h we easily conclude that among the multiples oj the reciprocal _,_,_...- there appear no i7itegers. Then I say that:
XXX X
M + [2x] + [3x]+...+
+
'MMl\
f^H-...+
= nh.
Proof. — In the series [x] + [2x] + [3a:] +• • -I^^], which we set equal to Q, all the terms from the first up to and including the
1
X
the
are manifestly zero, the following terms up to and including
are equal to 1, and the following up to equal to 2 and so on. Hence we have-
term are
fi= ox i
-
Ix
J
+ix{
2 xj
1
X
+2x{
3 xj
2
X
. (r
4]
r^i
+3X
xj
\_xj
= hn —
«'-.i-[^
+h
n —
l
X
-
'I
X
-
"3"
X
Q. E. D.
§6. Theorem. — // /e and p are positive odd numbers prime to each other, we have
\kl
r?^i
\^ki
4-
+
LpJ
IP \
LP J
+
p
+
'2p'
+
"3p" .k .
+...+ +...+
l p
-l)/2
[=
(fe-l)/2
(k-l)(p-l)
Proof. — Supposing that k < p we have —^ — < tj but >
— li — , and hence
■fe(p - l)/2"
k-\
118
SOURCE BOOK IN MATHEMATICS
From this it is clear that the theorem follows at once from the preceding one if we set
k p - I
- = X, — y— P 2
= n.
1
= h.
It is possible to prove in a similar way that if k is even and prime to p, then
+
'k' .p.
+
'2k' .P .
+
'3k'
lP .
+ ..
.+
'k(p- P
■ l)/2l
p
+
2p k
+
3p k
+ .
.+
kp/2 L k \
= k'-
- 1
However we will not prove this proposition as it is not necessary for our purpose.
§7. Now the main theorem follows from the combination of the last theorem with proposition VIII of paragraph 4. For if we designate by k and p any distinct, positive prime numbers^ and put
(K p) +
(P, k) +
r^"
\7k'
\^k'
+
+
+ .
.+
IP.
Lp .
I P _
p _k_
+
2p k
+
3p k
+ ..
.+
k(p - l)/2
P p(k - l)/2"
= u
= M
then it follows from §4, VIII, that L and M will always be even numbers. It follows from the theorem of §6 that
L + a; = ik. p) + (p.k) + (±^^^
Therefore, when {k — l)Cp — l)/4 is even, that is when one or both of the primes /j or p is of the form 4n + 1, then (p, k) and (k, p) are either both even or both odd. On the contrary when (k — l)(p — l)/4 is odd, that is when k and p are both of the form 4n + 3, then necessarily one of the numbers (k, p), (p, k) is even and the other odd. In the first case the relations of k to p, and of p to fe (as regards the quadratic character of one with respect to the other) are the same; in the second case they are opposite. Q. E. D.
• [In which] k and p should also be different from 2.
KUMMER On Ideal Numbers
(Translated from the German by Dr. Thomas Freeman Cope, National Research Fellow in Mathematics, Harvard University, Cambridge, Mass.)
Ernst Edward Kummer' (1810-1893), who was professor of mathematics in the University of Breslau from 1842 till 1855 and then in the University of Berlin until 1884, made valuable contributions in several branches of mathe- matics. Among the topics he studied may be mentioned the theory of the hypergeometric (Gaussian) series, the Riccati equation, the question of the convergency of series, tlie theory of complex numbers, and cubic and biquad- ratic residues. He was the creator of ideal prime factors of complex numbers and studied intensively surfaces of the fourth order and, in particular, the surfaces which bear his name.
In the following paper which appears in the original in Crelle's Journal filr die reine und angeivandte Matbematik (Vol. 35, pp. 319-326, 1847), Kummer introduces the notion of ideal prime factors of complex numbers, by means of which he was able to restore unique factorization in a field where the funda- mental theorem of arithmetic does not hold. Although Rummer's theory has been largely supplanted by the simpler and more general theory of Dedekind, yet the ideas he introduced were of such importance that no less an authority than Professor E. T. Bell is responsible for the statement that^ " Kummer's introduction of ideals into arithmetic was beyond all dispute one of the greatest mathematical advances of the nineteenth century." For the position of Kummer's theory in the theory of numbers, the reader is referred to the article by Professor Bell from which the above quotation is taken.
On the Theory Of Complex Numbers
(By Professor Kummer of Breslau.) (Abstract of the Beiicbten der Konigl. Akad. der Wiss. zu Berlin, March 1845.) I have succeeded in completing and in simplifying the theory of those complex numbers which are formed from the higher roots of unity and which, as is well known, play an important role in cyclotomy and in the study of power residues and of forms of higher degree; this I have done through the introduction of a peculiar kind of imaginary divisors which I call ideal complex
' For a short biographical sketch, see D. E. Smith, History of Matbematics, Vol. I, pp. 507-508, Boston, 1923.
- American Mathematical Monthly, Vol. 34, pp. 66.
119
120 SOURCE BOOK IN MATHEMATICS
numbers and concerning which I take the liberty of making a few remarks.
If a is an imaginary root of the equation a^ = 1, X a prime num- ber, and a, Oi, 02, etc. whole numbers, then /(a) = a + oia + 020:^ + . . • + a\^ia^~^ is a complex whole number. Such a complex number can either be broken up into factors of the same kind or such a decomposition is not possible. In the first case, the number is a composite number; in the second case, it has hitherto been called a complex prime number. I have observed, however, that, even though /(a) cannot in any way be broken up into com- plex factors, it still does not possess the true nature of a complex prime number, for, quite commonly, it lacks the first and most important property of prime numbers; namely, that the product of two prime numbers is divisible by no other prime numbers. Rather, such numbers /(a), even if they are not capable of decom- position into complex factors, have nevertheless the nature of composite numbers; the factors in this case are, however, not actual but ideal complex numbers. For the introduction of such ideal complex numbers, there is the same, simple, basal motive as for the introduction of imaginary formulas into algebra and analysis; namely, the decomposition of integral rational functions into their simplest factors, the linear. It was, moreover, such a desidera- tum which prompted Gauss, in his researches on biquadratic residues (for all such prime factors of the form 4m + 1 exhibit the nature of composite numbers), to introduce for the first time com- plex numbers of the form a -f b-\/—\.
In order to secure a sound definition of the true (usually ideal) prime factors of complex numbers, it was necessary to use the properties of prime factors of complex numbers which hold in every case and which are entirely independent of the contingency of whether or not actual decomposition takes place: just as in geometry, if it is a question of the common chords of two circles even though the circles do not intersect, one seeks an actual defini- tion of these ideal common chords which shall hold for all positions of the circles. There are several such permanent properties of complex numbers which could be used as definitions of ideal prime factors and which would always lead to essentially the same result; of these, I have chosen one as the simplest and the most general.
If p is a prime number of the form mX + 1, then it can be repre- sented, in many cases, as the product of the following X — 1 complex factors: p = /(a)-/(a^)-/(a^). . ./(a^"'); when, however, a
KUMMER 121
decomposition into actual complex prime factors is not possible, let ideals make their appearance in order to bring this about. If f(a) is an actual complex number and a prime factor of p, it has the property that, if instead of the root of the equation a^ = 1 a definite root of the congruence ^^ = 1, mod. p, is substituted, then /(^) = 0, mod. p. Hence too if the prime factor /(a) is contained in a complex number 4>(a), it is true that $(^) = 0, mod. p; and conversely, if $(^) = 0, mod. p, and p is factorable into X — 1 complex prime factors, then $(a) contains the prime factor /(a). Now the property $(^) = 0, mod. p, is such that it does not depend in any way on the factorability of the number p into prime factors; it can accordingly be used as a definition, since it is agreed that the complex number $(a) shall contain the ideal prime factor of p which belongs to a = ^, if $(^) = 0, mod. p. Each of the X — 1 complex prime factors of p is thus replaced by a congruence rela- tion. This suffices to show that complex prime factors, whether they be actual or merely ideal, give to complex numbers the same definite character. In the process given here, however, we do not use the congruence relations as the definitions of ideal prime factors because they would not be sufficient to represent several equal ideal prime factors of a complex number, and because, being too restrictive, they would yield only ideal prime factors of the real prime numbers of the form mX — 1.
Every prime factor of a complex number is also a prime factor of every real prime number g, and the nature of the ideal prime factors is, in particular, dependent on the exponent to which q belongs for the modulus X. Let this exponent be/, so thatg^ = 1, mod. X, and X — 1 = e-J. Such a prime number q can never be broken up into more than e complex prime factors which, if this decomposition can actually be carried out, are represented as hnear functions of the e periods of each set of / terms. These periods of the roots of the equation a^ = 1, I denote by rj, t/i, 772, . . .r)e-i; and indeed in such an order that each goes over into the following one whenever a is transformed into a'>', where 7 is a primitive root of X. As is well known, the periods are the e roots of an equation of the eth degree; and this equation, considered as a congruence for the modulus q, has always e real congruential roots which I denote by u, Ui, U2, . . .Ue_i and take in an order corresponding to that of the periods, for which, besides the con- gruence of the eth degree, still other easily found congruences may be used. If now the complex number c'77 + c/171 + Ca'Tjs 4-
122 SOURCE BOOK IN MATHEMATICS
. . . + c'e~iVe-u constructed out of periods, is denoted shortly by ^{rj), then among the prime numbers q which belong to the expo- nent /, there are always such that can be brought into the form
q = ^{ri)^r]i)Hm) ■ ■ -Hve-i), in which, moreover, the e factors never admit a further decomposi- tion. If one replaces the periods by the congruential roots corresponding to them, where a period can arbitrarily be designated to correspond to a definite congruential root, then one of the e prime factors always becomes congruent to zero for the modulus q. Now if any complex number J{a) contains the prime factor ^(v), it will always have the property, for 77 = u^t, tji = Uk+u 772 = Uk+2, etc., of becoming congruent to zero for the modulus q' This property (which imphes precisely / distinct congruence relations, the development of which would lead too far) is a permanent one even for those prime numbers q which do not admit an actual decomposition into e complex prime factors. It could therefore be used as a definition of complex prime factors; it would, however, have the defect of not being able to express the equal ideal prime factors of a complex number.
The definition of ideal complex prime factors which I have chosen and which is essentially the same as the one described but is simpler and more general, rests on the fact that, as I prove separately, one can always find a complex number ^(77), constructed out of periods, which is of such a nature that 'A ('?)'/' (171) "A ('72). . . yp{rie-i) (this product being a whole number) is divisible by q but not by q^. This complex number \l/(rj) has always the above- mentioned property, namely, that it is congruent to zero, modulo q, if for the periods are substituted the corresponding congruential roots, and therefore \p(r]) = 0, mod. g, for 77 = u, 771 = wi, 772 = M2, etc. I now set ^PiviJ^iv^) • • .\J/(ve~i) = "^(v) and define ideal prime numbers in the following manner: —
If /(a) has the property that the product /(a). ^(77r) is divisible by q, this shall be expressed as follows: J(a) contains the ideal prime factor of q which belongs to u = 77^. Furthermore, if f{a) has the property that J{(x).{'^(r]r)Y is divisible by g** but /(a) (^(77,))"+^ is not divisible by g^+S this shall be described thus: /(a) contains the ideal prime factor of q which belongs to u = 77r, exactly n times.
It would lead too far if I should develop here the connection and the agreement of this definition with those given by congru- ence relations as described above; I simply remark that the
KVMMER 123
relation; j{ot)'^{r)r) divisible by q, is completely equivalent to/ distinct congruence relations, and that the relation; /(a) (^(77,))" divisible by g", can always be entirely replaced by u-j congruence relations. The whole theory of ideal complex numbers which I have already perfected and of which I here announce the principal theorems, is a justification of the definition given as well as of the nomenclature adopted. The principal theorems are the following:
The product of two or more complex numbers has exactly the same ideal prime factors as the factors taken together.
If a complex number (which is a product of factors) contains all the e prime factors of q, it is also divisible by q itself; if, however, it does not contain some one of these 3 ideal prime factors, it is not divisible by q.
If a complex number (in the form of a product) contains all the e ideal prime factors ot q and, indeed, each at least fx times, it is divisible by g^.
If /(a) contains exactly m ideal prime factors of g, which may all be different, or partly or wholly alike, then the norm A//(«) = J{ot)f{a^) . . ./(a^~^) contains exactly the factor q"'^.
Every complex number contains only a finite, determinate number of ideal prime factors.
Two complex numbers which have exactly the same ideal prime factors differ only by a complex unit which may enter as a factor.
A complex number is divisible by another if all the ideal prime factors of the divisor are contained in the dividend; and the quotient contains precisely the excess of the ideal prime factors of the dividend over those of the divisor.
From these theorems it follows that computation with complex numbers becomes, by the introduction of ideal prime factors, entirely the same as computation with integers and their real integral prime factors. Consequently, the grounds for the complaint which I voiced in the Breslauer Programm zur Jubelfeier der Universitdt Konigsherg S. 18, are removed: —
It seems a great pity that this quality of real numbers, namely, that they can be resolved into prime factors which for the same number are always the same, is not shared by complex numbers; if now this desirable property were part oj a complete doctrine, the effecting oj which is CLS yet beset with great difficulties, the matter could easily be resolved and brought to a successful conclusion. Etc. One sees therefore that ideal prime factors disclose the inner nature of complex numbers, make them transparent, as it were, and show
124 SOURCE BOOK IN MATHEMATICS
their inner crystalline structure. If, in particular, a complex number is given merely in the form a -\- aia + 020^ + • • • + a\-ia^~^, little can be asserted about it until one has determined, by means of its ideal prime factors (which in such a case can always be found by direct methods), its simplest qualitative properties to serve as the basis of all further arithmetical investigations.
Ideal factors of complex numbers arise, as has been shown, as factors of actual complex numbers: hence ideal prime factors multiplied with others suitably chosen must always give actual complex numbers for products. This question of the combination of ideal factors to obtain actual complex numbers is, as I shall show as a consequence of the results which I have already found, of the greatest interest, because it stands in an intimate relation- ship to the most important sections of number theory. The two most important results relative to this question are the following:
There always exists a finite, determinate number of ideal complex multipliers which are necessary and sufficient to reduce all possible ideal complex numbers to actual complex numbers.^
Every ideal complex number has the property that a definite integral power of it will give an actual complex number.
I consider now some more detailed developments from these two theorems. Two ideal complex numbers which, w^hen muItipHed by one and the same ideal number, form actual complex numbers, I shall call equivalent or of the same class, because this investigation of actual and ideal complex numbers is identical with the classifica- tion of a certain set of forms of the X — 1st degree and in X — 1 variables; the principal results relative to this classification have been found by Dirichlet but not yet pubfished so that I do not know precisely whether or not his principle of classification coincides with that resulting from the theory of complex numbers. For example, the theory of a form of the second degree in two variables with determinant, however, a prime number X, is closely interwoven with these investigations, and our classification in this case coincides with that of Gauss but not with that of Legendre. The same considerations also throw great light upon Gauss's classification of forms of the second degree and upon the true basis for the diff"erentiation between Aequivalentia propria et impropria,^
1 A proof of this important theorem, although in far less generality and in an entirely different form, is found in the dissertation : L. Kronecker, De unilati- bus complexis, Berlin, 1845.
* [i. e., proper and improp>er equivalence.]
KUMMER 125
which, undeniably, has always an appearance of impropriety when it presents itself in the Disquisitiones aritbmeticae. If, for exam- ple, two forms such as ax^ + 2bxy + cy^ and ax^ — 2bxy + cy^, or ax^ + 2bxy + cy^ and cx^ + 2bxy + ay^, are considered as belonging to different classes, as is done in the above-mentioned work, while in fact no essential difference between them is to be found; and if on the other hand Gauss's classification must not- withstanding be admitted to be one arising for the most part out of the very nature of the question: then one is forced to consider forms such as ax^ + 2hxy + cy"^ and ax^ — 2bxy + cy^ which differ from each other in outward appearance only, as merely representative of two new but essentially different concepts of number theory. These however, are in reahty nothing more than two different ideal prime factors which belong to one and the same number. The entire theory of forms of the second degree in two variables can be thought of as the theory of complex numbers of the form x + y V -D and then leads necessarily to ideal complex numbers of the same sort. The latter, however, classify them- selves according to the ideal multipliers which are necessary and sufficient to reduce them to actual complex numbers of the form X + yy/D. Because of this agreement with the classification of Gauss, ideal complex numbers thus constitute the true basis for it. The general investigation of ideal complex numbers presents the greatest analogy with the very difficult section by Gauss: De compositione jorvxarum, and the principal results which Gauss proved for quadratic forms, pp. 337 and following, hold true also for the combination of general ideal complex numbers. Thus there belongs to every class of ideal numbers another class which, when multiphed by the first class, gives rise to actual complex numbers (here the actual complex numbers are the analogue of the Classis principalis).^ Likewise, there are classes which, when multiplied by themselves, give for the result actual complex numbers (the Classis principalis), and these classes are therefore ancipites;^ in particular, the Classis principalis itself is always a Classis anceps. If one takes an ideal complex number and raises it to powers, then in accordance with the second of the foregoing theorems, one will arrive at a power which is an actual complex number; if b is the smallest number for which (/(a))^ is an actual
^ [Principal class.]
^ [Dual, or of a double nature.]
126 SOURCE BOOK IN MATHEMATICS
complex number, then /(a), (/(q:))^ (/(a))^. . . (/(a))'' all belong to different classes. It now may happen that, by a suitable choice of /(a), these exhaust all existing classes: if such is not the case, it is easy to prove that the number of classes is at least always a multiple of h. I have not gone deeper yet into this domain of complex numbers; in particular, I have not undertaken an investi- gation of the exact number of classes because I have heard that Dirichlet, using principles similar to those employed in his famous treatise on quadratic forms, has already found this number. I shall make only one additional remark about the character of ideal complex numbers, namely, that by the second of the fore- going theorems they can always be considered and represented as definite roots of actual complex numbers, that is, they always take the form v$(a;) where $(«) is an actual complex number and b an integer.
Of the different applications which I have already made of this theory of complex number, I shall refer only to the application to cyclotomy to complete the results which I have already announced in the above-mentioned Programm. If one sets
{a, x) = X -\- ax" + aV' + . . . + aP-^.v"""',
where a^ = 1, x^ = 1, p = m\ + 1, and g is a primitive root of the prime number p, then it is well known that (a, x)^ is a complex number independent of x and formed from the roots of the equa- tion a^ = 1. In the Programm cited, I have found the following expression for this number, under the assumption that p can be resolved into X — 1 actual complex prime factors, one of which is
(a, x)^ = ±a''r'(a)-/'"K«')-rK«')- • .r^-'(a^-'), where the power-exponents mi, m2, m^, etc. are so determined that the general rn.K, positive, is less than X and k-mk = 1, mod. X. Exactly the same simple expression holds in complete generality, as can easily be proved, even when J(a) is not the actual but only the ideal prime factor of p. In order, however, in the latter case, to maintain the expression for (a, x)^ in the form for an actual complex number, one need only represent the ideal J(a) as a root of an actual complex number, or apply one of the methods (although indirect) which serve to represent an actual complex number whose ideal prime factors are given.
CHEBYSHEV (TCHEBYCHEFF) On the Totality of Primes
{Translated Jrom the French by Professor J. D. Tamarkin, Brown University, Providence, Rhode Island.)
Pafnuty Lvovich Chebyshev (Tchebycheff, Tcliebytcbeff) was born on May 14, 1821, and died on Nov. 26, 1894. He is one of the most prominent repre- sentatives of the Russian mathematical school. He made numerous important contributions to the theory of numbers, algebra, the theory of probabilities, analysis, and applied mathematics. Among the most important of his papers are the two memoirs of which portions are here translated:
1. "Sur la totalite des nombres premiers inferieurs k une limite donnee," Memoires presentes a l' Academic Imperiale des Sciences de St.-Petersbourg par divers savants et lus dans ses assemblees. Vol. 6, pp. 141-157, 1851 (Lu le 24 Mai, 1848); Journal de Mathematiques pures et appliquees, (1) Vol. 17, pp. 341-365, 1852; Oeuvres, Vol. 1, pp. 29-48, 1899.
2. "Memoire sur les nombres premiers," ifcici.. Vol. 7, pp. 15-33, 1854 (lu le 9 Septembre, 1850), ibid., pp. 366-390, ibid., pp. 51-70.
These memoirs represent the first definite progress after Euclid in the investigation of the function <^(x) which determines the totality of prime num- bers less than the given limit x. The problem of finding an asymptotic expression for 0(.v) for large values of x attracted the attention and efforts of some of the most brilliant mathematicians such as Legendre, Gauss, Lejeune- Dirichlet, and Riemann.
Gauss (1791, at the age of fourteen) was the first to suggest, in a purely
. X
empirical way, the asymptotic formula j — — for <t>ix). {Werke, Vol. Xi,
p. 11, 1917.) Later on (1792-1793, 1849), he suggested another formula
/*x fix . X .
I , > of which , is the leading terra (Gauss's letter to Encke, 1849,
J2 log X log X & V
Werke, Vol. II, pp. 44:4-44:7, 1876). Legendre, being, of course, unaware
of Gauss's results, suggested another empirical formula xi „ (Essai
sur la tbeorie des nombres, 1st ed., pp. 18-19, 1798) and specified the con- stants A and B as A = 1, B = —1.08366 in the second edition of the £5501 (pp. 394-395, 1808). Legendre's formula, which Abel quoted as "the most marvelous in mathematics" (letter to Holmboe, Abel Memorial, 1902, Corre- spondence, p. 5), is correct up to the leading term only. This fact was recog- nized by Dirichlet ("Sur I'usage des series infinies dans la theorie des nombres," Crelle's Journal, Vol. 18, p. 272, 1838, in his remark written on the copy pre- sented to Gauss. Cf. Dirichlet, Werke, Vol. 1, p. 372, 1889). In this note
127
128 SOURCE BOOK IN MATHEMATICS
X
to Gauss, Dirichlet suggested another formula ^ , The proof of these
results, although announced by Dirichlet, has never been published, so that Chebyshev's (TchebychefF's) memoirs should be considered as the first attempt at a rigorous investigation of the problem by analytical methods.
X
Chebyshev did not reach the final goal — to prove that the ratio 4>ix): -,
tends to 1 as a: —♦ 00. This important theorem was proved some 40 years later by Hadamard ("Sur la distribution des zeros de la fonction f (s) et ses consequences arithmetiques," Bulletin de la Societe Matbematique de France, Vol. 24, pp. 199-220, 1896) and by de la Vallee Poussin ("Recherches analy- tiques sur la theorie des nombres premiers," Annates de la Societe Scientifique de Bruxelles, Vol. 20, pp. 183-256, 1896), their work being based upon new ideas and suggestions introduced by Riemann ("tlber die Anzahl der Prim- zahlen unter einer gegebenen Grenze," Monatsbericbte der Berliner Akademie, pp. 671-680, 1859; Werke, 2nd ed., pp. 145-153, 1892).
Although Chebyshev did not prove this final theorem, still he succeeded in obtaining important inequalities for the function <i>{x), which enabled him to investigate the possible forms of approximation of </>(x) by means of expressions containing algebraically x, e', log x (Memoir 1, above) with a conclusion con- cerning the rather limited range of applicability of Legendre's formula. In the Memoir 2, Chebyshev obtains rather narrow limits for the ratio <^(x):
. — > which provide a proof for the famous Bertrand postulate: "If x ^ 2,
there is at least one prime number between x and 2x — 2."
Memoir 1 : On the Function which Determines the Totality OF Primes Less than a Given Limit
§L Legendre in his Tbeorie des nombres^ proposes a formula for the number of primes between 1 and any given limit. He begins by comparing his formula with the result of counting the primes in the most extended tables, namely those from 10,000 up to 1,000,000, after which he applies his formula to the solution of many problems. Later the same formula has been the object of investigations of Mr. Lejeune-Dirichlet who announced in one of his memoirs in Crelle's Journal, Vol. 18, that he had found a rigorous analytical proof of the formula in question.^ Despite the authority of the name of Mr. Lejeune-Dirichlet and the pro- nounced agreement of the formula of Legendre with the tables of primes we permit ourselves to raise certain doubts as to its
' Volume 2, p. 65 (3rd edition).
* [Naturally Chebyshev was unaware of the marginal notation made by Dirichlet in the copy of his paper presented to Gauss, to which we referred above.]
CHEBYSHEV (TCHEBYCHEFF) 129
correctness and, consequently, as to the results which have been derived from this formula. We shall base our assertion on a theorem concerning a property of the function which determines the totality of primes less than a given limit, — a theorem from which one might derive numerous curious consequences. We shall first give a proof of the theorem in question; after that we shall indicate some of its applications.
§2. Theorem 1. — // <t>{x) designates the totality of primes less than X, n is any integer, and p is a quantity > 0, the sum
log^x
<f>ix + 1) - <t>{x) ^
logx
■ 1+p
x-2 L
will have the property of approaching a finite limit as p converges to zero.
Proof. — We begin by establishing the property in question for the functions which are obtained by successive differentiations, with respect to p, of the three expressions
S^p-^' log''-X'°g(i-;i4)'
The summation over m is extended, here as well as later, over all integral values from m = 2 up to m = oo, while that over n is taken over primes only, likewise from ju = 2 up to ^i = <» . Consider the first expression. It is readily seen that^
^x^'dx = y— rz^ e-'x^dx,
g-rj^-1+pj^ = - I e'^x^dx, consequently
fm'
""xPdx
X"-
^ [The first of these formulas is obtained by expanding / x _ -ix in the
geometric series Se"™"", which, being multiplied by x'' and integrated termwise, yields the expression
The termwise integration can be readily justified.]
130 SOURCE BOOK IN MATHEMATICS
By virtue of this equation the derivative of any order n with respect to p of ^— yip^ will be equal to a fraction whose denomi-
■U.
nator is I e~''x''dx
n+l
and whose numerator is a polynomial in
I (^1 - ')«-'.v'log^W.r,. . . j] (— i-j - |)-'.v-log..v<;.v,
I e-^x''dx, I e-='x''\ogxdx, \ e-^xp[og-xdx,. . . \ e-'x" log" xc/.v.
But a fraction of this type, no matter whether n = 0 or ?7 > 0, approaches a finite limit at p — » 0; for, then the limit of the integral
e~'x'' dx is 1, and the remaining integrals have finite hmiting
values.^
This proves that the function V — ~ — - and its successive
derivatives remain finite when p — > 0. Consider now the function
logp- Xlog(^l -^i+p}
[(>-^.)(>-3^.)o-^)-r
It is known that
= 1 + _I_ + ± + _1 +
^ [The reasoning here is justified, since all the integrals in question are uni- formly convergent in p for 0 ^ p ^ A, A being any fixed positive constant.]
^ [This identity was established by Eulcr ("Varia; observationes circa series infinitae," Commentarii Academiae Scientiarutn Petropolitanoe, 9, pp. 160- 188, 1737 (Theorem 8, p. l74);Leonardi Euleri Opera Omnia, (1) 14, pp. 216- 244 (230). Euler introduces here what is now called Ricmann's f-function as defined by the series
f(p) = X"""' " > ^•
1/= 1 The use of this function made by Ricmann (loc. cil.) gave a most powerful impetus to the modern theory of functions of a complex variable.
The infinite product here is absolutely convergent since (1 — m~^~'')~' =
1 + i^Ti^ — — r) and the scries zli i+p 1 ) is absolutely convergent, as well
CHEBYSHEV (TCHEBYCHEFF) 131
whence, with the notation adopted above,
Hence
or else
logp-Xlog(i-^,) = log(i + X;S^V log p - X log (l - ;^,) = log [ 1 + P + (X;;;!^, - ;)
This equation shows that all the derivatives with respect to p of
log p - 2) log [l - j^^j
can be expressed in terms of a finite number of fractions whose denominators are positive integral powers of
1 +P +
^7711+" p)^'
and whose numerators are polynomials in p and the expression
X — — and its derivatives with respect to p. The fractions
of this type tend to finite hmits as p — > 0: the expression 1 + P +
I "V — ~ jp, which figures in the denominators of these fractions,
tends to 1 as p — > 0, since, as we have proved, the difference V— 7- remains finite; as to the numerators, they are poly-
nomials in V, — r- and its derivatives, and, since all these
functions tend to finite hmits as p -^ 0, the same will hold true for the numerators in question.
It remains to prove the same property for the derivatives of the function
X '°8 (i - ^) + Xj^,-
We observe first that its first derivative is S/x"^^" log /x.(l - fi-^-")-'^-
as the series S^"'"'', which is only a part of the absolutely convergent series Sm"'"". All these series and their derived series are also uniformly convergent for p > 0, which justifies the termwise differentiations in the following work.]
132 SOURCE BOOK IN MATHEMATICS
From this it is readily seen that the derivatives of higher order also can be expressed in terms of a finite number of expressions of the form
with p, q, r '^ 0. But, each expression of this type has a finite value for p ^ 0, since the function under the sign 2 is of order higher than 1 in 1/ju.
After it has been proved that the derivatives of the three expressions above tend to finite Hmits as p — > 0, the same property can be estabHshed for the expression
^[x'o^O -"-'-') +5:"-']+
|1 [log p - X log (l - .-')] + $^, {Xm-'-' - 5) which, after the differentiations are performed, reduces to /^ Iog"M _ Y Jog"~^ ^\
This result imphes our theorem above, since it is readily seen that the difference
•^ log" n -^ log""' m
is identical with
jc = » r
X
x=2
4>{x + 1) - <i>{x)
log" X
log X or, what is the same thing, with
'if [*(x + 1) - *(x)i!^^ - X !^.
X = 2 jc= 2
To prove this we have only to observe that the first term of
the difference above equals X ^^^ since the coefficient
Jog" X <i>{x -\- 1) — 0(x) of ———;bydefinitionof thefunction <^(x) reduces
X
to 1 or to 0 according as x is a prime or a composite number. The second term is transformed into ^ ^^^ by replacing x by m.'
^ [From the modern point of view the essence of Chebyshev's proof above lies in the fact that f(p) is analytic for all values of p ?^ 1 while it has a simple
pole at p = 1 with the residue 1, whence f (p) — -, — ^jr is an entire transcen- dental function. (Whittaker-Watson, Modem Analysis, 3rd edition, 1920, p. 26.)]
1
CHEBYSHEV {TCHEBYCHEFF) 133
This completes the proof of the theorem in question. §3. The theorem which has been proved above leads to many curious properties of the function which determines the totality of primes less than a given limit. We first observe that the difference
_1 p+ dx
log X J log X
for X very large is an infinitesimal of the first order in 1/x; conse- quently the expression
/_1 p+^ dx Xlog^x
yogx J^ logxj x^+f
will be of order 1 -\- p with respect to \/x.^ Hence the sum remains finite for p ^ 0. On adding this sum to the expression for which Theorem 1 holds true, we conclude that the expression
also remains finite as p — > 0. From this we can derive the follow- ing theorem.
Theorem 2. — The Junction 4){x) which designates the totality of primes less than x, satisfies infinitely many times, between the limits X = 2 and x = «>, each of the inequalities
*w > £i^, - isf^ ^"<i *w < J" li + \sh'
no matter how small is the positive number a and, at the same time, how large is n.
Proof. — We shall restrict ourselves to the proof of one of these two inequalities; the second can be proved exactly in the same fashion. Take for instance the inequality /iN , s ^ C" dx , ax
To prove that this inequality is satisfied infinitely many times let us assume the contrary and examine the consequences of
^ [By this it is meant that the quotient of the difference in question by any power of 1/x less than (2 -f- p) tends to zero as x — * <« .]
134
SOURCE BOOK IN MATHEMATICS
this hypothesis. Let a be an integer greater than e" and, at the same time, greater than the greatest number which satisfies (1). With this assumption we shall have, for .v > a, the inequality
whence (2)
'^^'^ = £l^x-^r^^' i«g-^>"'
ax log" x'
J2 log X "^ log" X log X
0(x) -
But, if we admit inequalities (2), it will follow, in contradiction with the facts established above, that the expression
x = 2 L
4>{x + \) - 0(x)
r
log" X
•^ + ^ dx ' log X
will tend to + °o instead of converging to a finite limit as p — > 0. Indeed we can consider this expression as the limit of
0(.v+i) - <t>{x) - r^' ^•''
log.v
■l+P
log" X
— n:— as 5 — > 00 .
On assuming s > a, this can be presented under the form
</>(x + 1) - 0(x)
p + i dx
log" X
J''' + ^ dx _ log"x
log" X
(3) C+ X
x = a +
where
X = a
x = 2
remains finite for p ^ 0. On setting
u^ = 0(x) — I in the known formula
s s
2)"x(l'x+l - I'x) = UsVs+1 — UaVa+l " ^Vx{Ui - Ux_l),
dx
2 log ^'
Wx =
-1+p
a+l
we transform expression (3) into
,1+p
+
a + l
(/,(5 + 1) x=5
x = a+l riog"X log" (X - 1)
(x - 1)1+" _
CHEBYSHEV (TCHEBYCHEFF) which, in its turn, can be written as
C - U(a + 1) - I r^^ r-^^ + U(5 + 1)
135
I.
'' + ^ dx
log" 5
J^^ + i dx ]Iog"a r 2 log X J a'+p ■^['
r=a+lL
(/X_
1+p-
log (x - ^)
¥^.-;T^^'whereO<^<l. (x — 6)-+"
Let F denote the sum of the two first terms of this expression. Since, by virtue of condition (2), the third term is positive, we conclude that the expression above is greater than
x=s r
F+ X
<t>{x)
s:
' dx log.v
1 + p-
log" (x — d)
(x - 8)^+"
.». + ,L .. -"o-.. log(x-0)
The same conditions (2) show that the function under the sign S in the last expression remains positive within the hmits of sum- mation. Furthermore, we have, within the same hmits,
1°. 1 + P
2°.
>1-
since p > 0, x > a -{- I, 6 < I;
log (x — 0) ^ " log a
p dx a(x - 6)
J 2 log X log'* (x - e)
since, by the first of inequalities (2),
'' J2 log X log" X
while, by the second one, the derivative of -, — '- — 7 which equals "^ log" X ^
i 1 1 — i V is positive, whence,
log" X y log xy ^
ax
>
a (x - 6)
log" X log" (x — d) Hence our expression is greater than the sum
log" (x - e) _ (x - 6)^+" ~
^y^ a{x-d) / _ j^\ But this is obviously greater than
i+p
^-0-^)1,^-
1+p
136 SOURCE BOOK IN MATHEMATICS
which, for s — » « , reduces to
It is readily seen that the expression at which we have arrived tends to + =° as p — > 0. For, we have
— :rdx +00, e-^dx = 1.
0^-1 Jo
while both a and 1 — | are positive, the former by hypothesis
and the latter by the second of inequahties (2).
Thus, with the assumption made, it is assured that not only the sum
(/,(.v + l) - 4>(x) - ' "" ^""^'^'^
log x
but even a quantity which is less than this sum, tends to +oo, whence we conclude that the assumption in question is not admis- sible; this immediately proves Theorem 2.
§4. On the basis of the preceding proposition it will be easy now to prove the following theorem.
Theorem 3. — The expression -^ r ~ log -■^' can not hare a limit
0(x)
distinct from — 1 as x -^ oo .
Proof. — Let L be the limit as a: — > « of the difference — r^— log .v.
<t>{x)
Under this assumption we always can find a number N
x
so large that for x > iV the value of — r-r — log x will be within the
hmits L — t and L + e, e > 0 being as small as we please. For such values of x and e
But, by the preceding theorem, the inequahties
dx ax / V . r* dx . ax
, . C dx ax / N r* "X , ax
x
i
CHEBYSHEV (TCHEBYCHEFF) 137
are satisfied for infinitely many values of x, consequently also for values of x greater than N, for which inequalities (4) hold true. The inequalities (4), combined with those written above, imply
— log X > L — e.
r* dx __ ax J2 lo^ l^x
!:
dx ax
— logx < L +€,
'2
whence
L + 1 >
log x log" X
f
J log X log" X
^"^ dx ax
S
dx ax
log X log" X
Thus the absolute value of L + 1 does not exceed that of each of the expressions which figure in the right-hand members of the preceding inequalities. Furthermore, e can be made as small as we please by taking N sufficiently large, and the same will be true also of each of the quantities
X - {log X - 1) fr ^'^
s:
ax
dx _
log X log" X
for, it can be found by the principles of the differential calculus that their common limit for x = « is zero.
Thus it is shown that the limits between which the absolute value of L + 1 is included can be made arbitrarily small; hence L4-l=0orL=— 1, which was to be proved.
The fact established above concerning the limit of — 7-r — log x
for X = 00 does not agree with a formula given by Legendre for approximate computation of the totality of primes less than a given limit. According to Legendre the function 0(x) for x large
138 SOURCE BOOK IN MATHEMATICS
is expressed with a sufficient degree of approximation by the formula
X
<t>(x) =
logx - 1.08366
which gives for the limit of —^-^ — log .v the number — 1.08366
4>{x)
instead of —1,
§5. Starting from Theorem 2 it is possible to estimate the degree of approximation of the function 4>{x) by any other given function J{x). In what follows we shall compare the difference (x) — 0(x) with the expressions
XXX
log X log^ X log^ X ' ' '
To simplify the discussion we shall say that a quantity A is of
X . • V . . .
order j if, as x — > oo , the ratio of A to , — '- — is infinite for
log"* X log"* X
m > n and zero for m < n. We proceed now to prove the follow- ing theorem.
Theorem 4. — // tbe expression
has a finite [^0] or infinite limit as x — » oo, tbe Junction J(x) can not
X •
represent </)(x) up to terms oj order -. — ^^ inclusive.^
Proof. — Let L be the limit of the expression log"
n^^'^-£^)
as X ^ 00 . Since, by hypothesis, L is distinct from zero, it is either positive or negative. Assume L to be positive; our reasoning is readily apphed to the case of L < 0.
If L > 0 we can find a number N so large that for x > N the expression
remains always greater than a positive number /.
^ [This means to imply that the difference /(x) — <f>(x) can not be of order with m > n.]
log^x
CHEBYSHEV (TCHEBYCHEFF) 139
Hence, for x > N,
'■^^(^^^'-f--|-^)>'-
But, by Theorem 2, no matter how small a = 1/2 may be, the inequality
dx ax
log X log" X
(6) ■ 4>{x) < r v^. +
will be satisfied for infinitely many values of .v, which gives
p dx J log
/W- T~-<K^)-<P(^) +
ax
X log" x'
log" X on multiplying this by —^ — and observing that a = 1/2 we find
log" X
X
or, in view of (5)
J2 ^og^
< !^ f^(x) - 0(x)] + i
^°^" "" [/(-v) - <^(^-)] > I
X
Since 1/2 > 0 and the preceding inequality, as well as inequalities (5) and (6), are satisfied for infinitely many values of x, the hmit of
!^ [f(x) - 0(x)]
as X— > Qo can not be equal to zero. Then the difference /(x) — 0(x),
according to the agreement above, is either of order -. — '- — ^ ^ log" X
or of a lower order, which was to be proved.
On the basis of this theorem we can show that the formula of
X
Legendre, ? rruvvr^' for which the limit as x — > « of the
log X — 1.08366
expression
log^ X / X p dx \
X yog X- 1.08366 J2 logx^
equals 0.08366, can not represent 0(x) up to terms of order | — ^—
inclusive.
It is also easy to determine the constants A and B so that the
function -7—. ; — =5 will represent <i(x) up to terms of order -. — 5—
Alogx + B ^ -TK / i- log^x
140
SOURCE BOOK IN MATHEMATICS
inclusive. By the preceding theorem the constants A and B must satisfy the equation
X
lim
[^t
A log x-\-B On expanding we have
X I X B X
J2 ^ogA-y
= 0.
+
B^
A log X -{- B A log X A^ Iog2 X A^ log^ x while an integration by parts yields
J2 log X log X log2 X J2 log^ X The equation above then reduces to
+ C.
lim
log^ X /I X
B
+
B2
A log X A^ log^ X A^ log^ X log X Iog2 X J2 log^ X
-■■)l
= 0
or else to
li
im
(^_,),o,_(B+,) +
B2 1
A^' log X
...-2
log^ X
r
dx
- C
log^ X
= 0.
X f, log^ X X
On observing that all the terms beginning with the third con- verge to zero when x increases indefinitely, it is seen at once that
the preceding equation can not be satisfied unless -j — 1 = 0. -^+1=0. Hence A = I, B = -I.
Thus among all the functions of the form
A log X + B
only
log X — 1
can represent <^(x) up to terms of order
log^ X
inci
usive.
* [We omit §§6 and 7 of this memoir. In §6 Chebyshev proves by a method analogous to that used above that if <t>(x) can be represented up to terms of
order
inclusive by an expression algebraic in .v, log x, e', then </>(x)
log" X
can be represented also with the same degree of approximation by the expres- sion
X . \.x . . 1.2... (n - l).v
+ ...+
log" .V
log X log'' X
/dx . .
. by repeated integration by parts. §7 contains
CHEBYSHEV {TCHEBYCHEFF) 141
Memoir 2: Memoir on Prime Numbers^ §2. Let us designate by d{z) the sum of logarithms of all the primes which do not exceed z. This function equals zero when x is less than the smallest prime, viz. 2. It is not difficult to show that this function satisfies the following equation^
e{x) + d{x)y^ + ^(x)« + M
= Iogl.2.3...[x].
where the symbol [x] is used to designate the greatest integer contained in x.
To verify this equation we note that both its members are made up of terms of the form K log a, where a is a prime and K is an integer. In the left-hand member K is equal to the number of terms in the sequence
X. 2' 3'
(1) W'
which are not less than a, since the expression for d{z) will contain the term log a only in the case where z '^ a. As to the coefficient of log a in the right-hand member, it is equal to the highest power of a which divides 1.2.3...[^]- It is found however, that this power is also equal to the number of terms in the sequence (1) which are not less than a; for, the number of terms of the sequence
X X
an attempt (not rigorous) to prove the remarkable asymptotic relations
5) ~IogIogP + Ci. n(l --)~r-%
where Ci, d are fixed constants, P is any prime number, and the summation and product are extended over all primes /i ^ P.]
^ (We omit the introductory §1 of this memoir.]
* To abbreviate we write 0(x/n)"* instead of d[ (x/n)'"} .
142 SOURCE BOOK IN MATHEMATICS
which are not less than a, is equal to that of the terms of the sequence
1, 2, 3,...,[.v] which are divisible by a.
The same relationship exists between the number of terms of this sequence, which are divisible by a*, a"*, a*,. . .and the number of terms of the sequence
'""■ 0" 0)'
<*'■ 0)" 0)''
which are not less than a.
Hence both members of our equation are composed of the same terms, which proves that they are identical.
The equation just estabhshed can be presented as
(2) Hx) + 4^(-^ + ^(~\ + . . . = T{x)
on setting for abbreviation
... I d(z) + e(z)yi + 0(z)>3 + . . . = ^(z),
^^^ I log 1.2.3... [x] = T{x).
In apphcations of these formulas we shall observe that, in view of what has been said about the value of 6{z) when z < 2 the func- tion \p(z) vanishes when z < 2, and consequently, equation (2) will be valid in the limiting cases a: = 0, a: = 2 if we agree to take zero as the value of T{x) when x < 2.
§3. By means of this equation it is not difficult to find numerous inequahties which are satisfied by the function ^(x); those we shall use in this memoir are the following:
^(x) > Tix) + n^^] - T^^ - t(^^ - r(
30
Wx) - ^Q > r(.v) + t(^ - tQ - r(5) - r0).
To prove these inequalities we shall compute the value of
CHEBYSHEV (TCHEBYCHEFF) by means of (2), which leads to the equation
+^(35) + ^(235) + ^(570)+- • •
143
(4)
-r; -
- ri
•'©
+
-0-i3)-<3^)-- whose left-hand member reduces to
Ai, A2,. . .An. . .being numerical coefficients. Upon examining their values it is not difficult to establish that A„ = 1 if n = 30m + 1, 7, 11, 13, 17, 19, 23, 29, A„ = 0 if n = 30m + 2, 3, 4, 5, 8, 9, 14, 16, 21, 22, 25, 26, 27,
28, A, = -1 if n = 30m + 6, 10, 12, 15, 18, 20, 24, A„ = - 1 if n = 30m + 30.
Indeed, in the first case n is not divisible by any of the numbers 2, 3, 5, hence the term \p{x/n) figures only in the first fine in equa- tion (4). In the second case n is divisible by one of the numbers 2, 3, 5, hence, besides the term \l/{x/n) in the first line, the term — i/'(x/n) will be found in one of the last three lines, and, after reduction, the coefficient of ^}/{x/n) will become 0. In the third case n is divisible by two of the numbers 2, 3, 5. Hence the last three lines will contain two terms equal to —\p{x/n), while the first line contains i/'(x/n) with the plus sign, so that the result will be —yp{x/n). In the last case where n is divisible by 30 we arrive at the same conclusion, since the term +^{x/n) will figure in all the five lines, twice with the plus and three times with the minus sign.
Hence for n = 30m + 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 we find respectively
An = 1. 0, 0, 0. 0, -1, 1, 0, 0, -1, 1, -1, 1, 0, -1, 0, 1, -1, 1, -1,0,0, 1, -1,0,0,0,0, 1, -1,
144 SOURCE BOOK IN MATHEMATICS
which shows that equation (4) reduces to
where the terms of the left-hand member have the coefFicient 1 alternately with the plus and minus signs. Furthermore, since by nature of the function \l/(x) the series of the left-hand member is decreasing, its value will be included within the limits \}/(x) and \J/{x) — xpix/S). Hence, by the preceding equation, we shall have necessarily
^W - *Q i T(s) + 7(3!) - tQ - 70) - 70).
§4. Let us examine now the function T{x) which figures in these
formulas. On denoting by a the greatest integer contained in x,
which we shall assume to be ^ 1, we have from (3)
T{x) = log 1.2.3... a
or, what amounts to the same thing,
T{x) = log 1.2.3. . .a(a + 1) - log (a -|- 1).
But it is known that
log 1.2.3 . . . a < log \/27r + a log a — a -\- }y'2 log a +
log 1.2.3. . .a{a + 1) > log \/2r + (a + 1) log (a + 1) -
K2a,(a+1) +MIog(a + l); hence
T(x) < log \/2^ + a log a - a + K log a + M2a,
Tix) > log V2^ + (a + 1) log (a + 1) - (a + 1) - K log (a -t- 1)
and consequently
T{x) < log \/2ir -\- xlogx - X + }4 log X + K2»
T{x) > log -s/lw -\- xlogx — X — }^ log X,
since the inequalities
a ^ .V < a -t- 1, a ^ 1
obviously imply the conditions
X log X - X + 2 ^<^g ■^ + To - " ^°S tt — <^ + 2 ^^g '^ + 12"' X log X - X - M log X ^ (a + 1) log (a -f 1) - (a + 1)
- 1^ log (a + 1).
CHEBYSHEV (TCHEBYCHEFF) 145
The inequalities above concerning T(x) give Tix) + rf^ < 2 log v^ + ^ + l^.v log X - X log 30^0
^x + log X - 2 log 30, 31
T(x) + Tf^\ > 2 log \/2ir + ~x log X - X log 30^^<' -
30^
- log X + 2 log 30,
^0) + ^(l) + ^(0 < 3 log V2^ + ^ + l^x log X
- X log 2^^3>^5^^ - 1^^ + I Jog^ - 1 ^og 30,
- X log 2^^^3^^55^ _ l^x - I log X + ^ log 30.
On subtracting the last of these inequalities from the first and the third from the second we find
rw + r(^) - r0) - r0) - r(f) <a. + 1 log .
- \ log 1800;r + A
rw + t(^) - r0) - r(^) - r(^) > ^. - ^ ,„g,
, 1 , 4S0 3
+ 2'°8— -n'
where to abbreviate we have set
(5) A = log 2'^^-3^5''^30-^^'' = 0.92129202. . .
The analysis used in proving these inequalities assumes that
X ^ 30, since, in discussing T(x) we have assumed x ^ 1 and
after that we have replaced x successively by x/2, x/3, x/5 and
x/30. It is not difficult, however, to obtain formulas which can be
used for all values of x > 1, if we replace the preceding inequaHties
by simpler ones
rw + r(^) - r(^) - r0) - r(f) <a. + 1 logx, r(x) + r(^) , rg) - r(?) - r(^) > a. - ^ log . - i;
146 SOURCE BOOK IN MATHEMATICS
an examination readily shows that these inequalities are vahd for values of x between 1 and 30.
§5. On combining these inequahties with those derived above for the function '/'(x)(§3) we arrive at two formulas
i/'(r) > Ax - jlogx - \, \ly(x) - i^r^ j < Ax + 2 log x,
of which the first gives a lower limit for rpix).
As to the second formula, it will be used in assigning another limit for \p(x). For this purpose we observe that the function
J{x) = ^Ax + ^j^ Iog2 X + ^ log X
satisfies the equation
J(x)-jf^\ = Ax + I log X,
which, being subtracted from the inequahty
^{x) - xl^f^] < Ax + 2 log X
gives or else
Hx)-^(^^-Kx)-^j(f)<0
X\ r/X
HX)-KX) <rpr^\-JI^
On replacing x successively by x/6, x/6^. . .x/6"* in this formula we find
m - /w < ^@ - /@ < ■ ■ < ^(g^) - /(g^).
Assume now that m is the greatest integer which satisfies the
condition ^^1. Then x/6"'+' will be between 1 and 3^, while 6"'
rpiz) = 0 and —J(z) remains greater than 1 within the limits
z = 1, z = >^. Hence i/'(x/6'"+0 - /(x/6'"+i) < 1, and by the
preceding inequahties
^(x) -/(x) < 1.
Finally, on substituting the value of /(x) we have
Hx) < -^Ax + ^^^ Iog2 X + ^ log X + 1.
On the basis of the formulas just found it is not difficult to assign two limits including the value of ^(x).
CHEBYSHEV (TCHEBYCHEFF) 147
Indeed, we find from (3)
Hx) - ^(x)^ = d(x) + d(x)yi + dix)H +. . ., yP{x) - 2iA(x)H = dix) - [^(x)H _ e{xyi] - . . .
which shows that
(6) d{x) ^ Hx) - ^{x)^K dix) ^ Hx) - 2iA(x)H,
since the terms
0(.v)>3, d(x)H, . . . , ^(x)M - ^(x)M . . .
obviously are positive or zero. But we have found
yp{x) < ^Ax + ^ . ^ Iog2 X + ^ log X + 1,
\l/{x) > Ax — Y2 log X — 1, which gives
,A(x)>^ < |axH + ^^^-j^ Iog2 X + \ log X + 1.
i^{x)y^ > AxH _ I log X - 1, and consequently
4^{x) - ^{xyi < |ax - AxH + ^^J-g Iog2 X + ^ log X + 2.
rPix) - 2^p{x)y^ > Ax - ^AxV^ - g^ log^ X - ^ log X - 3
Hence, by (6),
( d(x) < f Ax - Axy^ + 3-j^ Iog2 X + ^ log X + 2 (7 J 5 4 log 6 ^ 2 ^
0(x) > Ax - ^Ax>^ - g^ log^ A- - ^ log X - 3.1
^ [We omit the concluding §§6-9 of the memoir. In §6 Chebyshev gives the proof of the Bertrand postulate, taking as the point of departure the obvious inequaHties
0(L) - 6(1) > m log /, 0{L) - 6(1) < m log L where m is the number of primes between / and L, and using the unequalities obtained above for 6(x). §7 contains a proof of the following remarkable
F(x) theorem: If for x sufFiciently large F(x) is positive and , is not increasing,
then the convergence of the series ^, • is a necessary and sufficient
condition for the convergence of the series SF(m). The proof is based up>on the simple transformation formula
l.L m = l ^
148 SOURCE BOOK IN MATHEMATICS
where the summation over n is extended over all primes, while that over m over all integers between the two given limits / and L. Thus the series
2 log 2 ^ 3 log 3 ' 5 log 5 ' ' • ' ' 2 log^ (log 2) ' 3 log^ (log 3)
^5IogMlog5) ^■"" are convergent while the series
1+1+1+ . __^+_i_+_L_ +
2^3^S^-" 2Iog2 ^3 log3 ^SlogS ^■••
are divergent. §§8 and 9 contain some applications of the above results to the approximate computation of sums of the form 2F(m) and, in the special case where F(x) = 1, to the computation of the totality of primes.]
NAPIER
On the Table of Logarithms
(Selections Made by Professor W. D. Cairns, Oberlin College, Oberlin, Ohio.)
John Napier (1550-1617), Baron of Merchiston, Scotland, has been given undisputed priority with regard to the publication of a table of logarithms and an account of their meaning and use. His work is the more important since, through improvements by himself, Henry Briggs, and others, it quickly became a system practical for purposes of calculation and nearly in the modern form. He published his system in 1614 in Mirifici logaritbmorum canonis descriptio and gave therein a description of the nature of logarithms and a table of his logarithms of the sines of angles for successive minutes. The present account is, however, taken from his Mirifici logaritbmorum canonis conslructio, which appeared posthumously in 1619 but which was written several years earlier than the Descriptio. Sufficient extracts are given, with the original numbers of the articles, to show his method of construction of the table, his definition of logarithms, and the rules for combining these.
The Descriptio was translated into English by Edward Wright under the title A Description of the Admirable Table of Logaritbmes and was published posthumously at London in 1616. The Constructio was translated into English by W. R. Macdonald (Edinburgh, Wm. Blackwood & Sons, Ltd., 1889). The following selections are taken from the latter work with the kind per- mission of the publishers, the numbers of the paragraphs being as in the original. Only the more important parts of the numbered paragraphs have been selected, there being sufficient to show Napier's method of constructing a logarithmic table. Upon the question of the invention of logarithms, see the articles on prosthaphaeresis (pp. 455 and 459).
1. A logarithmic table is a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space, by a very easy calculation ... It is picked out from numbers progressing in continuous proportion.
2. Of continuous progressions, an arithmetical is one which proceeds by equal intervals; a geometrical, one which advances by unequal and proportionally increasing or decreasing intervals.
16. If from the radius with seven ciphers added you subtract its 1000(X)00th part, and from the number thence arising its lOOOOOOOth part, and so on, a hundred numbers may very easily be continued geometrically in the proportion subsisting
149
150 SOURCE BOOK IN MATHEMATICS
between the radius and the sine less than it by unity, namely between 10000000 and 9999999; and this series of propor- tionals we name the First table.
Thus from the radius, with seven ciphers added for greater accuracy, namely, 10000000.0000000, subtract 1.0000000, you get 9999999.0000000; from this subtract .9999999, you get 9999998.0000001; and proceed in this way until you create a hundred proportionals, the last of which, if you have computed rightly, will be 9999900.0004950.
17. The Second table proceeds from the radius with six ciphers added, through fifty other numbers decreasing proportionally in the proportion which is easiest, and as near as possible to that subsisting between the first and last numbers of the First table
Thus the first and last numbers of the First table are 10000000.0000000 and 9999900.0004950, in which pro- portion it is difficult to form fifty proportional numbers. A near and at the same time an easy proportion is 100000 to 99999, which may be continued with sufficient exactness by adding six ciphers to the radius and continually subtract- ing from each number its own 100000th part; and this table contains, besides the radius which is the first, fifty other proportional numbers, the last of which, if you have not erred, you will find to be 9995001.222927.1
18. The Third table consists of sixty-nine columns, and in each column are placed twenty-one numbers, proceeding in the proportion which is easiest, and as near as possible to that sub- sisting between the first and last numbers of the Second table.
Whence its first column is very easily obtained from the radius with five ciphers added, by subtracting its 2000th part, and so from the other numbers as they arise.
In forming this progression, as the proportion between 10000000.000000, the first of the Second table, and 9995001.222927, the last of the same, is troublesome; there- fore compute the twenty-one numbers in the easy pro- portion of 10000 to 9995, which is sufficiently near to it; the last of these, if you have not erred, will be 9900473.57808. From these numbers, when computed, the last figure of each may be rejected without sensible error, so that others may hereafter be more easily computed from them.
1 (This should be 9995001.224804.]
NAPIER 151
19. The first numbers of all the columns must proceed from the radius with four ciphers added, in the proportion easiest and nearest to that subsisting between the first and the last numbers of the first column.
As the first and the last numbers of the first column are 10000000.0000 and 9900473.5780, the easiest proportion very near to this is 100 to 99. Accordingly sixty-eight numbers are to be continued from the radius in the ratio of 100 to 99 by subtracting from each one of them its hundredth part.
20. In the same proportion a progression is to be made from the second number of the first column through the second numbers in all the cohimns, and from the third through the third, and from the fourth through the fourth, and from the others respectively through the others.
Thus from any number in one column, by subtracting its hundredth part, the number of the same rank in the follow- ing column is made, and the numbers should be placed in order.
Remark: The last number in the Sixty-ninth column is 4998609.4034, roughly half the original number.
21. Thus, in the Third table, between the radius and half the radius, you have sixty-eight numbers interpolated, in the pro- portion of 100 to 99, and between each two of these you have twenty numbers interpolated in the proportion of 10000 to 9995; and again, in the Second table, between the first two of these, namely between 10000000 and 9995000, you have fifty numbers interpolated in the proportion of 100000 to 99999; and finally, in the First table, between the latter, you have a hundred numbers interpolated in the proportion of the radius or 10000000 to 9999999; and since the difference of these is never more than unity, there is no need to divide it more minutely by interpolating means, whence these three tables, after they have been completed, will suffice for com- puting a Logarithmic table.
Hitherto we have explained how we may most easily place in tables sines or natural numbers progressing in geometrical proportion.
22. It remains, in the Third table at least, to place beside the sines or natural numbers decreasing geometrically their logarithms or artificial numbers increasing arithmetically.
152 SOURCE BOOK IN MATHEMATICS
26. The logarithm of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which the radius began to decrease geometrically, and in the same time as the radius has decreased to the given sine.^
T
d
S
8
8
b
c
i
Let the line TS be the radius, and dS a given sine in the same line; let g move geometrically from T to d in certain determinate moments of time. Again, let bi be another line, infinite towards i, along which, from b, let a move arithmetically with the same velocity as g had at first when at T; and from the fixed point b in the direction of i let a advance in just the same moments of time up to the point c. The number measuring the line be is called the logarithm of the given sine dS.
27. Zero is the logarithm of the radius.
28. Whence also it follows that the logarithm of any given sine is greater than the difference between the radius and the given sine, and less than the difference between the radius and the quantity which exceeds it in the ratio of the radius to the given sine. And these differences are therefore called the limits of the logarithm.
0
T
d
5
8
8
8 c
b
1
1
Thus, the preceding figure being repeated, and ST being produced beyond T to o, so that oS is to TS as TS to dS.
' [To Napier the sine was a line, or the number measuring the line, as in the present-day line representation of functions of angles.]
NAPIER 153
I say that be, the logarithm of the sine dS, is greater than Td and less than oT. For in the same time that g is borne from 0 to T, g is borne from T to d, because (by 24) oT is such a part of oS as Td is of TS, and in the same time (by the definition of a logarithm) is a borne from 6 to c; so that oT, Td, and 6c are distances traversed in equal times. But since g when moving between T and o is swifter than at T, and between T and d slower, but at T is equally swift with a (by 26); it follows that oT the distance traversed by g moving swiftly is greater, and Td the distance traversed by g moving slowly is less, than be the distance traversed by the point a with its medium motion, in just the same moments of time; the latter is, consequently, a certain mean between the two former.
Therefore oT is called the greater limit, and Td the less limit of the logarithm which 6c represents.
29. To find the limits of the logarithm of a given sine.
By the preceding it is proved that the given sine being subtracted from the radius, the less limit remains, and that the radius being multiplied into the less limit and the product divided by the given sine, the greater limit is produced.
30. Whence the first proportional of the First table, which is 9999999, has its logarithm between the limits 1.0000001 and 1.0000000.
31. The limits themselves differing insensibly, they or anything between them may be taken as the true logarithm.
32. There being any number of sines decreasing from the radius in geometrical proportions, of one of which the logarithm or its limits is given, to find those of the others.
This necessarily follows from the definitions of arith- metical increase, of geometrical decrease, and of a loga- rithm ... So that, if the first logarithm corresponding to the first sine after the radius be given, the second logarithm will be double of it, the third triple, and so of the others; until the logarithms of all the sines are known. 36. The logarithms of similarly proportioned sines differ equally.
This necessarily follows from the definitions of a loga- rithm and of the two motions . Also there is the same ratio of equality between the differences of the respective limits of the logarithms, namely as the differences of the less
154 SOURCE BOOK IN MATHEMATICS
among themselves, so also of the greater among themselves, of which logarithms the sines are similarly proportioned.
38. Of four geometrical proportionals, as the product of the means is equal to the product of the extremes; so of their logarithms, the sum of the means is equal to the sum of the extremes. Whence any three of these logarithms being given, the fourth becomes known, ^
39. The difference of the logarithms of two sines lies between two limits; the greater limit being to the radius as the difference of the sines to the less sine, and the less limit being to the radius as the difference of the sines to the greater sine.^
47. In the Third table, beside the natural numbers, are to be written their logarithms; so that the Third table, which after this we shall always call the Radical table, may be made complete and perfect.
48. The Radical table being now completed, we take the numbers for the logarithmic table from it alone.
For as the first two tables were of service in the formation of the third, so this Radical table serves for the construction of the principal Logarithmic table, with great ease and no sensible error.
51. All sines in the proportion of two to one have 6931469.22 for the difference of their logarithms.^
52. All sines in the proportion of ten to one have 23025842.34 for the difference of their logarithms.
* [The modern theorem for the logarithm of a product does not hold here, since the logarithm of unity is not zero.]
^ [This is proved by the principle of proportion and of Article 36. This rule is used first in Articles 40-41 as an illustration to find the logaiithm of 9999975.5 from that of the nearest sine in the First table, 9999975.0000300, noting that the limits of the logarithms of the latter number are 25.0000025 and 25.0000000, that the difference of the logarithms of the two numbers by the rule just given is .4999712 and that the limits for the logarithm of 9999975.5 are therefore 24.5000313 and 24.5000288, whence he lists the logarithm as 24.5000300.
In Articles 41-45 he illustrates the fact that one may now calculate the logarithms of all the "proportionals" in the First, Second, and Third tables, as well as of the sines or natural numbers not proportionals in these tables but near or between them.]
^ [Napier obtains this result by first calculating the logarithm of 7071068, which is to the nearest unit, the square root of 50 X 10'^ and which is, to his "radius," the sine of 45°. By Article 39 its logarithm is 3465734.5, whence the result in Article 51.]
NAPIER 155
55. As the half radius is to the sine of half a given arc, so is the sine of the complement of the half arc to the sine of the whole arc.^
56. Double the logarithm of an arc of 45 degrees is the logarithm of half the radius.
57. The sum of the logarithms of half the radius and any given arc is equal to the sum of the logarithms of half the arc and the complement of the half arc. Whence the logarithm of the half arc may be found if the logarithms of the other three are given.
59. To form a logarithmic table. ^
1 [Only here does Napier begin to introduce angles into the construction of his tables. Napier proves Articles 55-57 by geometric principles and the preceding theorems concerning logarithms.]
^ [Napier's table is conocructed in quite the same form as used at present, except that the second (sixth) column gives sines for the number of degrees indicated at the top (bottom) and of minutes in the first (seventh) column, the third (fifth) column gives the corresponding logarithm and the fourth column gives the "differentiae" between the logarithms in the third and fifth columns, these being therefore essentially logarithmic tangents or cotangents. A reproduction of one page may be seen in Macdonald's translation, page 138.
DELAMAIN
On The Slide Rule
(Edited by Professor Florian Cajori, University of California, Berkeley,
California.)
The earliest publication describing a slide rule (an instrument differing from Gunter's scale, which had no sliding parts) was brought out in the year 1630 by Richard Delamain, a teacher of mathematics in London. It was a pamphlet of 30 pages, entitled Grammelogia^ and describing a circular slide rule. There is a copy in the Cambridge University Library in England. This tract has no drawing of the slide rule. During the next 2 or 3 years there were issued at least four undated new editions, or impressions, of the Grammelogia, with new parts added. The Cambridge University Library has a copy which is the same as the 1630 publication but with an appendix^ of 17 pages added. In the British Museum at London and in the Bodleian Library at Oxford, there are copies of another edition of 113 pages, which was published in 1632 or 1633, as is shown by its reference to Oughtred's book, the Circles oj Proportion of 1632. It has two title pages^ which we reproduce in facsimile.
> The full title of the Grammelogia of 1630 is as follows:
CTamelogia\or,\Tbe Matbemalicall Ring.\Sbewing {any reasonable Capacity that batb\ not Aritbmeticke) bow to resolve and tvoTke\all ordinary operations of Aritbmeticke.\And tbose wbicb are most difficult witb grealest\Jacilitie: Tbe extraction of Roots, tbe valuation oJ\ Leases, &c. Tbe measuring of Plaines\and Solids.\Witb tbe resolution oj Plaine and Spbericall\Triangles.\And tbat onely by an Ocular Inspection,\and a Circular Motion.\ Naturae secreta tempus ape7-t/.|London printed by John Haviland, 1630. 2 The appendix is entitled:
De la MainsI Appendix! Vpon his|MathematicaIl|Ring. Attribuit nullo (praescripto tempore) vitae|vsuram nobis ingeniique Deus.|London,| . . .The next line or two of this title page which probably contained the date of publication, were cut off by the binder in trimming the edges of this and several other pamphlets for bind- ing into one volume.
5 The first title page (engraved) is as follows:
Mirifica Logaritbmoru' Projectio Circularis. There follows a diagram of a circular slide rule, with the inscription within the innermost ring: Nil Finis, Motvs, Circvlvs vllvs Habet. The second title page is as follows:
Grammelogia\Or, tbe Matbematicall Ring.\Extracted from tbe Logarylbmes, and projected
Circular: Now published in tbe\inlargement tbereoj unto any magnitude jit jor use; shewing
any reason-\able capacity that bath not Aritbmeticke bow to resolve and worke.lall ordinary
operations oj Aritbmeticke:\And those tbat are 7nost difficult with greatest jacilitie, tbe extracti-]
on oj Rootes, tbe valuation oj Leases, &c. tbe measuring oj Plaines and Solids,\with tbe
resolution oj Plaine and Spbericall Triangles applied to tbe\Practicall parts oj Geometric,
Horologograpbie, Geograpbie\Fortijication, Navigation, Astronomic, ^c.\And tbat onely by
an ocular inspection, and a Circular motion. Invented and jirsl published, by R. Delamain,
Teacher, and Student oj tbe Matbematicks.\Naturae secreta tempus aperit.\
There is no date. There follows the diagram of a second circular slide rule, with the inscrip)-
tion witliin the innermost ring: Typus proiectionis Annuli adaucti vt in Conslusione Lybri
praelo commissi. Anno 1630 promisi. There are numerous drawings in the Grammelogia, all
156
DELAMAIN
157
In tlic Grammeloiiia of 1630, Dclamain, in an address to King Charles I, emphasizes tlie ease of operating with his slide rule by stating that it is "fit for use. . .as well on Morse backe as on Foot." Speaking "To the Reader," he states that he has "for many yeares taught the Mathematicks in this Towne" and made efforts to improve Gunter's scale "by some Motion, so that tlie whole body of Logaritlimes miglit move proportionally the one to the
other, as occasion required. This conceit in February last [1629] 1 struke upon, and so composed my Grammelogia or Aiathematicall Ring; by whicli only with an ocular inspection, there is had at one instant all proportionalls through
of which, excepting the drawings of slide rules on the engraved title pages were printed upon separate pieces of paper and then inserted by hand into the vacant spaces on the printed pages reserved for them. Some drawings are missing, so that the Bodleian Grammelogia diflers in this respect slightly from the two copies in the British Museum.
158 SOUliCli BOOK IN MATHIiMATICS
tin- said body ol Numbers." lie dates his preface "first of January, 1630." The term Gramniclaiiia is appHed to the instrument, as well as to the book. Delamain's description of this Gramniclogia is as follows:
The parts of the Instrument arc two Circles, tlie one movcal)ic, and the other fixed: The moveable is that unto which is fastened a small pin to move it by ; the other Circle may be conceived to be fixed; The circumference of the moveable Circle is divided into unequall parts, charactered with figures thus, 1. 2. 3. 4. 5. 6. 7. 8. 9. these figures doe represent themselves, or such numbers unto which a Cipher or Ciphers are added, and are varied as the occasion falls out in the speech oj Numbers, so 1. stands for 1. or 10. or 100.^ &c. the 2. stands for 2. or 20. or 200. or 2000., &c. the 3. stands for 30. or 300. or 3000.; &c.
"How to perform the Golden Rule" (tiie rule of proportion), is exijlained thus:
Seeke the first number in the mo\'eable, and bring it to the second number in the fixed, so right against the third number in the moveable, is the answer in the fixed.
If the Interest of 100. li. be 8. li. in the yeare, what is the Interest of 65. li. for the same time.
Bring 100. in the movable to 8. in the fixed, so right against 65. in the movable is 5.2. in the fixed, and so much is the Interest of 65. li. for the yeare at 8. li. for 100. li. per annum.
The Instrument not removed, you may at one instant right against any summe of money in the moveable, see the Interest thereof in the fixed: the reason of this is from the Defijiition oj Logarithmes.
Relating to the "resolution of Plaine and Sphericall Triangles," Delainain says:
If there be composed three Circles of equal thicknessc, A. B.C. so that the inner edge of D [should be B) and the outward edge of A bee answerably graduated with Logarithmall signes [sines], and the outward edge of B and the inner edge of A with Logarilhmes; and then on the backside be graduated the Logarithmall Tangents, and againe the Logarithmall signes oppositly to the former gradua- tions, it shall be fitted for the resolution of Plaine and Sphericall Triangles.
After twelve lines of further remarks (jn this |>oint, he adds:
DELAMAIN 159
Hence from the forme, I have called it a Ring, and Grammelogia by annoligie of a Lineary speech; which Ring, if it were projected in the convex unto two yards Diameter, or thereabouts, and the line Decupled, it would worke Trigonometrie unto seconds, and give proportionall numbers unto six places only by an ocular inspection, which would compendiate Astronomicall calculations, and be sufficient for the Prosthaphaeresis of the Motions: But of this as God shall give life and ability to health and time.
The patent and copyright on the instrument and book are as follows:
Whereas Richard Delamain, Teacher of Mathematicks, hath presented vnto Vs an Instrument called Grammelogia, or The Mathematical! Ring, together with a Booke so instituted, express- ing the use thereof, being his owne Invention; we of our Gracious and Princely favour have granted unto the said Richard Delamain and his Assignes, Privilege, Licence, and Authority, for the sole Making, Printing and Selling of the said Instrument and Booke: straightly forbidding any other to Make, Imprint, or Sell, or cause to be Made, or Imprinted, or Sold, the said Instrument or Booke within any our Dominions, during the space of ten years next ensuing the date hereof, upon paine of Our high displeasure, Given under our hand and Signet at our Palace of Westminster, the fourth day of January, in the sixth yeare of our Raigne.
OUGHTRED On the Slide Rule
(Edited by Professor Florian Cajori, University of California, Berkeley,
California.)
William Oughtred (1574—1660) was a clergyman living near London and intensely interested in mathematics. He taught mathematics at his residence, without compensation, to promising pupils. At one time, Oughtred had assisted Delamain in his mathematical studies. His Circles oj Proportion^ appeared in 1632, translated into English from his Latin manuscript by one of his pupils, William Forster. Forster wrote a preface in which he makes the charge (without naming Delamain) that "another. . .went about to pre- ocupate" the new invention. This led to verbal disputes and to the publica- tion by Delamain of the several new editions of the Grammelogia, describing further designs of circular slide rules and also stating his side of the controversy. Oughtred prepared an Epistle, in reply, which was published in the 1633 edition of his Circles of Proportion. Each combatant accuses the other of stealing the invention of the circular slide rule. After reading both sides of the controversy, we conclude that Oughtred invented the circular slide rule before the time when Delamain claimed to have made his invention, but it is not shown conclusively that the latter was dishonest; we incline to the opinion that he was an independent inventor. In 1633, Oughtred published the description of a rectilinear sHde rule, in the invention of which he has no rival.
Extracts from the Circles oJ Proportion 1 There are two sides of this Instrument. On the one side, as it were in the plaine oj the Horizon, is delineated the proiection of the Sphere. On the other side there are divers kindes of Circles, divided after many severall Waies; together with an Index to be opened after the manner of a paire of Compasses. And of this side we will speake in the first place.
' There are two title pages. The first is engraved and reads thus:
The|CircIes|of|Proportion|and|The Horizontali|Instrumcnt.|Both invented, and|the vses of both|Written in Latine by|Mr. W. O-lTransIated into English: and set forth]for the publique benefit by| William Forster. |London| Printed for Elias Allen makerlof these and all other Mathe:|maucal Instruments, and are tolbe sold at his shop ouer against! St Clements church with out Temple-barr.| 1632. T. Cecill Sculp| The second title page is:
The|Circlc|of| Proportion, land |The HorizontaIl|Instrvment.|Both invented, and the vses of both|written in Latine by that learned Mathe-|matician Mr W. 0.|Bvt|Translated into English: and set forth for|the publique benefit by William Forster, louer|and prac- tizer of the Mathemalicall Sciences. ILondon] Printed by Avg. Mathevves,|dwelling in the Parsonage Court, neere|St Brides. 1632. |
160
OUGHT RED 161
2 The First, or outermost circle is of Sines, from 5 degrees 45 minuts almost, vntill 90. Every degree till 30 is divided into 12 parts, each part being 5 min : from thence vntill 50 deg : into sixe parts which are 10 min : a peece : from thence vntill 75 degrees into two parts which are 30 minutes a peece. After that vnto 85 deg : they are not divided.
Oughtred's Circular Slide Rule, from his Circles of Proportions, 1632.
3 The Second circle is of Tangents, from 5 degrees 45 min : almost, untill 45 degrees. Every degree being divided into 12 parts which are 5 min : a peece.
4 The Third circle is of Tangents, from 45 degrees untill 84 degrees 15 minutes. Each degree being divided into 12 parts, which are 5 min : a peece.
5 The Sixt circle is of Tangents from 84 degrees till about 89 degrees 25 minutes.
162 SOURCE BOOK IN MATHEMATICS
The Seventh circle is of Tangents from about 35.Min : till 6 degrees.
The Eight circle is of Sines, from about 35 minutes til 6 degrees.
6 The Fourth circle is of Vnaequall Numbers, which are noted with the Figures 2, 3, 4, 5, 6, 7, 8, 9, 1. Whether you vnderstand them to bee single Numbers, or Tenns, or Hundreds, or Thousands, etc. And every space of the numbers till 5, is divided into 100 parts, but after 5 till 1, into 50 parts.
The Fourth circle also sheweth the true or naturall Sines, and Tangents. For if the Index bee applyed to any Sine or Tangent, it will cut thelrw