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
4S»\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 ultratechnical 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 sixteenthcentury 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 nontechnical 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 FerrariCardan 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 Socalled Pell Equation 214
Translated from the Latin by Edward E. Whitford John Wallis on General Exponents 217
Translated from the Latin by Eva M. Sanford Wallis ant) Newton on the Bino.mial Theorem for Fractional and
Negative Exponents 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 4rayed 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 AltshillerCourt Brianchon and Poncelet on the Ninepoint 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 NonEuclidean Geometry 351
Translated from the Latin by Henry P. Manning
LoBACHEVSKY ON NoNEuCLlDEAN GEOMETRY 360
Translated from the French by Henry P. Manning Bolyai ON NoNEuCLiDEAN 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. 311331, 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, 18621863.
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 lefthand 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.]