The relation of algebra to arithmetic is emphasized, the subject is treated topically, and each important point is touched at least twice. The book begins by showing the uses of algebra, employing such practical applications as are within the pupil's range of knowledge. When an interest has thus been awakened in the subject, the fundamental operations are presented with the simple explanations necessary to make the student independent of dogmatic rules. Throughout the book abundant oral and written drill exercises are provided. The work includes linear equations with two unknown quant.i.ties, and easy quadratics.

The leading features may be summarized as follows: (1) an arrangement in harmony with existing courses of study; (2) a presentation designed to awaken the interest of the pupils; (3) a topical arrangement for each half year, every important topic being repeated; (4) simplicity of explanations; (5) development of the relation of algebra to arithmetic both in theory and in applications; (6) emphasis laid on the importance of oral as well as written algebra.

GINN & COMPANY PUBLISHERS

BOOKS FOR TEACHERS

List price Allen: Civics and Health $1.25 Brigham: Geographic Influences in American History 1.25 Channing and Hart: Guide to the Study of American History 2.00 Hall: Aspects of Child Life and Education 1.50 Hodge: Nature Study and Life 1.50 Johnson: Education by Plays and Games .90 Johnson: What to Do at Recess .25 Kern: Among Country Schools 1.25 Mace: Method in History 1.00 MacVicar: The Principles of Education .60 Moral Training in the Public Schools 1.25 Prince: Courses of Studies and Methods of Teaching .75 Scott: Social Education 1.25 Tompkins: Philosophy of School Management .75 Tompkins: Philosophy of Teaching .75 Wiltse: Place of the Story in Early Education, and Other Essays.

A Manual for Teachers .50

FOR CLa.s.s RECORDS

Comings: Complete Record--Attendance and Scholarship Graded-School Edition .30 High-School Edition .30 Ginn and Company: Teacher's Cla.s.s Books No. I .30 No. II .40 Twenty Weeks' Cla.s.s Book .30

GINN AND COMPANY Publishers

TEXTBOOKS ON MATHEMATICS

FOR HIGHER SCHOOLS AND COLLEGES

List price Bailey and Woods: a.n.a.lytic Geometry $2.00 Beman and Smith: Academic Algebra 1.12 Beman and Smith: Higher Arithmetic .80 Beman and Smith: New Plane and Solid Geometry 1.25 Breckenridge, Mersereau, and Moore: Shop Problems in Mathematics 1.00 Byerly: Differential Calculus 2.00 Byerly: Integral Calculus (Second Edition. Revised and Enlarged) 2.00 Eisenhart: Differential Geometry of Curves and Surfaces 4.50 Faunce: Descriptive Geometry 1.25 Fine: College Algebra 1.50 Granville: Plane and Spherical Trigonometry and Tables 1.25 Granville: Plane and Spherical Trigonometry Without Tables 1.00 Granville: Plane Trigonometry and Tables 1.00 Granville: Logarithmic Tables .50 Granville and Smith: Elements of Differential and Integral Calculus 2.50 Hardy: a.n.a.lytic Geometry 1.50 Hardy: Elements of the Calculus 1.50 Hawkes: Advanced Algebra 1.40 Hawkes, Luby, and Touton: First Course in Algebra 1.00 Hedrick: Goursat's Mathematical a.n.a.lysis. Vol. I 4.00 Hedrick and Kellogg: Applications of the Calculus to Mechanics 1.25 Hooper and Wells: Electrical Problems 1.25 Lester: Integrals of Mechanics .80 Peirce, B. O.: Newtonian Potential Function (Third, Revised, and Enlarged Edition) 2.50 Peirce, J. M.: Elements of Logarithms .50 Peirce, J. M.: Mathematical Tables .40 Pierpont: Theory of Functions of Real Variables. Vol. I 4.50 Shepard: Problems in the Strength of Materials 1.25 Sloc.u.m and Hanc.o.c.k: Textbook on the Strength of Materials 3.00 Smith and Gale: Elements of a.n.a.lytic Geometry 2.00 Smith and Gale: Introduction to a.n.a.lytic Geometry 1.25 Smith and Longley: Theoretical Mechanics 2.50 Taylor: Elements of the Calculus (Revised and Enlarged Edition) 2.00 Taylor: Plane and Spherical Trigonometry 1.00 Taylor: Plane Trigonometry .75 Taylor and Puryear: Elements of Plane and Spherical Trigonometry 1.25 Wentworth: Advanced Arithmetic 1.00 Wentworth: a.n.a.lytic Geometry 1.25 Wentworth: College Algebra (Revised Edition) 1.50 Wentworth: Elementary Algebra 1.12 Wentworth: Higher Algebra 1.40 Wentworth: New School Algebra 1.12 Wentworth: Plane and Solid Geometry (Revised Edition) 1.25 Wentworth: Plane Geometry (Revised Edition); Solid Geometry (Revised Edition), each .75 Wentworth: Trigonometries (Second Revised Editions) (For list, see Descriptive Catalogue) Woods and Bailey: A Course In Mathematics Volume I 2.25 Volume II 2.25

GINN AND COMPANY PUBLISHERS

Notes

al-Mekk[=i] on a treatise on [.g]ob[=a]r arithmetic (explained later) called _Al-murshidah_, found by Woepcke in Paris (_Propagation_, p. 66), there is mentioned the fact that there are "nine Indian figures" and "a second kind of Indian figures ... although these are the figures of the [.g]ob[=a]r writing." So in a commentary by [H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i] (died in 1756) on the _Mokhta[s.]ar f[=i]'ilm el-[h.]is[=a]b_ (Extract from Arithmetic) by 'Abdalq[=a]dir ibn 'Al[=i]

al-Sakh[=a]w[=i] (died c. 1000) it is related that "the preface treats of the forms of the figures of Hindu signs, such as were established by the Hindu nation." [Woepcke, _Propagation_, p. 63.]]

which, of course, are interpolations. An interesting example of a forgery in ecclesiastical matters is in the charter said to have been given by St.

Patrick, granting indulgences to the benefactors of Glas...o...b..ry, dated "In nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei anno incarnationis ejusdem ccccx.x.x." Now if the Benedictines are right in saying that Dionysius Exiguus, a Scythian monk, first arranged the Christian chronology c. 532 A.D., this can hardly be other than spurious.

See Arbuthnot, loc. cit., p. 38.

[1] "_Discipulus._ Quis primus invenit numerum apud Hebraeos et aegyptios?

_Magister._ Abraham primus invenit numerum apud Hebraeos, deinde Moses; et Abraham tradidit istam scientiam numeri ad aegyptios, et docuit eos: deinde Josephus." [Bede, _De computo dialogus_ (doubtfully a.s.signed to him), _Opera omnia_, Paris, 1862, Vol. I, p. 650.]

"Alii referunt ad Phoenices inventores arithmeticae, propter eandem commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus sepulchrum est Lutetiae in comitio Maturinensi, refert ad Arabes." [Ramus, _Arithmeticae libri dvo_, Basel, 1569, p. 112.]

Similar notes are given by Peletarius in his commentary on the arithmetic of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed., p. 14): "La valeur des Figures commence au coste dextre tirant vers le coste senestre: au rebours de notre maniere d'escrire par ce que la premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont ete les premiers traffiquers de marchandise."

[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the Indians." [Waschke's German translation, Halle, 1878, p. 3.] Willichius speaks of the "Zyphrae Indicae," in his _Arithmeticae libri tres_ (Strasburg, 1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his _Le pratiche delle dve prime mathematiche_ (Venice, 1546, fol. 1). Woepcke is not correct, therefore, in saying ("Memoire sur la propagation des chiffres indiens," hereafter referred to as _Propagation_ [_Journal Asiatique_, Vol.

I (6), 1863, p. 34]) that Wallis (_A Treatise on Algebra, both historical and practical_, London, 1685, p. 13, and _De algebra tractatus_, Latin edition in his _Opera omnia_, 1693, Vol. II, p. 10) was one of the first to give the Hindu origin.

[3] From the 1558 edition of _The Grovnd of Artes_, fol. C, 5. Similarly Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [_De arte supputandi_, London, 1522, fol. B, 3.] Gemma Frisius, the great continental rival of Recorde, had the same idea: "Primum autem appellamus dexterum loc.u.m, eo qud haec ars vel a Chaldaeis, vel ab Hebraeis ortum habere credatur, qui etiam eo ordine scribunt"; but this refers more evidently to the Arabic numerals. [_Arithmeticae practicae methodvs facilis_, Antwerp, 1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing.

Even the modern Jewish writers claim that one of their scholars, M[=a]sh[=a]ll[=a]h (c. 800), introduced them to the Mohammedan world. [C.

Levias, _The Jewish Encyclopedia_, New York, 1905, Vol. IX, p. 348.]

[4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [_La prima parte del general trattato di nvmeri, et misvre_, Venice, 1556, fol.

9 of the 1592 edition.]

[5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [_Ain nerv geordnet Rechenbiechlin_, Augsburg, 1514, fol. 13 of the 1531 edition. The printer used the letters _rv_ for _w_ in "new" in the first edition, as he had no _w_ of the proper font.]

[6] Among them Glarea.n.u.s: "Characteres simplices sunt nouem significatiui, ab Indis usque, siue Chaldaeis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0 circulus, qui nihil significat." [_De VI. Arithmeticae practicae speciebvs_, Paris, 1539, fol. 9 of the 1543 edition.]

[7] "Barbarische oder gemeine Ziffern." [Anonymous, _Das Einmahl Eins c.u.m notis variorum_, Dresden, 1703, p. 3.] So Vossius (_De universae matheseos natura et const.i.tutione liber_, Amsterdam, 1650, p. 34) calls them "Barbaras numeri notas." The word at that time was possibly synonymous with Arabic.

[8] His full name was 'Ab[=u] 'Abdall[=a]h Mo[h.]ammed ibn M[=u]s[=a]

al-Khow[=a]razm[=i]. He was born in Khow[=a]rezm, "the lowlands," the country about the present Khiva and bordering on the Oxus, and lived at Bagdad under the caliph al-M[=a]m[=u]n. He died probably between 220 and 230 of the Mohammedan era, that is, between 835 and 845 A.D., although some put the date as early as 812. The best account of this great scholar may be found in an article by C. Nallino, "Al-[H)]uw[=a]rizm[=i]" in the _Atti della R. Accad. dei Lincei_, Rome, 1896. See also _Verhandlungen des 5.

Congresses der Orientalisten_, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey in the _Zeitschrift der deutschen Morgenland. Gesellschaft_, Vol. x.x.xIII, p. 224; Steinschneider in the _Zeitschrift der deutschen Morgenland.

Gesellschaft_, Vol. L, p. 214; Treutlein in the _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5; Suter, "Die Mathematiker und Astronomen der Araber und ihre Werke," _Abhandlungen zur Geschichte der Mathematik_, Vol. X, Leipzig, 1900, p. 10, and "Nachtrage," in Vol. XIV, p.

158; Cantor, _Geschichte der Mathematik_, Vol. I, 3d ed., pp. 712-733 etc.; F. Woepcke in _Propagation_, p. 489. So recently has he become known that Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter Arabes celebris Geometra, scripsit de figuris planis & sphericis."

[_Historia matheseos universae_, Leipzig, 1742, p. 438.]

In this work most of the Arabic names will be transliterated substantially as laid down by Suter in his work _Die Mathematiker_ etc., except where this violates English p.r.o.nunciation. The scheme of p.r.o.nunciation of oriental names is set forth in the preface.

[9] Our word _algebra_ is from the t.i.tle of one of his works, Al-jabr wa'l-muq[=a]balah, Completion and Comparison. The work was translated into English by F. Rosen, London, 1831, and treated in _L'Algebre d'al-Kh[=a]rizmi et les methodes indienne et grecque_, Leon Rodet, Paris, 1878, extract from the _Journal Asiatique_. For the derivation of the word _algebra_, see Cossali, _Scritti Inediti_, pp. 381-383, Rome, 1857; Leonardo's _Liber Abbaci_ (1202), p. 410, Rome, 1857; both published by B.

Boncompagni. "Almuchabala" also was used as a name for algebra.

[10] This learned scholar, teacher of O'Creat who wrote the _Helceph_ ("_Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum_"), studied in Toledo, learned Arabic, traveled as far east as Egypt, and brought from the Levant numerous ma.n.u.scripts for study and translation. See Henry in the _Abhandlungen zur Geschichte der Mathematik_, Vol. III, p.

131; Woepcke in _Propagation_, p. 518.

[11] The t.i.tle is _Algoritmi de numero Indorum_. That he did not make this translation is a.s.serted by Enestrom in the _Bibliotheca Mathematica_, Vol.

I (3), p. 520.

[12] Thus he speaks "de numero indorum per .IX. literas," and proceeds: "Dixit algoritmi: c.u.m uidissem yndos const.i.tuisse .IX. literas in uniuerso numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de opera quod fit per eas aliquid quod esset leuius discentibus, si deus uoluerit." [Boncompagni, _Trattati d'Aritmetica_, Rome, 1857.] Discussed by F. Woepcke, _Sur l'introduction de l'arithmetique indienne en Occident_, Rome, 1859.

[13] Thus in a commentary by 'Al[=i] ibn Ab[=i] Bekr ibn al-Jam[=a]l al-An[s.][=a]r[=i