give evidence of the homage that Education was forced to pay to Practical Life, at the sacrifice of matter better fitted to develop the mind of youth. With a view of supplying the information needed by merchants in business, arithmetics discussed such subjects as the United States Securities, the various rules adopted by the United States, and by the State governments on partial payments.

Authors and Books. The first arithmetics used in the American colonies were English works: Cocker, Hodder, Dilworth, "George Fisher" (Mrs. Slack), Daniel Fenning.' The earliest arithmetic written and printed in America appeared anonymously in Boston in 1729. Though a work of considerable merit, it seems to have been used very little; in early records we have found no reference to it; fifty years later, at the publication of Pike's Arithmetic, the former work was completely forgotten, and Pike's was declared to be the earliest American arithmetic. Of the 1729 publication there are two copies in the Harvard Library and one in the Congressional Library. In Appleton's Cyclopædia of American Biography its authorship is ascribed without reserve to Isaac Greenwood, then professor of mathematics at Harvard College, but on the title-page of one of the copies in the Harvard Library, is written the following: "Supposed that Sam! Greenwood was the author thereof, by others said to be by Isaac Greenwood." In 1788 appeared at Newburyport the New and Complete System of Arithmetic by Nicholas Pike (1743–1819), a graduate of Harvard College. It was intended for advanced schools, and contained, besides the ordinary subjects of that time, logarithms, trigonometry, algebra, and conic sections; but these latter subjects were so briefly treated as to possess little value. In the "Abridgment for the Use of Schools,"

1 See Teach. and Hist. of Math. in the U.S., pp. 12-16.
2 Ibidem, p. 14.

3 Ibidem, pp. 45, 46.

which was brought out at Worcester in 1793, the larger work is spoken of in the preface as "now used as a classical book in all the Newengland Universities." A recent writer1 makes Pike responsible for all the abuses in arithmetical teaching that prevailed in early American schools. To us this condemnation of Pike seems wholly unjust. It is unmerited, even if we admit that Pike was in no sense a reformer among arithmetical authors. Most of the evils in question have a far remoter origin than the time of Pike. Our author is fully up to the standard of English works of that date. He can no more be blamed by us for giving the aliquot parts of pounds and shillings, for stating rules for "tare and trett," for discussing the "reduction of coins," than the future historian can blame works of the present time for treating of such atrocious relations as that 3 ft. 1 yd., 5 yds. = 1 rd., 301 sq. yds. =1 sq. rd., etc. So long as this free and independent people chooses to be tied down to such relics of barbarism, the arithmatician cannot do otherwise than supply the means of acquiring the precious knowledge.

=

At the beginning of the nineteenth century there were three "great arithmeticians " in the United States: Nicholas Pike, Daniel Adams, and Nathan Daboll. The arithmetics of Adams (1801) and of Daboll (1800) paid more attention than that of Pike to Federal Money. Peter Parley tells us that in consequence of the general use, for over a century, of Dilworth in American schools, pounds, shillings, and pence were classical, and dollars and cents vulgar for several succeeding generations. "I would not give a penny for it" was genteel; “I would not give a cent for it" was plebeian.

Reform in arithmetical teaching in the United States did not begin until the publication by Warren Colburn, in 1821,

1 GEORGE H. MARTIN, The Evolution of the Massachusetts Public School System, p. 102.

of the Intellectual Arithmetic.1 This was the first fruit of Pestalozzian ideas on American soil. Like Pestalozzi, Colburn's great success lay in the treatment of mental arithmetic. The success of this little book was extraordinary. But American teachers in Colburn's time, and long after, never quite succeeded in successfully engrafting Pestalozzian principles on written arithmetic. Too much time was assigned to arithmetic in schools. There was too little object-teaching; either too much abstruse reasoning, or no reasoning at all; too little attention to the art of rapid and accurate computation; too much attention to the technicalities of commercial arithmetic. During the last ten years, however, desirable reforms have been introduced.3

2

"Pleasant and Diverting Questions"

In English and American editions of Dilworth, as also in Daniel Adams's Scholar's Arithmetic we find a curious collection of "Pleasant and diverting questions." We have all heard of the farmer, who, having a fox, a goose, and a peck of corn, wished to cross a river; but, being able to carry only one at a time, was confounded as to how he should take

1" WARREN COLBURN's First Lessons have been abused by being put in the hands of children too early, and has been productive of almost as much harm as good."-REV. THOMAS HILL, The True Order of Studies, 1876, p. 42.

2 The teacher who has been accustomed to the modern erroneous method of teaching a child to reason out his processes from the beginning may be assured this method of gaining facility in the operations, before attempting to explain them, is the method of Nature; and that it is not only much pleasanter to the child, but that it will make a better mathematician of him."-T. HILL, op. cit., p. 45.

3 For a more detailed history of arithmetical teaching, see Teach. and Hist. of Math. in the U.S.

4 Seventh Ed., Montpelier, Vt., 1812, p. 210.

them across so that the fox should not devour the goose, nor the goose the corn. Who has not been entertained by the problem, how three jealous husbands with their wives may cross a river in a boat holding only two, so that none of the three wives shall be found in company of one or two men unless her husband be present? Who has not attempted to place three digits in a square so that any three figures in a line may make just 15? None of us, perhaps, at first suspected the great antiquity of these apparently new-born creatures of fancy. Some of these puzzles are taken by Dilworth from Kersey's edition of Wingate. Kersey refers the reader to "the most ingenious" Gaspard Bachet de Méziriac in his little book, Problèmes plaisants et délectables qui se font par les nombres (Lyons, 1624), which is still largely read. The first of the above puzzles was probably known to Charle magne, for it appears in Alcuin's (?) Propositiones ad acuendos juvenes, in the modified version of the wolf, goat, and cabbage puzzle. The three jealous husbands and their wives were known to Tartaglia, who also proposes the same question with four husbands and four wives.1 We take these to be modified and improved versions of the first problem. The three jealous husbands have been traced back to a MS. of the thirteenth century, which represents two German youths, Firri and Tyrri, proposing problems to each other.2 also the following: Firri says: "There were three brothers in Cologne, having nine vessels of wine. The first vessel contained one quart (amam), the second 2, the third 3, the fourth 4, the fifth 5, the sixth 6, the seventh 7, the eighth 8, the ninth 9. Divide the wine equally among the three brothers, without mixing the contents of the vessels." This

1 PEACOCK, p. 473.

The MS. contains

2 DR. S. GÜNTHER, Geschichte des mathematischen Unterrichts im deutschen Mittelalter, Berlin, 1887, p. 35.

question is closely related to the third problem given above, since it gives rise to the following magic square demanded by

that problem.

2 7

6

9 5 1

Magic squares were known to the Arabs and, perhaps, to the Hindus. To the Byzantine writer, Moschopulus, who lived in Constantinople in the early part of the fifteenth century, appears to be due the introduction into Europe of these curious and ingenious products of mathematical thought. Mediæval astrologers believed them to possess mystical properties and when engraved on silver plate to be a charm against plague.1 The first complete magic square which has been discovered in the Occident is that of the German painter, Albrecht Dürer, found on his celebrated wood-engraving, "Melancholia."

4 3

8

Of interest is the following problem, given in Kersey's Wingate: "15 Christians and 15 Turks, being at sea in one and the same ship in a terrible storm, and the pilot declaring a necessity of casting the one half of those persons into the sea, that the rest might be saved; they all agreed, that the persons to be cast away should be set out by lot after this manner, viz. the 30 persons should be placed in a round form like a ring, and then beginning to count at one of the passengers, and proceeding circularly, every ninth person should be cast into the sea, until of the 30 persons there remained only 15. The question is, how those 30 persons ought to be placed, that the lot might infallibly fall upon the 15 Turks and not upon any of the 15 Christians?" Kersey lets the letters a, e, i, o, u stand, respectively, for 1, 2, 3, 4, 5, and gives the verse

From numbers' aid and art,
Never will fame depart.

1 For the history of Magic Squares, see GUNTHER, Vermischte Untersuchungen, Ch. IV. Their theory is developed in the article " Magic Squares " in JOHNSON's Universal Cyclopædia.