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everything said, and was forever denying and distinguishing upon trifles, to the disturbance of all conversation."

This assertion of Franklin that all mathematicians he had met were insufferable from their trifling and captious spirit, has been extensively quoted by opponents of the mathematical sciences. It was quoted by Goethe, and afterwards by Sir William Hamilton, the metaphysician, when he was engaged in a controversy with Whewell, the celebrated author of the History of the Inductive Sciences, on the educational value of mathematical studies. Hamilton attempted to prove the startling proposition that the study of mathematics not only possessed no educational value, but was actually injurious to the mind. He must have experienced exquisite pleasure in finding that Franklin, the greatest physical philosopher of America, had made a statement to the effect that all mathematicians he had met were "forever denying and distinguishing upon trifles."

We shall not speak of this controversy, except to protest against any general conclusion being drawn from Franklin's experience of the captiousness of mathematicians. Take, for examples, David Rittenhouse and Nathaniel Bowditch, who were early American mathematicians, and, like Godfrey, self-taught men. Though Franklin's statement may be true in case of Thomas Godfrey, it is most positively unjust and false when applied to the other two scholars. The biographers of David Rittenhouse are unanimous and explicit in their assertion that in private and social life he exhibited all those mild and amiable virtues by which it is adorned. As,to Nathaniel Bowditch, of whom we shall speak at length later on, we have the reliable testimony of numer ous writers that he was a man remarkable for his social virtues, modest and attractive manners, and Franklinian common sense.

Mention should be made here of Benjamin Banneker, the self-taught "negro astronomer and philosopher," born in Maryland, who became noted in his neighborhood as an expert in the solution of difficult problems, and who, with the use of Mayer's Tables, Ferguson's Astronomy, and Leadbeater's Lunar Tables, made creditable progress in astronomy, and calculated several almanacs. His first almanac was for the year 1792. The publishers speak of it as "having met the approbation of several of the most distinguished astronomers in America, particularly the celebrated Rittenhouse." Banneker sent a copy to Mr. Jefferson, then Secretary of State, who said in his reply, "I have taken the liberty of sending your almanac to Monsieur de Condorcet, secretary of the Academy of Sciences at Paris, and member of the Philanthropic Society, because I considered it a document to which your whole color had a right for their justification against the doubts which have been entertained of them."* Banneker was invited by Andrew Ellicott to accompany "the Commissioners to run the lines of the District of Columbia" upon their mission.

* History of the Negro Race in America, by George W. Williams, p. 386.

II.

INFLUX OF ENGLISH MATHEMATICS, 1776-1820.

The Revolutionary War bore down so heavily upon the educational work in both elementary and higher institutions, that many of them, for a time, actually closed their doors. The majority of students and professors of Harvard and Yale were in the Army, or were in some other way rendering aid to the national cause. The buildings of Nassau (Princeton) College were for a time used as barracks. The business of Columbia College in New York was almost entirely broken up. The professors and students of Rutgers College at New Brunswick, N. J., were sometimes compelled by the presence of the enemy to pursue their academical studies at a distance from New Brunswick. The operations of Brown University in Providence, R. I., were discontinued during part of the war, the college building being occupied by the militia and the troops of Rochambeau. At William and Mary College the exercises were suspended in 1781 for about a year, and the building was occupied at different times by both British and American troops. The walls of the college were "alternately shaken by the thunder of the cannon at Yorktown and by the triumphant shouts of the noble bands who had fought and conquered in the name of American Independence." Academies and primary schools were either deserted or taught by wo men and white-haired men too old to fight. That the philosophic pursuits of scientific societies should have sunk very low is not surprising. Fifteen years elapsed between the publication of the first and second volumes of the Transactions of the American Philosophical Society in Philadelphia.

In spite of the financial depression and poverty which existed immediately after the war, much attention was paid to education. While in 1776 there existed in the colonies only seven colleges, the number was increased to nineteen before the close of the eighteenth century. Acade mies and grammar schools were established, and a large number of text-books were put through the press. Even during the war the printing-press sent out an occasional school-book. Thus, in 1778, while the war was raging most fiercely, an edition of Dilworth's spelling book was printed, which contained in its preface the following patriotic passage: "At the beginning of the contest between the Tyrant and the States, it was boasted by our unnatural enemy, that, if nothing more, they could at least shut up our ports by their navy and prevent the importa

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tion of books and paper, so that in a few years we should sink down into barbarity and ignorance, and be fit companions for the Indians, our neighbors to the westward." These words, printed at the darkest period of the Revolutionary War, disclose a spirit far from submissive. The colonists were not quite ready to sink down into barbarity and ignorance. During the twenty-five years after the Declaration of Independence, more real progress was made in education than in the entire century preceding. Between 1776 and 1815 a large number of books on elementary and a few on higher mathematics were published in America. Many of them were reprints of English works, while others were compilations by American writers, modelled after English patterns. French and German authors were almost unknown. We may therefore call this the period of the "Influx of English Mathe. matics" into the United States. What little mathematics was studied in the colonies before the Revolution was, to be sure, gotten chiefly from English sources, but the scientific currents thither were then so very feeble and slow that we can hardly speak of an "influx."

ELEMENTARY SCHOOLS.

It is a significant fact that of the arithmetics used before the Revolution, but one work in the English language was written by an American author. It is equally significant that with the close of the great struggle for liberty, there began a period of activity in the production of new school-books. The second book devoted exclusively to arithmetic, compiled by an American author, and printed in the English language, was the New and Complete System of Arithmetic by Nicholas Pike, (Newburyport, 1788. )*

Nicholas Pike (1743-1819) was a native of New Hampshire, graduated at Harvard College in 1766, and was for many years a teacher and afterward a magistrate at Newburyport in Massachusetts. His arithmetic received the approbation of the presidents and professors of the leading New England colleges. A recommendation from Harvard professors contains the following timely remark: "We are happy to see so useful an American production, which, if it should meet with the encour agement it deserves, among the inhabitants of the United States, will save much money in the country, which would otherwise be sent to

It appears that Greenwood's Arithmetic, published nearly sixty years previously, was at this time not known to exist. Pike's Arithmetic was called the first American work of its kind. Dr. Artemas Martin has sent the writer the American Antiquarian, (Vol. IV, No. 12, New York, May, 1888) giving an account of Pike's book. It gives a letter written by George Washington at Mount Vernon, June 20, 1788, to Nicholas Pike, in which the former politely acknowledges the receipt of a copy of Pike's Arithmetic. We quote from the letter the following passage:

"Its merits being established by the approbation of competent judges, I flatter myself that the idea of its being an American production and the first of the kind which has appeared, will induce every patriotic and liberal character to give it all the countenance and patronage in his power."

Europe, for publications of this kind." Pike Pike's arithmetic passed through many editions, was long the standard mathematical manual in New England schools, and formed the basis for other arithmetics. It was a very extensive and complete book for that time. A large proportion of the rules were given without demonstration, while some were proved algebraically. In addition to the subjects ordinarily found in arithmetics, it contained logarithms, trigonometry, algebra, and conic sections, but these latter subjects were so briefly treated as to possess little value. After the appearance of Webber's, Day's, and Farrar's Mathematics for colleges, which elaborated these subjects at greater length, they were finally omitted in the fourth edition of Pike's Arithmetic, in 1822.

In 1788, when the first edition appeared, English money was still the prevalent medium of exchange in the United States. To be sure, Federal money was adopted by Congress as early as 1786, but previous to 1794 there was no United States coin of the denomination of a dollar. It was merely the money of account, based upon the Spanish dollar, which had long been in use in this country. Congress passed a law organizing a mint in 1792, but permitting the circulation of foreign. coins for three years, by which time it was believed the new coinage would be ready in sufficient amount. When dollars and cents began to replace pounds and shillings, it became desirable that the Federal currency be explained in arithmetics and taught in schools. In consequence of this, the sterling notation was changed to Federal in the third edition of Pike's arithmetic, which was brought out in Boston in 1808 by Nathaniel Lord. Similar changes were made in other arithmetics.*

Down to the year 1800, the only arithmetic written by an American, which enjoyed wide-spread and prolonged popular favor, was the one of Nicholas Pike. In 1800 appeared a second successful arithmetic, The School-master's Assistant, by Nathan Daboll, a teacher in New London

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Contemporaneously with Pike's Arithmetic there appeared in Philadelphia the Elementary Principles of Arithmetic, by Thomas Sarjeant. This book, as well as the Federal Arithmetic, or the Science of Numbers (Philadelphia, 1793), by the same author, had only an ephemeral reputation. John Gough's Treatise on Arithmetic in Theory and Practice, edited by Benjamin Workman (Boston, 1789), as well as Gough's American Accountant, or School-master's New Assistant, abridged by Benjamin Workman and revised by Patterson (Philadelphia, 1796), had a rather limited circulation. Nor did John Vinall's Arithmetic (Boston, 1792), enjoy better success. After having been a teacher in Newburyport for seventeen years, Vinall at last became writing-master in a school in Boston, his native city. He is said to have been coarse in speech and, like his book, unpopular. Gordon Johnson wrote an arithmetic (Springfield, 1792), which never had more than a passing local reputation. Somewhat more successful was the Introduction to Arithmetic (Norwich, Conn., 1793), by Erastus Root, a graduate of Dartmouth, for several years a teacher, and afterward an active politician and member of Congress.

Our list of arithmetics printed previously to the year 1800 includes the names of several other "quaint and curious volumes," which, after an ephemeral reputation,

(born 1750 and died 1818). This work passed through numerous editions. Though Daboll had to compete with Pike's Abridged Arithmetic and with the celebrated Scholar's Arithmetic of Daniel Adams, it nevertheless acquired an extensive popularity. The expression, "according to Daboll," came to be a synonym for "mathematical correctness." It pushed aside the less favorite works. The main element of popularity of Daboll's School-master's Assistant lay in the fact that it introduced Federal money immediately after the addition of whole numbers, and showed how to find the value of goods therein immediately after simple multiplication. This arrangement, says the author, may be of great advantage to many who perhaps will not have an opportunity to learn fractions. Decimal fractions were wisely made to precede vulgar fractions. In the "Recollections" by Peter Parley, of the town of Ridgefield, Conn., are found the following interesting remarks: "We were taught arithmetic in Daboll, then a new book, and which, being adapted to our measures of length, weight, and currency, was a prodigious leap over the head of poor old Dilworth, whose rules and examples were modelled upon English customs. In consequence of the general use of Dilworth in our schools for perhaps a century, 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."

Since the adherence to pounds and shillings came to be offensive to the people of the young republic, Mr. Hawley, in 1803, undertook to revise the work and alter all the problems to Federal currency. He called the new work "Dilworth's Federal Calculator," but after this change the book was so completely different from the original that the use of Dilworth's name in the title seemed hardly justifiable. Be that as it may, the Federal Calculator was not a success.

At the beginning of the nineteenth century there were three "great arithmeticians" in America, namely, Nicholas Pike, Nathan Daboll, and passed into forgetfulness, never to be resurrected to memory except by the curiosity of some inquiring lovers of "forgotten lore." To the above names we add: The American Tutor's Assistant, by John Todd, Philadelphia, third edition, 1797; Arithmetic by Zachariah Jess, Philadelphia, 1797; American Arithmetic, by David Cook, New Haven, Conn., 1799; The Usher, comprising arithmetic in whole numbers, Federal money, mensuration, surveying, etc., by Ezekiel Little, Exeter, 1799; Usher's Arithmetic, abridged, by Ezekiel Little, 1804; The American Accountant, by Chauncey Lee, Lansingburg, 1797; The American Accountant, by William Milne, New York, 1797, in which, instead of the answers to the problems, which were usually given, the author gave the remainders, after casting out the nines from the answers. A curious little volume is the following: "The Young Gentleman's and Lady's Assistant, containing Geography, Natural Philosophy, Rhetoric, Miscellaneous, to which is added a short and complete system of Practical Arithmetic, wherein the money of the United States of America is rendered easy to the perception of youth. The whole divided into small sections for the convenience of schools, by Donald Fraser, author of the Columbian Monitor, New York, 1706.”

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