only one year, we may imagine that the teaching was not of a very high order.

The first appointment to the Hollis professorship of mathematics and natural philosophy was that of Isaac Greenwood. He was the first to occupy a collegiate chair of mathematics in New England, but not the first in America, as is sometimes stated. This honor belongs to a professor at William and Mary College. Greenwood graduated at Harvard in 1721, then engaged in the study of divinity, visited England, and began to preach in London with some approbation.* He also attended lectures delivered in that metropolis on experimental philosophy and mathematics. In 1727 he entered upon his duties at Harvard. "In scientific attainments Greenwood seems to have been well qualified for his professorship." He made astronomical contributions to the Philosophical Transactions of 1728, and published in 1729 an arithmetic. That seems to have been the earliest arithmetic from the pen of an American author. This is all we know of Greenwood as a mathematician and teacher. Unfortunately he did not prove himself worthy of his place. We regret to say that the earliest professor of mathematics in the oldest American college was "guilty of many acts of gross intemperance, to the dishonor of God and the great hurt and reproach of the society." His intemperance brought about his removal from his chair in 1738.

On the dismissal of Greenwood, Nathaniel Prince, who had been tutor for thirteen years, aspired to the professorship. He was, says Elliot, superior "to any man in New England in mathematics and natural philosophy." But his habits being notoriously irregular, John Winthrop of Boston, was appointed in his stead. Winthrop graduated at Harvard in 1732, and was only twenty-six years old when he was chosen professor of mathematics and natural philosophy. He filled this chair for over forty years (until 1779) with marked ability. In mathematical science he came to be regarded by many the first in America.

If we could turn the wheel of time backward through one hundred and twenty revolutions, and then enter the lecture-room of Professor Winthrop and listen to his instruction, what a chapter in the his tory of mathematical teaching would be uncovered! But as it is, this history is hidden from us. We know only that during the early part of his career as professor," and probably many years before," the textbooks were the following: Ward's Mathematics, Gravesande's Philosophy, and Euclid's Geometry; besides this, lectures were delivered by the professors of divinity and mathematics.t

From this we see that some time between the years 1726 and 1738, Ward's Mathematics had been introduced, and Alsted's old Geometry had given place to the still older but ever standard work of Euclid. This is the first mention of Euclid as a text-book at Harvard. The in

Quincy's History of Harvard University, Vol. II, p. 14. + Peirce's History of Harvard, p. 237.

troduction of Gravesande's Philosophy is another indication of progress. Cravesande was for a time professor of mathematics and astronomy at the University of Leyden. He was the first who on the continent of Europe publicly taught the philosophy of Newton, and he thus contributed to bring about a revolution in the physical sciences. By the adoption of his philosophy as a text-book at Harvard we see that the teachings of Newton had at last secured a firm footing there. Ward's Mathematics continued for a long time to be a favorite text-book.*

It is probable that with the introduction of Ward's Mathematics, algebra began to be studied at Harvard. The second part of the Young Mathematician's Guide consists of a rudimentary treatise on this subject. It is possible, then, that the teaching of algebra at Cambridge may have begun some time between 1726 and 1738. But I have found no direct evidence to show that algebra actually was in the college curriculum previous to 1786.

Since Ward's Mathematics were used, to our knowledge, not only at Harvard, but also at Yale, Brown, and Dartmouth, and as a book of reference at the University of Pennsylvania, a description of the Young Mathematician's Guide may not be out of place.t

The first part treats of arithmetic (143 pages). Though very deficient according to modern notions, the presentation of this subject is superior to that in Dilworth's School-master's Assistant. It is less obscure.

According to ex-President D. Woolsey, the author of this book was the Ward who had been "president of Trinity College, Cambridge, and bishop of Exeter." (Yale College; A Sketch of its History, William L. Kingsley, Vol. II, p. 499.) Now, the only individual answering to this description is Seth Ward, the astronomer, whose time of activity preceded the epoch of Newton. We shall show that the book in question was not written by Seth Ward, but by John Ward, who flourished half a century later than Seth Ward and whose Young Mathematician's Guide was for a long time a popular elementary text-book in England. Wherever we have seen Ward's book mentioned in the curricula of American colleges it was always called "Ward's Mathematics." The baptismal name of the author was never given. This shows that there was only one Ward (either Seth or John) whose mathematical books were known and used in our colleges. Now, Benjamin West, professor of mathematics in Brown University from 1786 to 1799, published in the first volume of the American Academy of Arts and Sciences a paper "On the extraction of roots," in which he offers improvements on "Ward's" method. Now, I have seen a copy of Seth Ward's Astronomia Geometrica, but have found nothing in it on root extraction. One would hardly expect to find anything on it in Seth's "Trigonometry" or "Proportion." John Ward, on the other hand, treats of roots in his "Guide," and gives a "general method of extracting roots of all single powers." West takes two examples (two numbers, one of 14, the other of 18 digits) from "Ward," and shows how the required roots can be extracted by his method. But both these examples are given in John Ward's Young Mathematician's Guide. This evidence in favor of John Ward's book may be considered conclusive. Further information on “Ward's Mathematics" will be found in an article by the writer in the Papers of the Colorado College Scientific Society, Vol. I.

The copy which the writer has before him (Twelfth edition, London, 1771), was kindly lent him by Dr. Artemas Martin, of the U. S. Coast Survey, who has for years been making a collection of old and rare books on mathematics.

Like all books of that time, it contains rules, but no reasoning. What seems strange to us is the fact that subjects of no value to the beginner, such as arithmetical and geometrical proportion (i. e., progression), alligation, square root, cube root, biquadrate root, sursolid root, etc., are given almost as much space and attention as common and decimal fractions.

The second part (140 pages) is devoted to algebra. Ward had published a small book on algebra in 1698, but that, he says, was only "a compendium of that which is here fully handled at large." Like Harriot, he speaks of his algebra as "Arithmetick in species." This name is appropriate, inasmuch as he does not (at least at the beginning) recognize the existence of negative quantities, but speaks of the minus sign always as meaning only subtraction, as in arithmetic. A little further on, however, he brings in, by stealth, "affirmative" and "negative" quantities. The knowledge of algebra to be gotten from this book is exceedingly meagre. Factoring is not touched upon. The rule of signs. in multiplication is proved, but further on all rules are given without proof. The author develops a rule showing how binomials can be raised "to what height you please without the trouble of continued in.volution." He then says: "I proposed this method of raising powers in my Compendium of Algebra, p. 57, as wholly new (viz, as much of it as was then useful), having then (I profess) neither seen the way of doing it, nor so much as heard of its being done. But since the writing of that tract, I find in Doctor Wallis's History of Algebra, pp. 319 and 331, that the learned Sir Isaac Newton had discovered it long before." The subject of "interest" is taught in the book algebraically, by the use of equations.

Part III (78 pages) treats of geometry. In point of precision and scientific rigor, this is quite inferior. After the definitions follow twenty problems, intended for the excellent purpose of exercising the "young practitioner," and bringing "his hand to the right management of a ruler and compass, wherein I would advise him to be very ready and exact." Then follows a collection of twenty-four "most useful theorems in plane geometry demonstrated." This part is semi-empirical and semi-demonstrative. A few theorems are assumed and the rest proved by means of these. The theorem, "If a right line cut two parallel lines, it will make the opposite (i. e., alternate interior) angles equal to each other," is proved by aid of the theorem, that "If two lines intersect each other, the opposite angles will be equal." The proof is based on the idea that "parallel lines are, as it were, but one broad line," and that by moving one parallel toward the other, the figure for the former theorem reduces to that of the latter. The next chapter contains the algebraical solution of twenty geometrical problems.

Part IV, on conic sections (36 pages), gives a semi-empirical treatment of the subject. Starting with the definition of a cone, it shows how the three sections are obtained from it, and then gives some of their principal properties.

Part V (36 pages) is on the arithmetic of infinites. Judging from this part of the book, its author knew nothing of fluxions. The first edition appeared in 1707, after Newton had published the first edition of his Principia, in 1687, but his Method of Fluxious was not published till 1736, though written in 1671. Ward employs the method of integration by series of Cavalieri, Roberval, and John Wallis, and, thereby, finds the superficial and solid contents of solid figures. It does not appear that this part of the book was ever studied in American colleges. Ward's book met with favor in England. In the preface to the twelfth edition he says: "I believe I may truly say (without vanity) this treatise hath proved a very helpful guide to near five thousand persons, and not only so, but it hath been very well received amongst the learned, and (I have been often told) so well approved on at the universities, in England, Scotland, and Ireland, that it is ordered to be publicly read to their pupils."

In former times all professors of mathematics in American colleges gave instruction, not merely in pure mathematics, but also in natural philosophy and astronomy; and it appears that as a general rule these professors took more real interest and made more frequent attempts at original research in the fields of astronomy and natural philosophy than in pure mathematics. The main reason for this lies probably in the fact that the study of pure mathematics met with no appreciation and encouragement. Original work in abstract mathematics would have been looked upon as useless speculations of idle dreamers. The scien tific activity of John Winthrop was directed principally to astronomy. His reputation abroad as a scientist was due to his work in that line. In 1740 he made observations on the transit of Mercury, which were printed in the Transactions of the Royal Society. In 1761 there was a transit of Venus over the sun's disk, and as Newfoundland was the most western part of the earth where the end of the transit could be observed, the "province" sloop was fitted out at the public expense to convey Winthrop and party to the place of observation. He took with him two pupils who had made progress in mathematical studies. One of these, Samuel Williams, became later his successor at Harvard. In 1769 Winthrop had another chance for observing the transit of Venus, at Cambridge. "As it was the last opportunity that generation could be favored with, he was desirous to arrest the attention of the people. He read two lectures upon the subject in the college chapel, which the students requested him to publish. The professor put this motto upon the title page: Agite, mortales! et oculos in spectaculum vertite, quod hucusque spectaverunt perpaucissimi; spectaturi iterum sunt nulli." (Come, mortals! and turn your eyes upon a sight which, to this day, but few have seen, and which not one of us will ever see again.) The transit of 1769 was also observed in Philadelphia by David Rittenhouse, and in Providence by Benjamin West. These observations

* "John Winthrop," in the Biographical Dictionary by John Eliot, 1809.

were an important aid in determining the sun's parallax. Most gratifying to us is the interest in astronomical pursuits manifested in those early times. Expeditions fitted out at public expense, and private munificence in the purchase of suitable instruments, bear honorable testimony to the enlightened zeal which animated the friends of science.

In 1767 John Winthrop wrote his Cogita de Cometis, which he dedicated to the Royal Society, of which he had been elected a member. This was reprinted in London the next year, and gave him an extensive literary reputation.

In 1764 a calamity befell Harvard College. The library and philo sophical apparatus--the collections of over a century-were destroyed by fire. Among the books recorded as having been lost are the following: "The Transactions of the Royal Society, Academy of Sciences in France, Acta Eruditorum, Miscellanea Curiosa, the works of Boyle aud Newton, with a great variety of other mathematical and philo. sophical treatises." It is seen from this that, before the fire, books of reference in higher mathematics had not been entirely wanting.

John Winthrop died in 1779, and the robe of the departing prophet fell upon his former disciple, the Rev. Samuel Williams. Williams filled the mathematical chair for eight years. Having inherited from his master a love of astronomy, he frequently published observations and notices of extraordinary natural phenomena in the memoirs of the American Academy of Arts and Sciences. He occupied the mathematical chair at Harvard until 1788. Then he lectured at the University of Vermont on astronomy and natural philosophy for two years, and was subsequently minister at Rutland and Burlington, Vermont.

YALE COLLEGE.

Yale, the second oldest New England college, was founded in 1701, or sixty-three years after the opening of Harvard. During the first fifteen years it maintained a sort of nomadic existence. Previous to 1816. instruction seems to have been given partly at Saybrook and partly at Killingworth and Milford. Its course of instruction was then very limited. The mathematical teaching during the first years of its existence was even more scanty than in the early years at Harvard. Benjamin Lord, a Yale graduate of 1714, wrote in 1779 as follows in reply to inquiries by President Stiles: "As for mathematics, we recited and studied but little more than the rudiments of it, some of the plainest things in it. Our advantages in that way were too low for any to rise high in any branch of literature." Doctor Johnson, of the same class, says: Common Arithmetick and a little surveying were the ne plus ultra of mathematical acquirements." It appears from this that sur veying was taken at Yale, instead of the geometry which formed part

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* Vide Quincy's History of Harvard University, Vol. II, p. 481.

Yale Biographies and Annals, 1701-45, by Franklin Bowditch Dexter, pp. 115 and 116.