not receive that thorough "grinding" in the elements during the first years of college that they do now; on the contrary, no mathematics at all was taught except during the last year. The mathematical course began in the Senior year, and consisted of arithmetic and geometry during the first three-quarters of the year, and astronomy during the last quarter. Algebra was then an unknown science in the New World. It is interesting to notice that, in this original curriculum, the attention of each class was concentrated for a whole day upon only one or two subjects. Thus, Mondays and Tuesdays were devoted by the third year students exclusively to mathematics or astronomy, Wednesdays to Greek, Thursdays to "Eastern tongues," and so on. The importance attached to mathematical studies, as compared with other branches of discipline, may be inferred from the fact that ten hours per week were devoted to philosophy, seven to Greek, six to Rhetoric, four to Oriental languages, but only two to mathematics. According to these figures, Oriental languages were considered twice as important as mathematics. But we must remember that this course was laid out for students who were supposed to choose the clerical profession. For that reason, philosophical, linguistic, and theological studies were allowed to monopolize nearly the whole time, while mathematics was excluded almost entirely.

In what precedes we have measured the college work done in 1643 by the standards of 1889. Let us now compare it with the contemporaneous work in English universities. We may here premise that in the middle of the seventeenth century rapid progress was made in the mathematical sciences. In 1643, Galileo had just passed away; Cavalieri, Torricelli, Pascal, Fermat, Roberval, and Descartes were at the zenith of their scientific activity; John Wallis was a young man of twenty-seven, Isaac Barrow a youth of thirteen, while Isaac Newton was an infant feeding from his mother's breast. Though much original work was being done, especially by French and Italian mathematicians, the enthusiasm for mathematical study had hardly reached the universities. Some idea of the state of mathematics at Cambridge, England, previous to the appearance of Newton, may be gathered from a discourse by Isaac Barrow, delivered in Latin, probably in 1654, or eighteen years after the founding of Harvard College. In it occurs the following passage: "The once horrid names of Euclid, Archimedes, Ptolemy, and Diophantus, many of us no longer hear with trembling

Why should I mention the fact that by the aid of arithmetic, we have now learned, with easy and instantaneous work, to compute accurately the number of the very sands (themselves). And indeed that horrible monster that men call algebra many of us brave men (that we are) have overcome, put to flight, and (fairly) triumphed over; (while) very many (of us) have dared, with straight-along glance, to look into optics; and others (still), with intellectual rays unbroken, have dared to pierce (their way) into the still subtler and highly useful doctrine of dioptries."

From this it would seem that mathematical studies had been introduced into old Cambridge only a short time before Barrow delivered his speech. It thus appears that about 1636, when new Cambridge was founded in the wilds of the west, old Cambridge was not mathematical at all. In further support of this view we quote from the Penny Cyclopædia, article "Wallis," the following statement: "There were no mathematical studies at that time [when Wallis entered Emmanuel College in 1632] at Cambridge, and none to give even so much as advice what books to read. The best mathematicians were in London, and the science was esteemed no better than mechanical. This account is con. firmed by his [Wallis's] contemporary, Horrocks, who was also at Emmanuel and whose works Wallis afterwards edited." In a biography of Seth Ward, an English divine and astronomer, we meet with similar testimony. He entered Sidney Sussex College, Cambridge, in 1632. "In the college library he found, by chance, some books that treated of the mathematics, and they being wholly new to him, he inquired all the college over for a guide to instruct him that way, but all his search was in vain; these books were Greek, I mean unintelligible, to all the fellows in the college."

If so little was done at old Cambridge, then we need not wonder at the fact that new Cambridge failed to be mathematical from the start. The fountain could not rise higher than its source. It was not until the latter half of the seventeenth century that mathematical studies at old Cambridge rose into prominence. Impelled by the genius of Sir Isaac Newton, old Cambridge advanced with such rapid strides that the youthful college in the west became almost invisible in the distant

rear.

The mathematical course at Harvard remained apparently the same till the beginning of the eighteenth century. Arithmetic and a little geometry and astronomy constituted the sum total of the college instruction in the exact sciences. Applicants for the master's degree had only to go over the same ground more thoroughly. Says Cotton Mather: "Every scholar that giveth up in writing a system or synopsis or sum of logic, natural and moral philosophy, arithmetic, geometry, and astronomy, and is ready to defend his theses or positions, withal skilled in the originals, as above said, and of godly life to be dignified with the second degree."†

is fit

These few unsatisfactory data are the only fragments of information that we could find on the mathematical course at Harvard during the seventeenth century. The following note on the nature of the instruction given in physics is not without interest: Mr. Abraham Pierson, jr. (first rector of Yale College), graduated at Harvard in 1668. The college (Yale) possesses several of his MSS., "containing notes made by

Life of Right Reverend Seth, Lord Bishop of Salisbury, by Walter Pope. London, 1697, p. 9.

+Magnalia, Book IV, 128th ed., 1702.

him during his student life at Harvard on logic, theology, and physics, and so throwing light on the probable compass of the manuscript textbook on physics compiled by him, which was handed down from one college generation to another for some twenty-five years, until superseded by Clarke's Latin translation of Rohault's Traité de Physique. The Harvard notes on physics seem (from an inscription attached) to to have been derived in like manner from the teachings of the Rev. Jonathan Mitchel (Harvard College, 1647); they are rather metaphysical than mathematical in form, and it is even difficult to determine what theories of physical astronomy the writer held. Suffice it to say that he ranged himself somewhere in the wide interval between the Ptolemaic theory (generally abandoned one hundred years earlier) and the Newtonian theory (hardly known to any one in this part of the world until the eighteenth century). In other words, while recognizing that the earth is round, and that there is such a force as gravity, there is no proof that he had got beyond Copernicus to Kepler and Galileo." *

In this extract our attention is also called to the common practice among successive generations of students at that time of copying manu. script text books. As another instance of this we mention the manu script works, a System of Logic and a Compendium Physical, by Rev. Charles Morton, which (about 1692) were received as text-books at Harvard, "the students being required to copy them."

We shall frequently have occasion to observe that astronomical pursuits have always been followed with zeal and held in high estimation by the American people. As early as 1651 a New England writer, in naming the first fruits of the college," speaks of the "godly Mr. Sam Danforth, who hath not only studied divinity, but also astronomy; he put forth many almanacs," and "was one of the fellows of the college." Another fellow of Harvard was John Sherman. He was a popular preacher, an "eminent mathematician," and delivered lectures at the college for many years. He published several almanacs, to which he appended pious reflections. The ability of making almanacs was then considered proof of profound erudition. A somewhat stronger evidence of the interest taken in astronomy was the publication at Cambridge of a set of astronomical calculations by Uriah Oakes. Oakes, at that time a young man, had graduated at Harvard in 1649, and in 1680 became president pro tem. and afterwards president of Harvard College. In allusion to his size, he attached to his calculations the motto, "Parvum parva decent, sed inest sua gratia parvis." (Small things befit the small, yet have a charm their own.)

The preceding is an account of the mathematical and physical studies at Harvard during the seventeenth century. We now proceed to the eighteenth century. It appears that in 1700 algebra had not yet be

* Yale Biographies and Annals, 1701-1745, by Franklin Bowditch Dexter, p. 61. Quincy's History of Harvard University, Vol. 1, p. 70.

come a college study. The Autobiography of Rev. John Barnard * throws some light on this subject. Barnard took his first degree at Harvard in 1700, then returned to his father's house, where he betook himself to studying. "While I continued at my father's I prosecuted my studies and looked something into the mathematics, though I gained but little, our advantages therefor being noways equal to what they have who now have the great Sir Isaac Newton and Dr. Halley and some other mathematicians for their guides. About this time I made a visit to the college, as I generally did once or twice a year, where I remember the conversation turning upon the mathematics, one of the company, who was a considerable proficient in them, observing my ig norance, said to me he would give me a question, which if I answered in a month's close application he should account me an apt scholar. He gave me the question. I, who was ashamed of the reproach cast upon me, set myself hard to work, and in a fortnight's time returned him a solution of the question, both by trigonometry and geometry, with a canon by which to resolve all questions of the like nature. When I showed it to him he was surprised, said it was right, and owned he knew no other way of resolving it but by algebra, which I was an utter stranger to." Though a graduate of Harvard, he was an utter stranger to algebra. From this we may safely conclude that in 1700 algebra was not yet a part of the college curriculum.

What, then, constituted the mathematical instruction at that time? Was it any different from the course given in 1643? Until about 1655, the entire college course extended through only three years; at this time it was lengthened to four years. We might have supposed that the mathematics formerly taught in the third year would have been retained as a study for the third or Junior year, but this was not the case. In the four-years' course, mathematics was taught during the last, or Senior year. Quincy, in his history of Harvard University (Vol. I, p. 441), quotes from Wadworth's Diary the list of studies for the year 1726. The Freshmen recited in Tully, Virgil, Greek testament, rhetoric, Greek catechism; the Sophomores in logic, natural philosophy, classic authors, Heerebord's Meletemata, Wollebins's Divinity; the "Junior sophisters" in Heerebord's Meletemata, physics, ethics, geography, metaphysics; while the "Senior sophisters, besides arithmetic, recite Alsted's Geometry, Gassendi's Astronomy in the morning; go over the arts towards the latter end of the year, Ames's Medulla on Saturdays, and dispute once a week." This quotation establishes the fact that ninety years after the founding of Harvard, the mathematical course was essentially the same as at the beginning. Arithmetic, geometry, and astronomy still constituted the entire course. Mathematics continued to be considered the crowning pinnacle instead of a corner-stone of college education; natural philosophy and physics were

Collections of the Mass. Hist. Soc., Third Series, Vol. V, pp. 177-243.

still taught before arithmetic and geometry. But we must observe that, in 1726, printed treatises were used as text-books in geometry and astron omy. We are not informed at what time these printed books were introduced. They may have been used as text-books much earlier than the above date. The authors of these books were in their day scholars of wide reputation. Johann Heinrich Alsted (1558-1638), the author of the Geometry, was a German Protestant divine, a professor of philoso phy and divinity at Herborn in Nassau, and afterwards in Carlsburg in Transylvania. In one of his books he maintained that the millenium was to come in 1694.

Pierre Gassendi (1592-1655), whose little astronomy of one hundred and fifty pages was used as a class-book at Harvard, was a contemporary of Descartes and one of the most distinguished naturalists, mathematicians, and philosophers of France. He was for a time professor of mathematics at the Collège Royal of Paris. What seems very strange to us is that nearly a century after the first publication of these books they should have been still in use and apparently looked upon as the best of their kind. Forty years after the publication of Newton's Principia an astronomy was being studied at Harvard whose author died before the name of Newton had become known to science. The wide chasm between the theories of Newton and those of Gassendi is brought to full view by the following quotation from Whewell's History of the Inductive Sciences (Third edition, Vol. I, p. 392): "Gassendi's own views of the causes of the motions of the heavenly bodies are not very clear. In a chapter headed Quæ sit motrix siderum causa,' he reviews several opinions; but the one which he seems to adopt is that which ascribes the motion of the celestial globes to certain fibers, of which the action is similar to that of the muscles of animals. It does not appear therefore that he had distinctly apprehended, either the continuation of the movements of the planets by the first law of motion, or their deflection by the second law."

The year 1726 is memorable in the annals of Harvard for the establishing of the Hollis professorship of mathematics. Thomas Hollis, a kind-hearted friend of the college, transmitted to the treasurer of the college the then munificent sum of twelve hundred pounds sterling, and directed that the funds should be applied to "the instituting and settling a professor of mathematics and experimental philosophy in Harvard College." To the game benefactor Harvard was indebted for the establishment of the professorship of divinity. Down to the commencement of the nineteenth century only one additional professor was appointed in the undergraduate department, namely, the Hancock professor of Hebrew, in 1765. Hence, it follows that almost all regular instruction was given by tutors. Previous to the establishment of the Hollis professorship the mathematical instruction was entirely in the hauds of tutors. Since almost any minister was considered competent to teach mathematics, and since tutors held their place sometimes for