trained and educated to the refined methods which he was introducing. The work of the survey had to be systematized. It continued under his direction until the time of his death, in 1843. He left the work well advanced between Narragansett Bay and Cape Henlopen, and the survey sufficiently organized in all its varied details. His course was, however, frequently criticised in Congress, and it was not always easy to get the necessary appropriations.

Mr. Hassler was very self-confident and independent. This was one cause of the occasional opposition to him. Though not conceited, he was conscious of his superiority over the great mass of men with whom he came in contact in Washington. The following anecdote is characteristic of him: At one time the cry of "retrenchment and reform" had become popular, and a newly appointed Secretary of the Treasury thought he could not signalize his administration more aptly than by reducing the large salary of the superintendent. He sent for Mr. Hassler and said, "My dear sir, your salary is enormous; you receive $6,000 per annum—an income, do you know, quite as large as that of the Secretary of State." "True," replied Hassler," precisely as much as the Secretary of State and quite as much as the Chief of the Treasury; but do you know, Mr. Secretary, that the President can make a minister of State out of anybody; he can make one even out of you, sir; but if he can make a Hassler, I will resign my place."

Hassler's successor was Alexander Dallas Bache, a great-grandson of Franklin and a graduate of West Point. He exercised a very marked influence over the progress of science among us. He graduated at the head of his class, and the great expectations that were then entertained of him have been fully realized. For eight years he devoted himself to physical science, while professor at the University of Pennsylvania, and gained a wide reputation. The Coast Survey made rapid progress under his management. Congress began to show better appreciation of this sort of work, and made more liberal appropriations. This enabled him to adopt a more comprehensive scheme. Instead of working only at one locality, as had been done previously, he was able to begin independent surveys at several places at once, each section employing its own base. He proposed eight sections, which number was increased on the annexation of Texas, and again on the annexation of California. Two of the most important improvements of modern geodesy were perfected and brought into use at the beginning of Bache's superintendency, namely, Mr. Talbott's method of determining latitudes and the telegraphic method of determining longitude. Various other refinements in every branch of work were introduced. Systematic observations of the tides, a magnetic survey of the coast, and the extension of the hydrographic explorations into the Gulf Stream were also instituted by Bache.

Having extended the scope of the Survey, Bache needed a greater number of assistants, but the supply was not wanting. Says Prof. T. 881-No. 3-19

H. Safford," he found available for its higher geodetic works a number of West Point officers, of whom T. J. Lee was one, and Humphreys, afterward chief engineer of the Army, another. One of the leaders in practical astronomy of the topographical engineers was J. D. Graham; and the work which had been done by that corps upon the national and State boundaries had familiarized a good many Army officers with field astronomy and geodesy.

"Bache, who had been out of the Army nearly twenty years employed his great organizing and scientific capacity in training the Coast Survey corps (including detailed Army officers) into practical methods for its various problems; and the connection between the West Point officers and the able young civilians, who are now the veterans of the Survey, was extremely wholesome.

"Lee prepared a work (Tables and Formulæ) which has served an excellent purpose in bridging the gap between theory and practice; especially for the last generation of West Point officers."

Graduates of West Point are now more closely employed in military and other public duty, and are no longer employed in the Coast Survey. The work of the Survey was interrupted by the Civil War. Soon after its close Bache died (1867). Benjamin Peirce, his successor in the superintendency, said of him: "What the Coast Survey now is, he made it. It is his true and lasting monument. It will never cease to be the admiration of the scientific world. * It is only necessary conscientiously and faithfully to follow in his foot-steps, imitate his example, and develop his plans in the administration of the Survey."

Under Peirce, the survey of the coasts was pushed with vigor, and it rapidly approached completion. He proposed the plan of connecting the survey on the Atlantic Coast with that on the Pacific by two chains of triangles, a northern and a southern one. This project received the sanction of Congress, and thus the plan of a general geodetic survey of the whole country was happily inaugurated.

Benjamin Peirce's successor on the Coast Survey was Carlile Pollock Patterson. He was a graduate of Georgetown College, Kentucky, and had for many years previous to this appointment, in 1874, been connected with the Survey as hydrographic inspector. Under him the extension of the Survey into the interior of our country, as inaugurated by Peirce, was continued. By the completion of this work this country will contribute its fair share to the knowledge of the figure of the earth, which has hitherto been derived entirely from the researches of other nations. On account of this extension, the name, "U. S. Coast Survey," was changed, in 1879, to "U. S. Coast and Geodetic Survey."

Patterson died in 1881, and Julius Erasmus Hilgard became his successor. Hilgard was born in Zweibrücken, Bavaria, came to this country at the age of ten, and at the age of twenty was invited by Bache to become one of his assistants on the Survey. Hilgard soon came to be

*Mathematical Teachings, p. 6.

recognized for great ability and skill, and rose to the position of assistant in charge of the Office in Washington. He held the superintendency from 1881 to 1885, when he resigned. His work consisted chiefly of researches and discussions of results in geodesy and terrestrial physics, and in the perfecting of the methods and instruments employed. The superintendency was next intrusted to Frank M. Thorn, who was succeeded in July, 1889, by T. C. Mendenhall, who now fills the office.

The work of the U. S. Coast Survey has been carried on with great efficiency from its very beginning, and reflects great credit upon America. In making the computations for the Survey, the method of least. squares for the adjustment of observations has found extended appli cation. Valuable papers on this subject by Bache and Schott have been printed in the reports of the U. S. Coast Survey. Charles A. Schott graduated at the Polytechnic School in Carlsruhe, came to this country in 1848, and has since that time been an efficient worker on the U. S. Coast Survey. He is now chief of the computing division.

It will be remembered that interesting researches on least squares had been made quite early in this country by Robert Adrain. Benjamin Peirce invented a criterion for the rejection of doubtful observations.t It proposes a method for determining, by successive approximations, whether or not a suspected observation may be rejected. Tables are needed for its application. Objections have been made to its use, because it "involves a contradiction of reasoning." The criterion is given by Chauvenet in his Method of Least Squares (1864), and has been used to some extent on the U. S. Coast Survey, but it has found no acceptance in Europe. Chauvenet gives an approximate criterion of his own for the rejection of one doubtful observation, which is derived, he says, "directly from the fundamental formula upon which the whole theory of the method of least squares is based." But this criterion, too, has been criticised as being "troublesome to use, and as based on an erroneous principle." Stone, in England, offered still another criterion for the rejection of discordant observations, but Glaisher pronounces it untrustworthy and wrong. No criterion has yet been given which has met with general acceptance. Indeed, Professor Newcomb considers it impossible that such a one should ever be invented. Says he (in his second paper mentioned below): "We here meet the difficulty that no positive criterion for determining whether an observation should or should not be treated as abnormal is possible. Several attempts have indeed been made to formulate such a criterion, the best known of which is that of Peirce."

* See reports for the years 1850, '55, '56, '58, '61, '64, '66, '67, '75.

+ Gould's Astronomical Journal, Vol. II, pp. 161-3.

See Prof. Mansfield Merriman's article in the Transactions of the Connecticut Academy, containing a list of writings relating to the method of least squares and the theory of the accidental errors of observation, which comprises 408 titles by 193 authors.

Valuable papers on least squares have been contributed in this coun. try by G. P. Bond,* of Harvard; Simon Newcomb, † C. S. Pierce, ‡ and Truman H. Safford.§ The text-books on this subject generally used in our schools are those of Chauvenet, Merriman, and T. W. Wright.

*"On the use of Equivalent Factors in the Method of Least Squares," Memoirs American Academy, Vol. VI, pp. 179–212.

+"A Mechanical Representation of a Familiar Problem," Monthly Notices of the Astronomical Society, London, Vol. XXXIII, pp. 573-4; "A Generalized Theory of the Combination of Observations so as to Obtain the Best Results," American Journal of Mathematics, Vol. VIII.

"On the Theory of Errors of Observations," Report U. S. Coast Survey, 1870, pp. 200-224.

§ "On the Method of Least Squares," Proceedings American Academy, Vol. XI.

IV.

THE MATHEMATICAL TEACHING AT THE PRESENT TIME.

The mathematical teaching of the last ten years indicates a "rup. ture" with antiquated traditional methods, and an "alignment with the march of modern thought." As yet the alignment is by no means rectified. Indeed it has but barely begun. The "rupture" is evident from the publication of such works as Newcomb's series of mathematical textbooks, recent publications on the calculus, the appearance of such algebras as those of Oliver, Wait, and Jones, Phillips and Beebe, and Van Velzer and Slichter; of such geometries as Halsted's "Elements" and "Mensuration;" of such trigonometries as Oliver, Wait, and Jones's; of Carll's Calculus of Variations; Hardy's Quaternions; Peck's and Hanus's Determinants; W. B. Smith's Co-ordinate Geometry (employing determinants); Craig's Linear Differential Equations.

Determinants and quaternions have thus far generally been offered as elective studies, and have formed a crowning pinnacle of the mathematical courses in colleges. It is certainly very doubtful whether this is their proper place in the course. It seems quite plain that the ele ments of determinants should form a part of algebra, and should be taught early in the course, in order that they may be used in the study of co-ordinate geometry. What place should be assigned to quaternions is not quite so plain. Prof. De Volson Wood introduces their elements in his work on co-ordinate geometry. The professors of Cornell have not taught quaternions directly for some years, but are convinced that most students derive more benefit by a mixed course in matrices, vector addition and subtraction, imaginaries, and theory of functions. The early introduction of determinants seems more urgent than that of quaterni

ons.

We think, however, that great caution should be exercised in incorporating either of these subjects in the early part of mathematical courses. Those universities and colleges which are, as yet, not strong enough to maintain a high and rigid standard of admission, and whose students enter the Freshman class with only a very meagre and superficial knowledge of the elements of ordinary algebra, would find the introduction of determinants and imaginaries as Freshman studies a hazardous innovation. One of the very first considerations in mathematical teaching is thoroughness. In the past the lack of thoroughness has poisoned the minds of the American youth with an utter dislike and bitter hatred of mathematics. Whenever a subject is not well understood, it is not liked; whenever it is well understood, it is generally liked.

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