The Net Advance of Physics RETRO:
The Struggles of a Self-Made Scientist

2013 January 21

It is 1872, and Asaph Hall III will soon be forty-three. He grew up on a small farm in the Berkshires, fell into poverty at thirteen when his father died, become a carpenter, taught himself simple mathematics to help with his trade, dreamt of becoming an architect. Then he had seen an item in the New York Tribune that changed his life: New York Central College at McGrawville in the Finger Lakes region was advertising what would now be called a work-study programme, a classical education for labourers.

Angelo Hall, biographer of his parents, would someday write: "Massachusetts educators [i.e., presumably, Horace Mann] would have us believe that a young man of twenty-five should have spent nine years in primary and grammar schools, four years more in a high school, four years more at college, and three years more in some professional school. Supposing the victim to have begun his career in a kindergarten at the age of three ... at twenty-five his education would be completed. He would have finished his education, provided his education had not finished him. Now at the age of twenty-four or twenty-five Asaph Hall 3rd only began serious study ..."

In some ways Central College was much like any other. Two years earlier, a student named William Austin had written a letter to his family describing life at a place which "begins to seem like home ... a beautiful section of country somewhat uneven but just enough to waken mankind to the romantic beauties of nature." It might have been written by a student of almost any school or era, but one line reveals it to be an anomaly for 1852: "The Ladies all room at the hall."

Central College was co-educational; it was also biracial, a foundation established by radical Baptist fundamentalists on the cutting edge of every movement for social change. (Here, perhaps, the modern reader may feel that the past is indeed a foreign country.) Astonishingly, even the faculty at Central College originally included Blacks, among them the mathematician Charles L. Reason. Western New York in the first half of the 1800s was impoverished, rural, and deeply divided, the "Burnt-Over Country" of religious sectarians, utopian revolutionaries, and tough, conservative farmers. In 1852 the hostility of the local community and the limits of tolerance even within the hearts of some liberal reformers had brought Central College to a crisis when one of the Black professors, William G. Allen, became engaged to a white student. A mob attacked Allen, who narrowly escaped lynching; the couple fled to England with help from Louisa May Alcott's uncle (the incident would later become the basis of one his niece's sensational short-stories, "M.L."); and the already-struggling college, derided in the Syracuse newspaper as a "nigger institution", began a precipitous decline which would lead to its closure in 1860.

This was the precarious institution of higher learning in which Asaph Hall had enrolled, and where he quickly established himself as top student in mathematics. Among his teachers was a student assistant, a suffragist in bloomers named Angeline Stickney who became his wife. A typical Baptist feminist zealot, she felt no shame at having attended what she dryly termed "an unpopular institution". She wrote back to the college after her departure:

"I expect to get the name that I have heard applied to all who come here, 'fanatic'... Let me live, let me die a fanatic. I will not seal up in my heart the fountain of love that gushes forth for all the human race ... [T]here are none here to say, 'thus far thou mayst ascend the hill of Science and no farther.' ... I can see my brother gathering those golden fruits [of knowledge], and ... there are none here to whisper, 'that is beyond thy sphere, thou couldst never scale those dizzy heights'; but, on the contrary, here are kind voices cheering me onwards." Nevertheless, as so often happened even in the most idealistic Nineteenth-Century circles, she soon abandoned her own promising scientific career to become, almost literally, her husband's muse.

The Halls had wandered the Midwest, peripatetic school-teachers on the edge of beggary. They returned to the East; Asaph was hired as an assistant at the Harvard Observatory for a wage too low to live on, and began calculating for almanacs on the side. During the Civil War his fortunes improved; he obtained a senior post at the Naval Observatory and once gave a distinguished visitor a tour of the heavens. A journalist would later write [ Popular Science 45, 833 (1894)]:

"A few nights later the trap- door opened again, and the same figure appeared. He told Prof. Hall that after leaving the observatory he had looked at the moon, and it was wrong side up as he had seen it through the telescope. He was puzzled, and wanted to know the cause, so he had walked up from the White House alone. Prof. Hall explained to him how the lens of a telescope gives an inverted image, and President Lincoln went away satisfied." Whether the story is true or one of the countless Lincoln legends, who can say?

But now it is 1872. Hall's greatest achievement, or from history's point of view his only achievement, is still five years ahead: urged on by Angeline after despairing of success, he will discover ... But his concern today is with theory, not with observation. How hard it is for an outsider to master mathematics! And how much of that difficulty comes from the books! He writes -- a Letter to the Editor:

[Nature 6, 351 (1872)]:

Hindrances to Students of Mathematics

It was the opinion of Dr. Samuel Johnson that everything ought to be persecuted in order that we may know whether it is worthy to live or not. There is, doubtless, a good deal of truth in this opinion, and the idea or the man that cannot endure and overcome a considerable amount of difficulty is of but little value. Still there must be a reasonable limit to persecutions and difficulties, and hence I hope that the praiseworthy efforts of the English mathematicians to improve their text-books of geometry will be successful. In considering such a matter as the improvement of text-books, an extensive knowledge of the experience of all classes of students will be valuable, and as many of the mathematical books profess to be written for those who are not fortunate enough to have a teacher, an account of the difficulties which such a one has experienced may be of some interest.

1. I place first among these difficulties the practice common to nearly all mathematical writers, of restricting the number of axioms or fundamental assumptions, making them fewer than they naturally are. It is worse than useless to attempt to prove something that is self-evident, or which is so nearly so that it is impossible to make any proof illustrate it. In all such cases it would be better to state frankly and clearly that we make an assumption, depending on observation to justify it. An example of this superfluous proof may be found in many of the books on rational mechanics, where we are told that a body cannot move out of the place [sic; should it read "plane"?] of the forces, because we know of no reason why it should move to one side rather than the other; therefore, &c. Of useless definitions we have an example in a popular work on arithmetic, where we are told that " time is the measurement of duration," and a few pages further on that "duration is a portion of time." Allied to this is the contemptible habit of those who explain, with kind condescension and with great detail, all insignificant matters, while at the same time they cover up or dodge by some such phrase as " it is evident " the really difficult points.

2. I do not object to a frequent and thorough application of the differential calculus in a text-book, and such an application seems to me better than the coarse processes under which this calculus is sometimes concealed; but there is a habit, common to young writers, of introducing forced and difficult demonstrations where more simple ones would be better. An illustration may be found in one of our best books on astronomy. In the first edition of this book the author gave a long and difficult demonstration of the well-known formula for the transformation of rectangular co-ordinates in a plane. The demonstration was made to depend on the solution of functional equations by means of the differential calculus, and is an awkward thing to place at the beginning of a text-book. In the second edition, having removed this demonstration and supplied its place by a simple one, the author has made the first chapter of his book the best synopsis of spherical trigonometry that I know of.

3. An error of the English text-books written by Cambridge men is, I think, the great number of examples given at the close of each chapter. At least one-half of these should be omitted. It is a great mistake to keep the student lingering over the never-ending questions of conic sections, of maxima and minima, &c., and to give him the habit of solving petty problems, when he should be led forward as soon as possible to the study of the memoirs of those who have created the science. In this connection it seems to me a mistake in treating the differential calculus to confine ourselves rigorously to the notion of a limit. Although the doctrine of limits may be the only logical foundation of this calculus, the student as he advances must soon become familiar with differentials, and it is well that he should make their acquaintance in his text-book.

4. A defect, perhaps of teaching rather than of text-book, is the ignorance of all American students of numerical and logarithm calculations, and from my slight observation I infer that such is the case also with English students. It is not uncommon to hear such calculations spoken of with contempt, but there is nothing that gives one a clearer idea of the meaning of analytical formulæ than to make a numerical application of them. In this matter it seems to me that the assistance of a teacher is of much more importance than in dealing with theoretical difficulties, since with these a student must generally be left to himself, while a little advice from a skilful computor will save the beginner much time and trouble.

5. Finally I mention, as a source of some confusion and perplexity to the student, the changes of notation and the introduction of new names. Some such change and inventions will be necessary with the progress of science, but any which tend to mar the symmetry of analytical expressions, and render less easy the reading of the great mass of mathematical literature that we already possess, should be avoided. To call a well-known function a "wonnumetonomy, " or a "subcontra-wonnumetonomy," does not of course endow it with any new properties, or make its discussion one whit easier, although we may gain a slight advantage in the way of brevity of reference. For my own part I hope that this introduction of words of thundering sound, and the calculation of almost interminable formulæ, for which no more ingenuity is required than for a numerical calculation, is only premonitory to the invention of a calculus of operations which shall furnish us with shorter and more powerful methods of investigation.


Washington, August 16

What Hall will discover in 1877. (Video by 20Bond09)