During my years as a graduate student at NYU’s Courant Institute of Mathematical Sciences (CIMS), I spent many a day in the CIMS library on the twelfth floor. From time to time I would take a break by looking for a book that was about mathematics, not an exposition of it.
Among my favorites was the work of Augustus De Morgan. De Morgan was a famed mathematician. He was also a wonderful writer.
He spent some of his spare moments debunking circle-squares and pi-finders, as shown by the following quote from the Wikipedia entry about him:
How can the sound paradoxer be distinguished from the false paradoxer? De Morgan supplies the following test:
“The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself… New knowledge, when to any purpose, must come by contemplation of old knowledge, in every matter which concerns thought; mechanical contrivance sometimes, not very often, escapes this rule. All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors and learned in what had been before them. There is not one exception.”
“I remember that just before the American Association met at Indianapolis in 1890, the local newspapers heralded a great discovery which was to be laid before the assembled savants — a young man living somewhere in the country had squared the circle. While the meeting was in progress I observed a young man going about with a roll of paper in his hand. He spoke to me and complained that the paper containing his discovery had not been received. I asked him whether his object in presenting the paper was not to get it read, printed and published so that everyone might inform himself of the result; to all of which he assented readily. But, said I, many men have worked at this question, and their results have been tested fully, and they are printed for the benefit of anyone who can read; have you informed yourself of their results? To this there was no assent, but the sickly smile of the false paradoxer”
He also wrote many anecdotes about mathematicians and scientists, Among my favorite is the story of how Charles Babbage, best known as the creator of the Analytical Engine, came to invent what is now known as the Cowcatcher:
a tale of two extraordinarily gifted and ill-fated British eccentrics whose
biographies might have been fabrications of Babbage’s friend Charles Dickens, if Dickens had been a
Like many contemporary software characters, these computer pioneers of the Victorian age attracted as
much attention with their unorthodox personal lives as they did with their inventions.
One of Babbage’s biographies is entitled Irascible Genius.. He was indeed a genius, to judge by what
he planned to achieve as well as what he did achieve. His irascibility was notorious. Babbage was
thoroughly British, stubbornly eccentric, tenaciously visionary, sometimes scatterbrained, and quite
wealthy until he sank his fortune into his dream of building a calculating engine.
Babbage invented the cowcatcher–that metal device on the front of steam locomotives that sweeps errant
cattle out of the way. He also devised a means of analyzing entire industries, a method for studying
complex systems that became the foundation of the field of operational research a hundred years
later. When he applied his new method of analysis to a study of the printing trade, his publishers were so
offended that they refused to accept any more of his books.
Undaunted, he applied his new method to the analysis of the postal system of his day, and proved that the
cost of accepting and assigning a value to every piece of mail according to the distance it had to travel was
far more expensive than the cost of transporting it. The British Post Office boosted its capabilities
instantly and economically by charging a flat rate, independent of the distance each piece had to travel–the
“penny post” that persists around the world to this day.
Babbage devised the first speedometer for railroads, and he published the first comprehensive treatise on
actuarial theory (thus helping to create the insurance industry). He invented and solved ciphers and made
skeleton keys for “unpickable locks”–an interest in cryptanalysis that he shared with later computer
builders. He was the first to propose that the weather of past years could be discovered by observing
cycles of tree rings. And he was passionate about more than a few crackpot ideas that history has since
proved to be nothing more than crackpot ideas.
His human relationships were as erratic as his intellectual adventures, to judge from the number of lifelong
public feuds Babbage was known to have engaged in. Along with his running battles with the Royal
Societies, Babbage carried on a long polemic against organ-grinders and street musicians. Babbage would
write letters to editors about street noise, and half the organ-grinders in London took to serenading under
Babbage’s window when they were in their cups. One biographer, B. V. Bowden, noted that “It was the
tragedy of the man that, although his imagination and vision were unbounded, his judgment by no means
matched them, and his impatience made him intolerant of those who failed to sympathize with his projects.”
Babbage dabbled in half a dozen sciences and traveled with a portable
He was also a supreme nit-picker, sharp-eyed and cranky, known to write outraged letters to publishers of
mathematical tables, upbraiding them for obscure inaccuracies he had uncovered in the pursuit of his own
calculations. A mistake in navigational table, after all, was a matter of life and death for a seafarer. And a
mistake in a table of logarithms could seriously impede the work of a great mind such as his own.
His nit-picking indirectly led Babbage to invent the ancestor of today’s computers. As a mathematician and
astronomer of no small repute, he resented the time he had to spend poring over logarithm tables, culling all
the errors he knew were being perpetuated upon him by “elderly Cornish Clergymen, who lived on seven
figure logarithms, did all their work by hand, and were only too apt to make mistakes.”
Babbage left a cranky memoir entitled Passages from the Life of a Philosopher–a work
described by computer pioneer Herman Goldstine as “a set of papers ranging from the sublime to the
ridiculous, from profundities to nonsense in plain bad taste. Indeed much of Babbage’s career is of this sort.
It is a wonder that he had as many good and loyal friends when his behavior was so peculiar.”
In Passages, Babbage noted this about the original inspiration for his computing machines:
The earliest idea that I can trace in my own mind of calculating arithmetical tables by machinery rose in this
One evening I was sitting in the rooms of the Analytical society at Cambridge, my head leaning forward on
the table in a kind of dreamy mood, with a Table of logarithms lying open before me. Another member,
coming into the room, and seeing me half asleep, called out, “Well, Babbage, what are you dreaming about?”
To which I replied, “I am thinking that all these Tables (pointing to the logarithms) might be calculated by
In 1822, Babbage triumphantly demonstrated at the Royal Astronomical Society a small working model of a
machine, consisting of cogs and wheels and shafts. The device was capable of performing polynomial
equations by calculating successive differences between sets of numbers. He was awarded the society’s
first gold medal for the paper that accompanied the presentation.
In that paper, Babbage described his plans for a much more ambitious “Difference Engine.” In 1823, the
British government awarded him the first of many grants that were to continue sporadically and
controversially for years to come. Babbage hired a master machinist, set up shop on his estate, and began
to learn at first hand how far ahead of his epoch’s technological capabilities his dreams were running.
The Difference Engine commissioned by the British government was quite a bit larger and more complex
than the model demonstrated before the Royal Astronomical Society. But the toolmaking art of the time
was not yet up to the level of precision demanded by Babbage’s design. Work continued for years,
unsuccessfully. The triumphal demonstration at the beginning of his enterprise looked as if it had been the
high point of Babbage’s career, followed by stubborn and prolonged decline. The British government finally
gave up financing the scheme.
Babbage, never one to shy away from conflict with unbelievers over one of his cherished ideas, feuded over
the Difference Engine with the government and with his contemporaries, many of whom began to make sport
of mad old Charley Babbage. While he was struggling to prove them all wrong, he conceived an idea for an
even more ambitious invention. Babbage, already ridiculously deep in one visionary development project,
began to dream up another one. In 1833 he came up with something far more complex than the device he
had failed to build in years of expensive effort.
If one could construct a machine for performing one kind of calculation, Babbage reasoned, would it be
possible to construct a machine capable of performing any kind of calculation? Instead of building
many small machines to perform different kinds of calculation, would it be possible to make the parts of
one large machine perform different tasks at different times, by changing the order in which the
Babbage had stumbled upon the idea of a universal calculating machine,
an idea that was to have momentous consequences when Alan Turing–another brilliant, eccentric British
mathematician who was tragically ahead of his time–considered it again in the 1930s. Babbage called his
hypothetical master calculator the “Analytical Engine.” The same internal parts were to be made to
perform different calculations, through the use of different “patterns of action” to reconfigure the order in
which the parts were to move for each calculation. A detailed plan was made, and redrawn, and redrawn
The central unit was the “mill,” a calculating engine capable of adding numbers to an accuracy of 50 decimal
places, with speed and reliability guaranteed to lay the Cornish clergymen calculators to rest. Up to one
thousand different 50-digit numbers could be stored for later reference in the memory unit Babbage called
the “store.” To display the result, Babbage designed the first automated typesetter.
Numbers could be put into the store from the mill or from the punched-card input system Babbage adapted
from French weaving machines. In addition, cards could be used to enter numbers into the mill and specify
the calculations to be performed on the numbers as well. By using the cards properly, the mill could be
instructed to temporarily place the results in the store, then return the stored numbers to the mill for later
procedures. The final component of the Analytical Engine was a card-reading device that was, in effect, a
control and decision-making unit.
A working model was eventually built by Babbage’s son. Babbage himself never lived to see the Analytical
Engine. Toward the end of his life, a visitor found that Babbage had filled nearly all the rooms of his large
house with abandoned models of his engine. As soon as it looked as if one means of constructing his device
might actually work–Babbage thought of a new and better way of doing it.
The four subassemblies of the Analytical Engine functioned very much like analogous units in modern
computing machinery. The mill was the analog of the central processing unit of a digital computer and the
store was the memory device. Twentieth-century programmers would recognize the printer as a standard
output device. It was the input device and the control unit, however, that made it possible to move beyond
calculation toward true computation.
This invention shows that Babbage, like my late friend, colleague and thesis advisor, Jacob T “Jack” Schwartz, had a very wide-ranging mind. He did not limit his skills, or his vision, to a single field, but worked in many fields.
This is also a reminder that it pays to attempt to expand one’s own reach. You should not be afraid to try new things, and to take risks in doing so, for that is the best way to grow, both professionally and personally.