Friday, February 17, 2012

Brook Taylor: Much More than a Series

Classes and research have been keeping me busy, so this post will also be on someone I have been tackling in my homework this week.  One of the main things that I have learned in graduate school thus far is that none of the equations we use are "correct."  They are all approximations of one sort or another, whether because we can't solve the real equation or because we can't take into account all of the interactions.  One of the most common tools for these approximations, when we have an equation but don't want to deal with it, is to use the Taylor expansion.  I hadn't given it or him much thought until this week, but they just keep popping up, so Taylor is this week's subject.

Modern chemistry seems to have developed in the 19th century.  That's when scientists finally agreed that atoms exist, and developed the modern concepts of energy and heat.  Mathematics, however, seems to have had a heyday in the 18th century based on the number of mathematical operators, functions, rules, etc. that have been named after the mathematicians of that century. These include Laplace, Lagrange, L'Hopital, Maclaurin, Euler, Gauss, Fourier, Legendre, and, of course (or else this interlude would be rather pointless), Brook Taylor.

Brook Taylor
In reading about Brook Taylor, I realized that, more than anyone I have discussed so far, I feel that I cannot do him justice.  This stems from two main causes: my lack of understanding of the finer points of mathematics and its history, and the number of interesting things that I discovered about Taylor.

Brook Taylor was an Englishman, born in 1685.  He went to St. John's College at Cambridge and studied mathematics, which was apparently quite popular in those days.  He began writing and publishing on mathematical subjects, but didn't publish soon enough after his discoveries to avoid trouble.  In 1708 he developed a solution to the problem of the center of oscillation.  I still haven't quite figured out what this is, but apparently it was a big deal.  He didn't publish his discovery, however, until 1713: De Inventione Centri Oscillationis.1

Meanwhile, Johann Bernoulli had independently come to the same discovery, and argued about precidence with Taylor.  In 1715 he published Methodus Incrementorum Directa et Inversa, which first introduced to the public what became known as Taylor's Theorem.  The work was also the first discussion of what came to be known as the calculus of finite differences, for more information on which you will have to ask a mathematician.  Taylor was not the first person to use the series, but he made the most general form of it.  Specific instances had already been used by Edmond Halley, Isaac Newton, Johann Bernoulli, and Johann Kepler.  The importance of the series was overlooked for many years, until it was pointed out by Joseph Lagrange in 1772.  Other problems that he solved in this book involved oscillations of a string and a change of variables formula.  He also write papers and letters on the subjects of magnetism, the movement of fluids, and logarithms.  His writing, however, suffered from a brevity that lead to confusion about what he actually meant, which led to his being under appreciated for all of the contributions that he made to mathematics.

In 1715 Taylor also published a work on linear perspective, followed in 1719 by New Principles of Linear Perspective, in both of which he used mathematics to explain linear perspective more generally than those before him had.  Bernoulli, with whom Taylor had already had heated arguments, declared that the book was "abstruse to all," especially artists.  Bernoulli's objections were so strong that Taylor wrote a reply in the Philosophical TransactionsApologia D. Brook Taylor, J V D. & R S. Soc. contra V. C J. Bernoullium, Math. Prof. Basileae. I think Bernoulli had a point, though, since Taylor's works on perspective contained no sketches, just written descriptions, and even when he wasn't writing about art, he had a tendency to be concise to the point of confusion.

Taylor had been elected a member of the Royal Society in 1712, and had sat on the committee which adjudicated between Newton and Leibniz on the issue of which had invented calculus (they sided with Newton).  After about 1715, Taylor began writing more philosophical papers, such as "On the Lawfulness of Eating Blood."  His final paper in the Philosophical Transactions was "An Account of an Experiment, Made to Ascertain the Proportion of the Expansion of the Liquor in the Thermometer, with Regard to the Degrees of Heat," published around 1721.  He seems to have focused more on domestic matters and his health after that time, for in 1721 he also married.  His father disapproved of his wife, which suggests that Taylor, for one, married for love.  When she died in childbirth two years later, however, he and his father became reconciled.  In 1729 (1725?) he married again, but she also died in childbirth.  Taylor died just one year later.

1. Most articles I found said that it wasn't until 1714 that he published it, but I think this is the article in question, and according to Jstor it was published in 1713. So that is what I'm going with.

References and further information

Brook Taylor, 1911 Encyclopedia Britanica
Brook Taylor, from someone at the University of St. Andrews
Brook Taylor, by Edward Irving Carlyle, Dictionary of National Biography, 1885-1900, vol. 55.
Dr. Brook Taylor's Principles of Linear Perspective, edited by Joseph Jopling, 1835.

Thursday, February 2, 2012

Diesel and His Engine

Rudolf Diesel
I've been hoping to find a scientist or engineer with an interesting story, and I think I found one. I was looking for information on how to synthesize monoglycerides, and discovered that the process is similar to making bio-diesel, which then begs the question (at least to me), what is diesel and why is it called that?

Rudolf Diesel invented the diesel engine, and thus in a remarkable fit of (probably) proper attribution, has his name attached to it. He is an interesting character, because he wanted to improve the efficiency of engines and change the world, a vision that I think few engineers really believe in today.

Diesel had a disjunct childhood.  He was born in Paris to Bavarian parents in 1858, but was sent to school in England in 1870 as a result of the Franco-Prussian War.  Less than a year later, he was sent to the Technical School in Augsburg, Germany.  He graduated from the Techincal University in that city in 1880, and began working with Carl von Linde (1842-1934) in Munich. Von Linde had recently developed a method for refrigeration using ammonia and was therefore very interested in the studies of heat.  In 1895, he even succeeded in liquefying air.1

Drawing from Diesel's apparatus for
converting heat into work,
US Pat. #542846
Working with von Linde, Diesel was able to work on a problem that he had begun considering when an undergraduate.  Steam engines were more efficient when large, so Diesel set out to develop an engine that would still be efficient when small. He was particularly interested in the ideal engine envisioned by Sadi Carnot (1796-1832) and descrived in 1824, called the Carnot cycle. At first, Diesel designed an engine similar to a steam engine that ran on ammonia, but, though the engine did work on a smaller scale than steam, he ran into problems like leakage.  He then considered a case in which the combustion of the fuel took place in a cylinder of the engine, rather than in a boiler.  Nikolaus Otto, a German engineer, had created the first marketed internal combustion engine in 1862, so this idea was not new.  What made Diesel's engine different was that it did not need a spark to ignite the fuel, but used higher compression ratios than the existing internal-combustion engines, leading to self-ignition. It was this isothermal combustion that set the diesel engine apart.

Diesel worked on models of the engine at the Augsburg-Nuremburg Engine Works with its financial backing and that of Krupp (a company that still exists today as ThyssenKrupp).  One of the greatest challenges was creating chambers that could withstand the large pressures that Diesel required for combustion. After four years of testing and various accidents, Diesel and his manufacturing aides created a working prototype engine in 1897. The engines got off to a rocky start.  Diesel tried to market his invention immediately, but there were still some kinks to work out.  Several accidents making dents in Diesel's profits from the patents he had taken out (see the list in references for more information).

Diesel had a larger vision for his engine than just making it more efficient. He thought that his engine could transform society. Since his engines worked on a smaller scale than the steam engines, they could be used by small craftsmen and help to counteract that increase in the scale of manufacturing resulting from the industrial revolution.  Diesel was part of a movement that believed that technology could save the world.  Rather than having the workers rise up as Marxism called for, he believed that technology could better the lot of workers and narrow the class divide, so such a revolution would not be necessary.  He did, however, believe in a form of communism in which workers would pool their resources for the greater common good.  He presnted his ideas in a 1903 book entitled Solidarismus: Natürliche wirtschaftliche Erlösung des Menschen (Solidarity: The Rational Economic Salvation of Mankind).  

In 1912, questions about whether Diesel actually invented the diesel engine came to a head.  Some people argued that credit needed to go to the factory assistants, rather than Diesel.  When a history of the diesel engine was to be published, Diesel preempted whatever it might say about him by presenting a paper explaining his development of the engine at the German Society of Naval Architects.  This might seem a strange place to give such a paper, but the main use of diesel engines at that point was in ships.

The following year, Diesel was crossing the English channel and went overboard during the night.  This incident led to much speculation about how he died, though the most likely explanation is that he committed suicide.  The most interesting story that I came across was that he was killed by the German secret service to prevent him from betraying secrets about submarines to the British.

1. Carl von Linde (back)

Holmgren, E. J., "Rudolf Diesel, 1858-1913" Nature 181, no. 4611 (1958), 737-738.
Bryant, Lynwood, "The Development of the Diesel Engine" Technology and Culture 17, no. 3 (Jul., 1976), 432-446.
Thomas, Donald Jr., "Diesel, Father and Son: Social Philosophies of Technology" Technology and Culture 17, no. 3 (Jul., 1978), 376-393.

List of Diesel's patents (back)
US Pat. #542846 Method of and Apparatus for Converting Heat into Work, filed August 26, 1892
US Pat. #608845 Internal-Combustion Engine, filed July 15, 1895
US Pat. #673160 Method of igniting and regulating combustion for internal-combustion engines, filed April 6, 1898
US Pat. #654140 Apparatus for Regulating Fuel-Supply of Internal-Combustion Engines, filed September 10, 1898

US Pat. #736944 Internal-Combustion Engine, filed November 1, 1899
US Pat. #RE11900 Internal-Combustion Engine, filed July 3, 1900
US Pat. #708029 Internal-Combustion Engine, filed January 18, 1901
US Pat. #873926 Longitudinally-Displaceable Car-Body for Motor-Vehicles, filed January 25, 1908