Monday, August 30, 2010

The Slide Rule: Mathematical Relic?

Supposedly superseded by the electronic pocket calculator back in 1974, are the days of the slide rule as a viable mathematical computational aid finally over?

By: Ringo Bones

While the electronic calculator and every similarly battery-operated model that superseded it since it was introduced back in 1974 had been decried by “old-school” math educators as the primary cause of the current “dumbing-down” of prospective math students, I sometimes wonder if those under-40-old-school-math-educators had ever experienced the joy of handling a slide rule. But is the slide rule still relevant in today’s world? Especially when prospective math students can easily install – if they chose to - scientific calculator apps in their PDAs and other hand-held computers?

A slide rule is a mechanical analog computer, a mechanical device consisting of moveable scales, arranged to slide opposite each other, by means of which certain mathematical operations may be carried out quickly. Also known colloquially as a “slip-stick”, it is used primarily for multiplication and division and also for “scientific” functions such as roots, logarithms and trigonometry. More often than not, slide rules do not generally perform addition or subtraction.

The working principle behind the slide rule was made possible when John Napier invented the concept of logarithm, his treatise of the subject being published in 1614. In 1632, William Oughtred, an English clergyman and teacher, arranged two logarithmic scales – which he invented back in 1622 – to form the first slide rule. The slide rule in its modern form – before being superseded by the modern pocket calculator – was developed by Amédée Mannheim, a French artillery officer, in 1850.

Amédée Mannheim’s version of the slide rule consists of a 10-inch rule, with three parts: the stock, or two fixed parallel rules, each with a scale on its inner edge; the slide, a single rule, moving between them, having a scale on each outer edge corresponding to the fixed scale to which it is adjacent; and the cursor, a square glass with a hair line which may be moved the length of he rule as an aid to reading it.

Before the advent of the pocket calculator, the slide rule was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow through the 1950s and the 1960s even as digital computing devices were being gradually introduced. Famed aerospace engineer Clarence “Kelly” Johnson was purported to have used the slide rule in designing the U2 and SR-71 Blackbird spy planes. High fidelity audio equipment engineered using slide rules during the Golden Age of Stereo – i.e. 1950s to the 1960s – were famed for their warm and smooth natural sound. By around 1974, the pocket-sized electronic scientific calculator introduced by Texas Instruments and other competing consumer electronics manufacturers, largely made the slide rule obsolete and most suppliers exited the business around 1976.

The only saving grace of the slide rule – if you’re fortunate enough to find one in a garage sale or antiques shop being offered at a keen price - over competing electronic calculating devices is that it doesn’t require batteries. And also, more “mature” old-school math teachers praise the slide rule for keeping math students on their toes in its use thus making their users smarter. To me, the slide rule or slip-stick’s primary advantage over the electronic pocket calculator is that the slide rule is 100% water proof and dust proof – as in desert conditions dust proof.

Given that the “modern” slide rule was developed by a French artillery officer named Amédée Mannheim, nobody has yet pitted the slide rule side-by-side against those dedicated palm-held ballistic computer. Like Knights Armament Company’s KAC Bullet Flight 2.0.0 described as a palm-held ballistic computer designed to provide quick firing solutions in the field. I wonder how this palm-held push-button firing solution fares well against an early 1960s era slide rule when providing firing solutions for a 123-grain Lapua Scenar round with a ballistic coefficient of 0.547 and a muzzle velocity of 2,600 feet-per-second.

Saturday, August 21, 2010

Seki Kówa: Japan’s Forgotten Mathematical Genius?

Credited for independently discovering his version of integral calculus and determinants in 17th Century isolationist era Japan; Was Seki Kówa Japan’s forgotten mathematical genius?

By: Ringo Bones

Fortunately for us in the West, Japanese intellectuals was still “literate” enough to jot down their musings for posterity even in the harshly despotic and isolationist regimes of the Hideoshi and Tokugawa shogunates of 16th and 17th Century era Japan. It probably wasn’t until Commodore Matthew Perry imposed his “Gunboat Diplomacy” back in 1853 and 1854 that finally allowed the curious West to scrutinize what Japanese intellectuals managed to discover during their country’s isolationist period. And with the Meiji Restoration period of 1868 to 1912, the Western world finally uncovered that the Japanese development in mathematics was as advanced that of in Europe.

Enter Seki Kówa, traditionally credited in Japan for discovering his version of the integral calculus during the 17th Century. His yenri or circle principle, which was documented back in 1670, uses a series of triangles to measure the area of a circle. Like that used by 17th Century contemporary calculus discoverers in Europe, Isaac Newton and Gottfried Wilhelm von Leibniz.

Seki Kówa also independently discovered the mathematical principle of determinants back in 1683. While in Europe, calculus pioneer Gottfried Wilhelm von Leibniz also independently discovered the principle of determinants 10 years later – in 1693. While integral calculus in our day and age is indispensable in determining the volume of all manner of irregular shapes, such as airplane fuselages and oil storage tanks and the areas of curved surfaces to find the exact amount of sheet metal to use in a car body or the lifting surfaces of a modern jet. We – in the present - are still thankful for a now largely forgotten 17th Century Japanese mathematical genius named Seki Kówa for being curious enough for contributing his own ideas on integral calculus and other then still obscure mathematical concepts in the 1600s.

Alfred North Whitehead: Mathematical Unifier?

Inspired by the underlying commonality of the existing mathematics of his day, would Alfred North Whitehead be as successful as Rene Descartes in establishing a new branch of mathematics?

By: Ringo Bones

Ever since Rene Descartes made possible the happy marriage of curves and quantities – i.e. by merging all the arithmetic, algebra and geometry of ages past into a single technique – to produce analytic geometry, many a wannabe great mathematician had tried to create their own branch of mathematics by combining existing ones. During the latter half of the Victorian Era, none got closer than Alfred North Whitehead. But today’s kids would certainly ask who the heck is he?

Alfred North Whitehead (1861 – 1947) English mathematician and philosopher, was born at Ramsgate, the Isle of Thanet, Kent, on February 15, 1861, of a family of teachers and ministers. His father was an Honorary Canon of Canterbury. Upon entering Trinity College, Cambridge, Alfred North Whitehead devoted himself to mathematics. But by being a member of the Apostles Club – philosophy, literature, history, politics and religion were all subjects of intense discussion.

In 1885, Whitehead became a Fellow of Trinity and began teaching mathematics. His Treatise on Universal Algebra led to his election to the Royal Society in 1903. A decade of collaboration with the most brilliant of his former pupils – Bertrand Russell – resulted in the publication of the monumental Principia Mathematica consisting of three volumes between 1910 and 1913.

Alfred North Whitehead’s mathematical work that gained him fame was the 1898 publication of his pioneering work called A Treatise on Universal Algebra. Now called Abstract Algebra, it was his unfinished attempt to unify “the various systems of Symbolic Reasoning allied to ordinary algebra.” The first and only volume that was published is a detailed investigation of H. G. Grassmann’s Calculus of Extension – Ausdehnungslehre in German – which was first published in 1844 but insufficiently appreciated and George Boole’s Algebra of Logic. These mathematical works had attracted Whitehead’s attention by their bold extension of algebraic methods beyond the traditional realm of the quantitative.

Whitehead restated Grassmann’s calculus and employed it to unify a variety of geometries; thus the theorems of projective geometry were exhibited as consequences of the definitions of the calculus. Some, but comparatively few, additions to the superstructure of mathematics were included – resulting in a main achievement that was more a novel unification. But this was along relatively unorthodox lines, thus the work had little influence among mathematicians.

Returning to the great Principia Mathematica, the first portion of which is a deductive elaboration of formal logic from a few axioms; the remainder is a detailed deduction, from this alone, of the basic concepts and principles, first called cardinal arithmetic and then of the other recognized mathematical sciences; and many new sciences suggested by the broad definitions being laid down.

The whole is written in exact and elaborate symbolism, taken partly from Giuseppe Peano. The work is basic for students of the foundations of mathematics, in spite of the fact that the first portion, for which Russell was mainly responsible, was involved in difficulties which have challenged experts ever since, and that the fourth volume, which was to deal with geometry alone, and was to be written by Whitehead alone, never appeared. Thus his attempts to unify various geometries were seen nothing more by mathematicians – then and now – as nothing more that a novel unification.

Thursday, July 22, 2010

Is Prison A Good Place to Hone One's Math Skills?

From Jean Victor Poncelet to the imprisoned Tiananmen Square dissidents of 1989, is prison really a good place hone one’s mathematics skills?

By: Ringo Bones

Most prisoner of war inmates opt intricate scrimshaw carving or feeling sorry for themselves, but there are an exceptional few who used their time spent in captivity to develop and improve their existing math skills. Some have even managed to contribute indispensable facts to the still growing collective mathematical knowledge.

Captured during Napoleon’s disastrous Russian campaign of 1812, French mathematician Jean Victor Poncelet conquered the boredom of his prison camp by aligning a lot of disorganized non-Euclidian insights into a new branch of mathematics now known as projective geometry. Its aim is to study the properties of geometric shapes that stay unchanged when seen from a distance. An example of which is when Poncelet created a set-up during his captivity – a pyramid that contains a seemingly chaotic arrangement of colored cards. When the eye looks at this particular pyramid’s base, it sees an orderly pattern due to the angle at which “chaos” - or the chaotically arranged colored cards, is projected through space to the viewer’s eye. While some of his comrades are probably busied themselves carving intricate pieces of scrimshaw, Jean Victor Poncelet managed to create a new branch of mathematics now called projective geometry.

Professor Jackow Trachtenberg, a brilliant engineer who managed to invent a very handy set of mathematical shortcuts now known as the Trachtenberg Speed System of Mathematics during the years that he spent in captivity at Hitler’s concentration camps as a political prisoner. There’s even a Trachtenberg Mathematical Institute in Zurich, Switzerland established in honor of Professor Jackow Trachtenberg.

There also had been anecdotes during 1990 that some Chinese students who became political prisoners after their participation of the pro-democracy protests in Tiananmen Square back in 1989 have been making good use of their time spent in captivity as a political prisoner. Some have even tried to continue the unfinished work of Albert Einstein that he started since the1950s of formulating an equation that could unite the two very disparate systems of Quantum Mechanics and General Relativity. But is prison really an ideal place to hone one’s mathematics skills? Only time will tell.