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Napiers bones
- 1. Napier’s BonesNapier’s Bones An Adventure inAn Adventure in 1717thth Century ArithmeticCentury Arithmetic byby YASHNAYASHNA
- 2. John NapierJohn Napier John Napier, a 16th Century Scottish scholar, contributed a host of mathematical discoveries, among them the world’s first calculator. John Napier (1550 – 1617)
- 3. He is credited with creating the first computing machine, logarithms and was the first to describe the systematic use of the decimal point. Other contributions include a mnemonic for formulas used in solving spherical triangles and two formulas known as Napier's analogies. ““In computing tables, these large numbers may again be made still largerIn computing tables, these large numbers may again be made still larger by placing a period after the number and adding ciphers. ... In numbersby placing a period after the number and adding ciphers. ... In numbers distinguished thus by a period in their midst, whatever is written afterdistinguished thus by a period in their midst, whatever is written after the period is a fraction, the denominator of which is unity with as manythe period is a fraction, the denominator of which is unity with as many ciphers after it as there are figures after the period.”ciphers after it as there are figures after the period.”
- 4. High tech in the 17th century, was what we’d now call basic astronomical arithmetic calculations, all done by hand. It took Johannes Kepler (1571-1630) nearly 1000 large pages of dense arithmetic do discover the laws of planetary motion! Johannes Kepler (1571-1630) A typical page from one of Kepler’s notebooks
- 5. Napier’s BonesNapier’s Bones In 1617, the last year of his life, Napier invented a tool called “Napier's Bones” which reduces the effort it takes to multiply numbers. This was a time when few people could multiply beyond 5 (x) 5. “Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions... I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”
- 6. Napier’s bones were called that because they were often made of bone, ivory, silver, or wood. The were were universally popular and common until the late 1800s. Sometimes the Napier tables were engraved on rods in a case so that numbers could be “dialed in”.
- 7. Napier’s bones make multiplicationNapier’s bones make multiplication and division easier. Multiplicationand division easier. Multiplication and division are reduced to simpleand division are reduced to simple addition, although a pencil andaddition, although a pencil and paper are required.paper are required. This boxed set has ten rods, allowingThis boxed set has ten rods, allowing computations up to 100,000,000. Thecomputations up to 100,000,000. The left (or “index”)rod is fixed to theleft (or “index”)rod is fixed to the case. It is numbered from 1 to 9.case. It is numbered from 1 to 9. The movable rods are numbered atThe movable rods are numbered at the top. The numbers down themthe top. The numbers down them rods show the product of the numberrods show the product of the number at the top times the correspondingat the top times the corresponding numbers on the index rod.numbers on the index rod. Here the “3” rod shows three timesHere the “3” rod shows three times each of the numbers on the index rod.each of the numbers on the index rod.
- 8. The bones are easy to use. Multiplication and division are set up the same way. Set the problem up by laying down rods corresponding to the number being multiplied or divided. This setup shows the number 3579 which we will show being multiplied by 43.
- 9. This is the problem shown on our “paper bones.” The “3”, “5”, “7” and “9” strips are set up next to the index.
- 10. Using a pad of paper, we write down the individual products of 40 and 3 times each digit of 3579. The results are: Multiply by 40: 4 Times : 3 5 7 9 is 12 20 28 36 Adjust carries 1 4 3 1 6 --------------------------------------------------------- Shift to the right one decimal place and multiply by 3: 3 Times 3 5 7 9 is 9 15 21 27 Adjust carries 1 0 7 3 7 --------------------------------------------------------- Add results 1 5 3 8 9 7 (Adjust carries if necessary)
- 11. Divide 75159 by 3579: Set the rods of the divisor 3579, giving tables of 1 (x) 3579 to 9 (x) 3579 or 10 (x) 3579 to 90 (x) 3579. First digit: 10 (x) 3579 to 90 (x) 3579 Row 2: 20 (x) 71580 Row 3: 30 (x) 107370 meaning the dividend is between 20 and 30. The first digit is 2. Subtract: 75159 (-) 71580 3579 <- The remainder is now the dividend. Second Digit: The table now used as 1 (x) 3579 to 9 (x) 3579 Row 1: 3579, so the second digit is 1, and the solution is 21.
Editor's Notes
- JOHN NAPIER(1550-1617)Born at Merchiston, Edinburgh, Napier was a man of great intellect, persistence and will, known best for his development of logarithms in the field of mathematics. He had diverse interests outside his solitary and studious life, including treasure hunting, agricultural chemistry, and divination. He was also responsible for inventing the world&apos;s first computing device, nicknamed &apos;Napier&apos;s Bones&apos; on account of its design. Napier&apos;s bones were multiplication tables written on strips of wood or bones. The invention was used for multiplying, dividing, and taking square roots and cube roots. &apos;Constructio&apos;, which described the means by which he constructed the log tables, was published after his death. Napier is famous for creating mathematical logarithms, creating the decimal point, and for inventing Napier&apos;s Bones, a calculating instrument. John Napier - Inventor He proposed several military inventions including: burning mirrors that set enemy ships on fire, special artillery that destroyed everything within a radius of four miles, bulletproof clothing, a crude version of a tank, and a submarine-like device. John Napier invented a hydraulic screw with a revolving axle that lowered water levels in coal pits. The highlight of John Napier&apos;s life was the creation of logarithms and the decimal notation for fractions. His other mathematical contributions included: a mnemonic for formulas used in solving spherical triangles, two formulas known as Napier&apos;s analogies used in solving spherical triangles, and the exponential expressions for trigonometric functions.
- Johannes Kepler (1571-1630). Born on 27 December 1571 in Weil der Stadt, near Stuttgart, in a modest family. He graduated at age 20 from the University of Tuebingen, where he studied mathematics and astronomy under Michael Maestlin (1550-1631), an early supporter of the Copernican system. In 1594, while engaged in the final year of his studies in theology, he was given the chair of mathematics at Graz, where he became increasingly absorbed in astronomy. He was formally expelled from town in 1600 on account of his open adherence to the Protestant faith. He first came to Prague in 1599 to work as an assistant to Tycho Brahe, and upon Tycho&apos;s untimely death in 1601 inherited his massive stock of accurate planetary observations, as well as his job as Imperial Mathematician to Rudolf II. In 1612, following the downfall of Rudolf II he moved to Linz, in 1621 to Ulm, and in 1627 to Sagan. On the move again because of religious persecution, he fell ill, and died on 15 November 1630 in Regensburg. Through a quarter century of painstaking calculations Kepler brought the Copernican system to its modern form by replacing Copernicus circular heliocentric orbits by ellipses, with the Sun at one focus. The process through which he arrived at his justly famous Laws of Planetary Motion was often a contorted one, as Kepler&apos;s peculiar mixture of physical insight and mystical inclinations lead him to seek causes for the number and arrangement of planetary orbits, as opposed to constructing purely mathematical descriptions. His first such model involved the nesting of the five regular solids and was published in his 1596 Mysterium Cosmographicum. While never relinquishing this idea, in his 1619 Harmonices Mundi he also sought an explanation in terms of musical harmonies. Hidden deep in this work is the first statement of Kepler&apos;s so-called Third Law, establishing the proportionality of the square of planetary orbital periods to the cube of their mean distance to the Sun. Kepler&apos;s first two Laws of Planetary Motion were first adumbrated in his 1609 Astronomia Nova, but first laid out in detail together with his Third Law in book IV of his monumental work Epitoma astronomia Copernicanae, published between 1617 and 1621. The underlying physical explanation of his Laws would have to wait over half a century, until Isaac Newton provided the answer in terms of the theory of universal gravitation. In 1627 Kepler also finally published what was to be the crowning (but somewhat belated) achievement of Tycho Brahe&apos;s career: the Rudolphine Tables of planetary positions. These made full use of Tycho&apos;s store of accurate observations in conjunction with Kepler&apos;s new model for planetary orbits. On 28 May 1607 Kepler used his newly devised camera obscura to observe the solar disk and saw sunspot, which he mistook for a transit of Mercury, to the amazement of later astronomers who all agreed that of all people, Kepler really should have known better. Because of Kepler&apos;s position as Imperial Mathematician, his prompt and enthusiastic public endorsement of Galileo&apos;s telescopic discoveries did a lot to publicize the latter&apos;s fame in northern Europe. Kepler was a prolific author by any standards. Besides his astronomical books, he is (by some) credited with having written the first science fiction novel, his Somnium, published posthumously in 1634 and describing a voyage to the Moon. He wrote extensively on geometrical optics, and was the first to correctly sort out once and for all the production of real versus virtual images by mirrors and lenses. He is also said to have laid the foundations of cristallography in a little book on snowflakes written as a New Years gift to his patron Rudolf II in 1611. Bibliography: Caspar, M. 1959, Kepler, [1993 Dover reprint]. Beer, A., & Beer, P. (eds.) 1975, Kepler, vistas in astronomy vol. 18, Pergamon Press. Gingerich, O. 1989, Johannes Kepler, in The General History of Astronomy, vol. 2A, eds. R. Taton and C. Wilson, Cambridge University Press, pps. 54-78.
- Napier&apos;s Bones were invented in 1617 , when John Napier, a Scottish baron, published a book describing the device. Within a few years, it had spread throughout Europe and as far as China. Napier&apos;s Bones (so-called because they were often made of bone) were rods with multiplications tables on them. At the time, educated people often knew their multiplication tables only as far as 5 x 5.