Man searching for instruments to help him to calculate
Man has always searched for tools to help him count. As economical life developed, these tools became more and more necessary.
The early humans knew only three words to describe a counting result : 1,2 and 'a lot'. The latter meaning 'three' as well as any other number…
To indicate more precisely a quantity, they began to use twigs and pebbles. For each object or animal they exchanged, they threw a certain number of these pebbles or twigs together, and afterwards they counted them.
So, if somebody wanted to exchange five cows for chickens, and let's suppose that each cow is worth three chickens, they threw for each cow three pebbles together, then they took for each pebble a chicken.
Later they used tallysticks : long sticks in which they notched little incisions.( Each incision represented one unit )
THE PLUMMET BY THE EGYPTIANS
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Because the Egyptians made always big constructions, they needed a way to construct an angle of 90°.For this purpose, they made use of a plummet, because it always makes a perfect angle of 90° with the ground. ( They were also the ones that found out that the surface of an isosceles right-angled triangle can be calculated by the formule 'L´B/2', discovering that they had just to split the surface of a square. Later, they realised that this formula could be used for the surface of any triangle.)
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CALCULATION WITH THE FINGERS
Although men counted
already ages before the Romans with their fingers,
still most of the Romans and even men in the middle ages
counted this way.They even invented a way to multiply
with them! The fingers above the two who are lying against each other, represented a number from 1 to 2. (But now each hand apart !). They multiplied each hand apart (1´ 2) and then the two hands (2´ 2). At the end, they added the two numbers they became from the fingers above and beyond the touching fingers : 60+4 = 64 = 8´8 ! |
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THE QUIPU
In South America, the Incas(12th to 16th century) developed a way to calculate with ropes which were all tangled up. They called them 'quipus'.
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A 'quipu' consisted of a lot of ropes, in different colours, which were all tied up (partly parallel, partly from one common point) on a bigger one. They expressed numbers as well as sorts. The colours represented these sorts, the ropes were the numbers. |
| The knots on the end of a rope were the units, the knots above them the tens…And they had also already invented something for '0' : they left more space then normal between two knots ! To read such a quipu, they started with the highest numbers. There were also knots made of different ropes. Scientists today still aren't sure of there use, but they were prabably representing a multiplication. These quipus could only be read by certain people. |  |
Mainly because the colours represented something different in each quipu. In a quipu of the harvest, yellow could stand for wheat; but in a quipu of The Treasury, it stands for gold… The Incas made an inventory of everything they had, and for each sort there were special Incas who knew how to make a quipu for it. They had to teach this to their sons, so that their would always be someone who knew how to make that kind of quipu. We can say that the Inca-kingdom was ruled by the quipus !
THE ABACUS
| The abacus is an acient
calculating device made up of a frame of parallel wires
on which beads are strung. The method of calculating with a handful stones on a 'flat surface' (latin Abacus)was familiar to the Greeks and Romans, and used by earlier civilisations, possibly even in ancient Babylon. |
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But the most famous abacus is the 'SuanPan', which the Chinese used from 1200 A.D. It has two decks, an upper deck and a lower deck. The Chinese abacus is made of bamboo- sticks and ivory beads. Each rod on the upper deck has two beads, each rod on the lower deck has five beads. Such an abacus is called a 2/5 abacus. Each bead in the upper deck has a value of FIVE; each bead in the lower deck has a value of ONE. |
The beads are considered counted, when moved towards the beam that separates the two decks. The extreme-right column represents the column of the units; the next to the left represents the column of the tens. After 5 beads are counted in the lower deck, the result is 'carried' to the upper deck; after both beads in the upper deck are counted, the result (10) is then carried to the left adjacent column. You can add, subtract, divide and multiply with an abacus. The 2/5 model survived until about 1850 A.D., at which time it evolved into the 1/5 abacus until around 1930 A.D.
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The 1/4 abacus is the model actually preferred and manufactured in Japan; At the moment, the 1/5 models are very rare and the 2/5 models are difficult to find outside of China. The abacus is still used by shopkeepers in Asia and 'Chinatowns' in North America. Its use is still taught in certain schools in the Far East. In 1946 a contest between a Japanese Abacist(Kiyoshu Matzukai)and an Electronic computer was held for 2 days resulting in an unmistakable victory of the Abacist. |
THE SLIDE RULE
The slide rule was invented in 1622 by the English mathematician William Oughted. The French army officier Amédée Mannheim (1831-1906)devised a later version. Until the calculator was invented, engineers and scientists always used the slide rule. It is a mechanical instrument that is used to compute mathematical functions such as multiplication and division, involution and extraction of roots and some later models even computing exponential and trigonometric functions.
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Calculations are performed by moving two graduated scales over each other and reading the result with the aid of a travelling cursor. Slide rule operation is based on logarithmic scales which convert multiplication into addition. The most common slide rule consists of an horizontal sliding bar held between two fixed bars and a cursor which travels across the entire structure. Each bar contains scales representing different mathematical functions such as squaring a number or taking a logarithm. |
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You can ad 4 by 5 with 2
bars : put the zero of the second bar under the first number of your addition(4) on the first bar, then look at the second bar for your second number (5). Then the answer is the number above that number: 9 ! If you'd like to subtract 8 from 3, then you'll have to put the 3 of your second bar under the 8 of the first one, and then just look at the number above the zero from your second bar : 5 ! |
If you want to multiply and to divide, you need logarithms. A logarithm is the exponent of a number to a specified base.
EXAMPLES : 2log 8 = 3 and 3log 27 = 3
The first log tables (to base e) were published by the Scottish mathema- tician John Napier in 1614.'e' is an irrational number equal approximately to 2.7183. Base-e logarithms are called the natural or Napierian logarithms. Base-ten logarithms were introduced by the Englishman Henry Briggs (1561-1631) and Dutch mathematician Adriaen Vlacq (1600-1667). They are used frequently, and you can find them also on your calculator.
MECHANICAL CALCULATION
John Napier didn't only found out about the logarithms, he also invented a kind of a calculator, but in his most early stage.
With his bars, it was possible to move the columns of the multiplication table and to bring them next to each other in accordance with the numbers of the multiplied number. Moreover each section was diagonaly divided to distinguish the units from the tens. John Napier described the working of it in his Rhabdologia (1617). But it appeared that the Arabs had already invented such an instrument before John Napier did. Later this instrument became more manageable because of the use of a rotating cilinder. Presumable it was Kaspar Schott S.J.who made this improvement. |
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Ingrid Smet