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      <text>4.4.2 (History of exponentials and Logarithms)</text>
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      <text><![CDATA[<h3>History of exponentials and logarithms</h3>
<p>Recall that \(e\) is the number that governs all exponential growth. Despite the fact this notation only appeared with Euler in the 18th Century, mathematicians before this time still needed methods to solve exponential functions of the form \(a^x\). We have seen in our examination of logarithms that often the log of a number is not easy to deal with, it gives us a long string of decimals. We can deal with these easily now because we have computers and calculators, however in the past calculating such numbers was extraordinarily difficult and time consuming. In fact, until the advent of pocket computers, a calculator was actually a job title for many working in institutions such as a bank.</p>
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<p>Logarithms are intertwined with the history of Edinburgh. They were invented by John Napier (1550 - 1617) who was a mathematician who was from Merchiston, now within the city. Napier was a prominent polymath and was a prolific writer on many subjects ranging from mathematics to theology. One of his major advances was an early calculator for performing multiplications called <em>Napier's Bones</em> which can been seen below. He was also responsible for popularising the decimal point notation in mathematics. Napier began the publishing of seemingly endless tables of the logarithms of various numbers. These type of log-tables were used to calculate logs and power well into the 20th century. It is perhaps interesting that despite his invention of logs Napier never discovered the number \(e\). A frequent problem Napier had was how to scale his logarithms, but he never satisfactorily solved all of these issues because he didn't have the number \(e\) which would have to wait until centuries later. Logarithms has a profound effect on the scientific community. Kepler used them to describe the laws of planetary motion and they gave rise to important evidence for Issac Newton's theory of universal gravitation. In fact, Kepler published a letter congratulating and acknowledging the impact of Napier's discovery. Nowadays, Napier is honoured with a statue at the Scottish national portrait gallery and you can examine some of the calculators he invented and inspired at the National Museum of Scotland.</p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/napiers%20bones.jpg" alt="photo of Napier's bones calculator" width="349" height="232" /></p>
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<p>The history of exponentials and logarithms is intimately linked with some of the most influential and famous mathematicians throughout history. The natural exponential \(e\) was first described in the 17th century by Swiss mathematician Jacob Bernoulli. He realised that this number was appearing in problems where he was examining compound interest. We will examine this more closely ourselves in the next section. The notation \(e\), for the natural exponential was first proposed by Leonhard Euler (who we have met previously as he was prolific in a number of mathematical fields). Euler was writing a letter to Christian Goldbach in 1731 where he first used the notation \(e\) and the number was then published with this notation in \(1736\). Euler also managed to find the value of \(e\) to eighteen decimal places (remember that this was before calculators) using <em>continued fraction</em> expansions. Often \(e\) is called 'Euler's constant' in honour of him, and this has led many to think the notation \(e\) was chosen as it was the first letter of Euler's surname, however this is unsubstantiated. It may have been chosen just because Euler had run out of other notation.</p>]]></text>
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