Rough and ready measurements

To many students, the image of science is one of exactness and perfection. And yet, good scientists make rough estimates again and again, sometimes without ever making a precise measurement. It is important to teach students that rough measurements are respectable.

Of course, high precision is of the essence in many cases. A modern mass spectrograph must yield measurements of high precision if tiny mass-differences between one atomic nucleus and another are to be interpreted as energy-differences using E = mc².

Yet when Chadwick measured the nuclear charges of copper, silver and platinum, by alpha scattering in 1920, relatively rough measurements showed Rutherford's atomic model was correct. Chadwick showed that the nuclear charge (in electron units) is just equal to the atomic number, the number of the element in the periodic table, a series arranged in order of atomic masses. Those answers were suspected from the general pattern of theory and had to be whole numbers since a complete atom (of nucleus plus outside electrons) is neutral. Much more precise measurements were neither needed, nor at the time, possible. Even before that, the first hint of atomic number measurements came in 1906, from Barkla's attempt to measure the number of electrons in a carbon atom by scattering X-rays. His measurements suggested a number of about 6 electrons per atom, in fact somewhere between 5 and 7, yet this rough estimate enabled the founding of atomic theory to proceed.

Galileo made the roughest measurements for his test of constant acceleration down an incline. He knew he was right in his simple summary of natural behaviour. He just wanted to convince some people by quoting an experiment.

Rough estimates are not just a misfortune peculiar to early, clumsy experimenters. They are the right thing in some parts of a growing science. Nuclear physicists and some cosmic ray physicists make very precise measurements. In other cases, they seek only a rough estimate to settle an essential point in the progress of their knowledge.

You cannot give the above examples to students if they do not know the science. In that case, the following may be some help.

"An invading army is about to go into a foreign land and the general wants to know the size of the enemy's forces. He learns that it is 18 000. Does it matter much to his plans if it is 19 000 or 15 000? What he wants to know is that it is about 18 000 and not 30 000. If he waits for his staff to carefully sift through reports and add up the guesses and check them and find that the enemy really has 18 473 men, then the general may set out too late to win the battle."

Other examples include:

• estimating how many snow ploughs are needed to clear a snowfall in the middle of the night;
• the Chancellor of the Exchequer makes a clever guess on the number of road vehicle licences which will be paid in the next year;
• a rough guess that the Sun is 300 000 times as massive as the Earth suffices to tell astronomers that the Earth is not massive enough to affect the orbit of the planet Venus, significantly.