The Seven Follies of Science [2nd ed.] / A popular account of the most famous scientific impossibilities and the attempts which have been made to solve them. To which is added a small budget of interesting paradoxes, illusions, and marvels

by John Phin

Fig. 3.

De Morgan says: "The following method of finding the circumference of a circle (taken from a paper by Mr. S. Drach in the 'Philosophical Magazine,' January, 1863, Suppl.), is as accurate as the use of eight fractional places: From three diameters deduct eight-thousandths and seven-millionths of a diameter; to the result, add five per cent. We have then not quite enough; but the shortcoming is at the rate of about an inch and a sixtieth of an inch in 14,000 miles."

For obtaining the side of a square which shall be equal in area to a given circle, the empirical method, given by Ahmes in the Rhind papyrus 4000 years ago, is very simple and sufficiently accurate for many practical purposes. The rule is: Cut off one-ninth of the diameter and construct a square upon the remainder.

This makes the ratio 3.16.. and the error does not exceed one-third of one per cent.

There are various mechanical methods of measuring and comparing the diameter and the circumference of a circle, and some of them give tolerably accurate results. The most obvious device and that which was probably the oldest, is the use of a cord or ribbon for the curved surface and the usual measuring rule for the diameter. With an accurately divided rule and a thin metallic ribbon which does not stretch, it is possible to determine the ratio to the second fractional place, and with a little care and skill the third place may be determined quite closely.

An improvement which was no doubt introduced at a very early day is the measuring wheel or circumferentor. This is used extensively at the present day by country wheelwrights for measuring tires. It consists of a wheel fixed in a frame so that it may be rolled along or over any surface of which the measurement is desired.

This may of course be used for measuring the circumference of any circle and comparing it with the diameter. De Morgan gives the following instance of its use: A squarer, having read that the circular ratio was undetermined, advertised in a country paper as follows: "I thought it very strange that so many great scholars in all ages should have failed in finding the true ratio and have been determined to try myself." He kept his method secret, expecting "to secure the benefit of the discovery," but it leaked out that he did it by rolling a twelve-inch disk along a straight rail, and his ratio was 64 to 201 or 3.140625 exactly. As De Morgan says, this is a very creditable piece of work; it is not wrong by 1 in 3000.

Skilful machinists are able to measure to the one-five-thousandth of an inch; this, on a two-inch cylinder, would give the ratio correct to five places, provided we could measure the curved line as accurately as we can the straight diameter, but it is difficult to do this by the usual methods. Perhaps the most accurate plan would be to use a fine wire and wrap it round the cylinder a number of times, after which its length could be measured. The result would of course require correction for the angle which the wire would necessarily make if the ends did not meet squarely and also for the diameter of the wire. Very accurate results have been obtained by this method in measuring the diameters of small rods.

A somewhat original way of finding the area of a circle was adopted by one squarer. He took a carefully turned metal cylinder and having measured its length with great accuracy he adopted the Archimedean method of finding its cubical contents, that is to say, he immersed it in water and found out how much it displaced. He then had all the data required to enable him to calculate the area of the circle upon which the cylinder stood.