This article was originally published in the Incorporated British Institute of Certified Carpenters’ journal (renamed the Institute of Carpenters in 1976) in October 1925.
As the Editor thought sufficiently well of my short article on the triangle to publish it, I am emboldened to offer another article for his criticism.
The circle is not a difficult geometrical conception and a representation of it can readily be made. For what purpose, and at what stage in the remote past, man originally wished to make this representation it is not possible to know, but, as Paine so well says of the triangle, the properties of the circle were existent before man represented a circle, in fact, the properties of this shape, as of all other shapes, date from the beginning of time.
Perhaps the first most striking thing about a circle is, that if its diameter is a commensurable length, that is a length that can be written down as so many units, or part of a unit, of length, then the circumference is an incommensurable length, that is a length which cannot be so written down.
The “rectification of the circle” that is the finding, by means of ruler and compasses, a straight line equal in length to the circumference of a circle, and the “squaring of the circle,” that is the finding the length of the side of a square equal in area to a circle—really one problem—occupied the minds of mathematicians for a very long time, and it required 2,000 years of mathematical progress to show, by ultimately discovering the nature of the number we generally call π, that the problem could not be solved.
As it can be shown that a circle is equal in area to a triangle whose base is equal to the circumference of the circle and height equal to the radius, it is obvious that if the circle can be “rectified,” then it can also be “squared.”
The circumference of a circle is 2, π × r and the area πr², where r is the radius of the circle. The number π lies between 3.14159 and 3.14160, but it cannot be completely represented by any number, its value has been obtained to 700 places of decimals and I would suggest to my readers that the finding of its value to, say, 701 places of decimals might provide a pleasant winter’s pastime. Before dismissing this part of the subject I would like to give a geometrical—or rather arithmetics—geometrical—method of obtaining π, the work of a 17th century mathematician.
Let O be the centre (Fig. 1.), and AB the diameter of a circle of unit radius. From O draw the line OC to meet the tangent at A in C, the angle AOC to be 30 degrees, from C mark off CD equal to 3 units. Then BD is a close approximation of π.
Having pointed out the “awkwardness” of this figure, let us consider one or two of its more easily understood properties.
If we fix two points A, B, near to the circumference and joined by straight lines to the centre O and to any other point P on the circumference, then if P is situated as in the diagram the obtuse angle AOB is twice the angle APB, and if P is situated on the circumference below AB, the reflex angle AOB is twice the angle APB.
In Fig. 3 if two points A, B, be taken on the circumference of a circle and they be joined to the point P1, P2, etc., as shown, then all the angles, such as AP, B, AP1B are equal. It is because of this property that we are able to obtain the soffit of a segmental arch, say, when the centre of the arc is inaccessible.
Another interesting property is shown in Fig. 4. If a tangent AB be drawn to the circle, having C as point of contact, and if through C any chord be drawn, then the angle between the tangent and chord is equal to the angle in the opposite segment; in this case, angle BCD equals angle CED. This property is of great use in the solving of a certain type of problem.
In Fig. 5 if in a circle two chords AB and CD be drawn intersecting at P, then AP multiplied PD will equal CP multiplied by PB. What is meant here by multiplying AP by PD is that one measures these two lines using the same unit for both and having obtained their lengths in terms of the chosen unit multiplies the numbers so found.
This property enables us to find the radius of, say, the soffit of a segmental arch if we know the span and rise of the arch. In Fig. 6, let ADB represent the soffit of the arch of span 12 feet and rise 2 feet, required to find the radius of this circle. Referring to the figure, the fact has just been stated that AP multiplied by PB equals CP multiplied by PD, but AP equals PB equals 6 feet and PD equals 2 feet, substituting these known numbers in the equation we get 6 multiplied by 6 equals CP multiplied by 2, from this we see that CP equals 18 feet, now I think it is obvious that CD is a diameter of the circle and its length is equal to CP plus PD equals 18 feet plus 2 feet equals 20 feet. Hence the radius is 10 feet.
This very remarkable property not only applies when the chords intersect at a point within the circle, but also when the chords are produced to meet at P when PB multiplied by PA equals PD multiplied by PC.
What happens when A and B coincide, that is when ABP becomes a tangent to the circle?
— G.A.
This article first appeared in the Incorporated British Institute of Certified Carpenters Journal renamed the Institute of Carpenters in 1976) in October1925. We then reprinted it in the December 2014 issue of Cutting Edge. As part of the IOC membership, IOC members receive quarterly editions of Cutting Edge magazine and access to all back issues online.
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