349 
1888 .] Dr E. Sang on the Diameter of a Circle. 
algebra, and witb the advanced branches of trigonometry ; must 
know how to divide an arc whose tangent is an aliquot part of the 
radius, into smaller arcs whose tangents have the same character ; 
and while studying these chapters in mathematical science we 
shall have often used our knowledge of this very ratio of which we 
are in search. The real utility of this formula lies in its enabling 
us to extend the approximation to a great number of decimal places. 
Even although the labour be not overwhelming, we can hardly 
regard as other than useless, the toil of computing and verifying 
the value of tt to upwards of five hundred places. 
But a knowledge of this ratio is needed by artificers of all kinds, 
for in every branch of workmanship we have rounds as well as 
squares, and hence the determination thereof has almost come to be 
a social problem. In the opinion of many, the “ squaring of the 
circle ” is hedged in by insuperable difficulties, and one writer even 
appeals to divine aid for guidance in the matter. Among the 
multitudes who use and believe in this mysterious number 3T416, 
not one in a thousand has attempted the verification, or even formed 
an idea as to how the verification should be gone about. 
The proceeding was in this way : — In and about a circle of known 
radius, regular polygons were described, and their areas or peri- 
pheries computed ; the ones being necessarily less, the others greater 
than the area or circumference of the circle. By doubling the 
number of the sides, polygons were got approximating nearer to 
each other, and, of course, to the circle, and by continuing this 
process of doubling, the results were brought to within some 
prescribed degree of exactitude. The requisite calculations are 
somewhat long and involved. 
In the later editions of his Elements of Geometry^ that is about 
the year 1816, John Leslie, then Professor of Mathematics in the 
University of Edinburgh, gave another view of the matter. Having 
assumed some known length for the boundary, he constructed a 
regular polygon and computed the radii of the inscribed and cir- 
cumscribed circles. Doubling and doubling the number of the 
sides, but still retaining the same total boundary, he brought the 
radii nearer and nearer to each other, until the difference was 
within the prescribed limit of accuracy, that is until the polygons 
did not differ perceptibly from each other or from a circle. By 
