The quadrature of the circle, the square root of two, and the right-angled triangle, by William Alexander Myers.

INTRODUCTION. 27
etc., alternately too small or too great, will fall within the known
limiits.
Here is another expression of the relation of the circle to the square
of the diameter, found by Lord Brunker about the same time.
The circle being one, the square is expressed by the following fraction carried to infinity:
2+25
2$-25
2+49
2+etc.
It will be seen that this fraction is such that the denominator is an
integer plus a fraction, whose denominator is 2 plus the square of one
of the odd numbers 1, 3, 5, 7, etc.; when brought to an end the limits
obtained are alternately in excess or too small.
Such was the knowledge of geometricians on this famous problem when
Newton and Leibnitz appeared on the arena. In 1682, Leibnitz gave
out in his Actes de Leipsig what he had discovered as early as 1673,
namely, that the square of the diameter being one, the area of the circle
is expressed by the infinite series 1 —+ + 1- 4- etc. It follows from
his discovery about the same time that the radius of the circle being unity
and the tangent of 'an arc t, this arc itself is t — t3-+ — t5- t7, etc.
If then the arc is 45~, the tangent t is equal to the radius or one.
Thus the arc of 45~ is 1- 1-+-l -- etc.; multiplying by 4 we shall
have the semi-circumference, which multiplied by the radius will give
the area of the circle equal 4 4-+-4 etc. the square of the diam4
eter being 4. Thus the square of the diameter being made unity, the
area of the circle will be 1-_+_-+- etc. to infinity.
The area can also be expressed by + + +T etc., viz.: by
adding together the two first terms, and the next two by two, or else
in this way, 1-22, — etc., where it is easy to see that the denominators are successively in the first the squares of 2, 6, 10, etc.,
diminished by unity, and in the second the squares of 4, 8, 12, etc.,
similarly reduced. But it must be conceded that these different series
do not converge rapidly enough to derive from them a value sufficientlyaccurate without the addition of a prodigious number of terms;
but Euler found a remedy.
The discoveries made by Newton, even before Leibnitz, had also
placed him in possession of various methods of expressing the circumference and the area of the circle, as also of segments by infinite
series.