which is interesting, as being the first recorded example of the determination, in a finite form, of the value of the ratio of two infinite products.
Lord Brouncker, being consulted by Wallis as to the value of the above expression, put it in the form of an infinite continued fraction, thus: 7r 1 1 ± 1 2 + 9 2+ 25 2+49 2 + etc.
In which 2 and the squares of the odd numbers appear. This formula has been employed to show that not only 7r, but its square, is incommensurable.
Perhaps the neatest of all the formulas which have been given for the quadrature of the circle is that of James Gregory for the arc in terms of its tangent—namely: 0 = tan. 0 — * tan. tan. — etc.
'This was appropriated by Leibnitz, and formed perhaps the first of that audacious series of peculations from English mathematicians which have for ever dishonored the name of a man of real genius.
If we notice that, by ordinary trigonometry, the arc whose tangent is unity (the arc IC of 45 or) falls short of four times the arc whose tangent is I by an angle whose tan 2r gent is we may easily calculate to any required number of decimal places by cal culating from Gregory's formula the values of the arcs corresponding to I and gat; as tangents. And it is, in fact, by a slight modification of this process (which was origin ally devised by Machin), that 7r has been obtained, by independent calculators, to GOO decimal places.
It is not yet proved, and it may not be true, that the area oreircurnference of a circle cannot be expressed in finite terms; if it can be, these must (of course) contain irrational quantities. The integral calculus gives, among hosts of others, the following very simple expression in terms of a definite integral: oo dx 2 —/ 1 + 0 Now it very often happens that the value of a definite integral can be assigned, when that of the general integral cannot; and it is not impossible, so far as is yet known, that the above integral may be expressed in some such form as 4T and 4/Y are irrational numbers. Such an expression, if discovered, would undoubtedly be hailed as a solution of the grand problem.
But this, we need hardly say, is not the species of solution attempted by "squarers." We could easily, from our own experience alone, give numerous instances of their help less absurdities, but we spare the reader, and refer him, for further information on this painful yet ridiculous subject, to prof. De Morgan's Budget of Paradoxes; and to the very interesting work of Montucla, Histoire des Ileekerehes sur la Quadrature du Gavle.