der its radius and circumference (see CIRCLE), so that the determination of the length of the circumference of a circle of given radius is precisely the same problem as that of the quadrature of the circle.

Confining ourselves strictly to the best ascertained steps in the history of the question, we remark that Archimedes proved that the ratio of the diameter to the circumference is greater than 1 to 3 or 34, and less than 1 to 348; the difference between these two extreme limits is less than the 1000th of the whole ratio. Archimedes' process depends upon the obvious truth, that the circumference of an inscribed polygon is less, while that of a circumscribed polygon is greater, than that of the circle. His calculations were extended to regular polygons of 96 sides.

Little more seems to have been done by mathematicians till the end of the 16th century, when P. Métius gave expression for the ratio of the circumference to the diameter as the fraction, which, in decimals, is true to the seventh significant figure inclusive; curiously enough it happens that this is one of the fractions which express in the lowest possible terms the best approximation to the required number. Métius seems to have employed, with the aid of far superior arithmetical notation, a process similar to that of Archimedes. Viete shortly afterwards gave the ratio in a form true to the tenth decimal place, and was the first to give, though of course in infinite terms, an exact formula. Designating, as is usual in mathematical works, the ratio of the circumference to the diameter T, Viete's formula is

1

{√ / Į × √ } + {√ į × √ ? + 1 √ } + 1/X etc.

Shortly afterwards, Adrianus Romanus, by calculating the length of the side of an equilateral inscribed polygon of 1073741824 sides, determined the value of π to 16 significant figures; and Ludolph Van Ceulen, his contemporary, by calculating that of the polygon of 36.893488147419103232 sides, arrived (correctly) at 36 significant figures. It is scarcely possible to give, in the present day, an idea of the enormous labor which this mode of procedure entails even when only 8 or 10 figures are sought; and when we consider that Ludolph was ignorant of logarithms, we wonder that a life time sufficed for the attainment of such a result by the method he employed.

The value of T was thus determined to

of its amount, a fraction of

1 3 X 1035 which, after Montucla, we shall attempt to give an idea, thus: suppose a circle whose radius is the distance of the nearest fixed star (250,000 times the earth's distance from the sun), the error in calculating its circumference by Ludolph's result would be so excessively small a fraction of the diameter of a human hair as to be utterly invisible, not merely under the most powerful microscope yet made, but under any which future generations may be able to construct.

These results were, as we have pointed out, all derived by common arithmetical operations, based on the obvious truth that the circumference of a circle is greater than that of any inscribed, and less than any circumscribed polygon.

They involve none of those more subtle ideas connected with limits, infinitesimals, or differentials, which seem to render more recent results suspected by modern "squarers." If one of that unhappy body would only consider this simple fact, he could hardly have the presumption to publish his 3.1250 or whatever it may be, as the accurate value of a quantity which, by common arithmetical processes, founded on an obvious geometrical truth, was several centuries ago shown to be the greater than 314159265358979323846264338327950288 and less than 314159265358979323846264338327950289.

We now know by far simpler processes its exact value to more than 600 places of decimals; but the above result of Van Ceulen is much more than sufficient for any possible practical application even in the most delicate calculations in astronomy.

Snellius, Huyghens, Gregory de St. Vincent, and others, suggested simplifications of the polygon process, which are in reality some of the approximate expressions derived from modern trigonometry. In 1668 the celebrated James Gregory gave a demonstration of the impossibility of effecting exactly the quadrature of the circle, which, although objected to by Huyghens, is now received as quite satisfactory. We may merely revert to the speculations of Fermat, Roberval, Cavalleri, Wallis, Newton, and others, as to quadrature in general; their most valuable result was the invention of the Differential and Integral Calculus, by Newton, under the name of Fluxions or Fluents. Wallis, however, by an П 2.4.4.6.6.8.8.10.10, etc., ingenious process of interpolation, showed that which 4 3.3.5.5.7.7.9.9.11, etc., is interesting as being the first recorded example of the determination, in a finite form, of value of the ratio of two infinite products.

Lord Bruncker, being consulted by Wallis as to the value of the expression, put it in the form of an infinite continued fraction, thus:

in which 2 and the squares of the odd numbers appear. This formula has been employed to show that not only, 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 tangentnamely, tan. 0 — tan. 30 + tan. 50-etc.

This was appropriated by Liebnitz, and formed, perhaps, the first of those audacious series of peculations from English mathematicians which have forever dishonored the name of a man of real genius.

If we notice that, by ordinary trigonometry, the arc, whose tangent is unity, (the arc of 45° or) falls short of four times the arc whose tangent is by an

π

4

angle whose tangent is, we may easily calculate

π

4

to any required number of decimal places by calculating from Gregory's formula the values of the arcs corresponding to and 25 as tangents. And it is, in fact, by a slight modification of this process (which was originally devised by Machin) that has been obtained by independent calculators to 600 decimal places. It is not yet proved, and it may not be true, that the area or circumference of a circle can not 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:

Now it very often happens that the value of a definite integral can be assigned when that of the general integral can not; and it is not impossible, so far as is yet known, that the above integral may be expressed in such form as

√ x + V ÿ

where x and 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 helpless absurdities, but we spare the reader and refer him, for further information on this painful yet ridiculous subject, to a recent series of papers, by Professor De Morgan, in the Athenæum, and to the very interesting work of Montucla, Histoire des Recherches sur la Quadrature du Cercle.

PART FIRST,

CONTAINING THE

GEOMETRICAL AND FINAL SOLUTION

OF THE

QUADRATURE OF THE CIRCLE,

BY AN ENTIRELY NEW METHOD, TOGETHER WITH AMPLE PROOFS OF

5

THE SAME.

(65)

QUADRATURE OF THE CIRCLE.

DEFINITIONS.

Of Lines, Angles, etc., see Plate 1.

1. Science is knowledge systematized.

2. Art is the skill with which the principles of a science are practically applied.

3. Quantity is anything which can be increased, diminished, or measured.

4. Mathematics is the science of quantity.

5. Geometry is that branch of mathematics which treats of the properties of extension and figure.

LINES.

6. A line has only one dimension, namely, length; without either breadth or thickness; lines are either straight or curved.

7. The extremities of a line are points.

8. A point has no dimensions, and therefore it has no size; and is usually represented to the eye by a dot, as at AA.

9. A straight line is the shortest distance between any two given points, as at B.

10. A curved line does not lie evenly between its extreme points; but constantly changes its course as at C.

11. A waved line is composed of curved lines as at E.

12. Parallel straight lines are in the same plane, but have no inclination towards one another; and being produced ever so far both ways can not meet, as at D.

13. Parallel curved lines if produced would form two concentric circles; that is, having a common center, as at F.

14. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is a right angle, and the straight line which stands on the other is called a perpendicular to it; IKL are perpendiculars to the line GH.