STADE

STADE, a fortified town of Hanover, capital of a province of the same name, in a marshy district on the

Squaring. prodigious. He seems to have valued highly this singular effort; for, in imitation of Archimedes, whose tomb was adorned with a sphere and cylinder, he directed that the ratio he had found might be inscribed on his tomb.

Snellius abridged greatly the labour of calculation; and although he did not go beyond Van Ceulen, yet he verified his result. Descartes also found a geometrical construction, which being repeated continually, gave the circumference, and from which he might easily have deduced an expression in the form of a series. Gregory of St Vincent distinguished himself also on this subject; he however committed a great error in supposing he had discovered the quadrature of both the circle and hyperbola. Gregory's mistake was the cause of a sharp controversy carried on between his disciples on the one side, and by Huygens, Mersenne, and Lestaud, on the other; and it was this that gave Huygens occasion to consider particularly the quadrature of the circle, and to investigate various new and curious theorems relating to it, which are contained in his Theoremata de Quadratura Hyperboles, Ellipsis, et Circuli, 1651; and in his work De Circuli Magnitudine inventa, 1654. In particular, he showed, that if $c$ denote the chord of an arch, and $s$ its sine, then the arch itself will be greater than $c\; \backslash times\; \backslash frac\{1\}\{2\}(c-s)$, but less than $c\; +\; \backslash frac\{4c+s\}\{2c+3s\}\; +\; \backslash frac\{1\}\{2\}(c-s)$; he also showed that the arch is less than the sum of $\backslash frac\{2\}\{3\}$ of its sine and $\backslash frac\{1\}\{3\}$ of its tangent.

James Gregory, in his Vera Circuli et Hyperbolæ Quadratura, gave several curious theorems upon the relations of the circle to its inscribed and circumscribed polygons, and their ratios to one another; and by means of these he found with infinitely less trouble than by the ordinary methods, and even by those of Snellius, the measure of the circle as far as 20 places of figures. He gave also, after the example of Huygens, constructions for finding straight lines nearly equal to arches of a circle, and of which the degree of accuracy was greater. For example, he found that if $A$ be put for the chord of an arch of a circle, and $B$ for twice the chord of half the arch, and $C$ be taken such that $A+B:B::B:C$, then the arch itself is

nearly equal to $\backslash frac\{8C+8B-A\}\{15\}$, but a little less, the error in

the case of a complete semicircle being less than its $\backslash frac\{1\}\{2\}$ part; and when the arch does not exceed $120^\backslash circ$, it is less than its $\backslash frac\{1\}\{2\}$ part; and finally, for a quadrant the error is not greater than its $\backslash frac\{1\}\{4\}$ part. And farther, that if $D$ be such that $A:B::B:D$, then the arch is nearly equal to

$\backslash frac\{12C+4B-D\}\{15\}$, but a little greater, the error in the semi-

circle being less than its $\backslash frac\{1\}\{4\}$ part, and in a quadrant less than its $\backslash frac\{1\}\{8\}$ part.

Dr Wallis gave, in his Arithmetica Infinitorum, a singular expression for the ratio of the circle to the square of its diameter. He found that the former was to the latter as 1 to the product

the fractions $\backslash frac\{3\}\{2\},\; \backslash frac\{5\}\{4\},\; \backslash frac\{7\}\{6\},\; \backslash frac\{9\}\{8\},\; \backslash dots$, being supposed infinite in number. The products being supposed continued to infinity, we have the ratio exactly; but if we stop at any finite number of terms, as must necessarily be the case in its application, the result will be alternately too great and too small, according as we take an odd or an even number of terms of the numerator and denominator.

An expression of another kind of the ratio of the circle to the square of the diameter was found by Lord Brounker. He showed that the circle being unity, the square of the diameter is expressed by the continued fraction

Schwinge, about 3 miles above its confluence with the Elbe, 22 miles west of Hamburg. It is surrounded by walls, and has been recently partly fortified, although the plan once entertained of making it a complete fortress is now given up. The chief buildings are three churches; a normal seminary; a gymnasium, which occupies the site of an ancient Augustinian convent; a house of correction, and others. Flannel and hosiery are manufactured, and a considerable trade is carried on. Stade was in ancient times the seat of independent counts, of whom we read in history as far back as 931; but it afterwards became incorporated with the possessions of the archbishops of Bremen. Pop. 7950.