122 QUADRANT angles in any plane. The use of quadrants has been for surveying and for making astronomi- cal observations, and especially in navigation for determining the meridian altitude of the sun, and through this the latitude of the ob- server. They have been constructed of a great variety of forms and dimensions adapted for their several uses ; but at present the interest attached to them is historical only, as they have been entirely superseded either by the sextant or the full circle. The former, of more portable form than the quadrant, by the use of two reflecting mirrors doubles the angle in- cluded between the direct and reflected line of light, and thus with an arc of 60 or one sixth of the circle includes a range of 120 ; while the circle, on account of the symmetry of its form and the completeness of its graduated arc all around, secures greater exactness in its read- ings, and is less liable to the introduction of any unsuspected source of error. Ptolemy made use of a quadrant for determining the obliquity of the ecliptic. Tycho Brahe had a large mural quadrant (so called from its being suspended upon an axis secured in a solid wall of ma- sonry) with which he observed altitudes, and also another on a vertical axis for measuring horizontal angles. The mural quadrants of that period were of 6 or 8 ft. radius, and for some time continued to be employed in the principal observatories. Sir Isaac Newton is said to have constructed a reflecting quadrant as early as 1672; but the first instrument of this character brought before the public was that afterward known as Hadley's, the in- vention of which was claimed by Godfrey, a mechanician of Philadelphia. This instrument, which has been in general use in navigation, is a graduated octant of 90 half degrees, reading as 90. With the radial bars at each extremi- ty of the arc it forms a triangular frame, which is made of convenient dimensions for holding in the hands. A movable radial bar or index revolves in the plane of the sector upon a pin passing through the centre. At the centre it carries a mirror, the face of which is perpen- dicular to this plane, and which in making an observation is turned toward the object, as the sun or a star, and at the other end it carries a vernier for subdividing the angles on the grad- uated limb. On the outer edge of the radial bar, back of the movable mirror, is the sight vane, which is directed across to a second mir- ror fixed upon the opposite bar, its plane per- pendicular to that of the bar, and its face so adjusted that a ray reflected from the first mir- ror to the second is transmitted from this to the eye at the sight vane. Only half of the glass of the second mirror, called the fore hori- zon glass, is silvered, and consequently rays passing through it from any object, as the hori- zon at sea, meet the eye in a direct line ; and if at the same instant, while the instrument is held to this position, the index is moved so as to bring the reflected image of the sun upon the silvered part of the glass and from this to QUADRATURE the eye, the reading of the vernier is the eleva- tion of the sun above the horizon. Various other appendages are introduced in the quad- rant, as a telescope for the sight vane, colored glasses for diminishing the intensity of the light, and a third mirror called the back hori- zon glass, with its sight vane, for taking a bark observation. (For Gunter's quadrant, see Gr.v- TEK.) In gunnery, the quadrant or gunner's square is a rectangular frame with a graduated arc between the two limbs. One of the limbs is extended beyond the arc, so as to be set into the mouth of the piece, the elevation of which it is to measure. A plummet suspended from the point of meeting of the two arms marks by the intersection of its line on the graduated arc the degree of elevation. QUADRATURE, the finding of a square equal in area to that of any given figure. No math- ematical problem has excited so great interest as the quadrature of the circle, or the deter- mination of a square of the same area. As it is proved that the area of a circle is equal to that of a right-angled triangle, the altitude of which is the radius of the circle and the base its circumference, and as the side of the square of equal surface with the triangle is a mean proportional between the height and half the base of the triangle, the problem would bo solved if the circumference could be imme- diately calculated from the radius which is known. Thus the question of the quadrature of the circle is reduced to finding the propor- tion between the diameter and circumference. Archimedes undertook the solution of the prob- lem on tha principle of calculating the periph- eries of two polygons of many sides (as 96), one circumscribed about the circle and the other inscribed, between which must lie the circumference of the circle. He thus found that the ratio of the diameter to the circum- ference lay between 1 : 8| and 1 : 3f ?, and he adopted the former, which is also expressed 7 : 22. The Hindoos at some early period, certainly before any improvement was made upon this result in Europe, obtained the pro- portion 1,250 : 3,927, or 3-1416, which is much more exact than that of Archimedes. Ptolemy gives 3-141552, which is not quite so correct. In modern times the first great step in extend- ing this calculation was made by Peter Metius, a Hollander, and was published by his son Adrian Metius. By calculating from polygons of about 1,536 sides he found that the propor- tion was less than 3 T y,r and greater than 3-JW ; and presuming that the mean of these was nearer the truth than either limit, he happily hit thus by chance on a near approximation, and determined a ratio convenient for practical purposes, and easy to recollect from its terms being made up of successive pairs of the first three odd numbers, viz.: 113 : 355. The error involved in this expression in a circle of 1,000 miles circumference is less than one foot. Lu- dolph van Ceulen (or Keulen), another Holland- er, in 1590, about the same time that Metius
