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and fractions, after a more easie and exact forme then in former time hath beene set forth, 8vo, 1552.—This work went through many editions, and was corrected and augmented by several other persons; as first by the famous Dr. John Dee; then by John Mellis, a schoolmaster, 1590; next by Robert Norton; then by Robert Hartwell, practitioner in mathematics, in London; and lastly by R. C. and printed in 8vo, 1623.

3. The Castle of Knowledge, containing the Explication of the Sphere bothe Celestiall and Materiall, and divers other things incident thereto. With sundry pleasaunt proofes and certaine newe demonstrations not written before in any vulgare woorkes. Lond. folio, 1556.

4. The Whetstone of Witte, which is the seconde part of Arithmetike: containing the Extraction of Rootes: the Cossike Practise, with the rules of Equation: and the woorkes of Surde Nombers. Lond. 4to, 1557.— For an analysis of this work on Algebra, with an account of what is new in it, see pa. 79 of vol. 1, under the article Algebra.

Wood says he wrote also several pieces on physic, anatomy, politics, and divinity; but I know not whether they were ever published. And Sherburne says that he published Cosmographiæ Isagogen; also that he wrote a book, De Arte faciendi Horologium; and another, De Usu Globorum, & de Statu Temporum; which I have never seen.

RECTANGLE

, in Geometry, is a right-angled parallelogram, or a right-angled quadrilateral figure.

If from any point O, lines be drawn to all the four angles of a Rectangle; then the sum of the squares of the lines drawn to the opposite corners will be equal, in whatever part of the plane the point O is situated; viz, . For other properties of the Rectangle, see Parallelogram; for the Rectangle being a species of the parallelogram, whatever properties belong to the latter, must equally hold in the former.

For the Area of a Rectangle. Multiply the length by the breadth or height.—Otherwise; Multiply the product of the two diagonals by half the sine of their angle at the intersection.

That is, AB X AC, or AD X BC X 1/2 sin. [angle]P = area. A Rectangle, as of two lines AB and AC, is thus denoted, AB X AC, or AB . AC; or else thus expressed, the Rectangle of, or under, AB and AC.

Rectangle

, in Arithmetic, is the same with product or factum. So the Rectangle of 3 and 4, is 3 X 4 or 12; and of a and b is a X b or ab.

RECTANGLED

, Right-angled, or RECTANGULAR, is applied to figures and solids that have at least one right angle, if not more. So a Right-angled triangle, has one right angle: a Right-angled parallelogram is a rectangle, and has four right angles. Such also are squares, cubes, parallelopipedons.

Solids are also said to be Rectangular with respect to their situation, viz, when their axis is perpendicular to their base; as right cones, pyramids, cylinders, &c.

The Ancients used the phrase Rectangular section of a cone, to denote a parabola; that conic section, before Apollonius, being only considered in a cone having its vertex a right-angle. And hence it was, that Archimedes entitled his book of the quadrature of the parabola, by the name of Rectanguli Coni Sectio.

RECTIFICATION

, in Geometry, is the finding of a right line equal to a curve. The Rectification of curves is a branch of the higher geometry, a branch in which the use of the inverse method of Fluxions is especially useful. This is a problem to which all mathematicians, both ancient and modern, have paid the greatest attention, and particularly as to the Rectification of the circle, or finding the length of the circumference, or a right line equal to it; but hitherto without the perfect effect: upon this also depends the quadrature of the circle, since it is demonstrated that the area of a circle is equal to a right angled triangle, of which one of the sides about the right angle is the radius, and the other equal to the circumference: but it is much to be feared that neither the one nor the other will ever be accomplished. Innumerable approximations however have been made, from Archimedes, down to the mathematicians of the present day. See Circle and Circumference.

The first person who gave the Rectification of any curve, was Mr. Neal, son of Sir Paul Neal, as we find at the end of Dr. Wallis's treatise on the Cissoid; where he says, that Mr. Neal's Rectification of the curve of the semicubical parabola, was published in July or August, 1657. Two years after, viz in 1659, Van Haureat, in Holland, also gave the Rectification of the same curve; as may be seen in Schooten's Commentary on Des Cartes's Geometry.

The most comprehensive method of Rectification of curves, is by the inverse method of fluxions, which is thus: Let ACc be any curve line, AB an absciss, and BC a perpendicular ordinate; also bc another ordinate indefinitely near to BC; and Cd drawn parallel to the absciss AB. Put the absciss AB = x, the ordinate BC = y, and the curve AC = z: then is Cd = Bb = x. the fluxion of the absciss AB, and cd = y. the fluxion of the ordinate BC, also Cc = z. the fluxion of the curve AB. Hence because Ccd may be considered as a plane right-angled triangle, , or ; and therefore ; which is the fluxion of the length of any curve; and consequently, out of this equation expelling either x. or y., by means of the particular equation expressing the nature of the curve in question, the fluents of the resulting equation, being then taken, will give the length of the curve, in finite terms when it is