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line, centre, protractor, angle, diameter and draw

PBOTIIYIZUM, (from rrpoOvpov) a porch at the outer door of a house, or portal.

PIZOTIZ.ACTOi;, (from the Latin prot•actus, to draw out) an instrument used in surveying, by which the angles taken in the field with a theodolite, circumferentor, or the like, are plotted, or laid down, on paper.

This protractor consists of a semicircular limb of brass, silver, horn, or the like, divided into 180°, and subtended by a diameter, in the middle of which is a little notch, or lip, called the centre of the protractor.

For the convenience of reckoning both ways, the degrees are numbered from the left hand towards the right, and front the right hand towards the left.

But this instrument is made much more commodious by transferring the divisions on the same circumference to the edge of a ruler, whose side is parallel to the diameter, which is easily done by laying a ruler on the centre, and the several divisions on the semi-circumference, and marking the inter sections of that ruler on the line : so that a ruler with these divisions marked on three of its sides, and numbered both ways, as in the protractor (the fourth, or blank side repre senting the diameter of the circle) is of the same use as a protractor, and much better adapted to a case.

On the limb of the protractor are sometimes also placed numbers, denoting the angles at the centres of regular polygons : thus, against the number 5, denoting the sides of a pentagon, is found 7:2, the angle at the centre of a pentagon, &c.

The uses of this instrument are, 1. To lay down an angle of any given quantity, or number of degrees, at any point, and with any given line.

Krample.—Lny the centre of the protractor on the given point, and the diameter of the protractor on the given line : make a mark against the given degree (say 50) on the limb of the protractor ; through which, from the given point, draw a line, and it will give the angle required.

'2. To find the quantity of a given angle.—Lay the centre of the protractor on the point of the angle, and the diameter on one of the lines forming it. The degree of the limb cut by the other line (viz. 50) is the number of degrees of the angle required.

3. To inscribe any given regular polygon (a pentagon for example) in a circle.—Lay the centre and diameter of the circle, and make a dot against the number of degrees of the angle at the centre, viz. '72. Through this dot and the centre of the circle draw a line, cutting the circumference of the circle. To the point of intersection. from the point where the diameter cuts the circumference, draw a right line ; this line will be a side of the pentagon, which, being taken in the compases, and set oil' as often as it will ? in the circum ference, will points, which, being connected by lines, will form the pentagon required.

4. To describe any regular polygon (e. gr. an octagon) on a given line.—Subtract the angle at the centre, which the pro tractor gives (say 45(') from 180 ; and the remainder, 135, will be the angle included between two sides of the octagon, one-half of which is ti7k. Applying then the diameter of the protractor over the given line, with the centre over one extreme ; make a dot against 67;4, to which, front the centre, draw a line : apply the protractor to the other end of the line, with the centre over the extreme, and there set oil' another angle of Oly. From the point where the two lines thus drawn intersect, as a centre, describe a circle with the interval of the given line.. The given line will be one side of the octagon ; and this set off as often as it will go in the circumference thus drawn, will give points, which, being con ne•ted, will form the octagon required.