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duction probably came from some German work, it has thence acquired its name. It consists of a square and a straight rule, the edge of the square is moved in use by one hand against the rule, which is kept steady by the other, the edge having been previously set to the given line: its use and construction is obvious from fig. 1, plate 3. Simple as it is in its principle, it has undergone some variations, two of which I shall mention; the one by Mr. Jackson, of Tottenham; the other by Mr. Marquois, of London.

Mr. Jackson's, jig. 6, plate 3, consists of two equal triangular pieces of brass, ivory, or wood, ABC, DEF, right angled at B and E; the edges AB and AC are divided into any convenient number of equal parts, the divisions in each equal; BC and DE are divided into the same number of equal parts as A C, one side of D E F may be divided as a pro

tractor.

To draw a line G H, fig. V, plate 2, parallel to a given line, through a point P, or at a given distance.

Place the edge DE upon the given line I K, and let the instrument form a rectangle, then slide the upper piece till it come to the given point or distance, keeping the other steady with the left hand, and draw the line GH. By moving the piece equal distances by the scale on BC, any number of equidistant parallels may be obtained.

If the distance, fig. X, plate 2, between the given and required lines be considerable, place A B upon IK, and EF against AC, then slide the pieces alternately till D E comes to the required point; in this manner it is easy also to construct any square or rectangle, &c.

From any given point or angle P, fig. S, plate 2, to let fall or raise a perpendicular on a given line. Place cither of the edges AC upon GP, and slide AB upon it, till it comes to the point P, and draw P H.

To divide a dine into any proposed number of equal parts, fig. T, plate 2. Find the proposed number

in the scale B C, and let it terminate at G, then place the rules in a rectangular form, and move the whole about the point G, till the side D E touches H; now move D one, two, or three divisions, according to the number and size of the required divisions, down B C, and make a dot at I, where D E cuts the line for the first part; then move one or more divisions as before, make a second dot, and thus proceed till the whole be completed.

Of Marquois's parallel scales. These consist of a right angled triangle, whose hypothenuse, or longest side, is three times the length of the shortest, and two rectangular scales. It is from this relative length of the hypothenuse that these scales derive their peculiar advantages, and it is this alone that renders them different from the common German parallel rule: for this we are much indebted to Mr. Marquois.

What has been already said of the German rule applying equally to those of Mr. Marquois's, I shall proceed to explain their chief and peculiar excellence. On each edge of the rectangular rule are placed two scales, one close to the edge, the other within this. The outer scale, Mr. Marquois terms the artificial scale, the inner one, the natural scale: the divisions on the outer are always three times longer than those on the inner scale, as, to derive any advantage from this invention, they must always bear the same proportion to each other, that the shortest side of the right angled triangle does to the longest. The triangle has a line near the middle of it, to serve as an index, or pointer; when in use, this point should be placed so as to coincide with the O'division of the scales; the numbers on the scales are reckoned both ways from this division; consequently, by confining the rule, and sliding the triangle either way, parallel lines may be drawn on either side of a given line, at any distance pointed out by the index on the triangle. The advantages of this contrivance are: 1. That the sight is greatly assisted, as the divisions on the outer

scale are so much larger than those of the inner one, and yet answer the same purpose, for the edge of the right angled triangle only moves through one third of the space passed over by the index. 2. That it promotes accuracy, for all error in the setting of the index, or triangle, is diminished one third.

Mr. Marquois recommends the young student to procure two rules of about two feet long, having one of the edges divided into inches and tenths, and several triangles with their hypothenuse in different proportions to their respective perpendiculars. Thus, if you would make it answer for a scale of twenty to an inch, the hypothenuse must be twice the length of the perpendicular; if a scale of 30 be required, three times; of 40, four times; of 50, five times, and so on. Thus also for intermediate proportions: if a scale of 25 is wanted, the hypothenuse must be in the proportion of 25 to 10 or 5 to 2; if 35, of 35 to 10 or 7 to 2, &c. Or a triangle may be formed, in which the hypothenuse may be so set as to bear any required proportion with the perpendicular,*

OF THE PROTRACTOR.

This is an instrument used to protract, or lay down an angle containing any number of degrees, or to find how many degrees are contained in any given angle. There are two kinds put into cases of mathematical drawing instruments; one in the form of a semicircle, the other in the form of a parallelogram. The circle is undoubtedly the only natural measure of angles; when a straight line is therefore used, the divisions thereon are derived from a circle, or its properties, and the straight line is made use of for some relative convenience: it is thus the parallelogram is often used as a protractor, instead of the semicircle, because it is in some cases more convenient, and that other scales, &c. may be placed upon it.

* Keith's improvement on Marquois's Scales, plate 38, fig. 2, 3, 4, and 5, will be hereafter described. EDIT.

The semicircular protractor, fig. 2, plate 3, is divided into 180 equal parts or degrees, which are numbered at every tenth degree each way, for the conveniency of reckoning either from the right towards the left, or from the left towards the right; or the more easily to lay down an angle from either end of the line, beginning at each end with 10, 20, &c. and proceeding to 180 degrees. The edge is the diameter of the semicircle, and the mark in the middle points out the centre. Fig. 3, plate 3, is a protractor in the form of a parallelogram: the divisions are here, as in the semicircular one, numbered both ways; the blank side represents the diameter of a circle. The side of the protractor to be applied to the paper is made flat, and that whereon the degrees are marked, is chamfered or sloped away to the edge, that an angle may be more easily measured, and the divisions set off with greater exactness.

Protractors of horn are, from their transparency, very convenient in measuring angles, and raising perpendiculars. When they are out of use, they should be kept in a book to prevent their warping. Upon some protractors the numbers denoting the angle for regular polygons are laid down, to avoid the trouble of a reference to a table, or the operation of 'dividing; thus, the number 5, for a pentagon is set against 72°; the number 6, for a hexagon, against 60°; the number 7, for a heptagon, against 51°; and so on.

Protractors for the purposes of surveying will be described in their proper place.

Application of the protractor to use. 1. A number of degrees being given, to protract, or lay down an angle, whose measure shall be equal thereto.

Thus, to lay down an angle of 60 degrees from the point A of the line AB, fig. 14, plate 3, apply the diameter of the protractor to the line A B, so that the centre thereof may coincide exactly with the point A; then with a protracting pin make a fine dot at

C against 60 upon the limb of the protractor; now remove the protractor, and draw a line from A through the point C, and the angle CAB contains the given number of degrees.

2. To find the number of degrees contained in a given angle BA C.

Place the centre of the protractor upon the angular point A, and the fiducial edge, or diameter, exactly upon the line A B; then the degree upon the limb that is cut by the line CA will be the measure of the given angle, which, in the present instance, is found to be 60 degrees.

3. From a given point A, in the line AB, to erect a perpendicular to that line.

Apply the protractor to the line A B, so that the centre may coincide with the point A, and the division marked 90 may be cut by the line AB, then a line DA drawn against the diameter of the protractor will be perpendicular to A B.

Further uses of this instrument will occur in most parts of this work, particularly its use in constructing polygons, which will be found under their respective heads. Indeed, the general use being explained, the particular application must be left to the practitioner, or this work would be unnecessarily swelled by a tedious detail and continual repetitions.

OF THE PLAIN SCALE.

The divisions laid down on the plain scale are of two kinds, the one having more immediate relation to the circle and its properties, the other being mere ly concerned with dividing straight lines.

It has been already observed, that though arches of a circle are the most natural measures of an angle, yet in many cases right lines were substituted, as being more convenient; for the comparison of one right line with another, is more natural and easy, than the comparison of a right line with a curve; hence

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