another are cutting the former in the points G, H, through which draw the right line GHC, cutting the former right line EFC in the point C; upon the point C as a centre with an extent equal to CA, CB, or CD, as radius, describe the sought circle.

PROBLEM X.

To divide a circle into 2, 4, 8, 16, or 32 equal parts.

Draw a diameter through the centre, dividing the circle into two equal parts; bisect this diameter by another, drawn perpendicular thereto, and the circle will be divided into four equal parts or quadrants; bisect each of these quadrants again by right lines Îrawn through the centre, and the circle will be divided into eight equal parts; and so you may continue the bisections any number of times. This problem is useful in constructing the mariner's compass.

PROBLEM XI.

To divide a given line into any number of equal parts Let it be required to divide the line AB into five equal parts. From the point A draw any line, AD, making an angle with the line AB; then through the point B draw a line, BC, parallel to AD; and from A, with any small opening in your compasses, set off a number of equal parts on the line AD, less by one than the proposed number (which number of equal parts in this example is 4); then from B, set off the same number of the same parts on the line BC; then join 4 and 1, 3 and 2, 2 and 3. 1 and 4, and these lines will cut the given line as required.

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CONSTRUCTION OF THE PLANE SCALE

1st. WITH the radiu3 you intend for your scale, describe a semicircle, ADB (Plate II fig. 1), and from the centre, C, draw CD perpendicular to AB, which will divide the semicircle into two quadrants, AD, BD; continue CD towards S, draw BT perpendicular to CB, and join BD and AD.

2dly. Divide the quadrant BD into 9 equal parts; then will each of these be 10 degrees; subdivide each of these parts into single degrees, and, if your radius will admit of it, into minutes or some aliquot parts of a degree greater than minutes.

3dly. Set one foot of the compasses in B, and transfer each of the divisions of the quadrant BD to the right line BD, then will BD be a line of chords.

4thly. From the points 10, 20, 30, &c., in the quadrant BD, draw right lines parallel to CD, to cut the radius CB, and they will divide that line into a line of sines which must be numbered from C towards B.

5thly. If the same line of sines be numbered from B towards C, it will become a line of versed sines, which may be continued to 180°, if the same divisions be transferred on the same line on the other side of the centre C.

6thly. From the centre C, through the several divisions of the quadrant BD, draw right lines till they cut the tangent BT; so will the line BT become a line of tangents. 7thly. Setting one foot of the compasses in C, extend the other to the several divisions, 10, 20, 30, &c., in the tangent line, BT, and transfer these extents severally to the right line, CS; then will that line be a line of secants.

8thly. Right lines drawn from A to the several divisions, 10, 20, 30, &c., in the quadrant BD, will divide the radius CD into a line of semi-tangents.

9thly. Divide the quadrant AD into eight equal parts, and from A, as a centre, transfer these divisions severally into the line AD; then will AD be a line of rhumbs, each division answering to 11° 15' upon the line of chords. The use of this line is for protracting and measuring angles, according to the common division of the mariner's compass. If the radius AC be divided into 100 or 1000, &c., equal parts, and the lengths of the several sines, tangents, and secants, corresponding to the several arcs of the quadrant, be measured thereby, and these numbers be set down in a table,* each in its proper column, you will by these means have a collection of numbers by which the several cases in trigonometry may be solved. Right lines, graduated as above, being placed severally upon a ruler, form the instrument called the Plane Scale 'see Plate II. fig. 2), by which the lines and angles of all triangles may be measured. AD right lines, as the sides of plane triangles, &c., when they are considered simply as such, without having any relation to a circle, are measured by scales of equal parts, one of which is subdivided equally into 10, and this serves as a common division to all the rest. In most scales, an inch is taken for a common measure, and what an inch is divided into is generally set at the end of the scale. By any common scale of equal parts, divided in this manner, any number less than 100 may be readily taken; but if the number should consist of three places of figures, the value of the third figure cannot be exactly ascertained, and in this case it is better to use a diagonal scale, by which any number consisting of three places of figures, may be exactly found. The figure of this scale is given in Plate II. fig. 3; its construction is as follows:

Having prepared a ruler of convenient breadth for your scale, draw near the edges thereof two right lines, af, cg, parallel to each other; divide one of these lines, as af into equal parts, according to the size of your scale; and, through each of these divisions draw right lines perpendicular to af, to meet cg; then divide the breadth into 10 equal parts, and through each of these divisions draw right lines parallel to af and : divide the lines ab, cd, into 10 equal parts, and from the point a to the first division

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* In Table XXIV. are given the sine and cosine to every minate of the quadrant, to five places of decimals.

The length of one of these equal parts at the end of the scale to which this description refers is ab

n the line cd, draw a diagonal line; then, parallel to that line, draw diagonal lines through all the other divisions, and the scale is complete. Then, if any number, consisting of three places of figures, as 256, be required from the larger scale, gd, you must place one foot of the compasses on the figure 2 on the line gd, then the extent from 2 to the point d will represent 200. The second figure being 5, count five of the smaller divisions from d towards c, and the extent from 2 to that point will be 250 Move both points of the compasses downwards till they are on the sixth parallel line below gd, and open them a little till the one point rests on the vertical line drawn through 2, and the other on the diagonal line drawn through 5; the extent then in the compasses will represent 256. In the same way the quantities 25.6, 2.56, 0.256, &c., are measured.

Besides the lines already mentioned, there is another on the Plane Scale, marked ML, which is joined to a line of chords, and shows how many miles of easting or westing correspond to a degree of longitude in every latitude. These several lines are generally put on one side of a ruler two feet long; and on the other side is laid down a scale of the logarithms of the sines, tangents, and numbers, which is commonly called Gunter's Scale; and, as it is of general use, it requires a particular description.

*As it would confuse the adjoined figure to describe on it the line of longitudes, it is neglected, but the construction is as follows; divide the line CB into 60 equal parts (if it can be done), and through each point draw lines parallel to CD, to intersect the arc BD; about B, as a centre, transfer the several points of intersection to the line of chords, BD, and then number it from D towards B, from 0 to 60, and it will be the line of longitudes, corresponding to the degrees on the line of chorda