the II parallels to the line C F; subdivide the first of these divisions A B and C D into 10 equal parts, and from the point C to the first division in the line A B, draw a diagonal line; and then lines parallel to this through each succeeding subdivision. Proceed similarly with the subdivision of the part of the scale on the left hand.

Then, if the larger divisions be reckoned as units, the first subdivisions will be tenths, and the second (marked by the diagonals upon the parallels) hundredths; but if we take each of the larger divisions to represent 10, then the first subdivisions will be units, and the second tenths; or if the larger divisions be hundreds, then will the first subdivisions be tens, and the second units; so that the value of the subdivisions depends on that of the larger divisions.

The numbers 376, 37·6, 3·76 may therefore all be expressed by the same extent of the compasses thus, setting one foot in the line marked 3 of the larger division, on the sixth parallel, and extending the other along the same parallel to the seventh diagonal, that distance will be the extent required; for if the three larger divisions be taken for 300, seven of the first subdivisions will be 70, which, upon the sixth parallel, taking in six of the second subdivisions for units, make the whole number 376: or if the three larger divisions be taken for 30, seven of the first subdivisions will be seven units, and the six subdivisions, upon the sixth parallel, will be six-tenths of a unit lastly, if the three larger divisions be esteemed as only three, then will the first subdivisions be seven-tenths, and the six second subdivisions be the six-hundredth part of a unit.

PROBLEM XI

To construct Lines of Chords, Rhumbs, Tangents, Sines, etc.

Describe a semicircle A D B with any convenient radius (Fig. 3, Plate I.), and from the centre C erect the perpendicular C D, continued at pleasure to F; through B draw B E parallel to C F; and draw the lines A D and D B. Divide the quadrant D B into nine equal parts, and with one foot of the compasses in B and the distances B 10, B 20, etc., transfer them to the line B D, which will be a LINE OF CHORDS.

Divide the quadrant A D into eight equal parts, and with one foot of the compasses in A, and the distance A 1, and A 2, etc., transfer them to the line A D, and it will be a LINE OF RHUMBS, containing eight points of the compass.

From the points 10, 20, 30, etc., in the arc B D, draw lines parallel to D C, which will divide the radius C B into a LINE OF SINES, reckoning from C to B, or of VERSED SINES, if it be numbered from B to C; which may be continued to 180, if the same divisions be transferred to the line C A, the other half of the diameter.

From the centre C draw lines through the several divisions of the quadrant D B until they cut the line B E, which will become a LINE OF TANGENTS.*

Transfer the distances between the centre C and the divisions on the line of tangents to the line D F, and these will give the divisions of the LINE OF SECANTS, which must be numbered from D towards F.

From A draw lines through the several divisions of the arc B D, and

* From the construction of the lines of chords, sines, and tangents, it is obvious that the chord of 60°, the sine of 90°, and the tangent of 45° are all equal to the radius of the circle.

they will divide the radius C D into a LINE OF SEMI-TANGENTS, which are to be marked with the corresponding figures of the arc D B.

Divide the radius A C into six equal parts; through each of these draw lines parallel to C D, intersecting the arc A D; then, with one foot of the compasses in A, and the distances of the arc A 50, A 40, etc., transfer these to the line A D, and it will give the divisions of the LINE Of Longitude.

If this line be laid upon the scale close to the line of chords, so that 60 on the line of longitude be opposite o on the chords, and any degree of latitude be counted on the chords, there will stand opposite to it, on the line of longitude, the miles contained in one degree of longitude in that latitude, the measure of a degree at the equator being 60 miles.

In the figure the divisions are given only to every tenth degree, and each point of the compass, which is sufficient to explain the method of construction; but in Fig. 4 these lines are graduated to degrees, and the rhumbs to quarters, and placed parallel, as exhibited on one side of a flat rule, which, with the scale of equal parts on the other side, constitutes the instrument called a PLANE SCALE,

Besides the lines already mentioned, there are frequently on the Plane Scale a few other lines, but these are only so many scales of equal parts, each having equal divisions of different lengths, for the more readily laying down lines and figures of different lengths and magnitudes.

PROBLEM XII

To make an Angle that shall contain any proposed Number of Degrees

NOTE.-Angles are measured or laid off by means of the Scale of Chords (see Fig. 4). But a brass semicircle, or transparent horn semicircle, is a very useful instrument for all chart purposes.

CASE ISt. When the given angle is right, that is, contains go degrees.

Draw the line A B, and from the scale take the extent of the chord of 60 degrees in the compasses; then set one foot of the compasses in A, and with the other describe the arc E D, and set off thereon, from E to D, the distance of the chord of 90°; through A and D draw the line A C, then will the angle BAC be a right angle. By this method a perpendicular may easily be raised on a given line, since the angle formed by one line that is perpendicular to another is always a right angle.

CASE 2nd. When the angle is to be acute; suppose one that shall contain 48 degrees.

Draw the line A B, and with one foot of the compasses in A (the chord of 60 degrees being taken as before), draw the arc E D, on which set off 48 degrees from E to D; through A and D draw the line A C; then will the angle B A C be made, containing 48 degrees, as was required.

CASE 3rd. When the angle is to be obtuse; suppose one that shall contain 126° 30'.

Draw the line A B, and from the point A, with the chord of 60° as before, draw the arc D E, and, as the divisions on the scale extend no further than 90°, first set off 90° from E to F; then set off the remainder, or excess above 90°, that is 36° 30', from F to D; through A and D draw the line A C, and the angle B A C will contain 126° 30'.

PROBLEM XIII

To measure a given Angle B AC

-126°30'

With one foot of the compasses in the angular point, and with the chord of 60 degrees, describe the arc D E (see the figures in Problem II.) cutting the legs in D and E; then the distance D E applied to the line of chords, from the beginning, will show the measure of the angle B A C, if it contain less than 90 degrees; but when the arc exceeds that quantity, take 90 degrees from the line of chords, and set it off from E to F; then measure the excess F D, and their sum will give the measure of the angle required. PROBLEM XIV

To describe a triangle of which the three sides shall be respectively equal to three given straight lines, two of these lines being greater than a third.

Let A B C be the three given lines. Take a straight line equal in length to A and call it D E. About D as a centre with a radius equal to B describe an arc of a circle, and about E as a centre with a radius equal to C describe another arc intersecting the former in the point. F. Join D F and E F, and FDE is the triangle required.

PROPOSITION

D

A

B

C

E

If a straight line falling upon two other straight lines makes the exterior angle equal to the interior and opposite upon the same side of the line, or makes the interior angles upon the same side together equal to two right angles, the two straight lines shall be parallel to one another.

Let the straight line E F, which falls upon the two straight lines A B, CD, make the exterior angle E G B equal to the interior and opposite angle GHD upon the same side; or make the two interior angles B.G H, G H D, on the same side, together equal to two right angles; A B is parallel to C D.

Because the angle E G. B is equal to

A

E

the angle G H D, and the angle E G B is equal to the angle A G H, the angle A G H is equal to the angle G H D, and they are alternate angles; therefore A B is parallel to C D. (Euclid 1-27).