cessary, and extend the compasses from the division upon the vernier scale, indicated by the third figure, to the subdivision indicated by the number remaining after performing the above subtraction.

Suppose it were required to take off the number 253.5. By extending the compasses from the third division of the vernier scale to the 32nd subdivision, the number 253 is taken off, as we have seen. To take off, therefore, 253-5, the compasses must be extended from one of these points to a short distance beyond the other. Again, by extending the compasses from the 4th division of the vernier scale to the 31st subdivision, the number 254 would be taken off. To take off 253.5, then, the compasses must be extended from one of these points to within a short distance of the other; and by setting the compasses so that, when one point of the compasses is set successively on the 3rd and 4th division of the vernier scale, the other point reaches as far beyond the 32nd subdivision as it falls short of the 31st, the number 253.5 is taken off. If the excess in one case be twice as great as the defect in the other, the distance represents the number 253, or 253.66; and if the excess be half the defect, the distance represents 2531, or 253.33. Thus distances may be set off with an accurately-constructed scale of this kind to within the threehundredth part of a primary division, unless these divisions be themselves very small.

We are not aware that a scale of this kind has been put upon the plain scales sold by any of the instrument makers; but, during the time occupied in plotting an extensive survey, the paper which receives the work is affected by the changes which take place in the hygrometrical state of the air, and the parts laid down from the same scale, at different times, will not exactly correspond, unless this scale has been first laid down upon the paper itself, and all the divisions have been taken from the scale so laid down, which is always in the same state of expansion as the plot. For plotting, then, an extensive survey, and accurately filling in the minutiæ, a diagonal, or vernier scale may advantageously be laid down upon the paper upon which the plot is to be made. A vernier scale is preferable to a diagonal scale, because in the latter it is extremely difficult to draw the diagonals with accuracy, and we have no check upon its errors; while in the former the uniform manner in which the strokes of one scale separate from those of the other is some evidence of the truth of both*.

* In Mr. Bird's celebrated scale, by means of which he succeeded in di

ON THE PROTRACTING SCALES.

The nature of these scales will be understood from the following construction (plate 1, fig. 1):

With centre o, and radius o A, describe the circle A B C D; and through the centre o draw the diameters a C, and B D, at right angles to each other, which will divide the circle into four quadrants, A B, B C, C D, and D A.

Divide the quadrant c D into nine equal parts, each of which will contain ten degrees, and these parts may again be subdivided into degrees, and, if the circle be sufficiently large, into minutes.

Set one foot of the compasses upon c, and transfer the divisions in the quadrant c D to the right line c D, and we shall have a scale of chords *.

From the divisions in the quadrant c D, draw right lines parallel to D o, to cut the radius o c, and, numbering the divisions from o, towards c, we shall have a scale of sines.

If the same divisions be numbered from c, and continued to A, we shall have a scale of versed sines.

From the centre o, draw right lines through the divisions of the quadrant c D, to meet the line c T, touching the circle at c, and, numbering from c, towards T, we shall have a scale of tangents.

Set one foot of the compasses upon the center o, and transfer the divisions in c T into the right line o s, and we shall have a scale of secants.

Right lines, drawn from a to the several divisions in the quadrant c D, will divide the radius o D into a line of semi tangents, or tangents of half the angles indicated by the numbers; and the scale may be continued by continuing the divisions from the quadrant c D, through the quadrant D A,

viding, with greatly-improved accuracy, the circles of astronomical instruments, the inches are divided into tenths, as in the scale described in the text, and 100 of these tenths are divided into 100 parts for the vernier scale.

* We give the constructions in the text to show the nature of the scales; but in practice a scale of chords is most accurately constructed by values computed from tabulated arithmetical values of sines, which computed values are set off from a scale of equal parts; and the circle is divided most accurately by means of such computed chords. The limits of our work forbid our entering further upon this interesting subject. All the other scales will also be most accurately constructed from computed arithmetical values, taken off by means of the beam compasses hereafter described, and corrected by the aid of a good Bird's vernier scale.

and drawing right lines from A, through these divisions, to meet the radius o D, produced.

Divide the quadrant A D into eight equal parts, subdivide each of these into four equal parts, and, setting one foot of the compasses upon A, transfer these divisions to the right line A D, and we shall have a scale of rhumbs.

Divide the radius A o into 60 equal parts, and number them from o towards A; through these divisions draw right lines parallel to the radius o B, to meet the quadrant A B; and, with one foot of the compasses upon A, transfer these divisions from the quadrant to the right line A B, and we shall have a scale of longitudes.

Place the chord of 60°, or radius*, between the radii o c and o B, meeting them at equal distances from the center; divide the quadrant C B into six equal parts, for intervals of hours, subdividing each of these parts into 12 for intervals of 5 minutes, and further subdividing for single minutes if the circle be large enough; and from the center o draw right lines to the divisions and subdivisions of the quadrant, intersecting the chord or radius placed in the quadrant, and we shall have a scale of hours.

Prolong the touching line T C to L; set off the scale of sines from c to L; draw right lines from the center o to the divisions upon c L, and from the intersections of these lines with the quadrant c B draw right lines parallel to the radius o c, to meet the radius o B, and we shall have a scale of latitudest.

Corresponding lines of hours and latitudes may also be con structed (as represented in our figure) more simply, and on a scale twice as large as by the preceding method, as follows:

With the chord of 45° set off from в to E, and again from B to F, we obtain a quadrant E F bisected in в; and, the chord of 60° or radius being set off from a, C, F, and E, this quadrant is divided into six equal parts. From the center o, draw straight lines through these divisions to meet the line touching the circle at B, and we shall have the line of hours.

From the point D, draw right lines through the divisions upon the line of sines o c, to meet the circumference в c, and * Chord of 60° is equal to radius. Euc. book iv. prop. 15, Cor.

The line of latitudes is a line of sines, to radius equal the whole length of the line of hours, of the angles, of which the tangents are equal to the sines of the latitudes. The middle of the hour line being numbered for three o'clock, the divisions for the other hours are found by setting off both ways from the middle the tangents of n. 15°, n. being the number of hours from three o'clock, that is, one for two o'clock and four o'clock, two for one o'clock and five o'clock, and three for twelve o'clock and six o'clock.

transferring these divisions from B, as a center to the chord B C, we shall have the corresponding line of latitudes.

It is not necessary that these scales should all be projected to the same radius; but those which are used together, as the rhumbs and chords, the chords and longitudes, the sines, tangents, secants, and semitangents, and, lastly, the hours and latitudes, must be so constructed necessarily. In the accom

panying diagram (plate 1, fig 2) we have laid down the hours and latitudes to a radius equal to the whole length of the scale, the other lines being laid down to the radius used in the foregoing construction.

The Line of Chords is used to set off an angle, or to measure an angle already laid down.

1st. To set off an angle, which shall contain 'D° from the point A, in the straight line Open the compasses to

A B.

the extent of 60° upon the
line of chords, which equals the
radius to which this line has A
been laid down (Euc. iv. prop.

B

15, Cor.), and, setting one foot upon A, with this extent describe an arc cutting A B in B; then, taking the extent of D from the same line of chords, set it off from в to c; and, joining a C, B A C is the angle required. Thus to set off an angle of 41°, having described the arc в C, as directed, with one foot of the compasses on B, and the extent of 41° on the line of chords, intersect B c in c, and join a c.

2nd. To measure the angle contained by the straight lines A B and A c already laid down. Open the compasses to the extent of 60° on the line of chords, as before, and with this radius describe the arc B c, cutting A B and A C, produced, if necessary, in the points в and c; then, extending the compasses from в to c, place one point of the compasses on the beginning, or zero point, of the line of chords, and the other point will extend to the number upon this line, indicating the degrees in the angle B A C. If, for instance, this point fall on the 41st division, or the first division beyond that marked 40 in the figure (plate 1, fig, 2), the angle B A C will contain 41°.

The Line of Rhumbs is a scale of the chords of the angles of deviation from the meridian denoted by the several points and quarter points of the compass, enabling the navigator, without computation, to lay down or measure a ship's course upon a chart Thus, supposing the ship's course to be

N.N.E. E. Through the point a, representing the ship's place upon the chart, draw the meridian A B, and with center A and distance equal to the extent of 60° upon the line of chords describe an arc cutting A B in B; then on the line of rhumbs take the extent to the third subdivision beyond the division marked 2, because N.N.E. is the second point of the compass from the north, and with one foot of the compasses on B describe an arc intersecting B C in c: join a c, and the angle B A C will represent the ship's course. On the other hand, if a

ship is to be sailed from the point A to a point on the line A c on a chart, draw the meridian A B, describe the arc B C with radius equal to chord of 60°, as before, and the extent from B to c, applied to the line of rhumbs, will give 2 pts. 3 qrs., denoting that the ship must be sailed by the compass N.N.E. E.

The Line of Longitudes shows the number of equatorial miles in a degree of longitude on the parallels of latitude in dicated by the degrees on the corresponding points of the line of chords. Example.-A ship in latitude 60° N. sailing E. 79 miles, required the difference of longitude between the beginning and end of her course. Opposite 60 on the line of chords stands 30 on the line of longitudes, which is, therefore, the number of equatorial miles in a degree of longitude at that latitude. Hence, as 30: 79 :: 60: 158 miles, the required difference of longitude.

The Lines of Sines, Secants, Tangents, and Semitangents are principally used for the several projections, or perspective representations, of the circles of the sphere, by means of which maps are constructed. Thus, the meridians and parallels of latitude being projected, the countries intended to be represented are traced out according to their respective situations and extent, the position of every point being determined by the intersection of its given meridian and parallel of latitude.

The plane upon which the circles are to be delineated is called the primitive, and the circumference of a circle, described with a radius, representing, upon the reduced scale of the drawing, the radius of the sphere, is called the circum ference of the primitive. Lines, drawn from all the points of the circles to the eye, by their intersection with the primitive form the projection.