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one of the subdivisions on the lowest line, is equal (Euc. vi. prop. 4) to one-tenth of a subdivision on the second line, to two-tenths of a subdivision on the third, and so on; so that this, and consequently each of the other diagonal lines, as it reaches each successive parallel, separates further from the perpendicular through the zero point by one-tenth of the extent of a subdivision, or one-hundredth of the extent of a primary division. Our figure represents the two diagonal scales. which are usually placed upon the plane scale of six inches in length. In one, the distances between the primary divisions are each half an inch, and in the other a quarter of an inch. The parallel next to the figures numbering these divisions must be considered the highest or first parallel in each of these scales to accord with the above description.

The primary divisions being taken for units, to set off the numbers 5.74 by the diagonal scale. Set one foot of the compasses on the point where the fifth parallel cuts the eighth diagonal line, and extend the other foot to the point where the same parallel cuts the sixth vertical line.

The primary divisions being reckoned as tens, to take off the number 46.7. Extend the compasses from the point where the eighth parallel cuts the seventh diagonal to the point where it cuts the fifth vertical.

The primary divisions being hundreds, to take off the number 253. Extend the compasses from the point where the fourth parallel cuts the sixth diagonal to the point where it cuts the third vertical.

Now, since the first of the parallels, of the diagonals, and of the verticals indicate the zero points for the third, second, and first figures respectively, the second of each of them. stands for, and is marked, 1, the third, 2, and so on, and we have the following

General Rule.-To take off any number to three places of figures upon a diagonal scale. On the parallel indicated by the third figure, measure from the diagonal indicated by the second figure to the vertical indicated by the first.

Vernier Scales.-The nature of these scales will be understood from their construction. To construct a vernier scale, which shall enable us to take off a number to three places of figures: divide all the primary divisions into tenths, and number these subdivisions, 1, 2, 3, &c., from the left hand towards the right throughout the whole extent of the scale.

Take off now, with the compasses, eleven of these subdivisions, set the extent off backwards from the end of the first

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primary division, and it will reach beyond the beginning of this division, or zero point, a distance equal to one of the subdivisions. Now divide the extent thus set off into ten equal parts, marking the divisions on the opposite side of the divided line to the strokes marking the primary divisions and the subdivisions, and number them 1, 2, 3, &c., backwards from right to left. Then, since the extent of eleven subdivisions has been divided into ten equal parts, so that these ten parts exceed by one subdivision the extent of ten subdivisions, each one of these equal parts, or, as it may be called, one division of the vernier scale, exceeds one of the subdivisions by a tenth part of a subdivision, or a hundredth part of a primary division. In our figure the distances between the primary divisions are each one inch, and, consequently, the distances between the subdivisions are each one-tenth of an inch, and the distances between the divisions of the vernier scale each one-tenth and one-hundredth of an inch.

To take off the number 253 from this scale. Increase the first figure 2 by 1, making it 3; because the vernier scale commences at the end of the first primary division, and the primary divisions are measured from this point, and not from the zero point *. The first thus increased with the second now represents 35 of the subdivisions from the zero point, from which the third figure, 3, must be subtracted, leaving 32; since three divisions of the vernier scale will contain three of these subdivisions, together with three-tenths of a subdivision. Place, then, one point of the compasses upon the third division of the vernier scale, and extend the other point to the 32nd subdivision, or the second division beyond the 3rd primary division, and laying down the distance between the points of the compass, it will represent 253, or 25.3, or 2.53, according as the primary divisions are taken as hundreds, tens, or units.

General Rule.-To take off any number to three places of figures upon this vernier scale. Increase

the first figure by one; subtract the third figure from the second, borrowing one from the first increased figure, if ne

* If the vernier scale were placed to the left of the zero point, a distance less than one primary division could not always be found upon the scale.

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 semitangents, 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 в 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 в, and we shall have a scale of latitudes †.

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 BC, 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.

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