SCALE

, a mathematical instrument, consisting of certain lines drawn on wood, metal, or other matter, divided into various parts, either equal or unequal. It is of great use in laying down distances in proportion, or in measuring distances already laid down.

There are Scales of various kinds, accommodated to the several uses: the principal are the plane Scale, the diagonal Scale, Gunter's Scale, and the plotting Scale.

Plane or Plain Scale, a mathematical instrument of very extensive use and application; which is commonly made of 2 feet in length; and the lines usually drawn upon it are the following, viz, |

1. The lines of equal parts are of two kinds, viz, simply divided, and diagonally divided. The first of these are formed by drawing three lines parallel to one another, and dividing them into any equal parts by short lines drawn across them, and in like manner subdividing the first division or part into 10 other equal small parts; by which numbers or dimensions of two figures may be taken off. Upon some rulers, several of these scales of equal parts are ranged parallel to each other, with figures set to them to shew into how many equal parts they divide the inch; as 20, 25, 30, 35, 40, 45, &c. The 2d or diagonal divisions are formed by drawing eleven long parallel and equidistant lines, which are divided into equal parts, and crossed by other short lines, as the former; then the first of the equal parts have the two outermost of the eleven pa- rallels divided into 10 equal parts, and the points of division being connected by lines drawn diagonally, the whole scale is thus divided into dimensions or numbers of three places of figures.

The other lines upon the scales are such as are commonly used in trigonometry, navigation, astronomy, dialling, projection of the sphere, &c, &c; and their constructions are mostly taken from the divisions of a circle, as follow:

Describe a circle with any convenient radius, and quarter it by drawing the diameters AB and DE at right angles to each other; continue the diameter AB out towards F, and draw the tangent line EG parallel to it; also draw the chords AD, DB, BE, EA. Then.

2. For the line of chords, divide a quadrant BE into 90 equal parts; on E as a centre, with the compasses transfer these divisions to the chord line EB, which mark with the corresponding numbers, and it will become a line of chords, to be transferred to the ruler.

3. For the line of rhumbs, divide the quadrant AD into 8 equal parts; then with the centre A transfer the divisions to the chord AD, for the line of rhumbs.

4. For the line of sines, through each of the divisions of the arc BE, draw right lines parallel to the radius BC, which will divide the radius CE into the sines, or versed sines, numbering it from C to E for the sines, and from E to C for the versed sines.

5. For the line of tangents, lay a ruler on C, and the several divisions of the arc BE, and it will intersect the line EG, which will become a line of tangents, and numbered from E to G with 10, 20, 30, 40, &c.

6. For the line of secants, transfer the distances between the centre C and the divisions on the line of tangents to the line BF, from the centre C, and these will give the divisions of the line of secants, which must be numbered from B towards F, with 10, 20, 30, &c.

7. For the line of semitangents, lay a ruler on D and the several divisions of the arc EB, which will intersect the radius CB in the divisions of the semitangents, which are to be marked with the corresponding figures of the arc EB.

The chief uses of the sines, tangents, secants, and semitangents, are to find the poles and centres of the several circles represented in the projections of the sphere.

8. For the line of longitude, divide the radius CD into 60 equal parts; through each of these, parallels to the radius BC will intersect the arc BD in as many points: from D as a centre the divisions of the arc BD being transferred to the chord BD, will give the divisions of the line of longitude.

If this line be laid upon the scale close to the line of chords, both inverted, so that 60° in the scale of longitude be against 0° in the chords, &c; and any degree of latitude be counted on the chords, there will stand opposite to it, in the line of longitude, the miles contained in one degree of longitude, in that latitude: the measure of 1 degree under the equator being 60 geographical miles. |

9. For the line of latitude, lay a ruler on B, and the several divisions on the sines on CE, and it will intersect the arc AE in as many points; on A as a centre transfer the intersections of the arc AE to the chord AE, for the line of latitude.

See also Robertson's Description and use of Mathematical Instruments.

Decimal, or Gunter's, or Plotting, or Proportional, or Reducing Scale. See the several articles.

, in Architecture and Geography, a line divided into equal parts, placed at the bottom of a map or draught, to serve as a common measure to all the parts of the building, or all the distances and places of the map.

In maps of large tracts, as kingdoms and provinces, &c, the Scale usually consists of miles; whence it is denominated a Scale of miles.—In more particular maps, as those of manors, &c, the Scale is usually of chains &c.—The Scales used in draughts of buildings mostly consist of modules, feet, inches, palms, fathoms, or the like.

To find the distance between two towns &c, in a map, the interval is taken in the compasses, and set off in the scale; and the number of divisions it includes gives the distance. The same method serves to find the height of a story, or other part in a design.

Front Scale, in Perspective, is a right line in the draught, parallel to the horizontal line; divided into equal parts, representing feet, inches, &c.

Flying Scale, is a right line in the draught, tending to the point of view, and divided into unequal parts, representing feet, inches, &c.

Differential Scale, is used for the scale of relation subtracted from unity. See Series.

Scale of Relation, in Algebra, an expression denoting the relation of the terms of recurring series to each other. See Series.

Hour Scale. See Hour.

, in Music, is a denomination given to the arrangement of the six syllables, invented by Guido Aratino, ut re mi fa sol la; called also gammut. It is called Scale, or ladder, because it represents a kind of ladder, by means of which the voice rises to acute, or sinks to grave; each of the six syllables being as it were one step of the ladder.

Scale is also used for a series of sounds rising or falling towards acuteness or gravity, from any given pitch of tune, to the greatest distance that is sit or practicable, through such intermediate degrees as to make the succession most agreeable and perfect, and in which we have all the harmonical intervals most commodiously divided.

The scale is otherwise called an universal system, as including all the particular systems belonging to music. See System.

There were three different Scales in use among the Ancients, which had their denominations from the three several sorts of music, viz, the diatonic, chromatic, and inharmonic. Which see.