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or ivory*, usually about 6 inches long, and 12 inches broad; and in its simplest form contains only two parallel lines on one of its faces, drawn in the direction of its length, and divided by small lines at right angles to the former, and at equal intervals of 1 inch, in., or some other unit of length agreed upon. Thus ABDC represents such an instrument, the two parallel lines on its face being divided into 6 equal parts, and the points of division marked 0, 1, 2, 3, 4, 5.

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The length of each of the portions so formed may be taken to represent one mile, or one yard, or other unit of length; and for the subdivision of the unit the first of them to the left is divided into so many equal parts, that each shall represent one of the denomination next inferior to that of the assumed unit, or some convenient number of them. Thus, if the Scale be one of feet, the subdivisions will be inches, that is, twelfths of the unit: if the Scale be one of yards, the subdivisions will be feet, or thirds of the unit; if of miles, the subdivisions will be furlongs, or eighths of the unit, &c. For general purposes, however, it is most convenient to subdivide the unit into tenths, as in our diagram above.

When this Scale is used, and the unit on it is an inch, as for example in laying down lines, or lengths, to a scale of a yard to an inch, suppose we want to find the line corresponding to 48 yards; put one foot of the compasses upon the point in the Scale numbered 4, and the other upon the number 8 in the subdivisions of the unit; then it is clear, that there is intercepted between the points of the compasses a length of 4 units and 8 tenths, that is, 4.8. And by considering the unit on the

* Ivory, though commonly used, is a bad material for the purpose, since its length varies with moisture; box is much better.



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cepted length will represent 4-8, or 48, or 480, or 48, or 048, respectively.

Ex. 1. Represent 118 ft. in a Scale of a foot to onetenth of an inch.

This quantity will measure on the Scale 118 tenths of an inch, or 11.8 inches, i. e. once the length of the Scale, (if its whole length be 6 inches,) and 5.8 inches more. So that, after taking the whole length of the Scale by the compasses, a second measurement must be taken, wherein one point will be exactly at the end of the Scale, on the figure 5, and the other upon the subdivision marked 8.

Ex. 2. What must one inch of the Scale represent, in order that '18 feet may be represented upon it, by the distance between the large division marked 1, and the subdivision marked 8?

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th of 1 foot+ ths of 1 foot; hence,



each of the large divisions must represent


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and the small divisions th of 1 foot; or, the Scale will


be one-tenth of a foot to an inch, or 1 foot to 10 inches.

If the quantity had been 018 feet, the Scale must be 1 inch to one-hundredth of a foot, or 100 inches to a foot.

N.B. The subdivisions of the unit may be other than tenths.

Thus, suppose them to be twelfths of an inch, each twelfth representing 1 foot; and let it be required to measure 43 ft. thereby.


Then, 43 ft. = 43 twelfths of an inch-3 hence the compasses must embrace 3 units and 7 twelfths; or, one point must be upon the larger division marked 3, and the other on the subdivision marked 7.

Nothing less than 1 foot could be laid down from this Scale. Also any large number of yards and feet,

If it be required to set off the angle, so that a proposed given straight line shall be one of the bounding lines which form the angle, then it is obvious that the first foot of the compasses must be placed upon the intersection of the arc with the given line.

Thus, if AB be a given straight line, and it is required to draw another line making an angle of 35° with the former, with centre A, and radius 60 from the line of chords, describe an arc cutting AB in C; then take 35 from the same scale, and set it off from C to D. 35°.

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Join AD, and CAD is an angle of

If the angle to be laid down be obtuse, since the scale of chords does not go beyond 90°, the angle must be divided into two, viz. 90°, and the excess above 90°; each of these being laid down separately, but contiguous, the sum of the two will plainly be the angle required.

2ndly. To measure an angle already laid down; let BAE be the given angle. With centre A, and radius 60 from the line of chords, describe an arc intersecting AB and AE in C and D. Then, with the compasses, take the length of the chord of CD, and apply it to the line of chords with one foot upon 0; the number which coincides with the other foot will be the numerical measure of the proposed angle in degrees.

251. It has been shewn that by means of the Plain, or Diagonal, Scale, diagrams or plans are reduced in any required proportion; and it is to be understood that similarly they may be enlarged, if needful, in any given proportion. Such results may also sometimes be conveniently obtained by means of the Proportional Compasses, or the Pantagraph, described in (177 and 178, Part II.). But in all cases the learner must bear in mind that the reduction or enlargement in question is according to

linear measure; that is, corresponding lines, and not areas, are in the stated proportion. Thus, if any diagram, or plan, in the form of a polygon, is to be reduced from, suppose, the scale of a yard to an inch, to the scale of a yard to one-fourth of an inch, a similar polygon is constructed, in which each side is one-fourth of the corresponding side of the former polygon; but, since by (92, Part 1.) the areas of similar polygons are to one another as the squares of any homologous sides, the area of the new polygon is not th, but

area of the given polygon.



16th, of the

In like manner, if in any diagram or plan, which is to be reduced or enlarged according to a certain scale, a circular area is found, the reduction or enlargement is effected by taking the radius according to the reduced or enlarged scale, and describing such an arc as will subtend the same angle at the centre. The circular arcs in the two diagrams will thus be in the stated proportion; but the areas, as in the case of rectilineal figures, will be to each other as the squares of the radii (see 93, Part 1.).


[In the following Exercises the Scale is decimally divided, except when it is otherwise stated.]

(1) Explain clearly the object of Scales in general; and exhibit the simplest form of Scale in common use. (2) Point out the difference between a Plain Scale and a Diagonal Scale, both as to form and power.

(3) What is the greatest error which can arise from measuring with an ordinary Diagonal Scale?

Ans. Less than 01.

(4) State the position of the feet of the compasses on a Diagonal Scale, when they include a length measured by the number 3.29.

(5) Explain the operations of laying down, from the same Diagonal Scale, the dimensions represented by the numbers, 187.5, 245·3, and 110.5.

(6) What alteration of the unit of measurement is necessary, to enable us, by the same interval between the



feet of the compasses, to indicate 329, 3.29, 32′9, and 329.

(7) On a Diagonal Scale, which is a foot in length, and divided into ten equal parts, how many inches and decimal parts of an inch would measure the several numbers, 327, 453, and 35?

Ans. 3.924 in., 5·436 in., 42 in.

(8) What is the length of a Scale, divided into 20 units, on which the number 18.5 measures 3 inches and 7 tenths? Ans. 4 inches.

(9) If the base of the diagonal compartment of a Scale be divided into 8, and the height into 10, equal parts, how many of the lowest measures on the scale are contained in one of the highest? Ans. 80.

(10) Suppose each of the first subdivisions of the primary unit in the scale (Ex. 9) to represent 3 inches, what will be the arithmetical measures of its smallest, and of its primary, divisions?

(1) Ans. ths of an inch. (2) Ans. 2 feet.

(11) What was the Scale used in the construction of a plan, upon which every square inch of surface represents a square yard? Ans. Scale of 3 feet to an inch.

(12) What is the Plain Scale on which a length of 4 ft. 10 in. measures exactly 45 inches?

(13) What is the Diagonal Scale, measuring inches, upon which 4 ft. 10 in. is represented by 2 inches? Ans. Scale of 2 feet to an inch.

(14) The ratio of one Scale is 2: 1, and of another 3:1, on which will a given length measure most?

(15) In what ratio will the scale length of a given line, as measured on a scale, whose ratio is 4: 1, exceed that of the same line as measured on another scale, whose ratio is 5: 1? Ans. 74: 6, or 5: 4.

(16) A draughtsman laid aside an unfinished plan, and after a while resumed his work, but found that he had forgotten the scale. How will he proceed to recover the lost scale?

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