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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.

1

4 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.).

QUESTIONS AND EXERCISES K.

[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

PART III.

6

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., 5436 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 4 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. 7: 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?

(17) A Land-Measuring Chain being 22 yards in length, make a diagonal scale of two chains to an inch, to shew feet.

(18) The chain, as before, being 22 yards, if a scale of 10 chains to an inch be taken, what will be the measure in acres of each square inch on the plan?

Ans. 10 acres.

(19) If each square inch on the plan were to represent 1 acre of surface, what would the scale be?

Ans. 10 chains to an inch.

(20) If a Scale be taken of 2 perches to an inch, and the base of the diagonal compartment be divided into 11 equal parts, whilst the height is divided into 12 equal parts, what will the smallest subdivision measure?

Ans. 3 inches.

LAND-SURVEYING.

252. One chief use of 'Geometry combined with Arithmetic' consists in the mapping, and measuring, of land, called Land-Surveying'.

The art of 'Land-Surveying' includes two branches:

1st. The laying down on paper a representation, or map, of an estate or parcel of ground to be surveyed.

2nd. The measuring in known units, as in square yards and feet, or in acres, roods, and perches, &c., the content or area of the land proposed.

The second operation can be performed without the first, that is, without producing an exact plan, or map; for we may draw a rough sketch of the land, and if certain linear measurements be correctly taken, we may then calculate the area without any further reference to the plan, except as to its general outline. Or, we may make the plan accurately correct to any given scale, and then measure the area from the plan, according to the methods

given before for measuring plane areas of any form, rectilineal, or circular.

Both the above methods will be here exhibited, and examples worked, with a view of teaching the principles, but not all the practical details, of Land-Surveying.

253. Το

small piece of land and measure, any map, bounded by straight lines, and considered as a plane surface.

1st. Let the plot be a triangle, as ABC, which can be traversed in any direction. Measure with a tape, on the ground, each of the sides AB, BC, CA.

Draw upon the paper a line, in any convenient direction, and on it lay off, by a scale, a length BC representing the arithmetical magnitude of the longest side BC. Next, thom the same scale take off the measured distances represented by AB, AC; and with these as radii, from Centres B and C, describe small intersecting arcs, to fix the true position of 4. Then join AB, AC, and 450 will be a correct map of the proposed plot of ANNING

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containing the right angle. The area will be half the product of those two sides.

2ndly, Let the plot be a parallelogram. Then, since it can be divided into two equal triangles by either of its diagonals, the mapping and measurement will be as before, except that when the sides of ABC have been laid down to the proposed scale, the remaining sides of the parallelogram are found by drawing lines parallel to AC, AB, respectively. And when the perpendicular AD has been determined as before, the area will be equal to the product of AD and BC.

If the parallelogram be rectangular, the perpendicular will coincide with one of the smaller sides, and the only measurements required will be those of any two adjacent sides.

Of course, if the plot be a square, it will only be necessary to measure a single side (222).

3rdly, Let the plot be a trapezium, that is, a quadrilateral with two of its sides parallel.

Measure one of the two sides that are not parallel and on any convenient line on the paper lay off, as before, by a scale, a length representing the arithmetical magnitude of that line. If the parallel sides be at right angles to this, draw two lines at right angles to the above base line from its extremities; measure the parallel sides, and lay off from the same scale their magnitudes upon these perpendiculars, join their extremities, and the plot is correctly mapped.

The area also is found, according to the method of Prob. 9, p. 217, by multiplying the numbers representing the magnitude of the base line and half the sum of the parallel sides.

But if the parallel sides be not at right angles to either of the other sides, then the area will be the product of half the sum of the parallel sides and the perpendicular distance between them; or it may be thought fit to treat the question as one of a general quadrilateral figure.

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