0.5

1 cm

8

ENGINEER AND MACHINIST’S DRAWING-BOOK.

ruler to another, are proportional to the lateral distances,

measured on the lines C A, Cl; and thus we may apply

any radius transversely to the line of chords to measure

or lay down any given or required angle ; and apply any

line transversely to the line of lines, to divide it in any

required proportions. The sector is therefore seen to be

of universal application, whilst the use of plain scales is

limited and special.

Plain Scales on the Sector.—On the outer edge of the

sector is usually given a decimal scale from 1 to 100; and

in connection with it, on one of the sides, a scale of inches

and tenths. These are identical with the lines on the

plain scale, previously mentioned, hut the latter are more

commodiously placed for use. On the other side we have

logarithmic lines of numbers, sines, and tangents; but

as these are more complicated than the ordinary plain

scales, we defer the consideration of them until we have

discussed the double scales.

Sectoral Double Scales.—These are respectively named

the Lines of Lines, Chords, Secants, Sines, and Tangents.

These scales have one line on each ruler, and the two lines

converge accurately in the central joint of the sector.

The Line of Lines.—This is a line of 10 primaries, each

subdivided into tenths, thus making 100 divisions. Its

use is, to divide a given line into

any number of equal parts; to give

accurate scale measures for the con-

struction of a drawing; to form any

required scale; to divide a given

line in any assigned proportion; and

to find third, fourth, and middle pro-

portionals to given right lines. The

scale can be applied to other pur-

poses ; but, if we take up those men-

tioned, they will be sufficient illus-

trations of its uses. Before entering

upon these propositions, we would

remark that a lateral distance is one

taken from the centre down either half of the scale ; and

a transverse distance is one measured across from scale to

scale. Thus (Fig. 16), a 1, a 2, a 3, &c., are lateral dis-

tances; and 1.1, 2.2, 3.3, &c., transverse distances.

1. To divide a given line into 8 equal parts. Take the

line in the compasses, and open the sector so as to apply

it transversely to 8 and 8, then the transverse from 1 to

1 will be the eighth part of the line. If the line is to

be divided into 5 equal parts, apply it transversely by the

compasses to 10 and 10, and the transverse of 2 and

2 is the fifth part. When the line is too long to fall

within the opening of the sector, take the half or the third

of it. Thus, if a line of too great length is to be divided

into 10 parts, take the half and divide into 5 parts ;

or if into 9 parts, take the third and divide into 3 parts.

And in other cases it may be necessary to divide the por-

tion of the line into the original number of parts, and set

off twice or thrice to obtain the required division of the

whole.

2. To use the Line of Lines as a scale of equal measure's.

Open the sector to a right angle, or nearly so, and obtain

dimensions by transverse measures from scale to scale,

taking care that the points of the compasses are directed

to the same division on both rulers. Thus, the transverse

measures to the primaries 1.1, 2.2, &c., will give any

denomination, as feet or inches, and similar measures to

the same subdivisions on both sides will give tenths.

3. To form any required scale—say, one in which 285

yards shall be expressed by 18 inches. Now, as 18 inches

cannot be made a transverse, take in the compasses 6

inches, the third part, and make it a transverse to the

lateral distance 95, which is the third of 285. The

required scale is then made; the transverse measures to

the primaries being 10 yards, and to the subdivisions so

many additional yards.

4. To divide a given line in any assigned proportion ;

say, a line of 5 inches in the proportion of 2 to 6. Take

5 inches in the compasses, and apply it to the transverse

of 8.8, the sum of the proportions; then, will the trans-

verse distances 2.2, 6.6, divide the given line as required.

5. To find a third proportional to the numbers 9 and 3,

or to lines 9 inches and 3 inches in length. Make 3

inches a transverse distance to 9.9 ; then take the trans-

verse of 3.3, and this measured laterally on the scale of

inches will give 1 inch. For 9:3:: 3:1.

6. To find a fourth proportional to the numbers 10, 7, 3,

or to lines measuring 10, 7, and 3 inches respectively.

Make 7 inches a transverse from 10 to 10, then the trans-

verse 3.3 will measure on the scale of inches 2-jL For

10 : 7 :: 3 : 2TV

7. To find a middle proportional between the numbers

4 and 9, or between 2 lines measuring 4 and 9 inches

respectively. To perform this operation the Line of Lines

on the one leg of the sector must first be set exactly at

right angles to the one on the other leg. This is done by

taking 5 of the primary divisions in the compasses, and

making this extent a transverse from 4 on one side to 3

on the other. For 3, 4, and 5, or any of their multiples,

form a right-angled triangle. The sector being thus ad-

justed, take in the compasses a lateral distance of 6 primaries

and 5 tenths, half the sum of the two lines or numbers,

and apply this measure transversely from 2 primaries and

5 tenths, half the difference, when the other point of the

compasses will reach the primary 6 on the opposite leg

of the sector. For 4 : 6 :: 6 : 9.

The Line of Lines is marked L on each leg of the

sector ; and it is to be observed that all measures are to

be taken from the inner lines, since these only run accu-

rately to the centre. This remark will apply to all the

double sectoral lines. With reference to some of the pre-

ceding operations by the Line of Lines, we may admit

that they are suggestive rather than practically useful.

They familiarize the young draughtsman with the capa-

bilities of scales, and offer him useful hints for the general

construction and management of lineal measures.

The Line of Chords.—The scale of chords on the sector

lias the same advantage over that on the plain scale, that

the line of lines has over the simply-divided single scales.

ENGINEER AND MACHINIST’S DRAWING-BOOK.

ruler to another, are proportional to the lateral distances,

measured on the lines C A, Cl; and thus we may apply

any radius transversely to the line of chords to measure

or lay down any given or required angle ; and apply any

line transversely to the line of lines, to divide it in any

required proportions. The sector is therefore seen to be

of universal application, whilst the use of plain scales is

limited and special.

Plain Scales on the Sector.—On the outer edge of the

sector is usually given a decimal scale from 1 to 100; and

in connection with it, on one of the sides, a scale of inches

and tenths. These are identical with the lines on the

plain scale, previously mentioned, hut the latter are more

commodiously placed for use. On the other side we have

logarithmic lines of numbers, sines, and tangents; but

as these are more complicated than the ordinary plain

scales, we defer the consideration of them until we have

discussed the double scales.

Sectoral Double Scales.—These are respectively named

the Lines of Lines, Chords, Secants, Sines, and Tangents.

These scales have one line on each ruler, and the two lines

converge accurately in the central joint of the sector.

The Line of Lines.—This is a line of 10 primaries, each

subdivided into tenths, thus making 100 divisions. Its

use is, to divide a given line into

any number of equal parts; to give

accurate scale measures for the con-

struction of a drawing; to form any

required scale; to divide a given

line in any assigned proportion; and

to find third, fourth, and middle pro-

portionals to given right lines. The

scale can be applied to other pur-

poses ; but, if we take up those men-

tioned, they will be sufficient illus-

trations of its uses. Before entering

upon these propositions, we would

remark that a lateral distance is one

taken from the centre down either half of the scale ; and

a transverse distance is one measured across from scale to

scale. Thus (Fig. 16), a 1, a 2, a 3, &c., are lateral dis-

tances; and 1.1, 2.2, 3.3, &c., transverse distances.

1. To divide a given line into 8 equal parts. Take the

line in the compasses, and open the sector so as to apply

it transversely to 8 and 8, then the transverse from 1 to

1 will be the eighth part of the line. If the line is to

be divided into 5 equal parts, apply it transversely by the

compasses to 10 and 10, and the transverse of 2 and

2 is the fifth part. When the line is too long to fall

within the opening of the sector, take the half or the third

of it. Thus, if a line of too great length is to be divided

into 10 parts, take the half and divide into 5 parts ;

or if into 9 parts, take the third and divide into 3 parts.

And in other cases it may be necessary to divide the por-

tion of the line into the original number of parts, and set

off twice or thrice to obtain the required division of the

whole.

2. To use the Line of Lines as a scale of equal measure's.

Open the sector to a right angle, or nearly so, and obtain

dimensions by transverse measures from scale to scale,

taking care that the points of the compasses are directed

to the same division on both rulers. Thus, the transverse

measures to the primaries 1.1, 2.2, &c., will give any

denomination, as feet or inches, and similar measures to

the same subdivisions on both sides will give tenths.

3. To form any required scale—say, one in which 285

yards shall be expressed by 18 inches. Now, as 18 inches

cannot be made a transverse, take in the compasses 6

inches, the third part, and make it a transverse to the

lateral distance 95, which is the third of 285. The

required scale is then made; the transverse measures to

the primaries being 10 yards, and to the subdivisions so

many additional yards.

4. To divide a given line in any assigned proportion ;

say, a line of 5 inches in the proportion of 2 to 6. Take

5 inches in the compasses, and apply it to the transverse

of 8.8, the sum of the proportions; then, will the trans-

verse distances 2.2, 6.6, divide the given line as required.

5. To find a third proportional to the numbers 9 and 3,

or to lines 9 inches and 3 inches in length. Make 3

inches a transverse distance to 9.9 ; then take the trans-

verse of 3.3, and this measured laterally on the scale of

inches will give 1 inch. For 9:3:: 3:1.

6. To find a fourth proportional to the numbers 10, 7, 3,

or to lines measuring 10, 7, and 3 inches respectively.

Make 7 inches a transverse from 10 to 10, then the trans-

verse 3.3 will measure on the scale of inches 2-jL For

10 : 7 :: 3 : 2TV

7. To find a middle proportional between the numbers

4 and 9, or between 2 lines measuring 4 and 9 inches

respectively. To perform this operation the Line of Lines

on the one leg of the sector must first be set exactly at

right angles to the one on the other leg. This is done by

taking 5 of the primary divisions in the compasses, and

making this extent a transverse from 4 on one side to 3

on the other. For 3, 4, and 5, or any of their multiples,

form a right-angled triangle. The sector being thus ad-

justed, take in the compasses a lateral distance of 6 primaries

and 5 tenths, half the sum of the two lines or numbers,

and apply this measure transversely from 2 primaries and

5 tenths, half the difference, when the other point of the

compasses will reach the primary 6 on the opposite leg

of the sector. For 4 : 6 :: 6 : 9.

The Line of Lines is marked L on each leg of the

sector ; and it is to be observed that all measures are to

be taken from the inner lines, since these only run accu-

rately to the centre. This remark will apply to all the

double sectoral lines. With reference to some of the pre-

ceding operations by the Line of Lines, we may admit

that they are suggestive rather than practically useful.

They familiarize the young draughtsman with the capa-

bilities of scales, and offer him useful hints for the general

construction and management of lineal measures.

The Line of Chords.—The scale of chords on the sector

lias the same advantage over that on the plain scale, that

the line of lines has over the simply-divided single scales.