5

of equal parts, D E will be the side of another square of

one-third the area. And if any number be brought to the

index, and the same number be taken by A B from a scale

of equal parts, D E will be the square-root of that number.

And in this latter case, D E will also be a mean propor-

tional between any two numbers, whose product is equal

to A B.

The line of solids expresses the proportion between cubes

and spheres. Thus, if the index be set at 2, and the dia-

meter of a sphere, or the side of a cube, be taken from a

scale of equal parts by A B, then will D E be a diameter

or side of a sphere or cube of half the solidity. And if the

slide be set to (say) 8, and the same number be taken from

a scale of equal parts, then will D E measure 2 on the same

scale, or the cube-root of 8.

The scale of lines and that of circles are those of most

value to the draughtsman. The first enables him to reduce

or enlarge in any required proportion ; and the second

gives him the side of the square or polygon, that can be

inscribed in a given circle. The instrument needs to be

used carefully, since its accuracy depends on the preser-

vation of the points. If both or either of these are broken,

or diminished in length, the proportions cease to be true.

In place of using the proportional compasses in setting off

a number of times, which would soon wear the points,

rather take the distance in the Dividers.

Beam Compasses. — The engineer has frequently to

measure and lay down distances, and to sweep with x’adii,

which the ordinaiy instruments cannot reach. In these

cases, and when extreme accuracy is necessary, he resorts

to the Beam Compasses, which are usually made of well-

seasoned mahogany, with a slip of holly or box-wood on

the face, to carry the scale. Two brass boxes with points

are fitted to the beam, one of which moves freely to take

in any required distance, and the other is connected with

a slow-motion screw working in the end of the beam, and

can thus be adjusted with extreme delicacy to any measure

or radius. Reading-plates, on the vernier principle, sub-

divide the divisions of the scale on the beam, and by them

any measure of three places of figures is taken with extreme

truth. Referring to Fig. 7, we proceed to describe more

{Fig. 7.)

particularly. C C is the mahogany beam, whose length i

may be taken at pleasure, although it is not advisable to

extend it beyond four or five feet, lest it bend by its own

weight; a a is the strip of holly or box-wood, on which

the scale is engraved; B is the brass box, which moves

freely along the beam, and is secured in position by the

clamp-screw F; A is the other brass box, made fast to the

slow-motion screw D, which works in the end of the beam,

and winds it into or out of the box A, to obtain perfect

adjustment; and d b are the vernier scales, or reading-

plates.

Before describing the method of setting the instrument,

we must explain, in few words, the nature of a vernier

scale. Take any primary division of a scale, and divide

it into ten parts, then take eleven such parts and divide

the line which they form into tenths likewise; this last

then becomes a vernier or reading-scale. The primary

division is (say) 100, its subdivision 10, and the excess of

the vernier division 1; so that if the scale and vernier are

placed parallel and close to each other, a distance or measure

may be read accurately to the unit of three places of figures.

We illustrate by a diagram (Fig. 8), which shows the ver-

nier attached to the scale ab of the ordinary barometer.

Here a b is divided into inches and tenths of

inches; and c d is the vernier, consisting of eleven

subdivisions of a b, divided into tenths. Now

the zero, or commencement of notation, on the

vernier is, in this case, adjusted to 80 inches on

the scale; and its division 10, coincides with 28

inches nine-tenths; hence every division of the

vernier is seen to be one and one-tenth of the

scale divisions. To read off, therefore, the hundredths ot

an inch that the zero of the vernier may be in advance of

a tenth, observe what division of the vernier coincides

most nearly with any division of the scale, and that will

indicate the hundredths. Thus, taking the adjustment of

the figure, the zero corresponds exactly with 30 on the

scale, and its division 10, with 28 and 9 tenths; and we

therefore read 30 inches. But if the zero were so posited

between 29 and 9 tenths, and 30, that the 8 of the vernier

should correspond exactly with a tenth of the scale, we

should read 29 inches, 9 tenths, and 8 hundredths. And

this is evident, for if zero be 8 hundredths in excess of a

tenth, it is only the eighth division of the vernier that

will be found to coincide exactly with a tenth of the scale.

To adjust the beam compasses for a distance or radius

of (say) 13 inches, 5 tenths, 3 hundredths, the box A is to

be moved by the screw D, until the zero of the vernier

corresponds with the zero of the beam, and is then to be

secured in position by the clamp E ; this done, the box B

is slid along the beam until the zero of its vernier coin-

cides with 13 inches 5 tenths; lastly, the box B is moved

by the slow-motion screw, and the third division of the

vernier brought to correspond with the third tenth of the

scale, which consequently adds 3 hundredths to the dis-

tance or radius previously taken. The point of the slide

or box F can be removed, and a pen or pencil substituted

with accurate adjustment. The beam compasses are seldom

employed, except when extreme accuracy is necessary.

On many occasions, curves of long radius are drawn by

means of slips of wood, one edge of which is cut to the

required circle.

Having described the various sorts of compasses in

ordinary use, it is unnecessary to do more than advert

briefly to some modifications and improvements in form

and detail. It has been thought an advantage to joint

both legs of the instrument, in order to bring them to a

(Fio, 8.)