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(by cor. to theo. 13.) will be equal, therefore if ABC be added to each, then ABCD = BEC.

PROB. XIX.

PL. 3. fig. 7.

To make a triangle DFH, equal to a given five-sided figure ABCDE.

Draw DA and DB, and also EH and CF, parallel to them (by prob. 8.) meeting AB produced in Hand F; then draw DH, DF, and the triangle HDF is the one required.

For the triangle DEA = DHA, and DBC= DFB (by cor. to theo. 13.) therefore by adding these equations, DEA+ DBC=DHA+DFB if to each of these ADB be added; then DEA+ ADB+ DBC=ABCDE=(DHA+ ABD+ DFB,= DHF.

PROB. XX.

PL. 3. fig. 8.

To project the lines of chords, sines, tangents and secants, with any radius.

On the line AB, let a semicircle ADB be described; let CDF be drawn perpendicular to this line from the centre C; and the tangent BE perpendicular to the end of the diameter; let the quadrants, AD, DB, be each divided into 9 equal parts, every one of which will be 10 degrees; if then from the centre C, lines be drawn through 10, 20, 30, 40, &c. the divisions of the quadrant BD, and continued to BE, we shall there have the tangents of 10, 20, 30, 40, &c. and the secants C 10, C 20, C 30, &c. are transferred to the line CF, by describing the arcs 10, 10: 20, 20: 30, 30, &c. If from 10, 20, 30,&c. the divisions of the quadrant BD, there

be let fall perpendiculars, let these be transferred to the radius CB, and we shall have the sines of 10, 20, 30, &c. and if from A we describe the arcs 10, 10:20, 20: 30, 30, &c. from every division of the arc AD; we shall have a line of chords. The same way we may have the sine, tangent, &c. to every single degree on the quadrant, by subdividing each of the 9 former divisions into 10 equal parts. By this method the sines, tangents, &c. may be drawn to any radius; and then, after they are transferred to lines on a rule, we shall have the scales of sines, tangents, &c. ready for use.

ATICAL

MATHEMATICAL

DRAWING INSTRUMENTS.

THE strictness of geometrical demonstration admits of no other instruments, than a rule and a pair of compasses. But, in proportion as the practice of geometry was extended to the different arts, either connected with, or dependent upon it, new instruments became necessary, some to answer peculiar purposes, some to facilitate operation, and others to promote accuracy.

As almost every artist, whose operations are connected with mathematical designing, furnishes himself with a case of drawing instruments suited to his peculiar purposes, they are fitted up in various modes, some containing more, others, fewer instruments. The smallest collection put into a case, consists of a plane scale, a pair of compasses with a moveable leg, and two spare points,

which may be applied occasionally to the compasses; one of these points is to hold ink; the other, a porte crayon, for holding a piece of black-lead pencil.

What is called a full pocket case, contains the following instruments.

A pair of large compasses with a moveable point, an ink point, a pencil point, and one for dotting; either of those points may be inserted in the compasses, instead of the moveable leg. A pair of plain compasses somewhat smaller than those with the moveable leg.

A pair of bow compasses.

A drawing pen with a protracting pin in the upper part.

A sector.
A plain scale.

A protractor.
A parallel rule.

A pencil and screw-driver.*

* Large collections are called, magazine cases of instruments f these generally contain

A pair of six inch compasses with a moveable leg, an ink point, a dotting point, the crayon point, so contrived as to hold a whole pencil, two additional pieces to lengthen occasionally one leg of the compasses, and thereby enable them to measure greater extents, and describe circles of a larger radius,

A pair of hair compasses.
A pair of bow compasses.

A pair of triangular compasses.

A sector.

A parallel rule,

A protractor.

A pair of proportional compasses, either with or without an adjusting screw.

A pair of wholes and halves.

Two drawing pens, and a pointril.

A pair of small hair compasses, with a head similar to those of the

bow compasses.

A knife, a file, key, and screw-driver, or the, compasses in one piece.

A small set of fine water colours.

To these some of the following instruments are often added.
A pair of beam compasses.

In a case with the best instruments, the protractor and plain scale are always combined. The instruments in most general use are those of six inches; instruments are seldom made longer, but often smaller. Those of six inches are, however, to be preferred, in general, before any other size; they will effect all that can be performed with the shortest ones, while, at the same time, they are better adapted to large work.

OF DRAWING COMPASSES.

Compasses are made either of silver or brass, but with steel points. The joints should always be framed of different substances; thus, one side or part, should be of silver or brass, and the other of steel. The difference in the texture and pores of the two metals causes the parts to adhere less together, diminishes the wear, and promotes uniformity in their motion. The truth of the work is ascertained by the smoothness and equality of the motion at the joint, for all shake and irregularity is a certain sign of imperfection. The points should be of steel, so tempered, as neither to be easily bent or blunted; not too fine and tapering, and yet meeting closely when the compasses are shut. As an instrument of art, compasses are so well known that it would be superfluous to enumerate the various uses; suffice it then to say, that they

A pair of gunners callipers.

A pair of elliptical compasses.

A pair of spiral ditto.

A pair of perspective compasses.

A pair of compasses with a micrometer screw,

A rule for drawing lines, tending to a centre at a great distance, A protractor and parallel rule.

One or more parallel rules.

A pantographer, or Pentagraph.

A pair of sectoral compasses, forming, at the same time, a pair of beam and calliper compasses,

are used to tranfer small distances, measure given spaces, and describe arches and circles.

If the arch or circle is to be described obscurely, the steel points are best adapted to the purpose; if it is to be in ink or black lead, either the drawing pen, or crayon points are to be used.

To use a pair of compasses. Place the thumb and middle finger of the right hand in the opposite hollows in the shanks of the compasses, then press the compasses, and the legs will open a little way; this being done, push the innermost leg, with the third finger, elevating, at the same time, the furthermost, with the nail of the middle finger, till the compasses are sufficiently opened to receive the middle and third finger;they may then be extended at pleasure, by pushing the furthermost leg outwards with the middle, or pressing it inwards with the four finger. In describing circles, or arches, set one foot of the compasses on the centre, and then roll the head of the compasses between the middle and four finger, the other point pressing at the same time upon the paper. They should be held as upright as possible, and care should be taken not to press forcibly upon them, but rather to let them act by their own weight; the legs should never be so far extended, as to form an obtuse angle with the paper or plane, on which they are used.

The ink and crayon points have a joint just under that part which fits into the compasses; by this they may be always so placed as to be set nearly perpendicular to the paper; the end of the shank of the best compasses is framed so as to form a strong spring, to bind firmly the moveable points, and prevent them from shaking. This is found to be a more effectual method than that by

a screw.

Two additional pieces are often applied to these compasses; these, by lengthening the leg, enable them to strike larger circles, or measure

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