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DBC are similar: thus the angle ABD=DBC, being both
right, the angle DAC is the complement of BDA to a right
angle (by cor. 2, theo. 5), and is therefore equal to BDC, the
angle ADC being a right angle as before; consequently (by
cor. 1, theo. 5) the angle ADB =DCB; wherefore (by
theo. 16),
AB : B D : : BD : BC
Or, A : BD: : BD : B.

- \PROBLEM XIV.
PL. 3. fig. 2.
To divide a right line AB in the point E, so that AE shall have the same
proportion to EB as two given lines C and D have.
Draw an indefinite blank line AF to the extremity of the
line AB, to make with it any angle ; lay the line C from A to
C, and D from C to D, and join the points B and D by the
line BD; through C draw CE parallel to BD (by prob. 8), so
is E the point of division.
For, by theo. 20, AC : CD :: AE : EB.
Or, C : D : : AE : : E.B.

PROBLEM XV.

PL. 3. fig. 3.

To describe a circle about a triangle ABC, or (which is the same thing) through any three points A, B, C, which are not situated in a right line.

By prob. 4. Bisect the line AC by the perpendicular DE, and also CB by the perpendicular FG, the point of intersection H of these perpendiculars is the centre of the circle required; from which take the distance to any of the three points A, B, C, and describe the circle ABC, and it is done.

For, by cor. to theo. 8, the lines DE and FG must each pass through the centre ; therefore their point of intersection H must be the centre.

SCHOLIUM.

By this method the centre of a circle may be found, by hav

ing only a segment of it given.

PROBLEM XVI.
PL. 3. fig. 4.
To make an angle of any number of degrees at the point A of the line AB,
suppose of 45 degrees. -
From a scale of chords take 60 degrees, for 60° is equal to
the radius (by cor, theo. 15), and with that distance from A as
a centre, describe a circle from the line AB; take 45 degrees,

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the quantity of the given angle, from the same scale of chords, and lay it on that circle from a to b; through A and b draw the line Abc, and the angle A will be an angle of 45 degrees, as required. If the given angle be more than 90°, take its half (or divide it into any two parts less than 90 and lay them after each other on the arc, which is described with the chord of 60 degrees; through the extremity of which and the centre, let a line be o and that will form the angle required, with the given e.

PROBLEM XVII.

p PL. 3. fig. 5.
To measure a given angle ABC.

If the lines which include the angle be not as long as the chord of 60° on your scale, produce them to that or a greater length, and between them so produced, with the chord of 60° from B, describe the arc ed; which distance ed, measured on the same line of chords, gives the quantity of the angle ABC, as required; this is plain from def. 17.

PROBLEM XVIII.
PL. 3. fig. 6.

To make a triangle BCE equal to a given quadrilateral figure ABCD.

Draw the diagonal AC, and parallel to it (by prob. 8) DE, meeting AB produced in E; then draw CE, and ECB will be the triangle required.

For the triangles ADC, AEC being upon the same base AC, and under the same parallel ED (by cor. to theo. 13), will be equal, therefore if ABC be added to each, then ABCD = BEC.

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64 MATHEMATICAL
PROBLEM 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 nine equal parts, every one of which will be ten 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, C20, 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 nine former divisions into ten 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.

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 portcrayon, for holding a piece of black-lead pencil.

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

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 screwdriver.”

* Large collections are called magazine cases of instruments; 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, a key, and screwdriver, 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.

A pair of gunners' callipers.

A pair of elliptical compasses.

A pair of spiral compasses.

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.

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 works.

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 their various uses; suffice it then to say, that they are used to transfer small distances, measure given spaces, and describe arches and circles. If an 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 forefinger. 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 fore-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

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