B

19. To ascertain the distance, geometrically, of any inaccessible object on an equal plane. Let it be required to find the distance between A and B, A being inaccessible; produce the line in the direction of A B to any point, as D; draw the line D d at any angle to the line A B; bisect the line D d, through which draw the line B b, making cb b equal to B c; draw the line dba; also through c, in the direction c A, draw the line ac A, intersecting the line dba; then b a equal a BA, the distance required.

20. Otherwise.

Prolong A B to any point D, making B C equal to C D; draw the line D a at any angle with D A, and the line Cb similar to B c; draw also the line DE F, which intersects the line Da; then ab a equal B A, or the distance required.

D

21. To find the proper position for an eccentric, in relation to the crank in a steam engine, the angle of eccentric rod, and travel of the valve, being given.

Draw the right line A B, as the situation of the crank at commencement of the stroke; draw also the line C d, as the proper given angle of eccentric rod with the crank; then from C as centre, describe a circle equal to the travel

of the valve; draw the line e ƒ at right angles to the line

C a, draw also the lines 1 1, and 2 2, parallel to the line ef; and at a distance from e f on each side, equal to the lap and lead of the valve, draw the angular lines C 1, C2, which are the angles of eccentric with the crank, for forward or backward motion, as may be required.

22. The throw of an eccentric, and the travel of the valve in a steam-engine, also the length of one lever for communicating motion to the valve, being given, to determine the proper length for the other.

On any right line, as A B, describe a circle A D equal to the throw of eccentric and travel of valve;

then from C as a centre, with a radius equal to the length of lever

C

D

given, cut the line A B, as at d, on which describe a circle, equal to the throw of eccentric or travel of valve, as may be required; draw the tangents B a, В a, cut ting each other in the line A B, and d B is the length of the lever as required.

Note. The throw of an eccentric is equal to the sum of twice the distance between the centres of formation and revolution, as a b, or to the degree of eccentricity it is made to describe, as c d. And

The travel of a valve is equal the sum of the widths of the two steam openings, and the valve's excess of length more than just sufficient to cover the openings.

c d

23. To inscribe any regular polygon in a given circle Divide any diameter, as A B, into

so many equal parts as the polygon is
required to have sides; from A and B
as centres, with a radius equal to the
diameter, describe arcs cutting each
other in C; draw the line C D through A
the second point of division on the
diameter e, and the line D B is one
side of the polygon required.

2

24. To construct a square upon a given right line.

From A and B as centres, with the radius A B, describe the arcs A c b, B c d, and from C, with an equal radius, describe the circle or portion of a circle a ed, A B, bc; from bd cut the circle at e and c; draw the lines A e, B c, also the line st, which completes the square as required.

25. To form a square equal in area to a given triangle.

t

B

Let A B C be the given triangle; let fall the perpendicular B d, and make A e half the height d B; bisect e C, and describe the semicircle en C; erect the perpendicular A s, or side of the square, then Astx is the square of equal area as required.

AC e A

26. To form a square equal in area to a given rectangle.

D

Let the line A B equal the length and breadth of the given rectangle; bisect the line in e, and describe the semicircle A D B; then from A with the breadth, or from B with the length, of the rectangle, cut the line A B at C,

A O

e

B

and erect the perpendicular C D, meeting the curve at D, and C D equal a side of the square required.

27. To find the length for a rectangle whose area shall be equal to that of a given square, the breadth of the rectangle being also given.

Let A B C D be the given square, and D E the given breadth of rectangle; continue the line B C to F, and

28. To bisect any given triangle.

Suppose A B C the given triangle; bisect one of its sides, as A B in e, from which describe the semicircle Ar B; bisect the same in r, and from B, with the distance Br, cut the diameter A B in v; draw the line v y parallel to A C, which will bisect the triangle as required.

T

B

A

29. To describe a circle of greatest diameter in a given triangle.

Bisect the angles A and B, and draw the intersecting lines A D, BD, cutting

each other in D; then from D as centre, with the distance or radii D C, describe the circle C e f, as required.

30. To form a rectangle of greatest surface, in a given triangle.

Let A B C be the given triangle; bisect any two of its sides, as A B, B C, in e and d; draw the line e d; also at right angles with the line e d, draw the lines ep, dp, and e ppd is the rectangle required.

20

DECIMAL ARITHMETIC.

DECIMAL ARITHMETIC is the most simple and ex plicit mode of performing practical calculations, on account of its doing away with the necessity of fractional parts in the fractional form, thereby reducing long and tedious operations to a few figures arranged and worked in all respects according to the usual rules of common arithmetic..

Decimals simply signify tenths; thus, the decimal of a foot is the tenth part of a foot, the decimal of that tenth is the hundredth of a foot, the decimal of that hundredth is the thousandth of a foot, and so might the divisions be carried on and lessened to infinity; but in practice it is seldom necessary to take into account any degree of less measure than a one-hundredth part of the integer or whole number. And, as the entire system consists in supposing the whole number divided into tenths, hundredths, thousandths, &c., no peculiarity of notation is required, otherwise than placing a mark or dot, to distinguish between the whole and any part of the whole; thus, 34-25 gallons signify 34 gallons 2 tenths and 5 hundredths of a gallon; 11.04 yards sig nify 11 yards and 4 hundredths of a yard, 16-008 shillings signify 16 shillings and 8 thousandth parts of a shilling; from which it must appear plain, that ciphers on the right hand of decimals are of no value whatever; but placed on the left hand, they diminish the decimal value in a tenfold proportion, for 6 signify 6 tenths; 06 signify 6 hundredths; and 006 signify 6 thousandths of the integer, or whole number.

REDUCTION.

Reduction means the construing or changing of vulgar fractions to decimals of equal value; also finding the fractional value of any decimal given.

Rule 1. Add to the numerator of the fraction any number of ciphers at pleasure, divide the sum by the denominator, and the quotient is the decimal of equivalent value.