EXAM. 4.

Find 200 on the line C, then move the slide until 1 on D, 200 on C, and 10 on D, cut the same number on the opposite lines, which you will find to be 5.84, the root required.

EXAM. 5.

Having found 1728 on the line C, move the slide until 1 on D, 1728 on C, and 10 on D, stand against the same number on the opposite lines, which in this case is 12, the root required.

EXAM. 6.

Find 1000 on the line C, then move the slide until 1 on D, 1000 on C, and 10 on D, cut the same number on the opposite lines, which you will find to be 10, the root required.

PART III.

Problems in Practical Geometry, for the exercise of the Learner.

PROBLEM I.
Solution.

Draw an indefinite right line with your pencil; and from any scale of equal parts, take 15 in your compasses, and transfer it to the right line; and you will have the given line AB, which you may then draw with ink. Bisect this line by Problem I. of the Guaging; and each part will measure 7 if the work. be right.

PROBLEM II.

Solution.

Draw two lines to form any angle ABC, at pleasure ; then bisect the angle by Problem II. of the Gauging; and the work will be done.

Draw the right Ene AB at pleasure, then take tenths of an inch in your compasses, and by proceed ing as directed in Case 2, Problem IIL. you will be a to draw the Fine CD, parallel to the line AB, a n quired.

PROBLEM IV.
Solution

Draw a right line at pleasure, then take in your compasses 18 inches, and lay it off upon this line;" and you will obtain the given line. Take 8 inches in your compasses, and lay it off from one end of the given line, and you will obtain the point at which the perpendicular must be erected. At this point, erect the perpendicular as directed in Problem IV, and the work will be completed.

PROBLEM V.
Solution.

Draw a right line at pleasure, then take 24 inches in your compasses, and lay it off upon this line, and you will obtain the base of the given triangle.

With 20 in your compasses, as a radius, and one end of the base as a centre, describe an arc; and with 16 in your compasses, and the other end of the base as a centre, describe another are, cutting the former arc; then proceed as directed in Problem VII.; and the work will be finished.

PROBLEM VI.

Solution.

Draw a right line at pleasure; then take 24 inches in your compasses, lay it off upon this line, and you will have the base of the triangle,

Take 14 in your compasses, and lay it off from one end of the base, and you will have the point, at which the perpendicular must be raised.

From this point, erect a perpendicular at pleasure by Problem IV.; and upon this line lay off 12 inches; proceed as directed in Problem VIII., and the work will be finished.

PROBLEM VII.

Solution.

Draw a right line at pleasure; then take 16 inches in your compasses, and lay it off upon this line, and you will obtain one side of the square; proceed as directed in Problem IX, and you will obtain the square required.

PROBLEM VIII.

Solution.

Having drawn a right line at pleasure, take 24 inches in your compasses, lay it off upon this line, and you will have the length of the rectangle.

At one extremity of the line already laid down, erect a perpendicular at pleasure; then take 12 inches in your compasses, lay it off upon this line, and you will have the breadth of the rectangle.

Proceed as directed in Problem X., and you will obtain the rectangle required.

PROBLEM IX.

Solution.

Draw a line at pleasure; take 24 inches in your com→ passes, and lay it off upon this line, and you will have the base of the regular rhombus.

Proceed as directed in Problem XI., and you will be able to complete the required figure.

PROBLEM X.

Solution.

Having drawn a right line at pleasure, take 16 in your compasses, and lay it off upon this line, and you will have the base of the irregular rhombus.

At one end of the base, erect a perpendicnlar, which make equal to 12 inches; and then by proceeding as directed in Problem XII., you will be able to construct the required figure.

PROBLEM XI.

Solution.

Draw a right line at pleasure, then take 36 in your compasses, and lay it off upon this line, and you will have the base of the regular rhomboid.

Take 18 in your compasses, and lay it off from one end of the base; and by proceeding as directed in Problem XIII., you will be able to complete the required figure.

PROBLEM XII.

Solution.

Having drawn a right line at pleasure, take in your compasses 28 inches, lay it off upon this line, and you will have the transverse diameter of the ellipse.

Bisect this diameter perpendicularly, and you will obtain the centre of the figure. From this centre, set off 8 inches both ways upon the bissecting line; and you will have the conjugate diameter. Then by proceeding as directed in Problem XV., you will be able to construct the ellipse required.

PROBLEM XIII.

Solution.

Take 14 inches in your compasses, and with this radius, describe a circle; then apply your compasses six times to the circle, and you will have the angular points of the hexagon. Join these points by right lines, and the figure will be completed. (See Problem XVII.)

PROBLEM XIV.

Solution.

Draw an unlimited line at pleasure; then take 60o in in your compasses, and with one end of this line as a centre, describe an arc. Take 42° 30' in the compasses, and set it off upon this arc; and by proceeding as directed in Problem XXIII., you will be able to complete the work.

PROBLEM XV.

Solution.

Draw the indefinite line AB; and with the chord of 60° in your compasses, describe the arc DE; set off 90° from D to C, and from C to G, set off the excess above 90°, which is 46° 45'. Draw the line AG, and GAB will be the required angle. (See Problem XXIV.)

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