PROBLEM I. PARALLEL LINES. Parallel lines are such as may be produced to any distance whatever, without meeting or inclining to each other. To draw a line parallel to a given line A B.* CASE 1. When the parallel line is to be at a given distance E. touch those arcs, without cutting them, and it will be the parallel line required. 1. Draw a line parallel to HK of a distance = L. 2. Draw another line, parallel to the original, of twice the distance of the length L. 3. Another, distant half the length of L. CASE 2. When the parallel line is to pass through a given point, as C. A * To perform these problems, the student should be provided with a case of mathematical instruments, the price of which varies according to the size and number of the instruments. case, sufficient for the ordinary purposes of the land-measurer, contains two pairs of compasses, three drawing pens, pencil-sweep, six-inch protractor and scale, and twelve-inch sector. C From the given point C, draw a line at pleasure, which shall cut the given line AB at r. A S With any radius, from r describe an arc s; and with the same radius, from C describe another arc o. Make the arc o equal to the arc s. A line drawn through C and o will be the line required. * Otherwise. From the point c, sweep an arc which shall touch the original line at o. With the same radius, and one foot in any other convenient part of the line AB, as at r, sweep Athe arc d. C d B From c, draw a line, touching the are d, and it will be the parallel line required.+ 1. By the last method, draw a line parallel to RS, * Euclid's Elements, Book I. Proposition 31. † Parallel lines are drawn with much facility by an instrument called a parallel ruler (fig. 1.). These rulers are of various lengths, Fig. 1. from one to three feet. The most convenient form of this ruler is when it is made to roll on two wheels fixed on one axis. which shall pass through the point a, below the given line. 2. Draw other lines parallel to RS, of double the distance a, above the line; by both the foregoing methods. PROBLEM II. BISECTED LINES. Any right line, as AB, may be bisected, i. e. divided into two equal parts, in the manner following: Set one point of a pair of compasses at the end of the line, at A, and with any extent greater than half the length of the line, A describe the arcs a and b, above and below the said line. a * b d B With the same extent, set one foot of the compasses in the end B; and describe other arcs, c and d, intersecting the former. A line drawn from one intersection to the other will divide the original line into two equal parts. EUCLID, I. 10.* * The same thing is readily performed by proportional compasses (fig. 2.). Those of seven or eight inches are of convenient length. Fig. 2. For Practice. -D 1. Divide the line CD into two equal parts. 2. Divide each of those parts into two equal parts. PROBLEM III. DIVIDED LINES. Any line, as AB, may be divided into three, or any other proposed number of equal parts. CASE 1. To divide it into five equal parts. From the end A of the given line, draw the line Ae, at pleasure; and from the other end, B, draw the line Bf parallel to Ae. From A towards e, set off one less than the required number of equal divisions, of any reasonable length. Do the same from B towards f, numbering the points of division, as in the figure. Draw lines from the corresponding numbers, intersecting the original line; and they will divide it, as required. CASE 2. It is proposed to divide the line AB into four equal parts. Draw the line CD, of any convenient length, greater than AB, and thereon, from C, set off as many equal parts as the given line is to be divided into; and number those parts 1, 2, 3. 4. Take C4 in the compasses, and, with that extent, upon C draw the arc d. With the same extent, and one foot on 4, draw the arc e. From the intersecting point F, draw FC and F4. Set off the length of the given line from F towards C and 4, on which draw AB. From F, to the given points, 1, 2, 3, 4, on the assumed line CD, draw the lines F1, F2, F3, F4. These lines will cut the line AB into the equal parts required. A For Practice. B 1. Divide the line AB into three equal parts, by the first method. 2. Divide the same into four equal parts. 3. By the second method, divide EG into five equal parts. 4. Divide the same into six equal parts. PROBLEM IV. PERPENDICULAR LINES. One line may be drawn perpendicular to another. |