Problem 203.

To join the extremities of any two given parallel lines AB, CD, by a pair of arcs, which shall touch each other, and the ends of the lines tangentially.

1. At C, erect a perpendicular CE (Pr. 2), and produce it to F, making EF equal to EA.

K

B

2. Bisect CF in G (Pr. 1), and through G, draw a line parallel to AB (Pr. 9) of indefinite length towards the left.

3. Mark off GH equal to AE. With centre H, and radius HA, describe the arc AK.

4. With centre G, and radius GK, describe the arc KC. Then the given parallel lines AB, CD, will be joined by the required pair of arcs AK, KC.

Problem 204.

To construct a circle which shall touch a given line AB in the given point C, and also a given smaller circle D.

1. Through the point C, draw a perpendicular EF (Pr. 2) of unlimited length.

2. Find the radius of the given circle D, and from EF cut off CE equal to it, and join DE.

3. Bisect DE in the point G (Pr. 1), and draw GH perpendicular to it, meeting EF in H; then H is the centre of the required circle.

4. From H, with radius HC, describe the required circle

E

A

H

which will touch the given line AB in the given point C, and also the given smaller circle D.

Problem 205.

To construct a circle which shall pass through two given points A and B, and shall touch a given line CD.

1. Draw the straight line BA, and produce it to meet DC produced in E.

2. Find a mean proportional between the lines BE and EA (Pr. 140), and from E on the line ED, mark off EF

E

C

D

equal to the mean proportional; then Fis the point in the given line CD, through which the required circle will touch it.

3. Through the three points B, A, F, describe the required circle BAF (Pr. 46).

Problem 206.

To draw a circle externally tangential to two given unequal circles A and B, and touching one of them in a given point C.

1. Find the centres of the given circles A and B (Pr. 45). 2. Join BC; and produce it indefinitely.

3. On BC produced, mark off CD equal to the radius of the larger circle A, and join AD.

E

A

4. From A, draw AE to meet DC produced in E, and making with DA an angle equal to ADE (Pr. 10); then E is the centre of the required circle.

5. From centre E, with radius EC, describe the required circle, which will be tangential to the two given circles A and B.

Problem 207.

To change any given rectilineal figure ABC into another rectilineal figure of equal area, but having one side more, &c.

Let the given figure ABC be a triangle.

1. Assume a point, D, as one of the angles of the four-sided figure to be obtained, and join AD.

2. Draw a line from A in the same direction as DC and parallel to it (Pr. 9).

3. Draw a line from C parallel to AD, and meeting AE in E; and join DE. Then the four-sided figure BAED will be equal in area to the given triangle ABC.

Next,

1. Assume a point, F, as one of the angles of the five-sided figure to be obtained, and join AF.

2. Draw a line from B parallel to AF (Pr. 9), and join EF. 3. Draw a line from A parallel to EF, and meeting BG in G, and join GF. Then the five-sided figure FGAED will be also equal in area to the triangle ABC.

NOTE. In the same manner, a figure of six, seven, &c., sides may be obtained, by assuming a new angular point in each case.

Problem 208.

To make a reduced copy of any given figure ABCD, making the given line EF correspond to AB.

1. Make angle FEG equal to BAD (Pr. 10), and angle EFG equal to ABD. Then EH will have a correct proportion to AD.

2. On the curve CB, mark any number of points, say two K, L.

3. From C, K, L, drop perpendiculars on AB (Pr. 3) to points 1, 2, 3.

4. Draw KM, LN, parallel to AB (Pr. 9).

5. Divide EF, EH, proportionally to the divisions on AB, AD (Pr. 16), as shown by the dotted lines in the figure.

6. From 4, 5, 6, erect perpendiculars (Pr. 2), and draw HO, 7P, 8R, parallel to EF (Pr. 9). Then HO corresponds to DC, and points P and R to points K and L.

7. Draw the curve OPRF, and EFOH will be a reduced copy of the figure ABCD.

NOTE. In a similar manner, an enlarged copy of any given figure may be made.

Problem 209.

To construct a common spiral,* the given diameter being AB. 1. Take any point C in AB as the eye of the required spiral.

6 44

HF DCE G K B

3

5

* A spiral is a curve which, as it revolves once or more round some fixed point called its centre, recedes regularly from that centre.

M