LESSONS IN 



that whi.-li hoa been described above; and by Problom 

 \ \ XVI. in the last lesson (page 49), which teaches as how to 

 i .< circumference of a circle through any three points 

 that are not in the same straight line, it is plain that we are 

 shown how to describe a circle about any given triangle ; for to 

 do this the oirolo must pass through the throe points in which 

 the three straight lines (which form the sides and angles of the 

 trkuitfle) meet. 



In making drawings of machines and machinery, the geome- 

 trical draughtsman will find it necessary 

 to know how to draw circles of different 

 radii touching each other with the 

 utmost nicety externally and internally, 

 and to draw tangents to two circles, 

 aa when it is desired to represent in 

 section the course of an endless band or 

 belt of leather which passes round two 

 y wheels or drums, and transmits motive 

 power from one to the other. This 

 brings us to the next two problems. 



PROBLEM XXXIX. To draw a circle 

 vith a given radius touching another 

 given circle externally or internally in a 

 given point. 



Let ABC (Fig. 58) be the given circle ; 

 it is required to draw two circles with 

 the given radius x, one touching the 

 given circle ABC internally in the given point A, and the other 

 externally in the given point c. 



First, let us take the circle that is to touch the given circle 

 internally in the point A. Find D, the centre of the given circle 

 ABC. Join A D, and produce it, if necessary, to meet the cir- 

 cumference of the given circle A B c in the point E. Along A B 

 set off A F equal to the given radius x, and from the centre F, 

 at the distance F A, describe the circle A o H, which touches the 

 given circle ABC internally in the given point A. 



To describe a circle with the given radius x, touching ABC 

 externally in the point c. Find D, the centre of the given 

 circle ABC. Join D c, and produce it indefinitely towards K. 

 Set off along c K the straight line C L equal to x, and from L 

 as centre, with the distance L c, describe the circle c M N, which 

 touches the given circle ABC 

 externally in the given point c. 



PROBLEM XL. To draw a 

 tangent to any two given circles. 



A straight line may bo drawn 

 touching any two given circles 

 either on one side of the straight 

 line that passes through and 

 joins the centres of the circles, 

 or crossing this line. First, let 

 ns take the case in which the 

 tangent to the given circles is 

 required to be on one side of 

 the straight lino that joins their 

 centres. 



Let A B c, D E F (Fig. 59) be the 

 given circles. Draw an indefinite 

 straight lino X Y, passing through 

 and joining the centres, a and 

 H, of the given circles, and also 

 passing through the point c in 

 the circumference of the circle 

 ABC, and the point F, in the cir- 

 cumference of the circle DBF. From the point F in the straight 

 line x Y, set off F K equal to o c, the radius of the circle ABC. 

 The remainder K H of the straight line F H, the radius of the 

 circle D E F, is manifestly the difference of the radii of the 

 given circles. With H K as radius, from the point H as centre, 

 describe the circle K L M, and from the point o, the centre of the 

 circle ABC, draw the straight line o M touching the circum- 

 ference of the circle K L M in the point M. Join H M, and 

 produce it to out the circumference of the circle D F E in the 

 point E, and through E draw E B parallel to o M. The straight 

 line E B is a tangent to the two given circles A B c, D B F. The 

 same result may be obtained by drawing o B through the point o, 

 parallel to H E, and joining the points B and E. The straight line 

 D E has bee* drawn as a tangent to the given circles ABC, 



Fig. 53. 



D B F on the rvjht of the straight line x T that pauses through 

 their centres. The dotted lines o L, u u, D A, and A o show 

 how the tangent may be drawn to the left of the straight line 

 X T. The straight lines A D, B E, with the arcs A x , D T B, 

 show the position of an endless band passing over the wheel* 

 or drums A B c, D B F. 



Now let it be required to draw a tangent to the given circles 

 A B c, D B F, crossing the indefinite straight line x T that psitnct 

 through their centres o and H. From the point c in the direc- 

 tion of r, set off along c T the straight line c equal to F H, 

 the radius of the circle D F B. The straight line o V is then 

 manifestly equal to the sum of the radii of the given nfrnioB, 

 being made up of o c, the radius of the circle ABC, and o V, 

 which has been made equal to F H, the radios of the circle 

 D F B. From the centre o, with the radius o X, describe the 

 arc o N P, and from H, the centre of the circle D F B, draw H o 

 touching the arc o N P in the point o. Join o o, cutting the 

 circumference of the circle A B c in the point Q ; and through 

 the point q draw q B para Id to H o. The straight line q B is 

 a tangent to the two given circles A B c, D B F. The same result 

 may be obtained by drawing u B through the point B, parallel 

 to o o, and joining the points q and B. The straight line Q B 

 has been drawn as a tangent to the given circles A B c, D E F, 

 crossing the straight line x Y that passes through their centres, 

 from left to right ; the dotted lines H P, P o, 8 T, and T H show 

 how the tangent may be drawn crossing x Y from right to left. 

 The straight lines q B, s T, with the arcs q A B s, TDBB, show 

 the position of a crossed endless band passing over the wheels 

 or drums A B c, D B F. 



The effect of crossing the endless band is to make the wheels 

 or drums over which it passes revolve in contrary directions. In 

 the first case, when the band forms tangents to the wheels on 

 both sides of the line that joins their centres, the wheels revolve 

 in the same direction, that is to say, by the action of the band the 

 wheels revolve, so that a point B, on the circumference of the 

 wheel A B c, is carried round towards A, and a point E on the cir- 

 cumference of the wheel D E F is carried round towards D, the 

 strap being supposed to move in the direction of the arrows 

 placed near the letters A, D, E, B. In the second case, when 

 the band forms tangents to the wheels crossing the line that 

 joins their centres, a point q in the circumference of the wheel 

 ABC would be carried round towards s, and a point T in the 

 circumference of the wheel DBF would bo carried round towards 

 B, the strap being supposed to move in the direction indicated 

 by the arrows placed near the letters q, B, T, 8. 



The circles K L M , D E F are called concentric circles because 

 they are described from the same centre, H, and for the same 

 reason the arcs A c B, o N P are called concentric arcs. 



Fig. 59 suggests the method of drawing a circle of a given 

 radius to touch two given straight lines. Let L o, x o, pro- 

 duced indefinitely to a and 6, represent the two given straight 

 lines, and z the radius of the required circle. In o a take any 

 point u, and through u draw u w at right angles to o a. 

 Bisect the angle a o b formed by the straight lines o a, ob 

 (produced to meet in a if necessary) by the straight line o Y, 

 and set off along the straight line u w, u T equal to z. Then 

 through v draw v H parallel to o a, and meeting a Y in H. 

 Then from the point H as centre, with a radius equal to z, 

 describe the circle K L M . This circle touches the given straight 

 lines o a, o & in the points H and L. 

 PROBLEM XLI. To draw a tangent through any point in a 



yi:en arc, when it is inconvenient to determine the centre oj 



the circle of the circumference of which the given are it a part. 



Let A B c (Fig. 60) 

 be the given arc, and c 

 the given point through 

 which it is required to 

 draw a tangent to the 

 arc A B c. Through c /A 

 draw any straight line ' 

 or chord c A, cutting Fig. 80. 



the arc in the points A 



and c. Bisect A o in D, and through D draw D B at right angle* 

 to A c. Join B c, and at the point B in the straight line c B 

 make the angle c B E equal to the angle DC B or B c B. Then 

 through c draw the straight line x Y parallel to B B, The 

 straight line x Y is a tangent to the arc A B c, and it ia drawn 

 through the given point c, as required. 



