Fig. 146 - The Chord and Hight of a Segment of a Circle Being Given, to find the Center from which the Arc may be Struck.

15. To Find the Center from which a Given Arc is Struck by the Use of the Square. - In Fig. 143, let A

B C be the given arc. Establish the point B at pleas ure and draw two chords, as shown by A B and B C. Bisect these chords, obtaining the points E and D. Place the square against the chord B C, as shown in the engraving, bringing the heel against the middle point, D, and scribe along the blade indefinitely. Then place the square, as shown by the dotted lines, with the heel against the middle point, E, of the second chord, and in like manner scribe along the blade, cutting the first line in the point F. Then F will be the center of the circle, of which the are A B C is a part. This rule will be found very convenient for use in all cases where the radius is less than 24 inches in length. Fig. 145. - To Find the Center from which a Given Arc is Struck, by the Use of the Square.

16. The Chord and Hight of a Segment of a Circle being Given, to Find the Center from which the Arc may be Struck. - In Fig. 146, let A B represent the chord of a segment or arc of a circle, and D C the rise or hight. It. is required to find a. center from which an are, if struck, will pass through the three points A, D and 15. Draw A D and B D. Bisect A D, as shown, and prolong the line H L indefinitely. Bisect D B and prolong I M until it cuts H L, produced in the point E. Then E, the point of intersection, will be the center sought. It will be observed that by producing D C, and intersecting it by either H L or I M prolonged, the same point is found. Therefore, if preferred, the bisecting of either A D or D B may be dispensed with. A practical application of this rule occurs quite frequently in cornice work, in the construction of window caps and other similar forms, to lit frames already made. In the conveying of orders from the master builder or carpenter to the cornice worker, it is quite customary to describe the shape of the head of the frames which the caps are to lit by stating that the width is, for example. 36, inches, and that the rise is 4 inches. To draw the shape thus described, proceed as follows: Set off A B equal to 36 inches, from the center of which erect a perpendicular, D C, which make equal to 4 inches. Continue D C in the direction of E indefinitely. Draw A D, which bisect, as shown, and draw H L, producing it until it cuts D C prolonged, in the point E. Then with B as center and E D as radius, strike the are A D B.

17. To Strike an Arc of a Circle by a Triangular Guide, the Chord and Hight Being Given. - In Fig. 147. let AD be the given chord and B F the given bight. The first step is to determine the shape and size of the triangular guide. Connect A and F, as shown. From F, parallel to the given chord A D, draw F C, making it in length equal to A F, or longer. Then A F G. as shown in the engraving, is the angle of the triangular guide to be used. Construct the guide of any suitable material, making the angle of two of its sides equal to the angle A F G. Drive pins at the points A, F and D. Place the guide as shown. Put a pencil at the point F. Shift the guide in such a manner that the pencil will move toward A. keeping the guide at all times against the pins A and F. Then reversing, shift the guide so that the pencil at the point F will move toward D, keeping the guide during this operation against the pins F and D. By this means the pencil will be made to describe the arc A F D. It may be interesting to know that if the angle F of the triangular guide be made a, right angle, the arc described by it will be a semicircle. By these means, then, a steel square, may be used in drawing circles, as illustrated in Fig. 148, the pins being placed at A, B and C. Fig. 147. - To Strike an Arc of a Circle by a Triangular

Guide.

18. To Draw a Circle Through any Three Given Points not in a Straight Line. - In Fig. 149, let A. Dand F be any three given points not in a straight line. through which it is required to draw a circle. Conned the given points by drawing the lines A D and D E. Bisecl the line A D by F C. drawn perpendicular to it, as shown. Also bisect D F by the line G C, as shown. Then the point C, at which these lines meet, is the center of the required circle.

19. To Erect a Perpendicular to an Arc of a Circle, without having Recourse to the Center. - In Fig. 150, lei A D B be the are of a circle to which it is required to erect a perpendicular. With A as center, and with any radius greater than half the length of the given are, describe the are x x, and with B as center, and with the same radius, describe the arc y y, intersecting the arc first struck, as shown. Through the points of intersection draw the line F E. Then F E will be perpendicular to the arc, and if sufficiently produced will reach the center from which the arc A B is drawn. Fig. 148. - To Describe a Semicircle with a Steel Square.

20. To Draw a Tangent to a Circle or Arc of a Circle at a given Point without having Recourse to the Center. - In Fig. 151, let A D B be the arc of a circle, to which a tangent is to be drawn at the point D. With D as center, and, with any convenient radius, describe the arc A F B, cutting the given arc in the points A and B. Join the points A and B, as shown. Through D draw a straight line parallel to A B, as shown by F H, then F H will be the required tangent.

21. To Ascertain the Circumference of a Given Circle. - In Fig. 152, let A D B C be the circle, equal to the circumference of which it is required to draw a straight line. Draw any two diameters at right angles, as shown by A B and D C. Divide one of the four arcs, as, for instance, DB, into eleven equal pans, as shown.

From 9, the second of these divisions from the point B, let fall a perpendicular to A B, as shown by 9 V. To three times the diameter of the circle (A B or DC) add the length 9 F, and the result will be a eery close approximation to the length of the circumference. This rule, upon a diameter of 1 foot, gives a length of about 3/50 ths of an inch in excess of the actual length of the circumference.

22. To Draw a Straight Line Equal in Length to the Circumference of any Circle or of any Part of a Circle - Fig. 149, - To Draw a Circle Through any Three Given Points Not in a Straight Line. Fig. 150. - To Erect a Perpendicular to an Arc of a Circle. Fig. 151. - To Draw a Tangent to a Circle or Arc. Fig. 152. - To Ascertain the Circumference of a Given Circle. Fig. 153. - To Inscribe an Equilateral Triangle within a Given Circle. Fig. 154. - To Inscribe a Square within a Given Circle.

Various approximate rules, similar to the one given in the problem above, for performing these operations are known and sometimes used among workmen, but cannot be recommended here because in using them considerable time and trouble is required to obtain a result which is not accurate when obtained, thus rendering such methods impracticable. The simplest and most accurate method for obtaining the length of any curved line is as follows: Take between the points of the dividers a space so small that when the points of the dividers are placed upon the line, no perceptible curve shall exist between them, and, beginning at one end of the curve, step to the other end of the same, or so near the end that the remaining space shall be less than that between the points of the dividers, then beginning at the end of any straight line step off upon it the same number of spaces, after which add to them the remaining small space of the curve by measurement with the dividers. This will be found the quickest and most accurate of any method for the pattern cutters' use.

The most common rules in use for the construction of polygons, whether drawn within circles or erected upon given sides, are those which employ the straight-edge and compasses only. Other instruments may also be employed to great advantage, as will be shown further on, leaving the student to decide which method is the most suited to any case he may have in hand. Accordingly, the construction of polygons will be treated under three different heads arranged according to the tools employed.