(f g. From e measure to d, through this draw a line parallel to gtf; measure from c and dtog',g. Bisect the distance dc in d'; from ef as a centre, with d'g' as radius, describe the arcs joining the lines through d, c. In like manner,7 neasure from c to e and f: the points f, e " will be the centres of the arcs j )ining the lines.
Example 8, fig. 8, represents a projecting ' snug,' by which two parts may be joined by means of a bolt secured by a nut, passed through holes bored in each. Draw the line a b, and another at right angles
•scribe the curve. From / measure to e; a line drawn from this, parallel to ab, gives the end-line. The centre g (as also d) is found by trial on the copy, and the points transferred to corresponding parts on the board. The line dc represents one method of transferring them. 4 \ Example 9, fig. 9, represents a side view of a 1 pulley,9 or ' dram,' shewing the arms and centre. Draw any two lines corresponding to ab, c<L From g as centre, with gb as radius, describe the circle, and also the interior circle c/cf; from g with gh put in the small circle representing the diameter of the centre or eye of the wheel. From the lines 1, 1 with distance 1, 2 lay off on either side of all the centre-lines of the arms; next, from the points where the interior circle cuts these lines at the points cf, g', lay off on each side equal to half the thickness of the end of the arm as it joins the inside of wheel. Join the points thus obtained with those previously obtained on the centre of the wheel, as 2, 2.
Example 10, fig. 10, represents the plan of a circular cylinder or receptacle, the small circles shewing the position of the circular heads of the bolts used for attaching the cover to the main body of the receptacle. The method of finding the centres of the small circles is as follows: Draw any two lines ae, bd; from the point of intersection as centre, with radius ab, ac, describe circles; bisect the distance between these, as be, in the point /. From a as centre, with af as radius, describe a circle fed: the tres of the small circles will be found on this line. Find the position
to it. From a measure to b7 and put in the various horizontal lines and the base; from b measure to c, and parallel to ab draw a line from this point. From c measure to d; from d as centre with radius dc de-
of any faro of the circles, as fe or 4d; transfer these points to the board. In the copy the centres of four of the circles trill be found where the diameters ea, bd cut the circle drawn through/df. Count the number of circles between f and e, or € and d; divide the circular line passing through/, and between e and / or e and d, into as many equal parts as will give as many centres as there are circles in the copy: these points will be the centres of the circles.
Example 11, fig. 11, represents the plan of a small thumb-wheel attached to the head of a screw-bolt, by which it may be easily moved by means of the finger and thumb. From a with ab describe a circle, draw the diameter db; divide the semicircle db into four equal parts; from a draw lines through e, f; do the same on the other semicircle. From a measure to n; with an describe a circle: the points on the radial lines,
as n,¡¡where this intersects them, are the centres of the circles which terminate each radial arm. From a describe the small circle ac; from the points where this intersects the radial lines, as c, lay off on each side of these the distance coy join the points thus obtained on the circle aco with the extremities of the circular ends. Another way of joining the radial •arms to the centre or eye may be understood by inspection of the diagram in fig. 12, where J is the centre of the circle, part of which joins the arm with the centre.
Example 12, fig. 13. Draw any two lines corresponding to ag, dd in the copy; from the point of intersection c measure to the points h, g; through these draw lines parallel to dd. From A, g measure to mm, hk; join m h; put in, in like manner, the internal parallelogram lilt. From the point c, with radius ce, ca\ and ca, describe the circles as in the oopy.
Example 13,%. 14, represents plan of part of a 1 valve Opiate-1 From any centre a describe a circle ab, .and one within this, as ac; continue this last all round, the part from m to p being afterwards rubbed out when ; the drawing is finished and inked in. From a with ad put in part of a circle ede. From d measure to e, e, and through these draw lines to the points, as g, on each side /. On each side of the line ah measure to ' p and iTi, also from n to o; join ! mo. Put in the circles at n and h; join them fui in the drawing.
Example 14, fig. 15, represents the plan of a 1 lever.' Describe the circle a h, draw through a the diameter bad; from a measure to c; put in the circle cd. Bisect ac in e, and through this draw a line at right angle? to ad. In the copy take the points/(where ef intersects the curve) h and g9 where the curve hg touches or joins to the circles described from c and d. By means of these points (see the problem in the work on Practical Geometry, to find the centre of a curve, three points in that curve being given), the centre m will be found.
Example 15, fig. 16, represents the method generally employed of constructing the central part of a ' spur-wheel.' The circles c,/, and m are
; described from the centre d; the circle m is divided into as many equal parts as there are arms in the wheel, any central point of these, as w, being adopted as the datum-point from which to take the measurements. The space between any two of these arms, as ab9 is bisected, and a line, as dfy drawn. By measuring from /to e, g, the centres of the curves at e and^r will be obtained; the centre of the curve aft is also on the line df. Example 16, fig. 17, represents the plan of a pulley with curved arms. The method of describing these is explained in
Example 17, fig. 18. The first operation necessary to be done is to find in the copy, fig. 17, the centres of the circles forming the curves : these will easily be found by trial. Next draw two lines at right angles, as in fig. 18, intersecting in the point a corresponding with the centre c, ♦fig. 17. From a describe circles representing the rim and thé eye of the wheel in last figure. From c, in fig. 17, measure to the centré ft, from which the curve d is described, and from a, fig. 18, a circle a o : on this line the other centres, as e, fig. 17, will be found. In like manner, from the centre c measure to a, from which the curve as is described, and from a, fig. 18, describe the circle gh. From d measure to h, from h to f} and from / to\ g : these are the various centres. Or the curves next the eye may be drawn in first, and the curves with radius as be described, to meet these from the circle gh. In this example the arms are of uniform breadth ; where they get gradually less from the centre or eye of the pulley outwards, the method of describing them may be learned from
Example 18, fig. 19. The points from which the curves are drawn must be found, and corresponding points transferred to the paper, as in last example. Two circles, as d, o, will thus be obtained, in which the centres of the various curves will be found. Put in the circle representing the eye of the pulley, and draw a diameter aft; draw a line in the copy corresponding to this, and measure from ft to the point representing the centre of the circle from which the curve cc is drawn; transfer this to the copy, and from d with dc draw the curve cc; from c measure to f7 thus giving the breadth of arm at eye ; from /, with the radius of the curve f taken from the copy, cut the circle o in o ; from this point with same radius describe the curve fg. The various points denoting the centres of the curves are given in the circles, the points et being those where the curves join the circle or eye of the pulley.
Example 19, fig. 20, represents the bottom part of foot of a cast-iron framing. Draw a line cd; from c measure to a and b; through these ds&
draw lines perpendicular to cd; with ac from a describe the curve co. From b measure to e. Find the centre of the curve joining oe: it is/. Find by any of the methods already described the point m; join md by the curve.
Example 20, fig. 21, represents part of the frame-work forming the support for the bearings c in which vertical spindles revolve. Draw ab, ad; measure from a to d and c; draw ce at right angles to ad. From e measure to/, and from / draw to g parallel to ab; from a measure to h and m. The centre of the curve joining/m will be found at g on the line fg. The method of filling-in the drawing is shewn by one-half.
Example 21, fig. 22, represents the outline of side elevation of framing. Draw the line ab, and at right angles to it 2'd; measure from 2' to da, and to 3'. Through these points draw lines dd, cid, a'c'; join the points d, d by the part of the circle, as in the diagram. From 2' measure to f7 and draw the line tft; from f measure to t, t; from these points draw lines parallel to 2'd. From t measure to n; draw nnf, and from n, n', with radius nri', describe curves meeting, as in the drawing. From / measure to /, and draw hfh; from h, h with radius hh describe curves meeting in g on the line vv. The curves 5, 6 and 3 are described from the centres n\ w,and4,6 from centre h. The lines mm, oo are joined by curves described from the centre 8, which centre is found by describing arcs from the points m, o with any radius greater than half mo, and joining the intersection of these arcs by a line as in the copy. ¡pA
Example 22, fig. 23, is another outline representing the side elevation of framing. The curve k is described from the centre f on the centreline bf; the centre-lines of the other parts are^at m, e, d, and c. _ , fig. 24. I
Example 23, fig. 24, is another form of framing. The centre of the curve n, joining the lines from m, m, is at h, on the centre-line oh; the centres d, d are on the line drawn through c to h b, parallel to mm; the centre of the circle e is at
Example 24, fig. 25, represents the front elevation of a ( cross head9 and' side levers.' The centre-lines are ad, eh, vv. The plan is shewn i
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