FEa 13, lSSr>.] 



♦ KNOWLEDGE ♦ 



125 



just as our bookshelves were (Figs. 'JO and 30, Vol. VT. 

 pp. 479 and 180) ; ;^ in. gives a secure hold, and they may 

 further be glued and screwed (or nailed). As for the 

 length of the st^ps, we may make the one next the 

 ground 1 1) in. long, inside measurement, and the sixth one 

 connting from the bottom 1 ft. long, inside measure. This 

 spreads out the base of the step.*-', and makes them stand 

 more firmly and securely on the ground. In the front, I 

 may mention, the projecting part of the step extends 

 across the whole width, as shown in Fig. 38, A, the pi. in 

 of the step itself being shown at B. Perhaps I ought to 

 have said before this (whut, though, the figures will have 

 made quite clear) that the top and bottom of the sides >< s 

 must be cut on a bevel to the angle at which the steps 

 will stand when in use. For the top, I, wo shall want a 

 piece of inch deal Gin. wide and 11 in. long. A little 

 attention will show that this will overhang the top of the 

 completed .steps 1 iu. every way. To make a neat job the 



Fig. 38. 



u 



^s-m 



Fig. 39. 



sides should be morticed into this, though they may, if pre- 

 ferred, be secured in grooves in it by glue and screws. This 

 being accomplished, we fasten very firmly, by screwing, a 

 piece of inch board, 14 in. by 6 in. wide {h in Figs. 37 and 

 39), flush against the back of the sides and close up against 

 the top t. We mnst evidently bevel the top edges of this 

 to make it fit close against t. I have insisted on the firm 

 fixing of this board, b, because it is to it that we are about 

 to attach the frame which supports the steps, and which 

 we are immediately going to make. Fig. 39, A, shows the 

 frame completed. It tapers in precisely the same way as 

 the front of the steps — ie.,it measures 14 in. in outside 

 width at the top and 18 in. at the bottom. The rails, /• r, 

 are morticed into the styles, //, in the usual way. The top 

 rail, r, is G in. deep; the bottom ore, 2^ in. deep — this 

 being the width of the styles too. Note particularly the 

 .slight slope which must be given to the sides of these rails, 

 to make them fit true against the slightly diverging styles. 

 The bottom rail may be 1 ft. from the ground. The frame 

 should be attached to the steps by a pair of l^-in. "butts " 

 (h h, B, Fig. 39), screwed, as shown, into the backboard, b, 

 and the top rail, r. Finally, holes must be bored at c and 

 c', in the sides and in the styles of the frame, and pieces 

 of sash-cord passed through them and securely knotted at 

 each end. The length of these cords must be such that 

 they shall be strained quite tight when the hase of the 

 steps stands flat on the floor. 



And now for a more simple job still. Steps are almo.st 



always used indoors. For out-door work a ladder is handy, 

 and I will conclvKlo this jiapor with directions for making 

 one. First we want a clean, strainht lii-pole 18 ft long, 

 and 22 oak " stives," or " rounds " as they are called at 

 the timber-yards, where they cost a penny each. The pole 

 must bo marked oil" in 9 in. lengths, beginning 9 in. from 

 its thicker end, and bored right through at each maik with 

 a :,'in. auger. Bo sure and bore all these holes truly 

 square to the length of the pole. Then, with a sharp saw, 

 the pole is sawn exactly in a half from end to end, on a 

 line midway between the holes on each side of it. We 

 shall now have the two sides of the ladder flat on the 

 inside and rounded on the exterior. Into the holes in tho 

 flat side we drive our staves, which wo may first trim up 

 neatly with a spokcshave; and observe to make thoKo at 

 the bottom of tho ladder longer than those at the top, so as 

 to sjiread out the foot of our ladder and give it a firm base. 

 In the one we are making, tho bottom round may bo 11 in. 

 long, and the top one 9 in. in length, both inside measure, 

 which will give a very good taper. When the rounds are 

 all driven home the ends are sawn off outside of the sides, 

 chisel cuts made, and taper wedges of oak or other hard 

 wood dipped in very thin hot glue and driven hard home 

 in the cuts. Then our ladder is finished. 



CHATS ON 

 GEOMETRICAL MEASUREMENT. 



By Richakd A. Proctor. 



{ Continued from p. 85.) 



A. I .see that in dealing with curved areas you 

 up the surfaces you wi.-l 



of 



m 

 dividing 



use 

 1 to 



.see 

 varied ways 

 determine. 



M. Yes ; that is the great 'pleasure of the work— so 

 many ways of attacking problems are open to you. 



A. Will you give me some examples which seem to you 

 worth noting. 



^f. Here is one, taken from my book on the Cyclo'dal 

 Curves : — 



APD (Fig. 1) is a 

 half cycloid; ApB half 

 the generating circle ; 

 ABUT the enclosing 

 rectangle (as usual, i.e. 

 AT, D T tangents). 

 Draw LjuPM, IqQ in, 

 parallel to and equidis- 

 tant from A T and B D. 

 Then 7^ P = arc A j) ; 

 q Q=arc A (7=arc B ji. 



.•.;jP-)-(7 Q=arc A^j-farcB^) 



I 711. 



A p B = LM or 



Hence it we regard P;), r/Q, LM, Im as elementary rect- 

 angular areas, and suppose our p.irallels to begin from 

 coincidence with A T and B D, moving at equal rates till 

 they coalesce on C c, we see that the sum of the areas 

 swept over by the advancing parts p P and </ Q will always 

 be equal to either of the equal areas swept out by L M and 

 I m. Hence 



area A ;> P -f area B </ Q D = rect A M or rect B m, 

 and area Ap B D P=rect A c or B c 



= AT.AO = AC.arcApB = gene- 

 rating circle. 

 Hence area between cycloid and base = thrice the generating 

 circle, as before. 



A. That is very simple. You usually take rectangular 

 elements, I notice. 



