September 1, 18S6.] 



KNOWLEDGE 



337 



The next will have angles at m and b, foiirfi/ths oi ab 

 and BC from bc and ab re.spectively, and one-fourth further 

 than D from ba and bc respectively. 



And so we may pass on from angle to angle, getting 

 successive angles E, G, k, m, kc, 1, ^, f , 4, A'c, kc, of ab 

 from BC, and J , \, ^, ^, ikc, &c., further than d from ba ; 

 and successive angles F, H, L, b, kc, i, §, ^, i, kc, kc, 

 of BC from AB, and \, i, ^, \, kc, kc, further than D 

 from EC. 



The next longest either way will have angles at g and H, 

 further than d from ba and bc respectively. 



The next will liave angles at k and l, f of bc and ab 

 from BA and bc respectively ; and ^ further than d from 

 BA and BC respectively. 



The next will have angles at >[ and b, t of BC and ab 

 from ba and bc respectively, and \ further than D from 

 BA and bc respectively. 



And so we may pass on from angle to angle, getting suc- 

 cessive angles i, '^, J, i, kc, of bc and ab from ba bc, and 

 ■], ;V, }i, \, further than d from ba and BC respectively. 



Bj' a well-known property of the hyperbola these angles 

 all fall on a hyperbola fde through d, having ba and bc as 

 asymptotes. The points thu.s obtained fall nearer and 

 nearer together as the point d is approached. 



In the figure the parallelogram bd is rectangular ; but it 

 need not be. The construction and comments remain un- 

 changed if bd is rhomboidal. 



III. To inclose ten spaces by six ropes fastened to nine pegs. 



Draw two lines, ba, bc, as in fig. 4, preferably at right 

 angles. 



Divide ab into four parts, preferably equal at d, e, f, and 



A 



BC into four at c, ii, k. Put the nine pegs at the nine 

 points thu-s determined, and stretch the six ropes : — 



( 1 ) Fi om A to B ; 



(2) From b to c ; 



(3) From A to g : 



(4) From d to ii; 



(.i) From E to K ; and 



(G) From r to c. 

 In the same way, with four ropes and five pegs, three 

 spaces may be inclosed ; with five ropes and seven pegs, 

 six spaces ; [with six ropes end 9 pegs, ten spaces]; with 

 seven ropes and eleven pegs, fifteen spaces, kc, kc, and 

 generally — 



With (n-\-2) ropes and (2K-t-l)pegs ^('i-fl) spaces may be 



inclosed. 



If the figure is drawn symmetrically — that L", BA made 

 equal to bc — and both ba and bc divided into the same 

 number of equal parts, the lines drawn as AG, Dn, ek, kc, 

 all touch a parabola whose axis is the bisector of the 

 angle abc. 



If ba and bc are not equal, but divided equally, the lines 



drawn as ag, dh, ek. kc, all touch a parabola, whose axis 

 is parallel to a line drawn from b to the bisection of AC. 



The last two puzzles suggest excellent methods for 

 drawing hyperbolas and parabolas, on which we may here- 

 after touch. 



Here is a puzzle worth working at a little, the solution 

 being instructive as regards the linear and areal proportions 

 of similar figures. 



I^^. A farmer's property is in the form of a large ellipse, 

 the farmhouse occupying a space ichich may be neglected by 

 comparison with the farm, lying outside, but touching the 

 periphery. At his death the farm is to be divided equuliy 

 betireen the farmer's three sons, and so that each so7i shall 

 have access to the farmhouse, but his estate shall be inclosed 

 within a fence having the same length as the circumference of 

 the undivided far/n. 



The following puzzles depend on the properties of the 

 stereographic projection : 



V. Draw six circles intersecting each other in eight points 

 at angles of sixty degrees. 



VI. Draw twelve circles intersectimj each other in tvienty 

 points at angles of sixty degrees. 



THE NATURALIST'S LABORATORY. 



COXTEIBCTION III. 



'HE raison d'etre of a fireplace in the labo- 

 ratory is obviously to warm the room during 

 the colder periods of the year, but, like the 

 window, it may be made to subserve other 

 duties, and more notably to assist in the 

 thorough ventilation of the apartment. The 

 fireplace cannot, however, be constructed to 

 provide for perfect inlets of fresh air with economy, and 

 must therefore fulfil that function only as an adjunct to the 

 ventilating window in small rooms, and to additional special 

 inlets, such as those already described, for large apartments, 

 viz., Boyle's mica flap ventilator, the Sherringham ventilator, 

 and Lamb's air-box. Otherwise an adequate balance cannot 

 be maiatained between the fresh air introduced and the 

 vitiated air which is expelled, and it is only by an observ- 

 ance of this grand principle that efficient ventilation can be 

 secured. 



A good fireplace for the laboiutory, whether constructed 

 to burn solid, liquid, or gaseous fuel, ought to combine many 

 essential qualities. It should warm the room to the normal 

 summer temperature quickl}', and be capable of maintaining 

 that degree of warmth without much deviation either one 

 way or the other with a minimum of attention; it should 

 provide for the complete removal of the products of com- 

 bustion of the fuel in question ; it should afford means for 

 an inlet of fresh warmed air to assist in the thorough venti- 

 lation of the apartment; but, with all this, it should be 

 economical. It is only within the past few years that 

 this combination of requirements has been attained, and 

 with success pioportionate in each variety to the different 

 available forms of fuel. Thus, when wood, peat, coal, and 

 coke are burned in an open grate, a cheap and cheerftil * fire 

 is the result, and fireplaces made of ^.toneware, upon slow 

 combustion principle, with shallow fireclay fuel-holder.-. 



* " Cheerful " is a quality which is wholly dependent upon 

 custom. No one would think of producing a similar optical effect 

 in summer, nor after the transient twilight in the tropics, even 

 though such could be produced without raising the temperature, 

 e.g., by the incandescent electric light specia'ly modified. 



