March 1, 1886.] 



KNOVV^LEDGE ♦ 



167 



of AVhist bauds which he has suggested as tests of the methods of 

 leading likely to be followed by good players. Nothing can be 

 better than to deal with tests of this kind : because, in considering 

 what one would do in a test case, and noting what other players 

 would do, one learns the true principles which should govern Whist 

 in actual practice, depending as this does lai^ely on the considera- 

 tion of what other players are likely to do. 



A remark of mine (as " Five of Clubs ') about a rule touched on 

 by " Pembridge " iu regard to not leading from a long weak suit 

 without re-entering strength, has been misapprehended by the 

 able Whist-editor of the Australasian — though the fault is certainly 

 my own. I spoke of the rule " laid down by ' Pembridge ' " when 

 I ought to have said the rule " stated " by him. 



iAfrnit InbnUion^. 



[Ihe following paragraphs were left standing when the last 

 number of weekly Knowledoe came out, and appear now owing 

 to the change in publishing and printing arrangements. We do not 

 propose to continue the description of recent inventions, which are 

 unsuitable to Knowledge in its monthly form.] 



GRiCE'S PATENT EPICVCLIC ELLIPTOGEAPH. 



[Patent, No. 4,582]. — The Epicylic EUiptograph (Fig. 1) consists 

 of a cross frame, O, supported by uprights at each end, carrying at 

 the centre the operating liandle G, and the mechanism : which is 

 such, that while the handle, G, by means of a central pivot, gives the 

 inner arm, H, attached to it underneath, any given angular motion in 

 either direction, the wheel-work C D E F A R S S' B, by means of a 

 cannon pinion attached to the wheel F, and fitted round the central 

 pivot, gives the outer arm, L, an equal angular motion in an opposite 

 direction, which causes the pen, JI, fitted on the outer arm, L, to 

 move in an elliptical path. 



Fig. I. — Grace's Epicyclic EUiptograph, for describing ellipses 

 of various sizes and proportions, from a point upwards. Relative 

 diameters of the wheels : — A = 3 ; D = 1 : E and F = 2 : A R and 

 S = 2: SandB = 2i.) 



It can be set to describe ellipses of any given proportion, from a 

 straight line to a circle, and of any size, from a point up to twenty 

 inches by ten inches and over, by varying the length of the inner 

 arm, H, and the outer arm, L, by means of the fixing-screws 

 J K and N, and the graduated scale on the arms, so as to make 

 the inner arm, H, eqtial to one quarter the difference between the 

 major and minor axis of the required ellipse, and the outer arm, L, 

 equal to one quarter of their sum, or vice-versa, the inner arm, H, can 

 be made equal to one quarter their sum, and the outer arm, L, equal 

 to one quarter their difference. And after adjustment for the given 

 size and proportions, the required ellipses can be drawn, with their 

 axes, at any convenient angle with the cross frame 0. 



It is also made so that pairs of equal change-wheels of various 

 sizes can be fitted ; one on the under end of the cannon pinion of 

 wheel F, and the other on the up end of the pivot I, to vary the 

 length of the inner arm, H, instead of the permanent jointed 

 frame and wheels, T T ; and, by altering their relative size, the 

 instrument can be used to describe various symmetrical curves 

 and scrolls for ornamental purposes. 



The principle of the motion of the above elliptograph and its 

 adjustment can be proved mathematically. 



The elliptograph, having only a single cross-frame, is of smaller 



rclati\e size than the elliptographs constructed on the tramme^ 

 principle, which require a double cross-frame, equal in length to 

 the difference between the major and minor axes of the most 

 elongated ellipse they will describe, and it also requires a simpler 

 movement of the handle to operate it. 



The elliptograph, with or without change-wheels, can be ob- 

 tained of Mr. Harling, 40, Hattou-garden, E.G. 



Fig. 2. 



Proof. — In Figure 2, let JI I = the outer arm, L, of the ellipto- 

 graph, and G I = the inner arm, H. Let D E = major axis and 

 F = the minor axis of any ellipse, and let JI I be produced to meet 

 the minor axis, F O, in the point B. Let the vertical line, I W, be 

 drawn perpendicular to the major axis, D E, and the horizontal line, 

 I Z, be drawn perpendicular to the minor axis, F 0. Then, as by the 

 construction of the above epicyclic elliptograph, angle C I W always 

 equals angle IGZ, and the angle WIG also equals the alternate 

 angle IGZ, consequently the angle W I G equals the angle C I W, 

 and, therefore, the right-angled triangles G W I and C W I, having 

 the side W I in common, are similar and equal, and the hypotenuse 

 C I, equals the hypotenuse I G ; and it will be seen that the right- 

 angled triangle C W I is similar and equql to the right-angled 

 triangles I Z G and I Z B, as both the angles, IGZ and I B Z, 

 equal angle C I W, and their side, I Z, being equal to side W G, is 

 also equal to side C W. Therefore, their hypotenuses, viz. the lines 

 or part, I G - I G and I E, are equal in any position of the line or 

 arm, JI I, produced, and consequently the point C is always on the 

 same straight line as G W produced, viz. tlie major axis E D, and 

 also the end or point B is always on the line GZ, produced, viz. 

 the minor axis, F 0, produced or othei'wise, as the right angles of 

 the pair of triangles are adjacent. And as, by the princijjies of 

 ellipses, any point JI, on part B C, produced beyond C on the major 

 axis, will describe an ellipse, therefore, as part I JI represents the 

 outer arm of the above epicyclic elliptograph, the extremity of the 

 outer arm, L, of the above epicyclic elliptograph will describe an 

 ellipse. And as from the principles of ellipses the part C B must 

 equal the difference between half the major and minor axes of any 

 required ellipse, and as from above the parts I C, I G, and I B are 

 equal, therefore C B, or ,| (major — minor axis) equals the two parts, 

 C I and I B, and is, consequently, equal to twice I G, and conse- 

 quently I G. viz. the inner arm of the above epicyclic elliptograph, 

 equals half of half the difference between the given major and 

 minor axes: that is, one quarter the difference between the major 

 and minor axes, which is according to the rule for setting the inner 

 arm, H, of the elliptograph. And, also, as I G equals \ the difference 

 between the given major and minor axes, and as IG = I B, therefore 

 IB equals \ the difference between the given major and minor axes, 

 and as the whole line, JI 11, by principle equals half (he major axis, 

 therefore the part I JI, viz. the outer arm, L, of the above epicyclic 

 elliptograph equals J major axis — \ (major — minor axis) = \ major — ^ 

 major -^ \ minor axis, viz. \ (major -i- minor axis) which is also accord- 

 ing to the rule for setting the outer arm, L, of the above epicyclic ellip- 

 tograph. And from the above proof, showing that with the outer arm 

 JI 1 = ^ (major -I- minor axis) and the inner arm I G = J (major- 

 minor axis), and the angle C I W always = angle IGZ, any point 

 JI on C JI will describe an ellipse. It can also be shown that if the 

 inner arm H', viz., G I', be made = ^ (major -e minor axis) and the 

 outer arm L', viz., JI I', is made = ^ (major — minor axis), the point 

 JI on the shorter outer arm JI I' will also describe a similar and 

 equal ellipse. Let the longer inner arm I' G be drawn parallel and 

 equal to the longer outer arm JI I in any position as given above ; 

 then if the end II I' and I G be joined, JI I' G I will be a paral- 

 lelogram, and the side I G will represent the shorter inner arm as 

 given above, and it will be seen as follows that the shorter outer 

 arm of the above elliptograph, viz., M I', being set equal to the 

 side JI I' of the parallelogram M I' G I will also coincide with it. 



