March 20, 1885.] 



♦ KNOWLEDGE ♦ 



237 



than two ball-bearing j to the axle to reduce the friction by 

 preventing it from bending. Yet both these suggestions I 

 make with hesitation, as tliey niiy not be of much conse- 

 quence, tor the ingenious mikers have arranged a sjiiral 

 spring on the front fork, which absorbs some of the vibra- 

 tion of the small steering-wheel, and they have brought the 

 two ball-bearings so close together that tlie strain on the 

 axle can cause but little llexure of the short portion of the 

 axle between them. All the riders of this uuichine with 

 whom I have spoken have praised its performance highly, 

 and of course the workmanship, style, and finish are superb. 



The Quadrant, No. S, with bicycle-steering, differs prin- 

 cipally from the machine I have just described in having 

 a front steering-wheel, i!G in. in diameter. The driving- 

 wheels are made from 10 in. to 41 in. diameter to order. 

 The machine has a strong band-break on the bicycle steering- 

 hsmdle, and a reserve or safety-break which can bo applied 

 by the foot to the front wheel. This machine should be 

 freer from vibration than any other tricycle. The weight 

 of either of the three machines I have named made as a 

 special light roadster with break would be under GO lb. 



The well-named Pioneer made by Pausey is constructed 

 on the same general plan as the machines I have just 

 described, but the driving-wheels are only 3G in. diameter, 

 geared up to drive as 60. Owing to the small size of the 

 wheels, the weight of the machine is reduced to less than 

 50 lb. This is the first attempt known to me to introduce 

 a small geared-up tricycle, and I shall be greatly obliged 

 if any one who has ridden one of these machines will 

 inform me whether his experience of it is favourable, as T 

 should expect it to be. 



The Ceutr.il-Geared Eucker, with direct steering, is 

 very similar to the four machines I have described. The 

 great peculiarity is that the rod turned by the steering- 

 handles is attached directly to the centre of the front 

 Bteering-wheel, by a double joint known to mechanics as 

 Hook's joint 



This steering arrangement I have just had applied to my 

 own small central-geared Eucker, and very shortly I will 

 report on its performance. 



Once the best form for the tricycle has been found — and 

 I doubt if the tricycle of the future will vary much from 

 the lines of those I have described — we shall get the 

 machine improved in details. Only when this has been 

 carefully done can we hope to have machines which will 

 compare favourably with bicycles, the details of which have 

 been elaborated until there scarcely seems room left for 

 their further improvement. 



CHATS ON 



GEOMETRICAL MEASUREMENT, 



Bv EicHAKD A. Proctor. 



{Continued from p. 126.) 

 VOLUMES AND SURFACES OF SOLIDS. 



A. We have now done a good deal in the way of deter- 

 mining the areas of various conic and cycloidal areas, and I 

 think we have sufficiently tested the method of dividing 

 them up into numerous (I may say numberless) indefinitely 

 small portions. I would submit that it would now be well 

 to consider those surfaces, volumes, and so forth, which we 

 are more apt to meet with in ordinary life. I cannot say 

 that I ever had occasion for precise information about the 

 area of a parabola or hyperbola, neither do the cycloidal 

 curves come many times a day under my notice. But I 

 often want to know the approximate volume of globes, 



cylinders, cones, and so forth, because questions of interfst 

 (to me, at any rate) often depend on sucli knowledge. Only 

 the other day, for example, my fiiend .Stout was sptuking 

 of a cylindrical vat so many feet deep and so many binaii, 

 and the question aro-ie how m\ich the contents of that vat, 

 in beer (of known speL'ilic gravity) would weigh ; but I did 

 not remember the formula for the volume, and did not at 

 the moment see my way to determining it independently. 



M. There ought to have been no dilliculty on that ]ioint; 

 when you are dealing with spherical volumes and surfaces, 

 or with the volumes and surfaces of cones, spindles, and so 

 forth, you might not see your way to determine the correct 

 value, but the volume of a solid like a cylinder, which has 

 the same cross section all the way up (supposing it to stand 

 on a circular base) is obviously given by multiplying 

 together the numbers representing the height and the area 

 of the base. 



.1. Let us, however, consider these more familiar and 

 useful matters, at length. 



M. Willingly. As regards curved surfaces we shall not 

 have occasion for any knowledge beyond that which you 

 already have. But as regards S)lids it is necessary that 

 you should know what is not given in the parts of Euclid 

 usually read, — the relations between the volumes of 

 pyramids, prisms, and parallelopipeds. 



A. Why^ 



^f. Because we cannot deal generally with solids without 

 dividing them into their simplest elements, and as the tri- 

 angle with its three sides is a simpler element of an area 

 than the parallelogram with four, so the triangular prism 

 with nine edges and five faces is a simpler element of a 

 volume than the parallelepiped with twelve edges and six 

 faces, while the triangular pyramid with six edges and four 

 faces is a simpler element of volume even than the prism. 



A. Tell me, then, what are the relations necessary to be 

 known and understood 1 



M. Note, first, then, that the volumes of parallelopipeds 

 and prisms of all kinds are very simply dealt with. Paral- 

 lelopipeds are indeed prisms, and need not be separately 

 considered, were it not that the rectangular parallelopiped 

 is tiie form to which all volumes are most conveniently re- 

 duced, just as rectilinear areas are best represented by 

 rectangles. 



A. Shall we then first take the triangular prism ^ 



J/. We may treat this solid as triangles are dealt with 

 in the first book of Euclid, only we need not enter quite so 

 carefully into details as Euclid deemed it his duty to do. 

 Let A E C, a ec, be prisms on equal triangular bases ABC, 

 a be {B and C are supposed to be farther from the eye than 

 A C and b c), and between the same parallel jilanes, ABC, 

 a be, and L) E F, def. To compare their volumes, take 

 two planes close together and parallel to the parallel 

 planes between which the prisms stand. Let these two 

 close planes cut from the prisms the thin triangular slices 

 P Q R and p q r. Then these triangular slices are mani- 

 festly equal Ln volume. 



Fig. 1. 



A. I do not see that they are more obviously equal than 

 the prisms A E C and ace. They are both prisms, and 



