WHEEL. 



wheel would fall fn : the fame thing would alfo happen in 

 marfhy ground, where the fniall wheel would fink wholly in 

 the fame hole which the great one would fink into but in 

 part. The large wheel would alfo have the advantage of a 

 fmall one in rinng over eminences or rubs that occurred ; fo 

 that the former would go over rubs much higher than the 

 latter ; and indeed over any eminences, provided their height 

 be not equal to its femidiameter. Defaguliers has reduced 

 this matter to a mathematical calculation, in his Exp. Phil, 

 vol. i. p. 171, &c. 



A late writer has alfo proved, that a wheel of eight feet 

 diameter has fomewhat more than twice the advantage in 

 overcoming obilacles of a wheel of two feet ; and he found, 

 in praAice, that if it requires a certain power to draw a 

 carriage of a certain weight over a certain obftacle, with 

 wheels of any determinate diameter, it will reqiiire wheels 

 of four times that diameter, to draw the fame carriage over 

 the fame obftacle with half that power. This writer alfo 

 obferves, that, in the draught of carriages afcending inclined 

 planes, the moving power afts not only againft the vis in- 

 ertias, which is always equal to the abfolute gravity of the 

 load, but alfo againft its relative gravity, which increafes 

 with the inclination of the plane ; and with refpeft to car- 

 riages raifed on wheels, it is to be obferved, that the higher 

 the axle is removed from the plane, the farther is the centre 

 of gravity removed out of the perpendicular line of fup- 

 port ; fo that the lower the wheel, the lefs is the relative 

 gravity of the carriage. Hence he infers, that fuppofing 

 the friftion of two carriages of equal weight, but of dif- 

 ferent fized wheels, to be equal, the low-wheeled one would 

 be drawn up hill, on a fmooth plane, much more eafily 

 than the high-wheeled one ; though on a fmooth, horizontal 

 plane, the latter would be drawn more eafily than the for- 

 mer. On the contrary, in going down hill, the high- 

 wheeled carriage will be urged forward, by its relative gra- 

 vity, more than the low-wheeled one. Jacob, ubi fupra, 

 p. 63, &c. 



It appears, therefore, that the larger wheels are, the more 

 advantageous they are in proportion, provided that they are 

 not more than five or fix feet in diameter ; for when they ex- 

 ceed thele dimenfions, they become heavy ; or if they are 

 made light, their ftrength is proportionably diminiflied, and 

 the fpokes, being long, are more liable to break : befides, 

 horfes applied to fuch wheels, would be incapable of exert- 

 ing their utmoft ftrength, by having the axles higher than 

 their breafts, fo that they would draw downwards ; as in 

 fmall wheels the draught is made more difEcult, by the 

 horfes drawing upwards. 



It is obferved by Dr. Brewfter, in the appendix to his 

 edition of " Fergufon's Mechanics," that when the wheels 

 of carriages either move upon a level furface, or overcome 

 obftacles which impede their progrefs, they aA as mechani- 

 cal powers, and may be reduced to levers of the firft kind. 

 In order to elucidate this remark, which is of great import- 

 ance in the prefent difcuflion, let A be the centre, and B C N 

 the circumference of awheel 6 feet in diameter, and let the 

 impelling power P, which is attached to the extremity of a 

 rope A D P, paffing over the pulley D, aft in the hori- 

 zontal direftion A D. Then, if the wheel is not affefted 

 by friftion, it will be put in motion upon the level furface 

 M B, v/hc-n the power P is infinitely fmall. For fince the 

 whole weight of the wheel refts on the ground at the point B, 

 which is the fulcrum of the lever A B, the diftance of the 

 weight from the centre of motion will be nothing, and there- 

 fore the mechanical energy of the fmalleft power P, afting 

 at the point A, with a length of lever A B, will be infi- 

 nitely greit when compared with the refiftance of the 



weight to be raifed | and thk will be the cafe, howi^ver ftnall 

 the lever A B, and however great be the weight of the 

 wheel. But as the wheels of carriages are conftantly meet. 

 ing with impediments, let C be an obftacle fix inches liigh, 

 which the wheel is to furmount. Then the fpoke A C will 

 reprefent the \evir, C its fulcrum, A D the direftion of the 

 power ; and if the wheel weighs 100 pounds, we may repre- 

 fent it by a weight W, fixed to the wheel's centre A, or to 

 the extremity of the lever C A, and afting in the perpendi- 

 cular direftion A B, in oppofition to the power P. Now 

 the mechanical energy of the weight W to pull the lever 

 round its fulcrum in the direftion A E, is reprefented by 

 C E, while the mechanical energy of an equal weight P to 

 pull it in the oppofite direftion A F, is reprefented by C F ; 

 an equilibrium, therefore, will be produced, if the power P is 

 to the weight W as C E to C F, or as the fine is to the co- 

 fine of an angle, whofe verfed fine is equal to the height of 

 the obftacle to be furmounted ; for E B, the height of the 

 mound C, is the verfed fine of the angle B A C, and C E 

 is the fine, and C F the cofine of the fame angle. In the 

 prefent cafe, where E B is fix inches, and A B three feet, 

 E B, the verfed fine, will be 1666, &c. when A B is loCXD ; 

 and, confequently, the angle B A C will be 33° 33', and 

 C E will be to E F, as 52 to 83, or as 66 to 100. A 

 weight P, therefore, of 66 pounds, afting in a horizontal 

 direftion, will balance a wheel fix feet diameter, and 100 

 pounds in weight, upon an obftacle fix inches high ; and a 

 fmall additional power will enable it to furmount that ob- 

 ftacle. But if the direftion, A D, of the power, be in- 

 clined to the horizon, fo that the point D may rife towards 

 H, the line F C, which reprefents the mechanical energy of 

 P, will gradually increafe, till D A has reached the pofu 

 tion H A, perpendicular to A C, where its mechanical 

 energy, which is now a maximum, is reprefented by A C, 

 the radius of the wheel ; and fince E C is to C A as 53 to 

 1000, a little more than 53 pounds will be fufficient for 

 enabling the wheel to overcome the obftacle. 



Proceeding in this way , it will be found, fay s our author, that 

 the power of wheels to furmount eminences increafes with their 

 diameter, and is direftly proportional to it, when their weight 

 remains the fame, and when the direftion of the power is 

 perpendicular to the lever which afts againft the obftacle. 

 Hence we fee the great advantages which are to be derived 

 from large wheels, and the difadvantages which attend fmall 

 ones. There are fome circumftances, however, which con- 

 fine us within certain limits in the ufe of large wheels. When 

 the radius A B of the wheel is greater than D M, the height 

 of the pulley, or of that part of the horfe to which the rope 

 or pole D A is attached, the direftion of the power, or the 

 line of traftion A D, will be obhque to the horizon as A i^ 

 and the mechanical energy of the power will be only A e, 

 whereas it was reprefented by A E when the line of trac- 

 tion was in the horizontal line D A. Whenever the radius 

 of the wheel, therefore, exceeds four feet and a half, the 

 height of that part of the horfe to which the traces (hould 

 be attached, the line of traftion A D will incline to the ho- 

 rizon, and by declining from the perpendicular A H, its me- 

 chanical effbrt~will be diminifhed ; and fince the load refts 

 upon an inchned plane, the trams or poles of the cart will 

 rub againft the flanks of the horfe, ^ven in level roads, and 

 ftill more feverely in defcending ground. Notwithftanding 

 this diminution of force, however, arifing from the un- 

 avoidable obliquity of the impelling power, wheels exceed- 

 ing four and a half feet radius have ftill the advantage of 

 fmaller ones ; but their power to overcome refiftances does 

 not inqreafe fo faft as before. Hitherto we have fuppofed 

 the weight of the large and fmall wheels to be the fame, but 

 Z E 2 it 



