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Rh that she possesses the best roads, kept up to a certain extent by the cycle tax, whereby the cyclist acquires a certain official position and right; moreover cycles accompanied by their owners are conveyed without extra charge on the railways, and aid is given to the sport and pastime from public funds. In Belgium the cycle has worked a veritable revolution in the national life. The surface of the greater part of the country being loose and sandy, the roads have been paved, and this paving is so bad as to be impossible for light traffic. The cycle tax has consequently been devoted, first, to the construction of paths on which cyclists have equal rights with pedestrians, and secondly to the replacing of the paving by macadam. In this way alone cycling has proved of inestimable benefit to Belgium and Luxembourg. In the United States measures for securing good roads and side paths have been introduced in various states, mainly at the instigation first of cyclists and then of motorists, and in Great Britain the Roads Improvement Association has worked for the same end.

Each country also possesses an organization for the government of cycle racing; and although these unions, one object of which—usually the main one—is the encouragement of cycle racing and cycle legislation, boast an enormous

membership, their membership is often composed of clubs and not individuals. Among the most important are the National Cyclists’ Union of England and the Union Vélocipédique of France. These bodies are also bound together by the International Cyclists’ Association, which is devoted mainly to the promotion of racing and legislation connected with it all over the world. The National Cyclists’ Union, originally the Bicycle Union, which was the parent body of all, formed in February 1878, was the first to put up danger-boards, and also was early instrumental, alone and with the C.T.C., in framing or suggesting laws for the proper government and regulation of cycle traffic, notably in establishing its position as a vehicle in securing universal rights, in endeavouring, again in conjunction with the C.T.C., to increase facilities for the carriage of cycles on the railways, in securing the opening of parks, and in promoting many other equally praiseworthy objects. For a number of years, however, it has been more prominent as the ruling race-governing body. But cycle racing has fallen upon evil days. At one time cycle racing attracted a large number of spectators, but gradually it lost the public favour, or rather was ignored by the public because it became mainly an advertisement for cycle makers. The presence of the man, directly or indirectly, in the employ of, or aided by a maker, and the consequent mixing up of trade and sport, lowered racing not only in the public estimation, but in that of all genuine amateurs. There have always been a few amateurs who have raced for the love of the sport, but the greater number of prominent racing men have raced for the benefit of a firm, so much so that, at one time, an entire section of racing men were classed as “makers’ amateurs.” They did not confine themselves to the race track, but appropriated the public roads until they became a danger and a nuisance, and road-racing finally was abolished, though record rides, as they are called, are still indulged in, being winked at by the police and by the cycling authorities. The makers’ amateurs at least rode to win and to make the best time possible. But the scandal was so great that a system of licensing riders was adopted by the N.C.U., and if this did not effectively kill the sport, the introduction of waiting races did. There probably is considerable skill in riding two-thirds of a race as slowly as possible, and only hurrying the last part of the last lap, but it does not amuse the public, who want to see a fast race as well as a close finish. The introduction of pacing by multicycles and motors next took from cycle racing what interest was left. A motor race, in which the machines are run at top speed, is more exciting than the spectacle of a motor being driven at a rate which the cyclist can follow with the protection of a wind-shield. In America this system of proving what cyclists can do with racing machines was carried so far that in 1899 a board track was laid down on the Long Island railway for about 2 m. between the metals, and a cyclist named Murphy, followed a train, and protected by enormous wind-shields, succeeded in covering a mile in less than a minute in the autumn of 1900. Other cyclists have devoted themselves, at the instigation of makers, to the riding of 100 m. a day every day for a year. It would be difficult to say what advantage there is in these trials and contests. They are not convincing records, and only prove that some people are willing to take great personal risks for the benefit of their employers. E. Hale, during 1899–1900, covered 32,496 m. in 313 days. For many years also long-distance races, mostly of six days’ duration, have been promoted on covered tracks, and though condemned by all cycling organizations, they find a great deal of pecuniary support.

The cycle has also been taken up for military purposes. For this idea the British army is indebted to Colonel A. R. Savile, who in 1887 organized the first series of cycle manœvres in England. Since then military cycling has undergone a great development, not only in the country of its origin but in most others.

Cycling has produced a literature of its own, both of the pastime and of the trade. Owing to the enormous profits which, for several years, were obtained by cycle makers, a trade press appeared which simply lived by, and out of, its advertisers; and though each country has one or more genuine trade journals, the large proportion of these sheets have been worth, in a business aspect, as little practically as from a literary standpoint. On the other hand a vast mass of practical and unpractical, scientific and medical, historical and touring treatises and records have appeared, but mostly of a rather ephemeral character.

CYCLOID (from Gr. , circle, and  , form), in geometry, the curve traced out by a point carried on a circle which rolls along a straight line. The name cycloid is now restricted to the curve described when the tracing-point is on the circumference of the circle; if the point is either within or without the circle the curves are generally termed trochoids, but they are also known as the prolate and curtate cycloids respectively. The cycloid is the simplest member of the class of curves known as roulettes. No mention of the cycloid has been found in writings prior to the 15th century. Francis Schooten (Commentary on Descartes) assigns the invention of the curve to René Descartes and the first publication on this subject after Descartes to Marin Mersenne. Evangelista Torricelli, in the first regular dissertation on the cycloid (De dimensione cycloidis, an appendix to his De dimensione parabolae, 1644), states that his friend and tutor Galileo discovered the curve about 1599. John Wallis discussed both the history and properties of the curve in a tract De cycloide published at Oxford in 1659. He there shows that the cycloid was investigated by Carolus Bovillus about 1500, and by Cardinal Cusanus (Nicolaus de Cusa) as early as 1451. Honoré Fabri (Synopsis geometrica, 1669) treated of the curve and enumerated many theorems concerning it. Many other mathematicians have written on the cycloid—Blaise Pascal, W. G. Leibnitz, the Bernoullis, Roger Cotes and others—and so assiduously was it studied that it was sometimes named the “Helen of Geometers.” The determination of the area was the subject of many investigations and much controversy. Galileo attempted the evaluation by weighing the curve against the generating circle; this rough method gave only an approximate value, viz., a little less than thrice the generating circle. Torricelli, by employing the “method of indivisibles,” deduced that the area was exactly three times that of the generating circle; this result had been previously established in 1640 in France by G. P. de Roberval, but his investigation was unknown in Italy. Blaise Pascal determined the area of the section made by any line parallel to the base and the volumes and centres of gravity of the solids generated by revolving the curve about its axis and base. Before publishing his results he proposed these problems for public competition in 1658 under the assumed name of Amos Dettonville. John Wallis in England, and A. la Louère in France, accepted the challenge, but the former could only submit incorrect solutions, while the latter failed completely. Having established his priority, Pascal published his investigations, which occasioned a great sensation among his contemporaries, and Wallis was enabled to correct his methods. Sir Christopher Wren, the famous architect, determined the length of the arc and