QUARTER SQUARES. Among the labour-saving devices for multiplying is one which seems to be of Hindu origin and which may be stated as follows : to find the product of two numbers, take the difference between a quarter of the square of their sum and a quarter of the square of their difference. For example, 72. It depends upon the algebraic identity, If the com puter has a table of squares sufficiently large, the method has considerable value, and this is more evident if he has a table of quarter squares. The plan is given in various Arabic works, as in al-Karkhi's al-Kd frl-Hisab (c. 1020).
See J. Blater, Table des Quarts de carres de toes les nombres entiers de 1 a 200000 (1888). For a discussion of the subject, with a bibliogra phy and a description of related methods see H. Mehmke, Numer isches Rechnen, in W. F. Meyer, Encyklopiidie der mathematischen Wissenschaften, Band I., pp. A staff of wood from six to nine feet in length, used as a means of attack and defence; originally no doubt it was the cudgel or sapling with which many heroes are described by early writers as being armed. The quarter-staff at tained great popularity in England in the middle ages. It was usually made of oak, the ends often shod with iron, and it was held with both hands, the right hand grasping it one quarter of the distance from the lower end (whence the name) and the left at about the middle.
Egerton Castle (Schools and Masters of Fence) says that the staff was the "foil," or practice-substitute for the long sword or two-hander. In earlier times it may also have been used as a prac tice weapon for the spear and bill. In the prints illustrative of the life of Richard Beauchamp, earl of Warwick reproduced in Joseph Strutt's Manners, Customs, Arms, Habits, etc. of the Inhabitants of England, may be seen a combat between
two knights after they have splintered their lances and dis mounted, in which both are fighting with pointed staves about as long as a quarter-staff and held in the same manner. In the 17th century the staff was still popular in England.
At the present time the quarter-staff is used to a limited extent in military circles as a school for bayonet play. (See FENCING.) (A. R. H.) QUARTER TONES, in music. Although the semitone, or half tone, was formerly defined as "the smallest interval recognized in music" much smaller intervals have always been employed in oriental music (see INDIA), and in recent years attempts have been made by various musicians to introduce similar intervals, such as quarter tones, in Western music. Foremost among these has been the Czech composer and theorist, Alois Haba, who has written a number of works based on the quarter-tone system, though so far, it must be said, without securing for them much appreciation even at the hands of the best disposed listeners. To the contrary, the opinion expressed has usually been that the effect was too closely akin to that of instruments playing out of tune to be agreeable, although it may be that with further expe rience of the music it might produce a more favourable im pression. Another who theorized on the subject, and considered in cidentally the possibility of employing even smaller intervals than quarter tones, was the eminent pianist and composer Busoni, while John Herbert Foulds, an English composer, has done the like. (See HARMONY.)