SYMBOLS. 773 SYMMACHtrS. Of deduction. •.• (since). .•. (therefore). Of aggregation. (),[], {}' Of denominate nuniliers, 15" (degrees, minutes, 2d. (pounds, shillings. as in .$10, .'5° 4' seconds), £20 3s. pence), ewt. (100 lbs. ) , and various abbreviations. Of geometry. /:, ^» (angle, angles). X (perpendicular to). II (parallel to). ' — (congruent to). ^ (similar to) . =: (appi'oaches as a limit). A, A (triangle, triangles). (^'. ® (circle, circles). □, [£) ( square, squares ) . ul], dj (rectangle, rectangles). iZj, Gj (parallelogram, parallelograms). ^ ( a re ) . a (radians) . The question of the origin and development of mathematical symbols is a large one, and science has not .yet given satisfactory answers at manj" points. The probable origin of the remarkable digits 1 . . 9, is discussed in the article on Nu- merals. The origin of zero is unknown, there being no authentic record of its history before A.D. 400. The extension of the position s.ystem below unity is attributed to Stevin (1585), who called tenths, hiindredths, thousandths. . . . primes, sekondes, terxes, and wrote subscripts to denote the orders, thus 4.628 was written 4f„) 6(i) 2(,) 8(3). But RudoUT (1525) and Kepler (1571-1030) used the comma to set oft" the deci- mal orders, and Biirgi (1552-1632) and Pitiscus (1612) in their tables used the decimal fraction in the form 0.32, 3.2. Although the early Egyp- tians had s,vnibols for addition and equality, and the Greeks, Hindus, and Arabs symbols for equal- ity and for the unknown quantity, from earliest times mathematical processes were cumbersome for lack of proper symbols of operation. The expressions for such processes were either written out in full or denoted by word abbreviations. The later Greeks, the Hindus, and Jordanus in- dicated addition by juxtaposition ; the Italians usually denoted it by the letter P or p with a line drawn through it to distinguish it as an operation, but their sj-mbols were not uniform. Pacioli, for example, sometimes used p and some- times e, and Tartaglia commonly expressed the operation by 9. The German and English alge- braists introduced the sign -1-, but spoke of it as si<int(ni addilo7-u>n and first employed it only to indicate excess. Subtraction was indicated by Diophantus by the symbol •/•. The Hindus used a dot, while the Italian algebraists denoted it by M or m with a line drawn through the letter. The symbols m and de were, however, used by Pacioli. The German and English algebraists were the first to use the present symbol and de- scribed it as sifinum siibtructorum. The symbols + and — appeared first in print in an arith- metic of Widmann (1489). The svnibol X for 'times' is due to Oughtred (1631). To Rahn (1659) is due the ]uesent sign -~ for division; Harriot (1631) used a period to indicate multi- plication, and Descartes (1637) used juxtapo- sition. Leibnitz in 1088 employed the sign -^ to denote multiplication and — to denote di- vision. Division among the Arabs was desig- nated variously by a — h, a/h, —, but Clairaut
(1760) made familiar the form a : b. Descartes made popular tlie notation a" for involution and 'allis defined the negative exponent. The s.yni- bol of equality, =r, is due to Recorde (1557), and the symbols >■, <;, for greater than and less than, originated with Harriot (1631). ViOta (1591) and Girard (1029) introduced various symbols of aggregation. The symbol 00 for in- finity was first em])loyed by Wallis in 1655. The symbols of diflercntiatiou dx and of integration I as used in calculus, are due to Leibnitz, as is also the symbol ■ — . for similarity, as used in geometry. The symbolism ^, f, V, as used in theory of functions, is due to Abel. Consult Cantor, Vorlesuiiiirii i'tbcr Oeschichte dcr Matheinatik (2d ed., Leipzig, 1900). SYME, Sim, James (1799-1870). A noted Scotch surgeon, born in Fife, and educated at the Universitv of Edinburgh. He was lecturer (1823-32) and professor (1833-48) of surgery at his alma mater, after having served as dem- onstrator under Listen. He was the inventor of the mackintosh waterproof cloth. He was clini- cal professor from 1829 to 1833 in Minto House Hospital, which he founded at his own ex- pense. He was one of the ablest teachers and operative surgeons of the age. He de- vised resection of the joints, Syme's ampu- tation of the foot, and excision of the lower jaw. He was the author of many valuable works, in- cluding The Excision of Diseased Joints (1831) ; Principles of Surgcrij ( 1832) ; the same, to which is added Diseases of the Rectum (1866) ; Pathol- o'jy and Praetice of Suryery (1848) ; Stricture of the Urethra and Fistula in Perinea (1849). Consult Paterson, Memoir (Edinburgh, 1874). SYMINGTON, sim'ing-ton. William (1763- 1831). A British inventor, born at Leadhills. In 1786 he constructed a working model of a steam road-carriage, and afterwards patented .a steam-engine in which he obtained rotary motion by chains and ratchet wheels. In 1788 he and Patrick ]Iiller used an engine constructed on the lines of this patent to propel a small pleas- ure boat on Dalswinton Loch. In the following 5'ear the}' experimented on a larger scale on the Forth and Clyde Canal and succeeded in attain- ing a speed of seven miles an hour. As the type of engine used was imperfect, however, Syming- ton in 1801 patented another in which a piston rod guided by rollers was connected by a rod to a crank attached to the paddle-wheel shaft. In the following year he fitted out a boat called the Charlotte Dundas, which proved able to tow two barges a distance of 19i{> miles in six hours. The Duke of Bridgewater, Symington's patron, was so well pleased with the boat that he ordered eight others to be constructed. Unfortunately for the inventor, however, the Duke died short- ly afterwards, the order was canceled, and Sym- ington was unable to finil another patron. SYMMACHTJS, stm'nia-kus. Pope. 498-514. He was bijrn in Sardinia, and was chosen to fill the vacancy left by the death of Anastasius II. A minority, howevpr, of the Byzantine faction, set up as a rival the archi]ircsbytpr Laiirentius. As a result of the schism, bloody encounters took
