Page:The American Cyclopædia (1879) Volume XIV.djvu/443

 KOPE 423 root of 16; 9 is the second power, or, as it is usually called, the square of 3, and 3 is the second or square root of 9. It will thus be seen that a root of a given number is a num- ber which being taken a certain number of times as a factor will produce the given num- ber. An arithmetical root of a number is in- dicated by the sign f with the number placed after it, and the number indicating the degree of the root placed above and before it ; thus ^16 is read "the fourth root of 16." The sign is a modification of the letter r, which was formerly used for this purpose. The second or square root is indicated by the sign alone, the figure 2 being omitted ; thus y 9 means the same as ^9. The first root of a number is the number itself, and therefore needs no sign. In the best modern works on algebra the sign ' is strictly limited to the designation of the arithmetical root of a quantity. II. In algebra the term root is used to denote any value of the unknown quantity in an equation, which being substituted for that quantity will satis- fy the equation; thus the equation x* 7x* + 27or 1 47a; + 26=0 can be satisfied by substi- tuting for x any one of four different values, two of which are real, viz., 1 and 2, and two are imaginary, viz. (2 + 3^1) and (2 3V 1). Any given number has only one arithmetical root of a given degree, but it has as many al- gebraical roots as there are units in the num- ber denoting the degree of the root ; thus 3 is the only arithmetical square root of 9, but algebraically 9 has two square roots, +3 and 3, because either of these will satisfy the equation * 2 =9. The arithmetical root is at the same time one of the algebraical roots. Algebraical roots are designated by means of fractional exponents (see EXPONENT) ; thus 9* may be considered as representing either + 3 or 3, and 1* means either 1 (its arithmetical value) or 1 or ^ 1 or y 1, either of which will satisfy the equation x*=l. "When the numerator of the fractional exponent is some number other than 1, it indicates that the root expressed by the denominator is to be raised to the power expressed by the nume- rator ; thus a* means the third power of any one of the five fifth roots of a. ROPE, a large cord, formed by twisting to- gether a collection of vegetable or animal fibres or metallic wires. The smaller cords are called twines and lines, and all are inclu- ded under the general name of cordage. The invention of ropes or cordage dates from the earliest times. The first ropes were probably made of the fibres of the inner bark of some kinds of trees or of grasses, and of thongs from the hides of animals. There are sculp- tures among the relics of the ancient Egyp- tians illustrating the manner of making ropes more than 4,000 years ago, and their most ancient records contain representations of well made ropes capable of transmitting the enor- mous power required in transporting their colossal statues and huge blocks of stone. It appears that they made use of flax, and also of the fibres of the date tree. The most cele- brated ropes known to history are the ca- bles used in the construction of the bridges of boats on which the army of Xerxes cross- ed the Hellespont (Herodotus, vii. 36). There were two bridges, and six cables were as- signed to each bridge. Two of the cables were of white flax, while four were of papy- rus. Both sets of cables were of the same size and quality, but the flaxen were the heav- ier, weighing not less than a talent the cubit. If we assume the talent to be equal to about 56 Ibs. and the cubit to be 22 in., the cables must have measured about 28 in. in circumfer- ence. The largest hemp cables ever made in the United States were 24 in. in circum- ference. The ancient Peruvians twisted to- gether the strong fibres of the maguey plant, forming them into cables as large as a man's body, used in the construction of the suspen- sion bridges by which their paved highways were carried over ravines and rivers. Many rude savages, especially among the islands of the Pacific and Indian oceans, are celebrated for making beautiful cordage. In modern times, among civilized nations, the principal materials for ropes are hemp, flax, manila hemp, plantain leaf, jute, and metallic wire. Hope making was regarded as a matter of great importance to the early American colonists, and efforts were made to introduce it with other branches of manufacture in Virginia, where the climate and soil were found to be favorable to the cultivation of hemp and flax ; but the culture was neglected for that of to- bacco. In New England it was regarded with more interest; crops were raised from seeds of the plant received in Salem in 1629, and in 1641 the general court of Massachusetts directed attention to the wild hemp which grew in the province, and was used by the Indians for making nets, mats, and lines. In the same year the manufacture of cordage was begun in Boston by John Harrison, and in 1662 in Charlestown by John Heyman. In Connecticut the government at Hartford in 1642 gave direction for the sowing of hemp and " for the better furnishing of the riv- er with cordage toward the rigging of ships." In the " History of Pennsylvania and West New Jersey," by Gabriel Thomas (London, 1698), there is a notice of large ropewalks in Philadelphia, several of which were owned by Joseph Wilcox. Hope Making. In the United States there are four principal kinds of rope in common use: hemp rope, made of the fibres of the cannabis saliva or hemp plant; manila rope, made of the fibres obtained from the leaves of the musa textilis, or wild plantain ; hide rope, made of long strips of green ox- hide ; and wire rope. The best hemp for rope making comes from Russia, the Eiga Rein be- ing the brand preferred. One reason assigned for tho superior quality of Russian hemp is the