Hence, assuming only accidental errors, f(x) = (c − x)2 + (s1 − 12x)2 + (s2 − 12x)2 = a minimum, or f1(x) = 2 (x − c) + 12x − s1 + 12x − s2 = 0, whence x = 13 (2c + s1 + s2), the most probable value.
The method of least squares is due to Legendre (1805), who introduced it in his Nouvelles methodes pour la determination des orbites des comètes. In ignorance of Legendre's contribution, however, an Irish-American writer, Adrain, editor of The Analyst (1808), first deduced the law. He gave two proofs, the second being essentially the same as Herschels (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. To him is due much of the honor of placing the subject before the mathematical world, both as to the theory and its applications.
For an introduction to the method of least squares, consult Comstock, Method of Least Squares (Boston, 1890); and for the history of the method, consult Merriman in the Transactions of the Connecticut Academy (1877, vol. iv., p. 151, with a complete bibliography).
LEATHER. (AS. leþer, OHG. leder, Ger. Leder, leather). The skin of an animal, dressed for use by some process which shall render its texture permanent in character. Untreated fresh skin is easily putrescible; dry skin is hard and horny and almost impenetrable to air. By converging a skin into leather, however, its nature is entirely changed and it is rendered practically imperishable, porous, and flexible. The skin of an animal consists of two layers: the outer or epidermis, which has no blood-vessels and is hard and horny, and the inner true skin or corium, which is made up of gelatinous fibres. This inner layer or true skin is the basis of leather, and the process of leather-making consists of applying to this skin certain substances which shall enter into combination with the gelatin in such a manner as to produce the desired characteristics of durability, penetrability, and flexibility. Three methods of accomplishing this have been practiced from very early times: Tanning, in which the gelatin is combined with tannin or tannic acid, or, by a much later process, with chromium salts. Tawing, in which the gelatin is combined with certain mineral salts. Shamoying, in which the leather is combined with oil or fatty substances.
Historical Development. Probably, the original process of curing skins was that of simply cleaning and drying. Then, the use of smoke, sour milk, various oils, and the brains of the animals themselves was found to improve the texture of the leather. Later it was discovered that certain astringent barks and vegetables effected permanent changes in the texture of skins, and stopped decay. This knowledge was possessed by the ancient Egyptians, for engravings on their tombs depict the process of tanning. In China specimens of leather have been discovered in company with other relics that prove them to be over 3000 years old. The Romans used leather which they tanned with oil, alum, and bark. The earliest explorers of America found the Indians wearing skins prepared with buffalo-dung, oil, and clay. No improvement in the general methods of preparing leather took place from the most primitive times until about 1790, when the use of lime, to loosen the hair, was introduced. By 1825 English tanners were attempting to introduce new methods by which the tanning process could be shortened. One of the pioneers in these experiments was John Burridge, the inventor of the ‘barkometer,’ an instrument for determining the strength of tanning-liquors.
Tanning Industry in the United States. The first tannery in America was built in Virginia, in 1630. A few years later a second one was established in Lynn, Mass. The tanning industry was well represented among the early settlers of Massachusetts, for it is recorded that no fewer than 51 tanners had come over to the new colony before 1650. There was great demand for their labors, for skins accumulated so rapidly that in 1640 it was found necessary to pass a law “that every hide and skin should be dried before it corrupts, and sent where they may be tanned and dressed.” The tanning industry was also encouraged throughout the colonies by many laws forbidding the exportation of untanned leather. Tanneries flourished everywhere, and by 1810 their annual output was $20,000,000. See paragraph Statistics.
Manufacturing Processes. The hides of commerce are brought to the tanneries in four different forms: either they are simply ‘green’ or ‘fresh’ hides, direct from the slaughter-houses, or, in ease they have been shipped from a long distance, they are wet-salted, dry-salted, or simply dried. The preliminary process of preparing the hides for tanning differs somewhat with the condition in which they are received, salted and dried hides requiring much more thorough cleansing and softening than green hides. The process also differs somewhat in preparing sole-leather, harness-leather, and dressed leather. The first step is to soak the skins or hides in water, to soften them, after which every vestige of adherent flesh is scraped from the inside. They are then laid in heaps for a short time, and afterwards hung in a heated room, by which means a slight putrefactive decomposition is started and the hair becomes so loose as to be easily detached. This process of unhairing, called ‘sweating,’ is mostly followed in America for making sole-leathers, while the process of liming or loosening the roots by the milk of lime is used for dressed leather; but in Great Britain milk of lime is used for depilation of all leathers. The process may be hastened by use of sulphuric or other acid. Hides or skins intended for dressing purposes, as for shoes, coaches, harness, or bookbinding, after the hair is taken off by the lime, have to be submitted to a process called bating for the purpose of reducing the swelling or thickening occasioned by the introduction of the lime, and for cleansing the skin from grease and other impurities. This is effected by working the skin in a decoction of pigeon's or dog's dung and warm water. This process does something more than cleanse the leather; it effects a marked change in its texture, reducing it to an extremely flaccid condition. If the old method of tanning is followed, the hides after unhairing are placed in the tan-pits, with layers of oak-bark or other tanning materials between them, and when as many layers of hides and bark are arranged as the pit will hold, water is let in, and the hides are allowed to remain for an indefinite period to be acted on by the tanning
