LEASE AND KELEASE. 53 «;utrj' of llie U-nant ii]kiii the land before he be- came capable of taking llie reversion by the deed of release. This diilieully was obviated by the coiislruetioii put >y the eourts upon the famous Statute of Uses (27 lien. Vlll., c. 10), which per- mitted the creation of a complete leasehold es- tate without the necessity of an entry by the tenant, by the device of conveying the land to bis •use' for a year. The statute in cpu'stion, l)y cxi- cuting the use, i.e. by transferring to the tenant a legal title coextensive with his use, or equi- table title, vested the possession in him and thus rendered him instantly capable of taking the landlord's estate by release. The two instru- ments of lease and of release could thus be executed in quick succession, and, later, the two acts became simultaneous and were merged in one and the same instrument. This process of lease and release was the usual mode of convey- ance in England for three hundred years, and jirevailed in the United States as well, until superseded by the simpler conveyance by deed of grant, which is now in vogue in both countries. See Ukud; Cham'; Coxveva.nce. LEASEHOLD. In English law, the technical <lescription of au estate for years. It arises upon a lease, and constitutes a valid title, or estate, in the premises for the ])eriod described. It may be for any period of time, however brief or long, whether for a week or for a thousand years, and is subject to no restriction excepting that the limit of its duration shall be definitely fixed. If a conveyance of land be indefinite it is not a leasehold, even though it be measured in years. Owing to the circumstances of its origin, the leasehold estate is classified as personal jiropcrty, which, upon the death of the tenant, or lease- holder, passes to his executor or administrator, and not, like real property, to his heir. Ancient- ly such an interest was not regarded as property at all ( the feudal conce])tion of estates in land being eontined to the class of interests known as freeholds), but as a mere contract right, enforce- able only against the lessor or owner of the land, and against him only by an action for dam- ages. But in the course of time, partly as the result of statutes and partly through a growing recognition of the importance of leasehold inter- «9ts, the lessee came to be protected by a va- riety of actions, of which the action of ejectment was the most important, whereby he might re- cover the land itself either from his lessor or from any other intruder. ITie right of the lessee thus became a true estate in the land, strictly amilogous to the freehold estates previously rec- ognized, hut it was now too late to secure its recognition as inheritable real property. As the right of action for breach of contract, which was all that the lessee formerly had, passed to hisex- ectitor. so the leasehold estate which developed out of that contract right has continued to do to the present time. It is distinguished from other personal property hy the phrase "chattel real.' This contract origin of the leasehold is respon- sible, also, for some of the advantages which it enjoys over the freehold. Not only may it be created and assigned with less difficulty and for- mality, as by parol or. at the most, by a simple writing, while a deed is requisite to the creation or transfer of a freehold : but it has always been capable of being created so as to take effect at a future time, which, in the ease of freeholds, was not possible at the common law. LEAST SQUARES. Technically, the proper mode of creating an estate for years is by a lease, or demise, fol- lowed by the entry of the lessee. Any form of words showing the intention to create the relation of landlord and tenant will sullice. See Estate; Ekeeholu; Uv.nulokd am) Tenant ; Lease. LEAST SQUARES, Method of. An applica- tion of the theory of probabilities (q.v.) to the ileduction of the most probable value from a number of observations, each of which is liable to certain accidental errors. Th<- methods by which this is done may be understood from a, single example. Let it be found that a given bar has, at the temperatures of 20°, 40°, 50°, and CO" C, respectively, the lengths 1000.22, 1000.05, 1000.90, and 1001.05 millimeters; and let it be required to ascertain the coellicient of linear expansion, i.e., the amount of linear ex- jiansion per degree of temperature. If ?„ de- notes the length of the bar at 0° ('., c the co- ellicient of linear expansion, and It the length of the bar at ^° C, then l„+ t c = 1,. Substitut- ing respectivelv 20 and 40 for (, and the corre- sponding values of h, we get /„ -|- 20c = 1000.22 and l„ + 40c = 1000.05. Solving these two equations for /„ and c, we obtain: /„ = 999.79, and c = 0.215. But if these values of !„ and are then substituted in the equaticms correspond- ing to ? = 50 and 1 = 00, we find, respectively, ?.^ = 1000.87, L= 1001.08, instead of the ex- perimental figures 1000.90 and 1001.05. The( dillerence between the 1000.87 and the 1000.90, — 0.0.3, is called the resiilual of the tliird equai tion, while -|- 0.03 is obviously the residual of the fourth equation. In the same way we might solve the first and fourth equations and obtain /„= 999.80, c = 0.0208. in which case the residuals of the second and third equations would be + 0.02. + 0.00. Other combinations of the given equations would give other residuals, and the small<'r the resid- uals the closer the probable a])])roximation. It can he shown analytically and exjierimentally that in a series of observations affected by acci- dental errors, errors whose law of recurrence is such that in the long run they are as often positive as negative, the number of errors of a given magnitude is a function of that magnitude. This particular function is f{x) =■- hir-'i--'^', where h is a constant for all observations of a series, and ir and e have their usual meanings. The distribution of residuals follows this law, which is represented graphically hy the curve i/ — /itt-^c-'"'". If a; = 0, y = /w-^, and therefore varies directly as h ; but as X becomes very large, y becomes very small. That is, the number of errors of very small magnitude is relatively large, and the number of errors of very large magnitude is small. It has further been found that the sum of the squares of the residuals. 2ar, varies inversely as h. and hence when h is largest. 2j-" is small- est; in other words, that the most probable val- ues of the tinknowns are those which make So;' a minimum. From this is derived the name Mrtliod of Lcnf!t Sqnnrcs. For example, suppose a circumference, or. bi- sected by a diameter, is measured and found to be c, and the two semicircumferenccs are also measured and found to be .5,. s™. and wc are re- quired to find the most probable value of as. The residuals are c — x, s, — ia-, s, — ir.
