or RESHT, a town of Persia, and the capital of the province of Ghilan. It is a large town, with a considerable trade, and well-furnished shops and warehouses. It is so enveloped in trees, that no idea can be formed of it taken in any one point of view. Exclusive of the bazaars, which occupy a considerable space in the centre of the town, it is composed of small compartments, divided for the most part by narrow and obscure alleys. There are not many principal streets; and only some of them are paved, whilst others have merely a little gravel thrown in the centre for the water to run on, and some are left unfashioned in any way, except in having a raised pathway at the sides for foot-passengers, which is of great convenience in wet weather. The bazaars or shops, of which there are said to be 1200, are extensive, regular, clean, and well kept. They are well paved, but not entirely covered in from the weather. Instead of arched or domed roofs, extending from one side of the street to the other, the roofs only project nearly to the centre of the street, covering the shops and foot-passengers. There is in this town a bustle and hum of business, more than is seen in most of Persian towns. Natives of various countries are seen passing and repassing on the pavement below; and persons of every occupation, busily following their several callings, animate the scene, and enliven it with their gay costumes. This town swarms with beggars of all descriptions, beyond what can easily be conceived. They are covered with leprosy, and cutaneous disorders the most revolting. The opium-eaters are among the most miserable of these objects. The religious mendicants, or fakeers and dervishes, are also very numerous, and the most impudent of all. There were formerly Russians, Armenians, and Jews in this city; but they were forced away by the bigotry and intolerance of the inhabitants, who reviled every foreigner that professed a different creed from their own. Reshid is one of the most considerable entrepôts on the Caspian for exchanging the commodities of Persia with those of Astracan, and is the chief mart for silk, the staple produce of the province. The rate of interest for money is from four to twelve per cent. per month; but this last rate is seldom paid. Fraser, who visited this place in 1822, was at pains to ascertain the amount of the population; and from every inquiry he could make, he estimated the male inhabitants at between thirty and forty thousand; and, considering other circumstances, the appearance of the markets and other public places, he estimates the whole population at eighty thousand. Reshid is an inland town, and all its maritime trade is carried on through the port of Ezellee. Long. 49. 50. E. Lat. 37. 20. N.
RESIDUAL Analysis, a calculus invented by Mr Landen, and proposed as a substitute for the method of fluxions. The design of it was to avoid introducing the idea of motion, and of quantities infinitely small, into mathematical investigation. The residual analysis accordingly proceeds by taking the difference of the same function of a variable quantity in two different states of the said variable quantity, and denoting the relation of this difference to the difference between the two states of the said variable quantity. This relation being first generally expressed, is next considered in the case when the difference of the two states of the variable quantity is = 0; and by that means it is obvious that the same thing is done as when the function of a variable quantity is assigned by the ordinary methods.
The evolutions of the functions, considered in this very general view, require the aid of a new theorem, discovered by Mr Landen, and remarkable for its simplicity and great extent. It is, that if \( x \) and \( v \) are any two variable quantities
\[ \frac{x^n - v^n}{x - v} = x^{n-1} + \frac{v}{x} + \frac{v^2}{x^2} + \cdots + (m) \]
where \( m \) and \( n \) are any integer numbers.
This theorem is the basis of the calculus, and from the expressions \( x^n - v^n \), and \( x - v \) having the form of what algebraists denominate residuals, the inventor gave to his method the name of the residual analysis.
Mr Landen published the first account of this method in 1758, which he denominated a Discourse concerning the Residual Analysis. The first book of the analysis appeared in 1764, and contained an explanation of the principles of the new calculus, with its application to problems of the direct method of fluxions; the second book, which solved several problems of the inverse method, was never published.
Residual Figure, in Geometry, the figure remaining after the subtraction of the less from the greater.
Residual Root is a root composed of two members only connected by the sign — or minus. Thus, \( a - b \), or \( 5 - 3 \), is a residual root; and is so called, because its true value is no more than the residue or difference between the parts \( a \) and \( b \) or 5 and 3, which in this case is 2.