He solved quadratic equations both geometrically and algebraically, and also equations of the form x 2 "+ax n +b=o; he also proved certain relations between the sum of the first n natural numbers, and the sums of their squares and cubes.
Many of the names of the rich residential streets and squares in the west have associations with the various owners of the properties; but Mayfair is so called from a fair held on this ground in May as early as the reign of Charles II.
In the top squares of the slips the ten digits are written, and each slip contains in its nine squares the first nine multiples of the digit which appears in the top square.
With the exception of the top squares, every square is divided into two parts by a diagonal, the units being written on one side and the tens on the other, so that when a multiple consists of two figures they are separated by the diagonal.
I shows the slips corresponding to the numbers 2, o, 8, 5 placed side by side in contact with one another, and next to them is placed another slip containing, in squares without diagonals, the first nine digits.
It has the usual rectangular plan, with several pretty squares and straight, clean, well-paved streets.
There are several handsome squares and public gardens, adorned with statues, trees and shrubbery.
This class of representation ceases when the number of squares exceeds eight.
The cases of two, four and six squares had been given by K.
Jacobi and that of three squares by F.
There are many fine streets and squares and some handsome public monuments, notably among the last the fountain on the market square surmounted by a statue of Charlemagne, the bronze equestrian statue of the emperor William I.
Of the squares, the principal is the Friedrich-Wilhelmplatz, on which lies the Elisenbrunnen with its colonnade and garden, the chief resort of visitors taking the baths and waters.
The principal squares are Cathedral, Santa Ana, Bolivar and Lesseps.
The chapel, hall and residential buildings surrounding the squares within, are picturesque, but of later date.
The general type of problem is to find two, three or four numbers such that different expressions involving them in the first and second, and sometimes the third, degree are squares, cubes, partly squares and partly cubes, &c. E.g.
Tannery); To find four numbers such that, if we take the square of their sum any one of them singly, all the resulting numbers are squares (III.
29); To find three squares such that their continued product added to any one of them gives a square (V.
No number of the form 4n+3, or 4n - I, can be the sum of two squares), and goes on to a d, practically, the condition stated by Fermat, "and the double of it [n] increased by one, when divided by the greatest square which measures it, must not be divisible by a prime number of the form 4n - 1," except for the omission of the words "when divided.
After the great fire of 1872 it became possible, in the reconstruction of the business district, to widen and straighten its streets and create squares, and so provide for the traffic that had long outgrown the narrow, crooked ways of the older city.
In one of the squares there is a bronze statue, "The Hewer," by G.
The central and older portion of the city is laid out in squares surrounding a public Green of 16 acres, which was in former days the centre of religious and social life.
It is sometimes assumed that this is measured perfectly by the standard deviation,' which is obtained by taking the squares of the differences between the average and the individual prices, summing them and extracting the square root.
It contains several squares and boulevards, a large public garden, and many handsome public and private edifices.
Its port was revived, and since that time piers and quays have been constructed, and spacious squares and broad regular streets have been laid out.
Neighbourhood parks, playgrounds and squares were increased to 80, embracing 2,908 acres.
The hypothetical radicals of acids, were denoted by squares enclosing the initial letter of the base; an alkali was denoted by a triangle, and the particular alkali by inserting the initial letter.
In that case we need simply cover the map with a network of squares - the area of each of which has been determined with reference to the scale of the map - count the squares, and estimate the contents of those only partially enclosed within the boundary, and the result will give the area desired.
Instead of drawing these squares upon the map itself, they may be engraved or etched upon glass, or drawn upon transparent celluloid or tracing-paper.
There is an entire underground city with several storeys of larger and smaller streets, squares and cross ways, cut out of the rock; at the intersection of the cross ways FIG.
The Clos St Jean, a pleasant park, lies to the north-west, and squares and open spaces are numerous.
(2) The words of the text are ranged in squares in such a manner as to be read either vertically or boustrophedon.
For example, the class of rationals whose squares are less than 2r satisfies the definition of a real number; it is the real number A I 2.
Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers.
The town has been built without the slightest regard to regularity; the streets are even more intricate and winding than those in most other Eastern towns, and with the exception of the bazaars and some open squares, the interior is little else than a labyrinth of alleys and passages.
Leonardo gave as solution the numbers 11 i 4 4, 16, 9 4 - 7 4 and 6197 T, - the squares of 3,, 41'v and 2, 7; and the method of finding them is given in the Liber quadratorum.
The battle is chiefly notable for the steadi- donor, ness with which the allied right, covered by the Light Division in squares, changed position in presence of the French cavalry; and for the extraordinary feat of arms of Captain Norman Ramsay, R.H.A., in charging through the French cavalry with his guns.
The largest of the public squares in Hamburg is the Hopfenmarkt, which contains the church of St Nicholas (Nikolaikirche) and is the principal market for vegetables and fruit.
Honolulu has other parks, a fine Botanical Garden, created by the Bureau of Agriculture, several public squares, several hospitals, a maternity home, the Lunalilo Home for aged Hawaiians, an asylum for the insane, several schools of high rank both public and private - notably Oahu College on the E.
And, dividing by y1y2...ym, the discriminant of f is seen to be equal to the product of the squares of all the differences of any two roots of the equation.
Stern, on terrestrial magnetism by Goldschmidt, and on the method of least squares by K.
(1556-1598), and was modelled on that of Antwerp. The streets of the city are regular and broad; there are three fine squares or plazas.
The opening up of the wide thoroughfare of Chambers Street, on the site of College Wynd and Brown and Argyll Squares, cleared the precincts of unsightly obstructions and unsavoury neighbours.
The device known as the method of least squares, for reducing numerous equations of condition to the number of unknown quantities to be determined, had been adopted as a practically convenient rule by Gauss and Legendre; but Laplace first treated it as a problem in probabilities, and proved by an intricate and difficult course of reasoning that it was also the most advantageous, the mean of the probabilities of error in the determination of the elements being thereby reduced to a minimum.
Thus a comparison of numbers occurring in a table of squares 1 2 =1 112=121 2 2= 4 122=144 3 2 =9 132=169 suggests the formula (A+a)2=A2+2Aa+a2.
There are many beautiful private buildings, broad and well-paved streets, numerous squares and public gardens.
If we put for shortness 7 for the quantity under the last circular function in (I), the expressions (i), (2) may be put under the forms u sin T, v sin (T - a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin T and cos T in the expression u sin T +v sin (T - a), so that I =u 2 +v 2 +2uv cos a, which becomes on putting for u, v, and a their values, and putting f =Q .
It is mainly a rich residential quarter; the most fashionable part is found in the south, in the vicinity of Cavendish and Portman Squares, but there are numerous fine houses surrounding Regent's Park and in the north-western district of St John's Wood.
In one of the public squares is a Dominican church built in 1538.
North London is as a whole residential: Hackney, Islington and St Pancras consist mainly of dwellings of artisans and the middle classes; while in Hampstead, St Marylebone and Paddington are many terraces and squares of handsome houses.
The outlying parts of the county to east, south and north are not lacking in open spaces, but there is an extensive inner area where at most only small gardens and squares break the continuity of buildings, and where in some cases old churchyards serve as public grounds.
In her own words, they were "good boards overhead, good boards all around, and a good window"--of two whole squares originally, only the cat had passed out that way lately.
They broke up two squares, your excellency.
Lines had been drawn on the grass, large squares like those used for wrestling, with a circle in the center.
Moonlight spilled over large buildings with triangular roofs into community squares abutting stacked parking lots.
Even the guy's boxers were neatly folded into squares that sat in the middle of the drawer.
Side of the Huatenay are two more fine squares, called the Cabildo and San Francisco.
Clermont has several handsome squares ornamented with fountains, the chief of which is a graceful structure erected by Bishop Jacques d'Amboise in 1515.
Aguilar "of the Frontier" was so named in the middle ages from its position on the border of the Moorish territories, which were defended by the castle of Anzur, now a ruin; but the spacious squares and modern houses of the existing town retain few vestiges of Moorish dominion.
Apart, and another similar system of lines at right angles to the first, thus dividing the silvered surface of the plate into squares 5 mm.
The houses are well built, the market-places and squares handsome and spacious.
Of f=0, :and notices that they become identical on substituting 0 for k, and -f for X; hence, if k1, k2, k 3 be the roots of the resolvent -21 2 = (o + k if) (A + k 2f)(o + k 3f); and now, if all the roots of f be different, so also are those of the resolvent, since the latter, and f, have practically the same discriminant; consequently each of the three factors, of -21 2, must be perfect squares and taking the square root 1 t = -' (1)ï¿½x4; and it can be shown that 0, x, 1P are the three conjugate quadratic factors of t above mentioned.
The tops of their heads had no hair, but were carved into a variety of fantastic shapes, some having a row of points or balls around the top, others designs resembling flowers or vegetables, and still others having squares that looked like waffles cut criss-cross on their heads.
The squares are cut out, so that the men stand in them firmly.
Or, if you choose to go farther, it will not be unwise, for I have found the increase of fair bait to be very nearly as the squares of the distances.
* So that squares matters.
In half an hour all was again in order, only the squares had become gray instead of black.