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numbers

numbers

numbers Sentence Examples

  • I don't remember the last three numbers but the first two were forty-five.

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  • You don't write a jejune collection of hodgepodge letters and numbers like these.

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  • She searched out other tip line numbers and the Warwick Police as well.

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  • "We put the numbers in a certain format that makes it easier for them to analyze," I said, trying to sound like I knew what I was talking about.

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  • But you don't need those numbers; I understand you've done your homework searching public records.

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  • And remember those three little numbers; 911.

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  • There is one set of numbers he changes for the time and he can set the location somehow by longitude and latitude.

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  • She stood up and bowed when she discovered eight different digits appeared, the numbers two through nine.

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  • Who has not asked himself how it is that the French were not all captured or destroyed when our three armies surrounded them in superior numbers, when the disordered French, hungry and freezing, surrendered in crowds, and when (as the historians relate) the aim of the Russians was to stop the French, to cut them off, and capture them all?

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  • We braced our ladder against the wall and played a game of numbers to see who would the first to climb.

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  • I knew from my in depth research, the number and location of all payphones bearing the area code and first two numbers dear Brenda provided me.

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  • The alarm clock beside her bed indicated 2:00 am in large orange numbers.

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  • When everyone was satisfied, they exchanged names and numbers.

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  • The site is indexed and assigned case numbers.

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  • Now, let's see what you've done with those numbers.

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  • Sheep production, mining volumes, births, crop production, weather patterns... it's all public information reports and numbers.

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  • What do you do with it... all these numbers?

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  • I only knew settings somehow incorporated GPS numbers.

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  • That's why he has so many precise numbers.

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  • She slowed to look at the room numbers.

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  • I assigned everyone a ring tone and put their phone numbers in it, he said, looking somewhat proud.

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  • There were three sets of numbers written in green ink on his palm.

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  • She snatched her satchel and dug out Linda and Traci's numbers.

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  • While there was a feeling of last-guy-in-turn-off-the-elevator, the Deans reluctantly agreed that some form of management was necessary to maintain order in the face of the ever-increasing numbers who wallowed in nature's wonders.

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  • Darkyn doesn't have the numbers to set up a facility like this in too many places.

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  • The summary of results might as well have been in a foreign language with the medical terminology, abbreviations and sprinkling of what seemed like random numbers.

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  • She couldn't read the writing, but she knew the numbers well enough to find the grids.

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  • Donnie had paid no attention to their conversation but having tired of his puzzle, picked up the notebook of letters and numbers and began to study it, turning the binder page by page.

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  • Do you think Annie was practicing her letters and numbers?

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  • The trouble with that theory is here we have letters and numbers.

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  • What do the numbers mean?

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  • There are a lot of places where numbers follow other numbers.

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  • How many different numbers and letters are there?

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  • "It still just looks like jumble of letters and numbers," Dean said as he read the first portion.

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  • Cynthia then began listing the different numbers that had been utilized.

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  • At first they tested to see if the numbers might be the vowels but that assumption didn't seem to work out.

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  • As much and as often as Annie wrote, the letters and numbers must have almost become a second language to her.

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  • The crowd had grown since Dean's Thursday trip, both in numbers of climbers and spectators.

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  • She writes itsy-bitsy numbers.

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  • We cannot call attention to ourselves until our numbers warrant our attack.

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  • I'm not very good with numbers.

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  • She reviewed the numbers that popped up on her screen.

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  • He read the numbers aloud again.

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  • Their numbers are far greater than anything we ever imagined.

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  • He pulled up the hood and strode into the forest, towards the castle.  Beneath the hood, he took in the numbers of demons present.  There weren't as many as he expected but far more than he could fight without his magic.

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  • Unfortunately, his efforts were all too often thwarted by a sympathetic judge or a system that could not find jail space for the numbers of criminals brought before it.

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  • He unobtrusively managed to locate all of the three renters of the adjoining rooms through their license plate numbers and was able to speak to two of them.

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  • You can bet the numbers are wrong.

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  • The riders quickly spread out, but because of the numbers there were always at least several in view.

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  • As all of the bikers were assigned numbers, Dean made a verbal note of each on a small handheld tape-recorder bor­rowed from his more official duties.

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  • The two men compared the cyclists' numbers each had listed as possible Byrne look-alikes.

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  • Fred checked the numbers on a master list of the tour's advance regis­trations.

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  • After I checked out the list I spent the afternoon at a rest stop squinting at a couple of a thou­sand bikers' numbers trying to spot him, but no luck.

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  • "I've got one of them phone cards," he said, taking the instrument and entering the req­uisite numbers, then handing the phone back to Dean.

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  • With each turn in the road he expected to see the summit but was only greeted with another long uphill climb until he lost track of the numbers.

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  • In time the trees began to thin and patches of old snow appeared in ever increasing numbers, tucked in dark crevices, left over from winter storms of months long past.

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  • No, but it's so cold up here lots of the bikers are wearing jack­ets and sweats that cover up their numbers.

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  • There was another concern—once the bikers hit the lower ele­vation and the heat of the afternoon, they would be shedding outer gear and perhaps identifying numbers with them.

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  • I enjoy working with numbers now and then.

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  • The women were of all ages while the men were either young or elderly, their numbers gutted by war.

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  • He gauged the watcher to be a good distance away, close enough to see their movement and numbers but far enough not to see their armament.

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  • The numbers atop the walls had been doubled, a sign of impending battle.

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  • Our numbers have plummeted the past year, and I don't understand the placement of our army.

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  • We would need to recall the armies from the south to bolster our numbers.

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  • "The numbers to the north are misstated," he said.

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  • The law of multiple proportions asserts that if two elements form more than' one compound, then the weights of the one element Law of which are found combined with unit weight of the other multiple in the different compounds, must be in the ratio of two propor or more whole numbers.

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  • In water and in ethylene experiment shows that 8 parts by weight of oxygen and 6 parts of carbon, respectively, are in union with one part of hydrogen; also, if the diagrams are correct, these numbers must be in the ratio of the atomic weights of oxygen and carbon.

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  • Fortunately, the compounds at first examined by the chemists engaged in verifying these laws were comparatively simple, so that the whole numbers referred to above were small.

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  • In a series of repetitions of the experiment, by different observers, the following numbers were obtained for the ratio of the copper in the two chlorides: 1.98, 1.97, 2.03, 2.003, the mean value being 1.996.

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  • necessary to determine the specific gravities of the various gases referred to some one of them, say hydrogen; the numbers so obtained giving the weights of the molecules referred to that of the hydrogen molecule.

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  • Cattle and sheep are produced in large numbers in some of the provinces, while in others mining forms the chief industry.

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  • Pigs and goats, however, with cattle, horses, asses and dogs, have been introduced, have multiplied, and in considerable numbers run wild.

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  • They were fanatical, and their tyranny drove numbers of their Jewish and Christian subjects to take refuge in the growing Christian states of Portugal, Castile and Aragon.

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  • Other elements of the problem there are none, except mere numbers and angles, which do not depend upon the fundamental measurements of space, time and mass.

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  • Although Mount Everest appears fairly bright at 100 miles' distance, as seen from the neighbourhood of Darjeeling, we cannot suppose that the atmosphere is as transparent as is implied in the above numbers; and, of course, this is not to be expected, since there is certainly suspended matter to be reckoned with.

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  • Pigs, sheep and goats are also kept in considerable numbers.

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  • AMICABLE NUMBERS, two numbers so related that the sum of the factors of the one is equal to the other, unity being considered as a factor.

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  • Amicable numbers were known to the Pythagoreans, who accredited them with many mystical properties.

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  • In Ireland the game took root very gradually, but in Ulster, owing doubtless to constant intercourse with Scotland, such clubs as have been founded are strong in numbers and play.

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  • The tribes of northern Albania, or Ghegeria, may be classified in seven groups as follows: - (1) The Mirdites, who inhabit the alpine region around Orosh to the south-east of Scutari - the most important of all in respect of numbers (about 17,000) and political independence.

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  • the Bulgarians conquered the southern portion of the country and Epirus as far as Khimara; under their powerful tsar Simeon (893-927), who defeated the Servians, they established their rule on the Adriatic littoral, except at Durazzo, which remained Byzantine, and colonized these regions in great numbers.

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  • Large numbers of horses, cattle, swine and poultry are reared.

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  • The other three methods he devised for the sake of those who would prefer to work with natural numbers; and he mentions that the promptuary was his latest invention.

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  • 2 In this treatise (which was written before Napier had invented the name logarithm) logarithms are called "artificial numbers."

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  • 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.

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  • The arrangement of the numbers on the rods will be evident from fig.

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  • Up to 1565 the national synod consisted of a minister with one or two elders or deacons from every church; after that date, to avoid overcrowding, its numbers were restricted to representatives from each provincial synod.

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  • Only in Jersey and Guernsey, whither large numbers of Huguenots had fled after the St Bartholomew massacre, was Presbyterianism fully permitted.

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  • A great and widespread revival marked the opening years of the century, resulting in marvellous increase of zeal and numbers.

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  • Whatever it may have been in remote geological periods, it is now extremely limited both in size and numbers.

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  • In 1878 the number of cattle was 12,000,000; of sheep, 65,000,000; and of horses, 4,000,000; in 1899 the numbers were - cattle, 25,000,000; sheep, 89,000,000; and horses, about 4,500,000.

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  • The existing system of taxation also presses heavily upon the provinces, as may be seen from the fact that the national, provincial and municipal exactions together amount to £7 per head of population, while the total value of the exports in 1898 was only L6 in round numbers.

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  • Rosas met the allies at the head of a body of troops fully equal in numbers to their own, but was crushingly routed, February 3rd, at Monte Caseros, about io m.

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  • 239,178 The Basques of Basses-PyrnCes go in considerable 442,777 numbers to the Argentine Republic, the inhabitants of 333,621 Basses Alpes to Mexico and the United States, and 429,812 there are important French colonies in Algeria and 315,199 Tunisia.

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  • The numbers of growers decreased from 139,000 in 1891 to 124,000 in 1905.

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  • (a) Normandy, Perche, Cotentin and maritime Flanders, where horses are bred in great numbers; (b) the strip of coast between the Gironde and the mouth of the Loire; (c) the Morvan including the Nivernais and the Charolais, from which the famous Charolais breed of oxen takes its name; (d) the central region of the central plateau including the districts of Cantal and Aubrac, the home of the famous beef-breeds of Salers and Aubrac.1 The famous pre-sal sheep are also reared in the Vende and Cotentin.

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  • Occupations.The following table, which shows the approximate numbers of persons engaged in the various manufacturing industries of France, who number in all about 5,820,000, indicates their relative importance from the point of view of employment:

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  • The Dogger Bank is frequented by numbers of French fishing-boats.

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  • They are composed of employers and workmen in equal numbers and are established by decree of the council of state, advised by the minister of justice.

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  • Ali Pasha's greater numbers enabled him to outflank his enemy.

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  • In 1816 he visited the continent, and first at Geneva and afterwards in Montauban (1817) he lectured and interviewed large numbers of theological students with remarkable effect; among them were Malan, Monod and Merle d'Aubigne.

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  • For the history of the Levites in the post-exilic and later ages, see the commentaries on Numbers (by G.

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  • Crystals have the form of small, sharply defined cubes of an oliveor grass-green colour, and occur together in considerable numbers on the matrix of the specimens.

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  • These fish frequent rocky shoals off the eastern coast and are caught in numbers outside Port Jackson for the Sydney market.

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  • It is difficult to believe that they at first arrived in such numbers as at once to overwhelm the Papuan population.

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  • There are words for the elementary numbers, one, two, three; but " four " is usually expressed by " two-two."

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  • The numbers of the native Australians are steadily diminishing.

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  • The failure of the crops was almost universal and large numbers of sheep and cattle perished for want of food.

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  • The House of Representatives was to consist of members chosen in the different states in numbers proportioned to their population, but never fewer than five.

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  • The result was a defeat, their numbers being reduced from 35 to 19; but a signal triumph was won for solidarity.

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  • Kummer, &c., on the theory of numbers.

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  • The main results of these researches, which occupied him from 1854 to 1864, are contained in his Report on the Theory of Numbers, which appeared in the British Association volumes from 1859 to 1865.

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  • The three subjects to which Smith's writings relate are theory of numbers, elliptic functions and modern geometry; but in all that he wrote an "arithmetical" made of thought is apparent, his methods and processes being arithmetical as distinguished from algebraic. He had the most intense admiration of Gauss.

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  • Thus, if x= horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as algebraical symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers.

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  • From the 12th to 22nd of July he was fighting continually, but finally, on the 31st of July, his army was annihilated by overwhelming numbers near Segesvár (Schassburg), Bem only escaping by feigning death.

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  • the flying fox; the python, the cobra, and many other varieties of snake, including the hamadryad; the alligator, the otter and the gavial, as well as countless kinds of squirrel, rat, &c., are found throughout the jungles of the peninsula in great numbers.

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  • The population of the peninsula numbers about 2,000,000, of whom about 600,000 inhabit the colony of the Straits Settlements, about 900,000 the Federated Malay States, about 200,000 the Malay State of Johor, and about 250,000 to 300,000 the remainder of the peninsula.

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  • The Sakai still inhabit in greatest numbers the country which forms the interior of Pahang, the Plus and Kinta districts of Perak, and the valley of Nenggiri in Kelantan.

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  • Though they now use metal tools imported by the Malays, it is noticeable that the names which they give to those weapons which most closely resemble in character the stone implements found in such numbers all over the peninsula are native names wholly unconnected with their Malay equivalents.

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  • He showed extraordinary energy, resource and military talent in stemming the advance of the royalists, who now followed up their victories by advancing into the association; he defeated them at Gainsborough on the 28th of July, and managed a masterly retreat before overwhelming numbers to Lincoln, while the victory on the 11th of October at Winceby finally secured the association, and maintained the wedge which prevented the junction of the royalists in the north with the king in the south.

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  • In December 16 J4 Cromwell despatched Penn and Venables with a fleet of thirty-eight ships and 2500 soldiers to the West Indies, their numbers being raised by recruits at the islands to 7000 men.

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  • Enormous numbers of ale-houses were closed - a proceeding which excited intense resentment and was probably no slight cause of the royalist reaction.

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  • From his pillar he preached and exercised a great influence, converting numbers of heathen and taking part in ecclesiastical politics.

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  • Among the inhabitants are numbers of Mahommedans, and there is a settlement of Falashas.

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  • Unlike any other buildings in Abyssinia, the castles and palaces of Gondar resemble, with some modifications, the medieval fortresses of Europe, the style of architecture being the result of the presence in the country of numbers of Portuguese.

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  • The whole district of Casas Grandes is further studded with artificial mounds, from which are excavated from time to time large numbers of stone axes, metates or corn-grinders, and earthern vessels of various kinds.

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  • numbers, and that passages may with impunity be written for thirty-two violins which no single player can execute clearly.

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  • Epidemics of cholera, which occurred during the years of scarcity and famine, also swept away large numbers.

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  • This fleet of cable ships now numbers over forty, ranging in size from vessels of 300 tons to 10,000 tons carrying capacity.

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  • Great numbers came to join the new order which responded so admirably to the needs of the time.

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  • Large numbers of quails are shot in the spring.

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  • The large numbers of emigrants, who are drawn chiefly from the rural classes, furnish another proof of poverty.

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  • Textile, building and mining industries show the highest percentage of strikes, since they give employment to large numbers of men concentrated in single localities.

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  • In 1871-1872 there were 375,947 scholars at the evening schools and 154,585 at the holiday schools, while in 1900-1901 these numbers had fallen to 94,510 and 35,460 respectively.

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  • The increase in the numbers rejected as unfit is accounted for by the fact that if only a small proportion of the contingent can be taken for service, the medical standard of acceptance is high.

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  • Within the cities and upon the open lands the Italians, in this and the next century, doubled, trebled and quadrupled their numbers.

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  • The events of 1820-1821 increased the agitation in Romagna, and in 1825 large numbers of persons were condemned to death, imprisonment or exile.

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  • the country districts, and to exclude from the franchise numbers of peasants and small proprietors who, though of more conservative temperament and of better economic position than the artizan population of the large towns, were often unable to fulfil the scholarship qualification.

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  • Instead of maintaining a firm policy, Giolitti allowed the movement to spread until, towards the autumn of 1893, he became alarmed and drafted troops into the island, though in numbers insufficient to restore order.

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  • The Dabormida division, unsupported by Albertone, found itself likewise engaged in a separate combat against superior numbers.

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  • The landlords on their part organized an agrarian union to defend their interests and enrolled numbers of non-union laborers to carry on the necessary work and save the crops.

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  • The ministerial majority was over three hundred, and although the Extreme Left was somewhat increased in numbers it was weakened in tone, and many of the newly elected reds were hardly more than pale pink.

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  • Danby therefore ordered a return from every diocese of the numbers of dissenters, both Romanist and Protestant, in order by a proof of their insignificance to remove the royal scruples.3 In December 1676 he issued a proclamation for the suppression of coffee-houses because of the "defamation of His Majesty's Government" which took place in them, but this was soon withdrawn.

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  • At the famous conference, which lasted from Monday the 15th to Tuesday the 23rd of June, the hostile barons were present in large numbers; on the other hand John, who rode over each day from Windsor, was only attended by a few followers.

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  • cylindrical or tapering from The numbers in square brackets [] refer to the bibliography at the end of this article; but when the number is preceded by the word Hydrozoa, it refers to the bibliography at the end of the articl Hydrozoa.

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  • name; never absent and usually present in great numbers (fig.

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  • It has been the habit of biologists to use the terms variation, selection, elimination, correlation and so forth, vaguely; the new school, which has been strongly reinforced from the side of physical science, insists on quantitative measurements of the terms. When the anatomist says that one race is characterized by long heads, another by round heads, the biometricist demands numbers and percentages.

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  • Dr Maitland (essay on" The Universal Ordinary ") thinks, but without very much foundation, that great numbers especially of the more important causes were tried before these delegates; although the records have largely perished, since they were the records of courts ' which were dissolved as soon as their single cause had been decided.

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  • Figurate numbers >>

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  • They appear to be present in large numbers in the soil, and to infect the Leguminous plant by attacking its root-hairs.

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  • They may occur on all parts, buds, leaves, stems or roots, as shown by the numerous species of Cynips on oak, Phylloxera on vines, &c. The local damage is small, - but the general injury to assimilation, absorption and other functions, may be important if the numbers increase.

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  • The numbers of Jewish families driven out of the country by Torquemada is variously stated from Mariana's 1,700,000 to the more probable 800,000 of later historians.

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  • The cat cemetery on the west side of the town consisted of numbers of large brick chambers, crammed with burnt and decayed mummies, many of which had been enclosed in cat-shaped cases of wood and bronze.

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  • In the retina the cones prevail in numbers over the rods, as in the mammals, and their tips contain, as in other Sauropsida, coloured drops of oil, mostly red or yellow.

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  • Their numbers vary from one pair to seven, and they are inserted either upon the middle portion of the bronchial semi-rings (Mesomyodi), or upon the ends of these semi-rings where these pass into the inner tympaniform membrane (Acromyodi).

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  • Ignoring the fact that some Oligomyodae are mesoand others acromyodian, they tried to combine two irreconcilable principles, namely, mere numbers against quality.

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  • The numbers of genera and species of birds are, of course, a matter of personal inclination.

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  • The records of his life and work are noticed in the articles Exodus, Numbers,.

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  • Of this halakhic Midrash we possess that on Exodus, called Mekhilta, that on Leviticus, called Sifra, and that on Numbers and Deuteronomy, called Sifre.

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  • (on Numbers), Debharim R.

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  • The great majority of the foreign population are Italians or Spaniards, with lesser numbers, in descending scale, of Brazilian, Argentine and French birth.

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  • As the non-privileged order increased in numbers, while the privileged order, as every exclusive hereditary body must do, lessened, the larger body gradually put on the character of the nation at large, while the smaller body put on the character of a nobility.

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  • It hampered the Brethren's progress in Germany, and explains the smallness of their numbers there.

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  • In Germany, therefore, the importance of the Moravians must be measured, not by their numbers, but by their influence upon other Christian bodies.

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  • On the 20th of October 1349 Clement published a bull commanding the bishops and inquisitors to stamp out the growing heresy, and in pursuance of the pope's orders numbers of the sectaries perished at the stake or in the cells of the inquisitors and the episcopal justices.

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  • Numbers of Beghards joined the Brethren of the Cross, and the two sects were confounded in the rigorous persecution conducted in Germany by the inquisitor Eylard Schoneveld, who almost annihilated the flagellants.

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  • The Nitidulidae are a large family with 1600 species, among which members of the genus Meligethes are often found in numbers feeding on blossoms, while others live under the bark of trees and prey on the grubs of boring beetles.

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  • b, Antenna of male; c, antenna swollen tail-ends black with the contained food-material, are often dug up in numbers in well-manured fields.

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  • This was owing to the fact that large numbers of the men engaged in agricultural pursuits during the summer temporarily move every year into the large industrial centres for the winter.

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  • m., drains 13 governments, the aggregate population of which numbers over 28,000,000.

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  • Europeans, the Germans only attain considerable numbers in European Russia.

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  • Considerable numbers of Germans, tradesmen and artisans, settled at the invitation of the Russian government in many of the larger towns as early as the 16th century, and to a much greater extent in the 18th century.

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  • Numbers were invited in 1762 to settle in S.

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  • During the closing years of the 19th century great numbers of Germans flocked into the industrial governments of Poland, namely, Piotrkow, Warsaw and Kalisz.

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  • Russia, and appearing in larger numbers only in the district of Rostov.

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  • Finally, in the Baltic provinces nearly all the land belongs to the German landlords, who either farm the land themselves, with hired labourers, or let it in small farms. Only one-fourth of the peasants are farmers, the remainder being mere labourers, who are emigrating in great numbers.

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  • Since their emancipation in 1861, the peasants of the central governments of Russia have in large numbers drifted away into the black earth zone, or have gone to the factories.

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  • Live-stock are diminishing in numbers all round: in the case of horses, from 21 per 100 inhabitants in 1882 to II per loo inhabitants in 1904; of cattle, from 31 in 1851 to 23 in 1882 and 27 in 1904; sheep, from 56 to 46 and 41 in the years named respectively; and pigs, from 13 to 9 and 10 respectively.

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  • Finland ponies are exported in large numbers.

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  • Poultryfarming is being more extensively engaged in, and vast numbers of eggs are exported.

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  • Polish proprietors settled in large numbers on the Cossack territory, and great efforts were made, with the assistance of the Jesuits, to bring the Orthodox population under papal authority.

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  • Workmen are killed and injured in this way, both while on duty and when going to and from their work; passengers, with or without right, go in front of trains at stations and at highway crossings at grade level; and trespassers are killed and injured in large numbers on railways everywhere, at and near stations, at crossings, and out on the open road, where they have no shadow of right.

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  • Its width depends on the numbers of tracks and their gauge; for a double line of standard gauge it is about 25 ft., a space of 6 ft.

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  • Push-and-pull shunting is simple, but it is also slow, and therefore efforts have been made at busy yards where great numbers of trains are dealt with to introduce more expeditious methods.

    0
    0
  • are plotted on the diagram, the reference numbers on which refer to the first column in the table.

    0
    0
  • All these have greatly declined in numbers, being profitably replaced by sheep. Land-birds are few in kind, and are mostly strays from South America.

    0
    0
  • As it was, although Parker said that Grindal "was not resolute and severe enough for the government of London," his attempts to enforce the use of the surplice evoked angry protests, especially in 1565, when considerable numbers of the nonconformists were suspended; and Grindal of his own motion denounced Cartwright to the Council in 1570.

    0
    0
  • and Heb.) as well as Buchanan Gray's instructive note in Numbers, p. 276.

    0
    0
  • In Ireland, in Cromwell's time, wolves were particularly troublesome, and said to be increasing in numbers, so that special measures were taken for their destruction, such as the offering of large rewards for their heads, and the prohibition (in 1652) of the exportation of "wolf-dogs," the large dogs used for hunting the wolves.

    0
    0
  • The active measures taken then and later reduced their numbers greatly, so that towards the end of the century they became scarce, but, as in the case of the sister island, the date of their final disappearance cannot now be ascertained.

    0
    0
  • Of native animals the varieties are few and the numbers of individuals small.

    0
    0
  • These first appeared in large numbers in the lower part of the Humboldt Valley in the summer of 1906, and in October and November 1907 it was estimated that they numbered on certain ranches from 8000 to 12,000 on every acre.

    0
    0
  • Charles, in a spirit of the most vindictive cruelty, had large numbers of Conradin's barons put to death and their estates confiscated, and the whole population of several towns massacred.

    0
    0
  • The legions at once joined him; numbers of Franks enlisted in his service; an increased and well-equipped fleet secured him the command of the neighbouring seas.

    0
    0
  • He took part in the siege of Yorktown, the battle of Fair Oaks, the seven days' battle before Richmond, and the battles of Antietam, Fredericksburg, where he was wounded, and Chancellorsville, where his brigade was reduced in numbers to less than a regiment, and General Meagher resigned his commission.

    0
    0
  • are the medicinal springs of Sibul, to which large numbers of patients from the neighbouring provinces come.

    0
    0
  • The following table gives the heats of neutralization of the commoner strong monobasic acids with soda: - Hydrochloric acid Hydrobromic acid Hydriodic acid Nitric acid Chloric acid Bromic acid Within the error of experiment these numbers are identical.

    0
    0
  • 14, 16 (the numbers are not inclusive), and reckons three deportations in the 7th (?

    0
    0
  • In spite of their desperate sallies, Jerusalem was surrounded by a wall, and its people, whose numbers were increased by those who had come up for the passover, were hemmed in to starve.

    0
    0
  • Large numbers fell in the actual fighting.

    0
    0
  • These academies were organized on both scholastic and popular lines; their constitution was democratic. An outstanding feature was the Kallah assemblage twice a year (in Elul at the close of the summer, and in Adar at the end of the winter), when there were gathered together vast numbers of outside students of the most heterogeneous character as regards both age and attainments.

    0
    0
  • This Smyrnan pretender not only proclaimed himself Messiah (c. 1650) but he was accepted in that role by vast numbers of his brethren.

    0
    0
  • After the Declaration of Independence, Jews are found all over America, where they have long enjoyed complete emancipation, and have enormously increased in numbers, owing particularly to immigration from Russia.

    0
    0
  • They have exercised an influence over distant neighbours, especially in Fiji, quite out of proportion to their numbers.

    0
    0
  • In 1904 the financial and legal administration was put into the hands of the British High Commissioner for the Western Pacific. The native king is assisted by a legislative assembly consisting, in equal numbers, of hereditary nobles and popular (elected) representatives.

    0
    0
  • Of the wild animals of Crete, the wild goat or agrimi (Capra aegagrus) alone need be mentioned; it is still found in considerable numbers on the higher summits of Psiloriti and the White Mountains.

    0
    0
  • A decimal system of numeration was used, with numbers going up to io,000.

    0
    0
  • For some time after his arrival complete tranquillity prevailed in the island, but the Moslem population, reduced to great distress by the prolonged insurrection, emigrated in large numbers.

    0
    0
  • In the second, which took place in the Church of St John and St Paul, and lasted three days, he undertook to refute innumerable errors in Aristotelians, mathematicians and schoolmen, to conduct his dispute either logically or by the secret doctrine of numbers, &c. According to Aldus, who attended the debate and published an account of it in his dedication to Crichton prefixed to Cicero's "Paradoxa" (1581), the young Scotsman was completely successful.

    0
    0
  • Large numbers of shad, blue fish, weak fish (squeteague), alewives, Spanish mackerel, perch, bass, croakers (Micropogon undulatus), mullet, menhaden, oysters and clams are caught in the sounds, in the lower courses of the rivers flowing into them, or in the neighbouring waters of the sea.

    0
    0
  • Farmers of the Piedmont Plateau formerly kept large numbers of horses and cattle from April to November in ranges in the Mountain Region, but with the opening of portions of that country to cultivation the business of pasturage declined, except as the cotton plantations demanded an increased supply of mules; there were 25,259 mules in 1850, 110,011 in 1890, 138,786 in 1900, and 181,000 in 1910.

    0
    0
  • There are only two genders and two numbers: the neuter gender is entirely wanting, and the dual number is not recognized in Syriac grammar, though there are plain traces of it in the language.

    0
    0
  • Many species of aphid migrate from one plant to another at certain stages in their life-cycle when their numbers have very largely increased, and F.

    0
    0
  • Two festivals are held in the town, a less important one in October, the other, on the 24th and 25th of May, unique for its gathering of gipsies who come in large numbers to do honour to the tomb of their patroness Sara, contained in the crypt below the apse.

    0
    0
  • from Montes Claros, in the valley of the Verde Grande River, and attracted large numbers of miners.

    0
    0
  • On the 28th Sir Harry Smith, with a view to clearing the left or British bank, attacked him, and after a desperate struggle thrice pierced the Sikh troops with his cavalry, and pushed them into the river, where large numbers perished, leaving 67 guns to the victors.

    0
    0
  • He was severe, but just and impartial, and strove to effect necessary reforms by reducing the numbers of the Janissaries, improving the coinage, and checking the state expenditure.

    0
    0
  • The saw-fishes, Pristidae, the electrical rays, Torpedinae, and ordinary rays and skates, are also found in considerable numbers.

    0
    0
  • The ancient Sacae, or Scyths, are recognized in the Aryan population, who may be found in great numbers and in their purest form in the more inaccessible mountains and glens of the central highlands.

    0
    0
  • From choice or compulsion large numbers settled in Egypt in the time of the Ptolemies, and added an appreciable element to Alexandrine culture, while gradual voluntary emigration established Jewish communities in Syria, Asia Minor, Greece and Italy, who facilitated the first spread of Christianity.

    0
    0
  • At present the largest numbers are to be found in the eastern parts of Europe.

    0
    0
  • Large flocks of sheep are kept, both for their flesh and their wool, and there are in the province large numbers of horned cattle and of pigs, Geese and goose feathers form lucrative articles of export.

    0
    0
  • numbers.

    0
    0
  • The gonads are, moreover, limited and fixed in numbers, and are practically invariably attached to the intersegmental septa, usually to the front septum of a segment, more rarely to the posterior septum.

    0
    0
  • The testes vary in numbers of pairs.

    0
    0
  • Four (Ozobranchus) to six (Glossiphonia) or ten (Philaemon) are common numbers.

    0
    0
  • Two of the docks are for the accommodation of the fishing fleet, which, consisting principally of steam trawlers, numbers upwards of 500 vessels.

    0
    0
  • imparted a self-reliant enthusiasm to his countrymen, formed them into an army, and organized them as a political community; his mountaineer infantry, though limited in numbers, proved desperately courageous; his cavalry was daring and ubiquitous.

    0
    0
  • The population was greatly reduced in numbers, and much of the land was left uncultivated.

    0
    0
  • They contain, however, a fragment of a separate tract on Polygonal Numbers.

    0
    0
  • The difference in form and content suggests that the Polygonal Numbers was not part of the larger work.

    0
    0
  • The "Porisms" quoted are interesting propositions in the theory of numbers, one of which was clearly that the dif f erence between two cubes can be resolved into of two cubes.

    0
    0
  • Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.

    0
    0
  • But the bulk of the work consists of problems leading to indeterminate equations of the second degree, and these universally take the form that one or two (and never more) linear or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x.

    0
    0
  • 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.

    0
    0
  • To find three numbers such that the product of any two added to the sum of those two gives a square (III.

    0
    0
  • 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.

    0
    0
  • 22) To find two numbers such that their product their sum gives a cube (IV.

    0
    0
  • With one symbol for an unknown, it will easily be understood what scope there is foradroit assumptions, for the required numbers, of expressions in the one unknown which are at once seen to satisfy some of the conditions, leaving only one or two to be satisfied by the particular value of x to be determined.

    0
    0
  • The book is valuable also for the propositions in the theory of numbers, other than the "porisms," stated or assumed in it.

    0
    0
  • Religion.-Most of the Russians and the Georgians belong to the Orthodox Greek Church (over 4,000,000 in all); but considerable numbers (estimated at nearly 122,000, though in reality probably a good many more) are Nonconformists of different denominations.

    0
    0
  • Large numbers of mulberry trees are planted for rearing silkworms, especially in Kutais, Erivan, Elisavetpol (Nukha) and Baku (Shemakha); the groves occupy nearly 150,000 acres, and the winding of the silk gives employment to large numbers of the population.

    0
    0
  • The palace is surrounded by gardens and ornamental waters - to the north the Jardin de l'Orangerie, to the south the Jardin Anglais and the Parterre, between which extends the lake known as the Bassin des Carpes, containing carp in large numbers.

    0
    0
  • Immense numbers of ducks and geese were reared.

    0
    0
  • Meanwhile large numbers of landowners were forced to adopt one of two alternatives.

    0
    0
  • The numbers of live stock in the United Kingdom are shown at five-yearly intervals in Table XII.

    0
    0
  • relative proportionsas distinguished from the actual numbers -in which stock are distributed over the several sections of the United Kingdom do not vary greatly from year to year.

    0
    0
  • The numbers of cattle (both fat and store) weighed at scheduled places in 1893 and 1905 2 were respectively 7.59 and 18% of those entering those markets.

    0
    0
  • The numbers for Scotland are greater throughout than those for England, 72% of the fat cattle entering the scheduled markets in Scotland in 1905 2 having been weighed, while in England the proportion was only 20 7 0.

    0
    0
  • At Deptford, for example, large numbers of cattle and sheep which thus arrive - mainly from Argentina, Canada and the United States - are at once slaughtered, and so furnish a steady supply of fresh-killed beef and mutton.

    0
    0
  • shows the number of cattle, sheep and pigs shipped from Ireland into Great Britain in each of the fifteen years 1891-1905, the numbers of horses similarly shipped being also indicated.

    0
    0
  • The universal attack that has been made upon this pest has, however, largely decreased its numbers.

    0
    0
  • The conditions which are peculiar to the modern world are the large numbers we have to deal with, the vast and fairly homogeneous areas in which justice is administered and property secured, and the enormously increased facilities for transport and communication.

    0
    0
  • Whether such large numbers have the character of the " economic man " of the early economists matters very little.

    0
    0
  • Sensations, he argued, thus being representable by numbers, psychology may become an "exact" science, susceptible of mathematical treatment.

    0
    0
  • Naval preparations went on apace at all the dockyards, and numbers of flat-bottomed boats were built or repaired at the northern harbours.

    0
    0
  • These numbers are valuable as an exhibition not so much of events as of the feelings of the Parisian people; they are adorned, moreover, by the erudition, the wit and the genius of the author, but they are disfigured, not only by the most biting personalities and the defence and even advocacy of the excesses of the mob, but by the entire absence of the forgiveness and pity for which the writer was afterwards so eloquently to plead.

    0
    0
  • 1 In April 1792 Desmoulins founded with Stanislas Freron a new journal, La Tribune des patriotes, but only four numbers appeared.

    0
    0
  • On the 7th of January 1 794 Robespierre, who on a former occasion had defended Camille when in danger at the hands of the National Convention, in addressing the Jacobin club counselled not the expulsion of Desmoulins, but the burning of certain numbers of the Vieux Cordelier.

    0
    0
  • He had exhibited in the numbers of the Vieux Cordelier almost a disregard of the death which he must have known hovered over him.

    0
    0
  • The eggs of locusts may remain for years in the ground before hatching; and there may thus arise the peculiar phenomenon of some species of insect appearing in vast numbers in a locality where it has not been seen for several years.

    0
    0
  • To enable the reader to compare the several groups of Nitzsch with the families of L'Herminier, the numbers applied by the latter to his families are suffixed in square brackets to the names of the former; and, disregarding the order of sequence, which is here immaterial, the essential correspondence of the two systems is worthy of all attention, for it obviously means that these two investigators, starting from different points, must have been on the right track, when they so often coincided as to the limits of what they considered to be, and what we are now almost justified in calling, natural groups.'

    0
    0
  • By the numbers prefixed it would look as if there should be four new members of this order; but that seems to be due rather to a slip of the pen or to a printer's error.

    0
    0
  • These houses clustered round the churches which now began to be built in considerable numbers, and formed the various contrade of the city.

    0
    0
  • The scuole were divided into the six scuole grandi, so called from their numbers, wealth and privileges, and the scuole minori or fraglie, which in most cases were associated with an art or craft.

    0
    0
  • Foxes are still found in considerable numbers in suitable habitats; opossums, skunks and raccoons are plentiful in some parts of the state; and rabbits and squirrels are still numerous.

    0
    0
  • Large numbers of horses, cattle and' sheep are bred, the cattle being famous.

    0
    0
  • There are large numbers of private schools, in art, music and academic studies.

    0
    0
  • numbers on the mud flats at the mouths of rivers in the tropics, skipping about by means of the muscular, scaly base of their pectoral fins, with the head raised and bearing a pair of strongly projecting versatile eyes close together.

    0
    0
  • Considerable numbers of mackerel are taken off Norfolk and Suffolk in May and June, and also in September and October.

    0
    0
  • There can be no doubt that they enter the North Sea from the English Channel, and return by the same route, but others travel round the north of Scotland and appear in rather small numbers off the east coast of that country.

    0
    0
  • No doubt large numbers are devoured by insectivorous birds, mammals and reptiles, but the mortality due to them and other foes sinks into insignificance beside that caused by the persecution of hymenopterous insects of the families Ichneumonidae and Pompilidae, especially of the latter, many species of which systematically ransack the country for spiders wherewith to feed their young in the breeding season.

    0
    0
  • It sheds its " forms " (as the buds are called), blooms, and even half-grown bolls in great numbers.

    0
    0
  • with less easy terms for payment than were usual in Manchester, prevented any great numbers from departing from the beaten track.

    0
    0
  • Colenso, and learned to regard the prophetic narrative of Genesis, Exodus, and Numbers as older than what was by the Germans denominated Grundschrift (" Book of Origins").

    0
    0
  • Twothirds of the matter had been contributed by the editor, and the two stout volumes in which the numbers were collected contained the best political thought which had for long appeared in Germany.

    0
    0
  • Bears, wild boars, hares, wolves, foxes and wild cats are very common, and in the north sables are found in great numbers.

    0
    0
  • This bird is exported in large numbers to northern China, where it is much prized on account of its extraordinary power of imitation.

    0
    0
  • In such immense shoals do these fish appear in some of the smaller streams that numbers are squeezed out on to the banks and there perish.

    0
    0
  • The native Christians suffered; the pilgrims of the West found their way made still more difficult, and that at a time when greater numbers than ever were thronging to the East.

    0
    0
  • The second, led by Peter himself, passed safely through Hungary, but suffered severely in Bulgaria, and only attained Constantinople with sadly diminished numbers at the end of July.

    0
    0
  • Little driblets of men might indeed be added to the numbers of the Franks; but the great bodies of crusaders either perished in Asia Minor, as in I ioi and 1147, or found themselves thwarted and distrusted by the native Franks.

    0
    0
  • Crusades appear to have been dignified by numbers when they followed some crushing disaster - the loss of Edessa in 1144, or the fall of Jerusalem in 1187 - and were led by kings and emperors; or when, like the Fourth and Fifth Crusades, they achieved some conspicuous success or failure.

    0
    0
  • In the second place, the Mongols of the 13th century were not as yet, in any great numbers, Mahommedans; the official religion was "Shamanism," but in the Mongol army there were many Christians, the results of early Nestorian missions to the far East.

    0
    0
  • In the steppe-land and in the southern trans-Jordanic districts are numbers of true Arabs, mostly belonging to the great Anazeh family, which has been coming northwards from Nejd in detachments since the 13th century.

    0
    0
  • It was chiefly owing to his skill and courage as a parliamentary debater and his tact as a leader that the party held its own and constantly increased in numbers during the great struggle with the Prussian government.

    0
    0
  • Ducks, wild turkeys, bears and wild cats (lynx) are found, but in decreasing numbers.

    0
    0
  • About 722 he visited Hesse and Thuringia, won over some chieftains, and converted and baptized great numbers of the heathen.

    0
    0
  • A vigorous campaign against monasticism took place; the monasteries were closed, and many of them pulled down or converted into barracks; monks and nuns were compelled to marry, and exiled in large numbers to Cyprus; the literary and artistic treasures were sold for the benefit of the imperial treasury.

    0
    0
  • Numbers of statues - among them a series of draped and richlycoloured female figures - masterpieces of painted pottery, only equalled by the Attic vases found in Magna Grecia and Etruria, and numerous bronzes, were among the treasures of art now brought to light.

    0
    0
  • Owing to the numbers and activity of its institutions, both native and foreign, for the prosecution of research and the encouragement of classical studies, Athens has become Scientific once more an international seat of learning.

    0
    0
  • 3 7 tions numbers some distinguished scholars among its members, and displays great activity in the conduct of excavations.

    0
    0
  • The harbour, in which ships of all nations may be seen, as well as great numbers of the picturesque sailing craft engaged in the coasting trade, is somewhat difficult of access to larger vessels, but has been improved by the construction of new breakwaters and dry docks.

    0
    0
  • As their fur is an important article of commerce, large numbers are annually killed, being either trapped or speared at the mouths of their holes.

    0
    0
  • The concluding feast does not seem to refer to tabernacles per se, but to be distinct from it, as is shown by the break in the descending series of the sacrifices of bullocks as given in Numbers.

    0
    0
  • Near the Manikarnika ghat is the well held to have been dug by Vishnu and filled with his sweat; great numbers of pilgrims bathe in its venerated water.

    0
    0
  • But besides an immense resort to Benares of poor pilgrims from every part of India, as well as from Tibet and Burma, numbers of rich Hindus in the decline of life go there for religious salvation.

    0
    0
  • equal changes in temperature and pressure occasion equal changes in equal volumes of all gases and vapours - Avogadro deduced the law: Under the same conditions of temperature and pressure, equal volumes of gases contain equal numbers of molecules; and he showed that the relative weights of the molecules are determined as the ratios of the weights of equal volumes, or densities.

    0
    0
  • He numbers the carbon atoms placed at the corners of a hexagon from i to 6, and each side in the same order, so that the carbon atoms i and 2 are connected by the side 1, atoms 2 and 3 by the side 2, and so on.

    0
    0
  • According to the law of Avogadro, equal volumes of different gases under the same conditions of temperature and pressure contain equal numbers of molecules; therefore, since the density depends upon the number of molecules present in unit volume, it follows that for a comparison of the densities of gases, the determinations must be made under coincident conditions, or the observations reduced or re-computed for coincident conditions.

    0
    0
  • To reduce these figures to a common standard, so that the volumes shall contain equal numbers of molecules, the notion of molecular volumes is introduced, the arbitrary values of the crystallographic axes (a, b, c) being replaced by the topic parameters' (x, ?i, w), which are such that, combined with the axial angles, they enclose volumes which contain equal numbers of molecules.

    0
    0
  • List Of Wagner'S Works The following are Wagner's operas and music-dramas, apart from the unpublished Die Hochzeit (three numbers only), Die Feen, and Das Liebesverbot (Das Liebesverbot was disinterred in 1910).

    0
    0
  • The size of the animals varies greatly, from forms a few millimetres in length to Gigantorhynchus gigas, which measures from 10 to 65 cms. The adults live in great numbers in the alimentary canal of some vertebrate, usually fish, the larvae are as a rule encysted in the body cavity of some invertebrate, most often an insect or crustacean, more rarely a small fish.

    0
    0
  • All these numbers take no account of the troops left behind in Macedonia, 12,000 foot and 1,500 horse, according to Diodorus.

    0
    0
  • Of the numbers and constitution of the Antigonid fleet we know nothing.'

    0
    0
  • His goods were confiscated, his aged mother turned into the street and numbers of other members of the clan in Rome were arrested, while Giuffre Borgia led an expedition into the Campagna and seized their castles.

    0
    0
  • The map produced on this large scale numbers over 5000 sheets, and is used as a basis for the geological surveys carried on in several of the states of Germany.

    0
    0
  • It was only the chronologists and historians of the church who, following Julius Africanus, made use of apocalyptic numbers in their calculations, while court theologians like Eusebius entertained the imperial table with discussions as to whether the dining-hall of the emperor - the second David and Solomon, the beloved of God - might not be the New Jerusalem of John's Apocalypse.

    0
    0
  • with the duke of Burgundy; finally, they investigated and judged numbers of private cases, lawsuits between prelates, members of religious orders and holders of benefices, thus themselves falling into one of the serious abuses for which they had most blamed the court of Rome.

    0
    0
  • Manning, and small numbers of British and Boer mounted infantry, Indian and African troops were employed, while an Abyssinian force held the line of the Webi Shebeli.

    0
    0
  • Here vast numbers of ducks, geese, swans and pelicans resort every year.

    0
    0
  • The pirates sold great numbers of slaves at Delos, where was the chief market for this kind of wares; and these sales went on as really, though more obscurely, after the successful expedition of Pompey.

    0
    0
  • Though the Roman slaves were not, like the Spartan Helots, kept obedient by systematic terrorism, their large numbers were a constant source of danger.

    0
    0
  • As correct a notion as can be obtained of the numbers annually exported from the continent about the year 1790 by traders cf the several European countries engaged in the traffic is supplied by the following statement: - " By the British, 38,000; by the French, 20,000; by the Dutch, 4000; by the Danes, 2000; by the Portuguese, 10,000; total 74,000."

    0
    0
  • The circumstances of their subsequent life on the plantations were not favourable to the increase of their numbers.

    0
    0
  • One cause which prevented the natural increase of population was the inequality in the numbers of the sexes; in Jamaica alone there was in 1789 an excess of 30,000 males.

    0
    0
  • The slaves were overworked now that fresh supplies were stopped, and their numbers rapidly decreased.

    0
    0
  • As these waste places have been gradually brought under the plough, in England and Scotland particularly, the haunts and means of subsistence of the linnet have been curtailed, and hence its numbers have undergone a very visible diminution throughout Great Britain.

    0
    0
  • It is the home of great numbers of the working classes of Sydney and some of the largest factories and most important docks are situated here.

    0
    0
  • Being the starting-point of the "overland route" to India, and the residence of the chief foreign consuls, it quickly acquired a European character and attracted not only Frank residents, but great numbers of Greeks, Jews and Syrians.

    0
    0
  • According to the Arda Viraf, I, 2, Zoroaster taught, in round numbers, some 300 years before the invasion of Alexander.

    0
    0
  • Large numbers of geese and poultry are kept.

    0
    0
  • The species of each genus are then arranged either systematically or alphabetically in separate covers of stout, usually light brown paper, or, if the genus be large, in several covers with the name of the genus clearly indicated in the lower left-hand corner of each, and opposite it the names or reference numbers of the species.

    0
    0
  • The Pythagorean theory of numbers, Neoplatonic ideas of emanation, the Logos, the personified Wisdom, Gnosticism - these and many other features combine to show the antiquity of tendencies which, clad in other shapes, are already found in the old pre-Christian Oriental religions.

    0
    0
  • According to Numbers xxi.

    0
    0
  • These are simply parts of the sea which have escaped the filling-in process carried on by the great river and the lesser streams. A second class, called " ox-bow" lakes, large in numbers but small in area, includes ordinary cut-off meanders along the Mississippi and Red rivers.

    0
    0
  • Bats in prodigious numbers, and some of them of extraordinary size, inhabit the many caves of the island; more than twenty species are known.

    0
    0
  • What are numbers?

    0
    0
  • Also in addition to the cardinal numbers there are the ordinal numbers: the fifth apple and the tenth pear claim thought.

    0
    0
  • A critical defence of them would require a volume.1 Cardinal Numbers.

    0
    0
  • The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and 13, satisfying the conditions (1) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in common, and then by defining by reference to a and (3 two other suitable classes whose cardinal numbers are defined to be respectively the required sum and product of the cardinal numbers in question.

    0
    0
  • Cardinal numbers form a class.

    0
    0
  • If s is any class and zero is a member of it, also if when x is a cardinal number and a member of s, also x-}-I is a member of s, then the whole class of cardinal numbers is contained in s.

    0
    0
  • If a and b are cardinal numbers, and a+I =b+1, then a =b.

    0
    0
  • For there is no self-contained science of cardinal numbers.

    0
    0
  • They are excellent principles of the highest value, but they are in no sense the necessary premisses which must be proved before any other propositions of cardinal numbers can be established.

    0
    0
  • Thus, corresponding to the cardinal numbers 2, 3, 4.

    0
    0
  • there are the ordinal numbers 2, 3, 4....

    0
    0
  • But the definition of the cardinal number of a class applies when the class is not finite, and it can be proved that there are different infinite cardinal numbers, and that there is a least infinite cardinal, now usually denoted by o where to is the Hebrew letter aleph.

    0
    0
  • These relation-numbers are the infinite ordinal numbers.

    0
    0
  • - Rational numbers and real numbers in general can now be defined according to the same general method.

    0
    0
  • If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n X x = m X y.

    0
    0
  • The arithmetic of rational numbers is now established by means of appropriate definitions, which indicate the entities meant by the operations of addition and multiplication.

    0
    0
  • The real numbers, which include irrational numbers, have now to be defined.

    0
    0
  • A real number is a class (a, say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of a.

    0
    0
  • But now the irrational real numbers have all made their appearance.

    0
    0
  • The arithmetic of real numbers follows from appropriate definitions of the operations of addition and multiplication.

    0
    0
  • Real numbers with signs (+or -) are now defined.

    0
    0
  • The addition and multiplication of these "signed" real numbers is suitably defined, and it is proved that the usual arithmetic of such numbers follows.

    0
    0
  • Such a number is a "one-many" relation which relates n signed real numbers (or n algebraic complex numbers when they are already defined by this procedure) to the n cardinal numbers I, 2..

    0
    0
  • The sum of two complex numbers x i e l +x 2 e 2 + ...

    0
    0
  • But an indefinite number of definitions of the product of two complex numbers yield interesting results.

    0
    0
  • Each definition gives rise to a corresponding algebra of higher complex numbers.

    0
    0
  • We will confine ourselves here to algebraic complex numbers - that is, to complex numbers of the second order taken in connexion with that definition of multiplication which leads to ordinary algebra.

    0
    0
  • The product of two complex numbers of the second order - namely, l e l +x 2 e 2 and y i e l +y 2 e 2, is in this case defined to mean the complex (x i y i - x 2 y 2)e i +(x i y 2 +x 2 y 1)e 2.

    0
    0
  • This is exactly the same reason as that which has led mathematicians to work with signed real numbers in preference to real numbers, and with real numbers in preference to rational numbers.

    0
    0
  • For example, the application of the theory of cardinal numbers to classes of physical entities involves in practice some process of counting.

    0
    0
  • Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers.

    0
    0
  • The application of the theory of real numbers to physical quantities involves analogous considerations.

    0
    0
  • The compactness of the series of rational numbers is consistent with quasi-gaps in it - that is, with the possible absence of limits to classes in it.

    0
    0
  • Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers.

    0
    0
  • But among the real numbers all classes have limits.

    0
    0
  • Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers.

    0
    0
  • Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.

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  • (1st ed., 1887), both tracts translated into English under the title Essays on the Theory of Numbers (Chicago, 1901).

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    0
  • numbers, bringing with them wealth and also slave labour..

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    0
  • while a class of peasant proprietors was created; its numbers being increased by the custom that, if any man reclaimed a piece of waste land, it became his own property after ten years.

    0
    0
  • The garrison numbers about 20,000 Austrian troops, and there are 7100 native troops.

    0
    0
  • About half a mile to the north-west of the village is the Menec system, which consists of eleven lines, numbers 874 menhirs, and extends a distance of 3376 ft.

    0
    0
  • Ertoghrul first camped at Jessin, east of Erzerum; a second appeal to Ala-ud-din was more successful - the numbers of the immigrants had become too insignificant for their presence to be a source of danger.

    0
    0
  • The regular troops comprised also armourers (jebeji), from 6000 to 8000 men, and six squadrons of cavalry; these were recruited in the same way as the Janissaries, and their numbers were raised by Murad III.

    0
    0
  • The tide of success now turned again in favour of the Turks, who recaptured Karansebes and Lippa, and at Lugos exterminated by the weight of overwhelming numbers an Austrian force under Field-marshal Count Friedrich von Veterani (1630-1695), the hero of many victories over the Turks, who was killed in the battle.

    0
    0
  • Meanwhile the sultan's whole efforts were directed towards the reform of the country; the newly-instituted militia was in every respect a success; it grew in numbers, and hopes were entertained that it would gain popularity.

    0
    0
  • These four groups, however, were of unequal importance, and thanks to this arrangement the English, although weakest in point of numbers, were able to exercise the same influence in the council as if they had formed a fourth of the voters.

    0
    0
  • He had meanwhile received (false) information of a British landing at Boulogne, and he was seriously deceived as to the numbers of Napoleon's forces.

    0
    0
  • Hohenlohe had determined to drive the French into the ravine at daybreak, but had no idea as to the numbers in front of him.

    0
    0
  • The archduke Charles was the foremost amongst many workers who had realized that numbers were absolutely needed to confront the new French methods.

    0
    0
  • With these numbers it was impossible to attain the high degree of individual efficiency required for the old line tactics, hence they were compelled to adopt the French methods of skirmishers and columns, but as yet they had hardly realized the increased density necessary to be given to a line of battle to enable it to endure the prolonged nervous strain the new system of tactics entailed.

    0
    0
  • The crops being still green, and nothing else available as forage for the horses, an epidemic of colic broke out amongst them, and in ten days the mounted arms had lost upwards of one-third of their strength; men died of sunstroke in numbers, and serious straggling began.

    0
    0
  • The allies, however, continued to retreat, but unfortunately Vandamme, with his single corps and unsupported, issued out of the mountains on their flank, threw himself across their line of retreat near Kulm, and was completely overwhelmed by sheer weight of numbers (29th).

    0
    0
  • If, however, he was weak in numbers, he was now again operating in a friendly country, able to find food almost everywhere and practically indifferent as to his communications.

    0
    0
  • Napoleon continued to build line-of-battle ships in numbers from Venice to Hamburg, but only in order to force the British government to maintain costly and wearing blockades.

    0
    0
  • Three or four days' journey east and southeast of Besha are the encampments of the Bani Kahtan, one of the most ancient tribes of Arabia; their pastures extend into the adjoining district of Nejd, where they breed camels in large numbers, as well as a few horses.

    0
    0
  • In his Practica geometriae plain traces of the use of the Roman agrimensores are met with; in his Liber abaci old Egyptian problems reveal their origin by the reappearance of the very numbers in which the problem is given, though one cannot guess through what channel they came to Leonardo's knowledge.

    0
    0
  • The Liber abaci, which fills 459 printed pages, contains the most perfect methods of calculating with whole numbers and with fractions, practice, extraction of the square and cube roots, proportion, chain rule, finding of proportional parts, averages, progressions, even compound interest, just as in the completest mercantile arithmetics of our days.

    0
    0
  • They teach further the solution of problems leading to equations of the first and second degree, to determinate and indeterminate equations, not by single and double position only, but by real algebra, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown quantity being called res and its square census.

    0
    0
  • 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.

    0
    0
  • Arabian authors already had found three square numbers of equal difference, but the difference itself had not been assigned in proposing the question.

    0
    0
  • Fibonacci's series is a sequence of numbers such that any term is the sum of the two preceding terms; also known as Lame series.

    0
    0
  • They kept horses (though in small numbers), sheep and goats, but no traces of their rearing horned cattle have yet been found.

    0
    0
  • Swine are reared in large numbers.

    0
    0
  • The rapid settling of the state drove its native fauna, which comprised buffalo, deer, moose, bear, lynx and wolves, in great numbers into the northern sections, westward into Dakota, or across the Canadian border.

    0
    0
  • The extraordinary numbers of utilizable water-powers, the unusual transport facilities affording ample means of reaching the great markets, and finally the proximity to the raw materials of manufacture, have made Minnesota of great importance as a manufacturing state.

    0
    0
  • It became clear that only very rough estimates of the numbers of planktonic organisms in a volume of sea-water as large as (say) 10 cubic metres could be made, but that these estimates could nevertheless be trusted to show very marked regional and seasonal differences.

    0
    0
  • Palafox (20,000) was near Saragossa and observing Sanguessa; Castanos with the victors of Baylen" In this account of the war the losses and numbers engaged in different battles are given approximately only; and the former include killed, wounded and missing.

    0
    0
  • Massena, superior in numbers and over-confident, made a direct attack upon the heights on the 27th of September 1810: his strength being about 60,000, while that of the Allies was about 50,000, of whom nearly half were Portuguese.

    0
    0
  • Many solutions in which the transport numbers vary at high concentration often become simple at greater dilution.

    0
    0
  • In such salts as potassium chloride the ions seem to be simple throughout" a wide range of concentration since the transport numbers for the same series of concentrations as those used above run Potassium chloride 0.5 1 5, 0.515, 0.514, 0.513, 0.509, 0.508, 0.507, 0.507, 0.506.

    0
    0
  • The results go to show that, where the existence of complex ions is not indicated by varying transport numbers, the observed velocities agree with those calculated on Kohlrausch's theory.

    0
    0
  • The freezing points and vapour pressures of solutions of sugar are also in conformity with the theoretical numbers.

    0
    0
  • Allowing for incomplete ionization the general concordance of these numbers with the theoretical ones is very striking.

    0
    0
  • These numbers agree with those indicated by theory, viz.

    0
    0
  • The ratios of the coagulative powers can thus be calculated to be i: x: x 2, and putting x =32 we get I: 32: 1024, a satisfactory agreement with the numbers observed.4 The question of the application of the dissociation theory to the case of fused salts remains.

    0
    0
  • It must be remembered that, the so utions not being of quite the same strength, these numbers are not strictly comparable, and that the experimental difficulties involved in the chemical measurements are considerable.

    0
    0
  • Nevertheless, the remarkable general agreement of the numbers in the four columns is quite enough to show the intimate connexion between chemical activity and electrical conductivity.

    0
    0
  • From these numbers we can, by help of the equation, calculate the conductivity of the acids for any dilution.

    0
    0
  • By experiments on diffusion this constant has been found by Scheffer, and the numbers observed agree with those calculated (HC1= 2.30, HNO 3 = 2 22, NaCI = I II).

    0
    0
  • A partial exception may perhaps be made in the case of cocoa, when the two plants are placed not too closely in about equal numbers.

    0
    0
  • Vast numbers of small lakes stud the Vitim and upper Selenga plateaus; the lower valley of the latter river contains the Goose Lake(Gusinoye).

    0
    0
  • According to Nordenskjold r it numbers only twenty-nine species of mammals, of which seven are marine and seventeen or eighteen may be safely considered as living beyond the forest limit.

    0
    0
  • The Turkish stock of the Yakuts in the basin of the Lena numbers 227,400.

    0
    0
  • Arithmetical groups, connected with the theory of quadratic forms and other branches of the theory of numbers, which are termed "discontinuous," and infinite groups connected with differential forms and equations, came into existence, and also particular linear and higher transformations connected with analysis and geometry.

    0
    0
  • anv, where the first suffixes are the natural numbers I, 2, 3, ...n taken in order, and a, 0, y, ...

    0
    0
  • v is some permutation of these n numbers.

    0
    0
  • This permutation by a transposition of two numbers, say a, 13, becomes 0, a, 7, ...

    0
    0
  • Y ...a n v, the summation being for all permutations of the n numbers, is called the determinant of the n 2 quantities.

    0
    0
  • permutations of the n numbers I, 2, 3, ...

    0
    0
  • In the latter case the sign is determined by -I raised to the same power as before, with the exception that Tux., replaces Tusk; but if one of these numbers be even the other must be uneven; hence A ik = - Ais� rk Moreover aik a,, aikarsAik +aisarkAis Aik, rk aik ars rs where the determinant factor is giyen by the four points in which the deleting lines intersect.

    0
    0
  • p 1 p 2 ...p n), it is necessary to bring under view other functions associated with the same series of numbers: such, for example, as P P3 P2 P4 Pn -2 /, /, /, F i a i 1 a 2 Fi a 1 a 2 ...

    0
    0
  • aPn 1 2 3 ' the summation being for all permutations of the indices which result in different terms. The function is written (plp2p3�4n) for brevity, and repetitions of numbers in the bracket are indicated by exponents, so that (p1p1p2) is written (p1p 2).

    0
    0
  • The weight of the function is the sum of the numbers in the bracket, and the degree the highest of those numbers.

    0
    0
  • A table may be formed expressing the k expressions Pa l), P(2),...P(1) as linear functions of the k expressions (m"`'sm�2sm�3s...), s =1, 2, ...k, and the numbers BSc occurring therein is 2s 3s possess row and column symmetry.

    0
    0
  • It is now to be remarked that the partition (/,A.1/2)1/42/A38...)can be derived from (m"13m�2sm"`38...) 1 2 3 is 2s 3s by substituting for the numbers mi., m 231 m 331 ...

    0
    0
  • certain partitions of those numbers (vide the definition of the specification of a separation).

    0
    0
  • The weight of the function is bipartite and consists of the two numbers Ep and Eq; the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number Ep, Eq.

    0
    0
  • It will be apparent that there are four numbers associated with a covariant, viz.

    0
    0
  • Dolmens, however, occur in great numbers in Tunisia and the province of Constantine.

    0
    0
  • The sum of the corresponding numbers multiplied by 332600 gives Log.

    0
    0
  • The most likely distance of the sun may be stated in round numbers as 9 3,000,000 miles.

    0
    0
  • VII, Usually considered to be the tergum of the genital somite, but suggested by Pocock to be that of the other [According to the system of numbering explained in the text, if VII is the tergum of the praegenital somite (as is probable) it should be labelled Prg without any number, and the somites VIII to XIII should be lettered 1 to 6, indicating that they are the six normal somites of the mesosoma; whilst XV to XVIII should be replaced by the numbers 7 to 12 - an additional suppressed segment (making up the typical six) being reckoned to the metasomatic fusion.] (From Lankester, Q.

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    0
  • Authorities cited by numbers in the text.-1.

    0
    0
  • The settlement in Flying Fish Cove now numbers some 250 inhabitants, consisting of Europeans, Sikhs, Malays and Chinese, by whom roads have been cut and patches of cleared ground cultivated.

    0
    0
  • The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Leonhard Euler's works and Adrien Marie Legendre's Theory of Numbers), which Riemann read, mastered and returned within a few days.

    0
    0
  • He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial differential equations, and Jacobi's on analytical mechanics and higher algebra.

    0
    0
  • In the theory of numbers he furnished solutions of many of P. Fermat's theorems, and added some of his own.

    0
    0
  • Here, on the 31st of July 1849, the Hungarian army under Bern was defeated by the overwhelming numbers of the Russian General Liiders.

    0
    0
  • They include oysters, crabs of great size, and a small mussel, found in enormous numbers.

    0
    0
  • Of the governments which were parties in these several cases Great Britain heads the list in point of numbers, the United States of America being a good second.

    0
    0
  • The former as a consequence won large numbers of supporters who were drawn by the possibility it afforded of adopting an attractive faith which did not involve a rupture with the religion of Roman society, and consequently with the state.

    0
    0
  • All the rivers are richly stocked, and valuable fishing grounds are to be found along the coast, especially that of southern Bahia and Espirito Santo where the garoupa (Serranus) is found in large numbers.

    0
    0
  • The sea-coast, bays and tide-water rivers are still fringed with mangrove, and on the sandy shores above Cape Frio grow large numbers of the exotic cocoa-nut palm.

    0
    0
  • Along the coast the best known fisheries are among the Abrolhos islands and in the shallow waters of Espirito Santo, where the garoupa, pargo and vermelho (species of Serranus) abound in great numbers.

    0
    0
  • Numbers of English artisans and shipbuilders, Swedish ironfounders, German engineers and French manufacturers sought fortunes in the new country, and diffused industry by their example.

    0
    0
  • Then, having withdrawn to its own quarter, it was suddenly attacked by the infuriated citizens (noveschi and dodicini), who broke into houses and workshops and put numbers of the inhabitants to the sword without regard for age or sex.

    0
    0
  • Meanwhile the monte of the nine, the chief promoters of the revolution of 1480, were exposed to the growing hatred and envy of their former allies, the monte del popolo, who, conscious of their superior strength and numbers, now sought to crush the noveschi and rise to power in their stead.

    0
    0
  • The bibliography of Lebeuf's writings is, partly, in various numbers of the Bibliotheque des ecrivains de Bourgogne (1716-1741).

    0
    0
  • In addition to the industries connected with the shipping, large numbers of hands are employed in the government railway works, where the locomotives and rolling stock used by the state railways are manufactured.

    0
    0
  • Insects abound in great numbers, the most troublesome and destructive being the tick (Ixodes natalensis), which infests the pasturage, and the white ant (Termes mordax).

    0
    0
  • The new administration found it hard to please the Dutch farmers, who among other grievances resented what they considered the undue favour shown to the Kaffirs, whose numbers had been greatly augmented by the flight of refugees from Panda.

    0
    0
  • The sugar industry soon became of importance, and the planters were compelled to seek for large numbers of labourers.

    0
    0
  • This ideal lot, from the native point of view, drew such numbers of immigrants from disturbed districts that with the natural increase of population in thirty years the native inhabitants increased from about ioo,000 to fully 350,000.

    0
    0
  • But the improved quality of the live stock is more worthy of notice than the growth in numbers.

    0
    0
  • Large numbers of horses are exported annually, principally to Austria, Germany, Italy, France and Rumania.

    0
    0
  • They are exported in large numbers (408,000 in 1905), almost exclusively to Austria.

    0
    0
  • Thus the Kumanian colonists, mostly pagans, whom he settled in vast numbers on the waste lands, threatened to overwhelm the Christian population; while the numerous strongholds, which he encouraged his nobles to build as a protection against future Tatar invasions, subsequently became so many centres of disloyalty.

    0
    0
  • It was a mere rump, for wholesale executions had thinned its numbers and the reconquered countries were not represented in it.

    0
    0
  • But a nation that for a thousand years had maintained its individuality in the midst of hostile and rival races could not be expected to allow itself without a struggle to be sacrificed to the force of mere numbers, and the less so if it were justified in its claim that it stood for a higher ideal of culture and civilization.

    0
    0
  • The development of symbolic algebra by the use of general symbols to denote numbers is due to Franciscus Vieta (Francois Viete, 1540-1603).

    0
    0
  • The harmony between arithmetical and geometrical measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance.

    0
    0
  • The treatment of equations of the second and higher degrees introduces imaginary and complex numbers, the theory of which is a special subject.

    0
    0
  • Hence we can separate the numbers from the common unit, and replace 3 lb - I - 5 lb - 7 lb +2 lb by (3+5-7+2) lb, the additions and subtractions being then performed by means of an addition-table.

    0
    0
  • (v.) The use of the solidus / separating two numbers is for convenience of printing fractions or fractional numbers.

    0
    0
  • We must count with something; and the successive somethings obtained by the addition of successive units are in fact numerical quantities, not numbers.

    0
    0
  • When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical.

    0
    0
  • The extension of the methods to fractional numbers is part of the establishment of the laws governing these numbers (� 27 (ii.)).

    0
    0
  • Expressed Equations.-The simplest forms of arithmetical equation arise out of abbreviated solutions of particular problems. In accordance with � 15, it is desirable that our statements should be statements of equality of quantities rather than of numbers; and it is convenient in the early stages to have a distinctive notation, e.g.

    0
    0
  • combine the separate quantities or numbers expressed in terms of x as unit into a single quantity or number so expressed, obtaining sx =10.

    0
    0
  • Use of Letters in General Reasoning.-It may be assumed that the use of letters to denote quantities or numbers will first arise in dealing with equations, so that the letter used will in each case represent a definite quantity or number; such general statements as those of �� 15 and 16 being deferred to a later stage.

    0
    0
  • (iii.) The general statement of the laws of operation of fractions is perhaps best deferred until we come to fractional numbers, when letters can be used to express the laws of multiplication and division of such numbers.

    0
    0
  • (a) The commutative law and the associative law are closely related, and it is best to establish each law for the case of two numbers before proceeding to the general case.

    0
    0
  • Thus any pair of adjoining numbers can be interchanged, so that the numbers can be arranged in any order.

    0
    0
  • (All the numbers are, of course, pb +qd supposed to be positive.) 27.

    0
    0
  • Negative Quantities and Fractional Numbers.

    0
    0
  • We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.

    0
    0
  • (v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (�� 41, 44)� (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

    0
    0
  • such that 7 X 1 s = is., and can thence pass to the idea of fractional numbers.

    0
    0
  • When, however, we come to the equation x 2 --- 5, where we are dealing with numbers, not with quantities, we have no concrete facts to assist us.

    0
    0
  • When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X.

    0
    0
  • While mensuration is concerned with the representation of geometrical magnitudes by numbers, graphics is concerned with the representation of numerical quantities by geometrical figures, and particularly by lengths.

    0
    0
  • Graphic methods also enter into the consideration of irrational numbers (� 65).

    0
    0
  • Elementary Algebra of Positive Numbers.

    0
    0
  • (ii.) By means of the commutative law we can collect like terms of a monomial, numbers being regarded as like terms. Thus the above expression is equal to 6a 5 bc 2, which is, of course, equal to other expressions, such as 6ba 5 c 2.

    0
    0
  • 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.

    0
    0
  • The principles of arithmetical transformation follow from those stated in �� 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities.

    0
    0
  • Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit.

    0
    0
  • A°a°, and the successive numbers in the (n+i)th line are the coefficients of A n a O An-lal ..

    0
    0
  • we obtain I, 4, 9, � � � This suggests that, if u n is the sum of the first n odd numbers, then u n = n 2 .

    0
    0
  • - The numbers denoted by n (r) in � 41 are the binomial coefficients shown in the table in � 40; n (r) being the (r+ i) th number in the (n+ i) th row.

    0
    0
  • They have arisen as the coefficients in the expansion of (A+a)"; but they may be considered independently as a system of numbers defined by (I) of � 41.

    0
    0
  • The individual numbers are connected by various relations, some of which are considered in this section.

    0
    0
  • In this table any number is equal to the sum of the numbers which lie horizontally above it in the preceding Column, And The Difference Of Any Two Numbers In A Column Is Equal To The Sum Of The Numbers Horizontally Between Them In The Preceding Column.

    0
    0
  • Constructed in this way, the successive diagonal rows, commencing with the first, give the figurate numbers of the first, second, third,.

    0
    0
  • (ii.) In the same way, if we multiply the table by c, the sum of the first n numbers in any column is equal to the nth number in the next following column.

    0
    0
  • do not contain n; and thence we can obtain the sum of the numbers found by putting n= 1, 2, 3,.

    0
    0
  • These numbers constitute an arithmetical progression of the rth order.

    0
    0
  • are certain numbers known as Bernoulli's numbers, and the terms within the bracket, after the first, have signs alternately + and -.

    0
    0
  • The values of the first ten of Bernoulli's numbers are B1= t, B2= 1, B3 =412, B4 =30, B5 =6 = 6 9 1 B7 = l, B =3 =4 4 fl, IV.

    0
    0
  • Negative Numbers and Formal Algebra.

    0
    0
  • The occurrence of negative quantities does not, however, involve the conception of negative numbers.

    0
    0
  • The graphic method, therefore, does not give any direct assistance towards the conception of negative numbers as operators, though it is useful for interpreting negative quantities as results.

    0
    0
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