AN EXCURSION IN FACTORING.

By M. A. BAILEY

About 275 B.C., Eratosthenes taught the sieve method of separating primes from composites. This method being fundamental to factoring, should introduce that subject wherever it is taught.

IU. Separate the primes from the composites through 30.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1/6 17 16 19 20 21 22 23 24 25 26 27 28 29 30

By crossing every second number after 2 we mark all the higher multiples of 2, by crossing every third number after 3 we mark all the higher multiples of 3, by crossing every fifth number after 5 we mark all the higher multiples of 5, by crossing every seventh number after 7 we mark all the higher multiples of 7, and so on. In the illustration it is unnecessary to continue the process after 7 because, as 7x7 is 49, a number greater than 30, every one of these numbers, when divided by a number greater than 7, must give a quotient less than 7, and all numbers less than 7 have been tried as factors. We have thus sifted the primes from the composites.

The sieve method furnishes the following rule for determining whether a number is prime. Test the divisibility of the number by the successive primes beginning with 2 until that prime has been tried whose square is greater than the number.

Ill. Is 101 a prime? Yes, it is not a multiple of 2, nor of 3, nor of 5, nor of 7, nor of 11.

Outside of the schoolroom there is little call for finding greatest common divisor, and, except in the addition and subtraction of fractions, the same is true of finding least common multiple. Outside of written tests, the least common denominator is usually less than 72 and should be found by inspection. To find the least common denominator of two fractions, compare the smaller denominator with the successive multiples of the larger until a multiple of the smaller is found. To find the least common denominator of more than two fractions, find the least common denominator of two of them, then the least common denominator of the result and a third, and so on.

Ill. Find the least common denominator of %, %, 142. 8 is not a factor of 12, but of 24; 9 is not a factor of 24, nor of 48, but of 72. 72 is the least common denominator.

We should give thanks that the following methods of finding greatest common divisor and least common multiple are no longer à la mode.

G.C.D. = 2x2 x3 = 12 L.C.M. = 2 × 2 × 3 × 2 × 3 = 72 The child who tries to use them is like the old woman with the infallible rule for testing eggs. "If the egg sinks in water it is either good or bad," said she, "and I declare to gracious I have forgotten which." In one case it is necessary to multiply the divisors alone, and in the other to multiply both divisors and quotients, but to remember which is which is beyond the ability of the average child. Consequently, it is found that about 40 out of a hundred will declare that 12 is the least common multiple of 24, 36 and 48; about 40 will declare that 72 is the greatest common divisor of 8, 9 and 12, and about 20 will give the right answer in both cases.

It is absurd to teach the Euclidean method of finding the greatest common divisor (divide the greater by the smaller, etc.) below the high school, and even there except in connection with algebra. The method is never understood by youngsters, nor would it be of value if it were, except possibly to win the approbation of teacher or examiner.

When the greatest common divisor and the least common multiple cannot be readily found by inspection the factoring method alone appeals to the intelligence. A skilful arrangement of the prime factors is of service. Arrange the prime factors of each number in such a way that each factor not found in a preceding number shall have a column to itself. To find the G. C. D. take a factor from each column which is complete; to find the L. C. M. take a factor from each column whether it is complete or incomplete.

Ill. Find the G.C.D. and L.C.M. of 198, 252, 324. 1982 x 3 x 3 x 11

X 2 X 7 X 2

X 3X 3

252 2 x 3 x 3 324 = 2 × 3 × 3 G.C.D. 2 × 3 × 3 = 18 L.C.M.2 × 3 × 3 × 11 × 2 × 7 × 3 × 3 = 24948

The arrangement of the prime factors is easily made. 1982 × 3 × 3 × 11. 252 contains the 2 of 198; write 2 in the first column and the quotient, 126, as scratch work; 126 contains the first 3 of 198, write 3 in the 2nd column and the quotient, 42, as scratch work; 42 contains the 2nd 3 of 198, write 3 in the 3rd column and the quotient, 14; 14 does not contain the 11 of 198, write its prime factors, 2 and 7, in separate columns; 324 contains the 2 of the 1st column, quotient, 162; 162 contains the 3 of the 3rd column, quotient, 54; 54 contains the 3 of the 3rd column, quotient, 18; 18 does not contain 11 of the 4th column, but 2 of the 5th, quotient, 9; 9 does not contain 7 of the 6th column; write its prime factors, 3 and 3, in separate columns.

The above arrangement aids also in finding the quotient of the L. C. M. by each number, as is necessary in the addition of fractions. Take a factor from each column not represented by the number.

conception. The shortest and most intelligible process is to find the prime factors of either numerator or denominator and to test the other term by these factors in succession.

Ill. Reduce 231523 to its lowest terms.

231 = 3 x 7 x 11; 523 is not a multiple of 3, nor of 7, nor of 11. The fraction is already in its lowest terms. Note. The above article is copyrighted by M. A. Bailey. Publications are enjoined against the use of the above arrangement of prime factors.

THANKSGIVING DAY MATERIAL

The Origin of Thanksgiving Day Jewish. Three thousand years ago the vintage or harvest celebration, called the Feast of the Tabernacles, was instituted among the Hebrews. For eight days the people ceased their work, to "eat, drink and be merry" according to the command "When ye have gathered in the fruit of the land, ye shall keep a feast unto the Lord, . . . and ye shall rejoice before the Lord your God seven days."

Greek and Roman. The harvest festival of ancient Greece, the greatest feast of all the year, was held in honor of Demeter, the goddess of the harvest. The Romans also worshipped this harvest deity under the name of Ceres. They went in processions to the fields, where they engaged in rustic sports, and crowned all their household gods with flowers.

English and Dutch. In England the autumnal festival was called the harvest home and was celebrated with many quaint customs. An old song with many variations still survives at the carrying home of the last load:

Harvest home! harvest home!

We've ploughed, we've sowed,
We've reaped, we've mowed,
We've brought home every load.

Hip, hip, hip, harvest home.

The Dutch celebrated their deliverance from the Spaniards on October third. This was their most popular festival except the Kirmess, and was kept as a religious and social holiday, though it soon degenerated into a day of merriment only. The chief dish at dinner was a Spanish hodge-podge, a stew of meat and vegetables.

The Pilgrims. The first authentic harvest festival was held by the Pilgrims in 1621. Shortly after their landing they had one day come upon some deserted Indian huts, in which they had found some baskets filled with corn. From this supply they planted their first year's crop, and throughout the spring and summer watched its growth most anxiously, realizing that upon this first harvest depended the prosperity of the little colony. To their great joy the harvest was bountiful; never in Holland nor in old England had they seen the like. For this they devoutly thanked God and made preparations for a feast. Hunters were sent out to procure the Thanksgiving dinner, and returned

The First English Thanksgiving in New York June 7, 1675, Governor Andros ordered: "That Wednesday ye 23d of this Instant month be appointed throughout ye government a day of Thanksgiving and Prayers to Almighty God for all His Past Deliverances and Blessings and Present Mercies to us, and to Pray ye continuance and Encrease thereof."

The First National Thanksgiving—1777

The immediate occasion was the surrender of General Burgoyne to General Gates. Thursday, the 18th of December, was designated by the Continental Congress and was duly observed by the army at Valley Forge.

In 1789 Washington as president of the United States issued his first proclamation for the observance of a day of Thanksgiving, setting apart Thursday, the 26th day of November. In 1795 he again appointed Thursday, November 26th, as a day to be observed for a general thanksgiving by the people of the United States.

Washington's proclamation was occasionally followed by similar papers from other presidents.

About 1830 the governor of New York appointed a day and other northern governors quickly followed suit.

In the South Thanksgiving Day was practically unknown until 1855, most of the citizens regarding it as a relic of Puritanic bigotry.

The celebration of the day was thus, for several years, though regularly observed, merely a state affair. In 1864 President Lincoln issued a proclamation, appointing the fourth Thursday of November with a view of having the day kept, thereafter, annually without interruption. The president's assassination, the next year, almost caused the suspension of his own rule, but after some persuasion President Johnson appointed the last Thursday in November; and since that time each president has followed his example. example. Thus we may say that since 1865, Thanksgiving Day has been an annual national holiday. It is the only religious festival celebrated in the United States by virtue of the authority of the civil government.

Give Thanks

For all that God, in mercy, sends;
For health and children, home and friends;
For comfort in the time of need,
For every kindly word and deed,
For happy thoughts and holy talk,
For guidance in our daily walk-

For everything give thanks!
For beauty in this world of ours,
For verdant grass and lovely flowers,
For song of birds, for hum of bees,
For the refreshing summer breeze,
For hill and plain, for stream and wood,,
For the great ocean's mighty flood—

REALIZING HISTORY

That Things Are no Worse, Sire From the time of our old Revolution, When we threw off the yoke of the King, Has descended this phrase to rememberTo remember, to say, and to sing; 'Tis a phrase that is full of a lesson;

It can comfort and warm like a fire; It can cheer us when days are the darkest: "That things are no worse, O my sire!"

"Twas King George's prime minister said it,
To the King, who had questioned, in heat,
What he meant by appointing Thanksgiving
In such days of ill-luck and defeat.
"What's the cause of your day of Thanksgiving?
Tell me, pray," cried the King in his ire.
Said the minister, "This is the reason-
That things are no worse, O my sire!"

-Helen Hunt Jackson.

A Thanksgiving Fable

It was a hungry pussy cat, upon Thanksgiving morn, And she watched a thankful little mouse that ate an ear of corn.

"If I ate that thankful little mouse, how thankful he should be,

When he has made a meal himself, to make a meal for me!

Then with his thanks for having fed, and his thanks for feeding me,

With all his thankfulness inside, how thankful I shall be!"

Thus mused the hungry pussy cat, upon Thanksgiving Day;

But the little mouse had overheard and declined (with thanks) to stay.

Thanksgiving

I thank Thee that I learn

-Oliver Herford.

(Continued from page 8)

"My dear," she said, "if I realize any more history to-day, I shall explode in a burst of sheer enthusiasm."

"Come away, quick, to something quieter," laughed Miss Dennett. So they visited Sleepy Hollow, where Emerson, Hawthorne, Thoreau and the Alcotts lie sleeping in the spot they loved so well. Later they enjoyed a quiet little dinner at the Thoreau House before they returned to Boston.

In the four days that remained, the wave of Miss Black's enthusiasm mounted higher and higher. They visited Bunker Hill, and cheerfully climbed the stairs in the tall monument which has no elevator service. In Cambridge, they stood beneath the elm where Washington took command of the American army, and they gazed with rapturous eyes at the homes of Longfellow and of Lowell in the same city. Faneuil Hall, the Old State House, with its wealth of relics, the new State House, the site of the Boston Tea Party, the Old South Meeting-house, the Hancock Mansion, King's Chapel and the ancient burying-grounds received the homage of these untiring pilgrims. Lastly, as a concession to modern days, they visited the Public Library and the Art Museum-"realizing by contrast," Miss Black called this.

At the end of the week, each girl's accountbook footed up as follows:

Fare to Boston and return..
Room rent

Meals

Trolley and boat fares. Carriage in Salem...

Admissions

$12.25

2.50

8.00

4.60

.50

1.25

$29.10

Not toil to spurn;

With all beneath the sun

It makes me one;

For tears, whereby I gain

Kinship with pain;

For Love, my comrade by the dusty ways,
I give thee praise-

-Emily Read Jones.

Teachers who go into the Alaskan service have some novel experiences. One interesting report describes the eruption of Matmai volcano, in western Alaska, which destroyed a native village and buried the country for a hundred miles around in volcanic ash. Three feet of pumice covered the ground where the village of Katmai formerly stood, and the natives had to flee for their lives. They were eventually rescued by the revenue cutter Manning and taken to a new site on Ivanoff bay, where the government has set them up in housekeeping. A school will probably be established in the new village.

There is no more difficulty about the pupils in Miss Black's grade "passing" in United States history. The wrinkles have smoothed out of her brow. That class is a model of interest, attention and results, since its teacher "realized history."

A traveling school of domestic science has been instituted in the department of Yonne, France. The school will make a stay of three months in any commune where an attendance of fifteen is guaranteed. Similar itinerant schools for domestic science exist in Ireland.

In the five states made from the northwest territory, the percentage of illiteracy is about three and a third. This is by the census of 1910. There was a decrease of one per cent since 1900. The illiteracy of the neighboring state of Iowa is but one and seven-tenths per cent. The reason for this difference is not clear at this distance.

THE OCTOBER MAGAZINES

Sarcasm

Arnold Bennett, in his series of impressions of "Your United States" in Harper's, reaches the matter of education.

One noon I was walking down the main road of a residential suburb when a number of young men and maids came out of a high school and unconsciously assumed possession of the street. It was a delightful sight. They were so sure of themselves, the maids particularly; so interested in themselves, so happy, so eager, so convinced that their importance transcended all other importances, so gently pitiful toward men and women of forty-five, and so positive that the main function of elders was to pay school fees, that I was thrilled thereby. I immediately desired to visit schools. Profoundly ignorant of educational methods, and with a strong distaste for teaching, I yet wanted to know and understand all about education in America in one moment. I failed, of course, in my desire partly from lack of knowledge to estimate critically what I saw, and from lack of time.

The Research Fever

The Great Discovery About Buxton, Century, is a take-off on the mad rush among some of our college faculties to engage in research. The professor of English is a born teacher, whose strong point is awakening enthusiasm for good literature among his students, and who has no bent towards research, which he says means in literature the discovery of facts. either disagreeable or unimportant. Goaded

on by his wife, however, he undertakes some investigations with unexpected results.

Robin Redbreast

The Robin and Its Treatment in the South, Craftsman, gives an interesting account of the campaign in Virginia, carried on largely through school children, which has resulted in the passage of a law for the protection of the robin.

The petition, which asked for the removal of the robin from the list of game birds, bore the name of practically every principal and student in the Virginia schools. The children have been taught to care for the birds as most important insect-destroyers and useful to the farmer in saving the trees and crops. The æsthetic side emphasized to the children is that birds, particularly robins, are beautiful and give joy through their color and songs, and have been the theme of poets and writers since the days of Aristotle, and that every good citizen should protect the birds.

In the same magazine is a story which reads. after the manner of a fairy tale of How a Schoolhouse Was Built by the Community.

The school had burned down in June and the principal appointed for the following year had asked the city not to rebuild it, but to allow him and the parents and pupils who would use it to build it. At first the board was amused, then interested, for this young man's argument that all should earn what was theirs had been convincing; he said that children did not appreciate the education now doled out to them; that while the state owed an education to its wards, the children in return owed the state a conscious citizenship. In the end the new principal-aged twenty-five -had won his point and the new Greenleaf School was to be built by its users, the community.

This was the first day on the job. Gangs of boys of all ages were busy with wheelbarrows and wagons cleaning up the débris. Three junkmen at the curb were buying whatever was offered for sale. Boys were driving teams which, I was told, were loaned by men who did contracting work and lived in the neighborhood. There was no talking; there was no confusion, only the unavoidable noise and jangle of scraping shovels and grinding wheels. By the end of the day the place was clear and ready for excavation. It was curious to see the idlers of the neighborhood-white and black-stand around a minute, then suddenly pitch in and get busy.

The next day the excavating was begun and the principal soon had the layout staked, and plows, shovels, picks, teams, barrows were busy. It was interesting to note the care that was taken in distributing the dirt so that it would not have to be handled more than once. Again there were no leaders, no noise of tongues, no singing, but there was much humming and quiet whistling. Day by day I saw the building grow, watched the boys get the brick and lime and cement from the car, and pull it around with a big oilburning tractor. It was amusing to see the policeman, over by the tracks, anxiously waiting for the end of his shift so he could shed his uniform and help unload the car. And the way that inanimate stuff was handled! Do you suppose bricks were smashed and cement spilled?-not a bit of it. You would have thought it vital that not a corner should be chipped, or even a paper bag torn. And when this habit of protection is instilled into the youth of the land, one of the many fibers of the roots of conscious citizenship has started to grow.

How to Tell College Timber

Who Should Go to College and Why, Review of Reviews, presents an argument for more flexible entrance requirements-"The way is made too easy for those who are drawn up, cog by cog, through twelve years of school attendance. It is too difficult for those who, though abundantly able and eager, have the misfortune to be obliged to walk part way instead of holding the seats in the educational car."

It is fallacious to argue, as college faculties are

prone to do, that the preparatory studies are the foundations of the college education and that any deficiencies therein will make the superstructure insecure. Intelligence is the foundation, and likewise the edifice. All studies, higher as well as lower, are but scaffolding, and if in attempting work on a third story some additional supports are needed below, it may not prove fatal or even dangerous to set them as required. The amount of such repair work that students can safely be entrusted to do will vary with their pewer and earnestness as intellectual builders.

School Girls of Fifty Years Ago

The fashionable girls' school of fifty years ago was a most interesting place apparently. "Looking backward to those busy, shining hours," writes Julia C. R. Dorr, in the October number of Harper's Bazar, "my first thought is, how we all studied! How eager we were! What keen delight we took in construing an intricate sentence or in solving a hard problem!"

There were about fifty scholars, or possibly seventyfive; and among them was a group of eight or ten bright young fellows who were fitting for college; preparing to enter as sophomores the coming autumn. What an ambitious lot we were, to be sure! I was the only girl in the "advanced Latin," and had the honor of a seat on one end of a long, narrow recitation bench, a little withdrawn, as was proper, from those stars of the first magnitude.

How many teachers did we have? Just one. I doubt very much if he was a marvel of learning, though I thought he was then. He was just out of college himself, and he had had no wide experience of books or men. But he had the rare gift of being able to stimulate and inspire his scholars, and kindle every latent spark of enthusiasm in their natures. Enthusiasm is a better word than ambition in this connection. Study was joyful labor, done for the pure love of it. It was its own end; not simply a means to some other end. The village academy of that day taught concentration if it taught nothing else. Study and recitation went on in the same room and at the same time.

Bad Air is Still, Warm Air

Leonard Hill, in Popular Science Monthly, presents the new theory of ventilation:

Experimental evidence is strongly in favor of my argument that the chemical purity of the air is of no importance. Analyses show that the oxygen in the worst-ventilated schoolroom, chapel or theater is never lessened by more than 1 per cent of an atmosphere; the ventilation through chink and cranny, chimney, door and window, and the porous brick wall, suffices to prevent a greater diminution. So long as there is present a partial pressure of oxygen sufficient to change the hemoglobin of the venous blood into oxyhemoglobin there can arise no lack of oxygen.

I conclude, then, that all the efforts of the heating and ventilating engineer should be directed toward cooling the air in crowded places and cooling the bodies of the people by setting the air in motion by means of fans. In a crowded room the air confined between the bodies and clothes of the people is almost warmed

up to body temperature and saturated with moisture so that cooling of the body by radiation, convection and evaporation becomes reduced to a minimum. The strain on the heat-regulating mechanism tells on the heart. The pulse is accelerated, the blood is sent in increased volume to the skin, and circulates there in far greater volume, while less goes through the viscera and brain. As the surface temperature rises the cutaneous vessels dilate, the veins become filled, the arteries may become small in volume, and the blood-pressure low, the heart is fatigued by the extra work thrown upon it. The influence of the heat stagnation is shown by the great acceleration of the pulse when work is done and the slower rate at which the pulse returns to its former rate on resting.

A Moderate Estimate

In the same magazine W. B. Pillsbury makes a fair summing up of Rousseau :

Enough of the psychology of Rousseau; our real question is, what did Rousseau contribute to psychology? This is somewhat difficult to answer. His specific contributions are practically nil. The psychology that he uses in his writings is varied. Passages in the Emile are evidently taken almost verbatim from Condillac, other passages he evidently owes to Descartes, while still others show the influence of Locke. In no place does he develop any important views of his own or even harmonize those that he borrowed. He had no followers in psychology. One can point to no one who was distinctively a psychologist that owes much to Rousseau. His strongest influence has been very recent and very indirect. Through his educational teachings that instruction should be based upon a knowledge of the child, he has perhaps had some small part in stimulating the studies of childhood that have been made in the past few decades.

Rousseau's greatest contribution to psychology is probably the raw material that he provided in his Confessions.

NEWS ITEMS

The largest public school building in Europe is the new continuation school in Vienna.

Last year Texas put up an average of two schoolhouses a day, the total bill being $3,000,000.

The educational expenditure for Scotland for the past fiscal year was $18,300,000, of which $840,000 was for continuation schools.

"More than one-half of our school population is trained in the rural schools. These schools are inadequate."-Superintendent E. T. Fairchild.

It costs an average of eighteen cents a school day per child to put the rising generation of this country through the elementary and high schools.

An investigation of the Prussian schools shows that an average of sixteen per cent of the pupils evade the prescribed physical exercises by physicians' certificates or other means.