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But I do lay claim to whatever merit should be accorded to me for persevering diligence in my profession. And I make the claim, not with a view to my own glory, but for the benefit of those who may read these pages, and, when young, may intend to follow the same career. Nulla dies sine line a. Let that be their motto. And let their work be to them as is his common work to the common laborer.—Anthony Troulope: Autobiography.

Nullification, Doctrine of. In the constitutional history of the United States this doctrine was that held by the ultra strict-constructionists (see Loose-constructionist). According to them, the Federal Union was a mere league of States, to which certain limited governmental powers had been delegated, ultimate sovereignty and all powers not expressly delegated remaining with the separate States; so that these latter might repudiate, each for itself, any general act of Congress which in its judgment exceeded the limits of the delegated powers strictly construed in favor of the States. An attempt was made in 1832 by the Legislature of South Carolina to "nullify" the United States tariff, held to be oppressive to the State and unconstitutional in that it went beyond the powers given to Congress to raise revenue by a tariff on imports, and embodied protective features in the interests of the manufacturing States and against those of the purely agricultural communities. Andrew Jackson's energetic measures, however, soon caused the repeal of the act of the South Carolina Legislature. He pronounced the act treasonable, and sent General Scott to Charleston to maintain the authority of the Federal government and aid the officials in enforcing the provisions of the act of Congress.

Numbers, Curiosities of. If it be true that figures won't lie, that they won't even equivocate, that two and two exhibit an unbending determination to make four and nothing but four, at least figures do often play strange pranks. They abound in paradoxes, and though a paradox is rightly defined as a truth that only appears to be a lie, yet the stern moralist, who hates even the appearance of evil, looks with scant favor upon a paradox. Luckily, we are not all so stern in our morality. Most of us welcome a little ingenious trifling, an amiable coquetting with the truth; we are willing that Mr. Gradgrind shall have the monopoly of hard facts; we like to find romance even in our arithmetic. And we don't have far to look.

There is the number nine. It is a most romantic number, and a most persistent, self-willed, and obstinate one. You cannot multiply it away or get rid of it anyhow. Whatever you do, it is sure to turn up again, as did the body of Eugene Aram's victim.

Mr. W Green, who died in 1794, is said to have first called attention to the fact that all through the multiplication table the product of nine comes to nine. Multiply by any figure you like, and the sum of the resultant digits will invariably add up as nine. Thus, twice 9 is 18; add the digits together, and 1 and 8 make 9. Three times 9 is 27; and 2 and 7 is 9. So it goes on up to 11 times 9, which gives 99. Very good. Add the digits together, 9 and 9 is 18, and 8 and 1 is 9. Go on to any extent, and you will find it impossible to get away from the figure 9. Take an example at random. Nine times 339 is 3051; add the digits together, and they make 9. Or again, 9 times 2127 is II34; add the digits together, they make 18, and 8 and I is 9. Or still again, 9 times 5071 is 45,639; the sum of these digits is 27; and 2 and 7 is 9:

This seems startling enough. Yet there are other queer examples of the same form of persistence. It was M. de Maivan who discovered that if you take any row of figures, and, reversing their order, make a subtraction sum of obverse and reverse, the final result of adding up the digits of the answer will always be 9. As, for example:

2941

Reverse, 1492

T449

Now, 1 + 4 + 4 + 9 = 18; and 1 -f 8 = 9.

The same result is obtained if you raise the numbers so changed to their squares or cubes. Start anew, for example, with 62; reversing it, you get 20. Now, 62 — 26 = 36, and 3+6 = 9. The squares of 26 and 62 are, respectively, 676 and 3844. Subtract one from the other, and you get 3168 == 18, and 1+8 = 9. So with the cubes of 26 and 62, which are 17,576 and 238,328. Subtracting, the result is 220,752 = 18, and 1 -1-8 = 9.

Again, you are confronted with the same puzzling peculiarity in another form. Write down any number, as, for example, 7,549,132, subtract therefrom the sum of its digits, and, no matter what figures you start with, the digits of the products will always come to 9. 7549132, sum of digits = 31. 3*

[table]

Now, you will see that the tens column reads down 1, 2, 3, 4, 5, 6, 7, 8 9, and the units column up 1, 2, 3, 4, 5, 6, 7, 8, 9.

Here is a different property of the same number. If you arrange in a row the cardinal numbers from 1 to 9, with the single omission of 8, and multiply the sum so represented by any one of the figures multiplied by 9, the result will present a succession of figures identical with that which was multiplied by 9. Thus, if you wish a series of fives, you take 5 X 9 = 45 for a multiplier, with this result:

12345679 45

61728395 493827i6_

555555555

A very curious number is 142,857, which, multiplied by 1, 2, 3, 4, 5, or 6, gives the same figures in the same order, beginning at a different point, but if multiplied by 7 gives all nines. Multiplied by 1 it equals 142,857; multiplied by 2, equals 285,714; multiplied by 3, equals 428,571 ; multiplied by 4, equals 571,428; multiplied by 5, equals 714,285; multiplied by 6, equals 857,142; multiplied by 7, equals 999,999. Multiply 142,857 by 8, and you have 1,142,856. Then add the first figure to the last, and you have 142,857, the original number, the figures exactly the same as at the start.

The number 37 has this strange peculiarity: multiplied by 3, or by any multiple of 3 up to 27, it gives three figures all alike. Thus, three times 37 will be in. Twice three times (6 times) 37 will be 222; three times three times (9 times) 37 gives three threes; four times three times (12 times) 37, three fours; and so on.

The wonderfully procreative power of figures, or, rather, their accumulative growth, has been exemplified in that familiar story of the farmer who, undertaking to pay his farrier one grain of wheat for the first nail, two for the second, and so on, found that he had bargained to give the farrier more wheat than was grown in all England.

My beloved young friend who love to frequent the roulette-table, do you know that if you began with a dime, and were allowed to leave all your winnings on the table, five consecutive lucky guesses would give you a million and a half of dollars, or, to be exact, $1,450,625.52?

Yet that would be the result of winning thirty five for one five times handrunning.

Here is another example. Take the number 15, let us say. Multiply that by itself, and you get 225. Now multiply 225 by itself, and so on until fifteen products have been multiplied by themselves in turn.

You don't think that is a difficult problem? Well, you may be a clever mathematician, but it would take you about a quarter of a century to work out this simple little sum.

The final product called for contains 38,589 figures, the first of which are 1442. Allowing three figures to an inch, the answer would be over 1070 feet long. To perform the operation would require about 500,000,000 figures. If they can be made at the rate of one a minute, a person working ten hours a day for three hundred days in each year would be twenty-eight years about it If, in multiplying, he should make a row of ciphers, as he does in other figures, the number of figures would be more than 523,939,228. This would be the. precise number of figures used if the product of the left-hand figure in each multiplicand by each figure of the multiplier was always a single figure, but, as it is most frequently, though not always, two figures, the method employed to obtain the foregoing result cannot be accurately applied. Assuming that the cipher is used on an average once in ten times, 475,000,000,000 approximates the actual number.

There is a clever Persian story about a wealthy Oriental who, dying, left seventeen camels to be divided as follows: his eldest son to have half, his second son a third, and his youngest a ninth. But how divide camels into fractions? The three sons, in despair, consulted Mohammed AH.

"Nothing easier," said the wise man. "I'll lend you another camel to make eighteen, and now divide them yourselves."

The consequence was, each brother got from one-eighth of a camel to onehalf more than he was entitled to, and Ali received his camel back again,— the eldest brother getting nine camels, the second six, and the third two.

There are many mathematical queries afloat whose object is to puzzle the wits of the unwary listener or to beguile him into giving an absurd reply. Some of these are very ancient, many are excellent Who, for example, has not at some period of his existence been asked, •' If a goose weighs ten pounds and half its own weight, what is the weight of the goose?" And who has not been tempted to reply on the instant, fifteen pounds? The correct answer is, of course, twenty pounds. Indeed, it is astonishing what a very simple query will sometimes catch a wise man napping. Even the following has been known to succeed:

"How many days would it take to cut up a piece of cloth fifty yards long, one yard being cut off every day?"

Or again:

"A snail climbing up a post twenty feet high ascends five feet every day, and slips down four feet every night: how long will the snail take to reach the top of the post?" Or again:

"A wise man having a window one yard high and one yard wide, and requiring more light, enlarged his window to twice its former size; yet the window was still only one yard high and one yard wide. How was this done?" This is a catch question in geometry, as the preceding were catch-questions in arithmetic,—the window being diamond-shaped at first, and afterwards made square. As to the two former, perhaps it is scarcely necessary seriously to point out that the answer to the first is not fifty days, but forty-nine; and to the second, not twenty days, but sixteen,—since the snail, who gains one foot each day for fifteen days, climbs on the sixteenth day to the top of the pole, and there remains.

Numbers have a legendary and mystic signification. It is not only the mathematician that has been fascinated by them. The poet, the philosopher, the priest, have pondered over their changeless relations to each other, have seen in mathematical truth the one thing absolutely fixed and sure, and have come to look upon numbers and their symbols as in some sort a revelation from on high, things to be dealt with reverently and awesomely. And so almost every number has been given an esoteric meaning.

The number one, as being indivisible, and as entering into all other numbers, was always a sacred number. The Egyptians made it the symbol of life, of mind, of the creative spirit.

Three, in the Pythagorean system, was the perfect number, expressive of beginning, middle, and end. From time immemorial greater prominence has been given to it than to any other number, save perhaps seven. And as the symbol of the Trinity its influence has waxed more potent in more recent times. It appears over and over again in the Old Testament and the New.

When the world was created we find land, water, and sky, sun, moon, and stars. Noah had three sons; Jonah was three days in the whale's belly; Christ three days in the tomb. There were three patriarchs,—Abraham, Isaac, and Jacob. Abraham entertained three angels. Job had three friends. Samuel was called three times. Samson deceived Delilah three times. Three times Saul essayed to kill David with a javelin. Jonathan shot three arrows on David's behalf. Daniel was thrown into a den with three lions for praying three times a day. Shadrach, Meshach, and Abednego were rescued from the fiery furnace. The Commandments were delivered on the third day. St. Paul speaks of Faith, Hope, and Charity, these three. Three wise men came to worship Christ with presents three. Christ spoke three times to Satan when tempted. He prayed three times before his betrayal. Peter denied him three times. Christ suffered three hours' agony on the cross. The superscription was in three languages, and three men were crucified. The third day Christ arose again, and appeared three times to his disciples. And so on, and so on. It were tedious to continue the enumeration.

In classic mythology the Graces and the Furies were three, the Muses were originally three, and Cerberus's three heads, Neptune's trident, the tripod of Delphi, are a few more instances of the sacred character of the number.

Who does not remember the three bears of nursery lore, the three feline infants who lost their mittens, the three wise men of Gotham who went to sea in a bowl, or the three finiking Frenchmen frying frogs, and recall the delight he felt in the story of the farmer's wife who vowed vengeance on the three hapless mice, or of Old King Cole with his "fiddlers three"? Then, when fairy-tales began to charm, who docs not recollect learning that the elfish creatures carried bows made of the ribs of a man buried where three lairds' lands meet? Those who followed Gulliver in his travels will call to mind that in the kingdom of Liliput the three great prizes of honor were fine silk threads, six inches long, in colors blue, red, and green; but perhaps every reader had not the opportunity of being fascinated by the German story which relates how a miller's daughter, wedded to a king, was ordered by him to spin straw into gold, and had it done for her by the dwarf Rumpelstilzchen, on condition that she gave him her first-born. She cried so bitterly that he promised to relent if she guessed his name in three days. Two days were spent in vain guesses, but the third the queen's servants heard a strange voice, singing "Little dreams my dainty dame Rumpelstilzchen is my name." The queen saved her child, and the dwarf killed himself with rage.

France, Belgium, Holland, and Italy all fly three national colors. The Turkish vizier has his standard ornamented with three horse-tails. The Prince of Wales's crest consists of three feathers. Indeed, the annals of heraldry revel in designs of a triplicate character, the three British lions being conspicuous. The original armorial ensign of the Isle of Man was a ship in full sail; but after the battle of RonaWsway Alexander III. substituted the present curious device, having probably taken it from the emblem of Sicily,—the ancient Trinacria found upon Greek vases. In 1363, Charles VI., it appears, reduced the Fleurs-de-Lis to three in number, from the mystic superstition of the Church. Every one familiar with University life knows what it is to drink copus, bishop, and cardinal. Ecclesiastical history is replete with such triads, as, for example, the Bell, Book, and Candle; the Triduum, or three days' prayer; the Pope's three crowns ; and "The Mystery of the Three Dons," a religious play which lasted three days.

Nay, do not life itself and nature proclaim the same truth? Have we not morning, noon, and night; fish, flesh, and fowl; water, ice, and snow; hell, earth, and heaven? The very lightning from heaven is three-forked. Life is divided into youth, manhood, and old age. The os sacrum, supposed to resist the action of water, fire, mill, or anvil, is triangular in shape. Man himself is said to be threefold,—body, soul, and spirit, or, as Laertes has it, a mortal part, a divine and ethereal part, and an aerial and vaporous part According to the Romans, man has a threefold soul,—the anima, or spirit, the umbra, and the manes; and, as was also the opinion of the Greeks, three Parcae, or Fates, arbitrarily controlled his birth, life, and death. Oculists affirm that our early progenitors were giants possessed of three eyes, the third eye being in the back of the head.

No wonder the witches in "Macbeth" ask, " When shall we three meet again?"

Four, as the first square, was highly revered by the Pythagoreans. They swore by it, but ten was the more holy as the symbol of the absolute. One plus two plus three plus four make ten, and four contains the smaller numbers. Therefore, since its contents made ten, it was sacred. Besides, four represented the four elements, the four cardinal points; it stood for equilibrium and for the earth.

Five was considered the number of dominion by knowledge. The pentagram, or Solomon's seal, was its symbol, and the Gnostic schools adopted it as their crest. It was much employed in incantations, and often was used as the symbol of man, who has five senses, five members,—head and four limbs, —five fingers, etc.

Six is a perfect number; its symbol is two triangles base to base; it represents equilibrium and peace.

Seven, which is composed of four, a good number, and three, a good number, has always been regarded as sacred and mystic; indeed, it rivals in popularity the number three.

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