文集 第 9 章

A snail was invited by a swallow to a dinner one league away. But in a day it could walk no more than a single inch of foot. Let whoever wishes tell, in how many days does that snail walk to the same dinner? A certain man walking along the road saw other men coming to meet him, and said to them: 'I wished that you were as many again as you are; and half of a half; and half of this number; then you would be one hundred together with me.' Let whoever wishes tell, how many were they who at first were seen by him? Two men walking along the road, seeing storks, said among themselves: 'How many are there?' Reckoning the number, they said: 'If there were as many again; and three times as many and half of the third, with two added, they would be one hundred.'

Let whoever can tell, how many were they that were at first seen by them? A certain man saw horses grazing in a field, and wished, saying: 'Would that you were mine, and that you were as many again, and half of a half; surely I would glory over a hundred horses.' Let whoever wishes discern, how many grazing horses did that man at first see? A certain buyer said: 'I wish to buy a hundred pigs with a hundred denarii; yet in such a way that boars are bought for ten denarii each, a sow for five denarii, but two piglets for one denarius.' Let whoever understands tell, how many boars, how many sows, and how many piglets there ought to be, so that in neither group the number is exceeded nor diminished? There were two merchants having a hundred shillings in common, with which they bought pigs.

Now they bought five pigs for two shillings, wishing to fatten them and again to sell them, and to make a profit in shillings. But when they saw it was not the time for fattening pigs, and they themselves were not able to feed them in the winter season, they tried by selling to make a profit if they could; but they could not, because they could not sell them for any more than they had been bought, that is, so that for five pigs they received two shillings. When they had observed this, they said to one another: 'Let us divide them.' But dividing and selling, just as they had bought, they made a profit. Let whoever is able tell, how many pigs there were at first; and let him divide and sell and make a profit, which he could not make from those sold together.

There is a dish which weighs thirty pounds or six hundred shillings, having in it gold, silver, brass, and tin. As much gold as it has, three times as much silver. As much silver, three times as much brass. As much brass, three times as much tin. Let whoever can tell, how much does it weigh in each kind? There is one cask which is filled with a hundred measures, each holding three modii; having three pipes. Of the number of modii, a third part and six run through one pipe; through another, a third part alone; through the third, only a sixth. Let whoever wishes now tell, how many sextarii would have run through each pipe. I have a cloak a hundred cubits long and eighty wide.

From it I wish to make little cloaks by portions, so that each portion may have five cubits in length and four cubits in width. Tell, I pray, wise one, how many little cloaks can be made from it? I have a piece of linen sixty cubits long, forty cubits wide. I wish to make portions from it, so that each portion may have six cubits in length and four in width, sufficient for sewing a tunic. Let whoever wishes tell, how many tunics can be made from it? If two men each take the other's sister in marriage, tell, I pray, by what kinship do their sons belong to one another?

A certain father of a household, dying, left an inheritance to his three sons: thirty glass jars, of which ten were full of oil, another ten half-full, and the third ten empty. Let whoever can, divide the oil and the jars, so that to each one of the three sons there may fall equally, as much of the glass as of the oil. A certain king ordered his servant to gather an army from thirty villages, in such a way that from each village he should take as many men as he had brought there. He himself, however, came to the first village alone; to the second with one other; now to the third three came. Let whoever can tell, how many men would have been gathered from the thirty villages. An ox that ploughs the whole day, how many footprints does it make in the last furrow?

I ask you to tell me how many furrows a man has made in his field, when at each end of the field he has made three turnings? Two men were driving oxen along the road, of whom one said to the other: 'Give me two oxen, and I shall have as many oxen as you have.' But the other said: 'You also give me two oxen, and I shall have double what you have.' Let whoever wishes tell, how many oxen there were, and how many each one had. There were three brothers who each had a single sister, and they had to cross a river (for each one of them had desire for the sister of his neighbour); coming to the river they found nothing but a small boat, in which no more than two of them could cross.

Let whoever can tell, how did they cross the river, so that not even one of the women among them was defiled? A certain man had to carry a wolf, a goat, and a bundle of cabbage across a river. And he could find no other boat except one which was able to carry only two of them. It had therefore been commanded to him that he should carry all these across wholly unharmed. Let whoever can tell, how was he able to carry them across unharmed? Concerning a man and a woman, each of whom had the weight of a loaded wagon, having two children who together weighed a wagon-load between them, they had to cross a river. They found a boat which could carry no more than a single wagon's weight. Let whoever thinks he can, make them cross, so that the boat is not sunk.

Concerning a male and female hedgehog having two young, weighing a pound, wishing to cross a river. There is a field which has two hundred feet in length and a hundred feet in width. I wish to put sheep there; yet in such a way that each sheep has five feet in length and four feet in width. Let whoever is able tell, I pray, how many sheep can be placed there? I. Here follows the solution concerning the snail. In one league there are fifteen hundred paces; seven feet make ninety inches. As many inches, so many days there were, which make two hundred forty-six years and two hundred ten days. II. The solution of the same proposition: Those who were at first seen by him were thirty-six. As many again, seventy-two. Half of a half, eighteen. And the half of this number is nine. Say therefore thus: seventy-two and eighteen make ninety.

Add nine, they make ninety-nine. Add the speaker, and you will have a hundred. III. The solution concerning the storks: twenty-eight and eighteen, and the third in this way; they make eighty-four. And half of the third makes fourteen. They are in total ninety-eight. With two added, a hundred appear. IV. The solution concerning the horses: there were forty horses grazing. As many again make eighty. The half of a half of this, that is, twenty, if it be added, makes a hundred. V. The solution concerning the buyer: make nine sows and one boar for fifty-five denarii; and eighty piglets for forty. Behold, ninety pigs. With the remaining five denarii, make ten piglets, and you will have the number of a hundred in both. VI. The solution concerning the pigs: at first there were two hundred fifty pigs, which were bought for a hundred shillings, as was said above, five pigs for two shillings:

for whether you say five times five, or five times fifty, you will count two hundred fifty. These being divided, one took a hundred twenty-five, the other likewise. One sold the poorer ones three always for a shilling; the other the better ones two for a shilling. Thus it came about that he who sold the poorer ones obtained forty shillings from a hundred twenty pigs. But he who sold the better ones obtained sixty shillings; because of the inferior thirty were always sold for ten shillings, but of the better twenty for ten shillings: and there remained to each five pigs, from which they were able to make a profit of four shillings and two denarii. VII. The solution: the gold weighs nine ounces; the silver three times nine ounces, that is, two pounds and three ounces.

The brass weighs three times two pounds and three ounces, that is, six pounds and nine ounces. The tin weighs three times six pounds, and three times nine ounces, that is, twenty pounds and three ounces. Nine ounces, and two pounds with three ounces, and six pounds with nine ounces, and twenty pounds with three ounces, united together, make thirty pounds. Likewise, in another way, by the shilling: the gold weighs fifteen silver shillings. The silver, three times fifteen, that is, forty-five. The brass, three times forty-five, that is, one hundred thirty-five. The tin, three times one hundred thirty-five, that is, four hundred five. Join four hundred five, and one hundred thirty-five, and forty-five, and fifteen; and you will find six hundred, which are thirty pounds. VIII. The solution: through the first pipe six hundred sextarii ran. Through the second, four hundred. Through the third, two hundred. IX. The solution.

Of four hundred, the eightieth part is five; and the hundredth, four. Whether therefore you take eighty times five, or a hundred times four, you will always find four hundred. So many cloaks there will be. X. The solution: the tenth part of sixty is six. But the tenth of forty is four. Whether therefore you take ten times the tenth of sixty, or ten times the tenth of forty, you will find a hundred portions, six cubits long and four cubits wide. XI. The solution of the same: for example: if I take the sister of my comrade, and he takes mine, and from us sons are begotten; then I am the paternal uncle of my sister's son, and she is the paternal aunt of my son. And by that kinship they belong to one another. XII. The solution: there are therefore three sons, and thirty jars.

Now of the jars, ten are full, ten half-full, and ten empty. Multiply three by ten; they make thirty. To each son come ten jars as his portion. Divide however by the third part, that is, give to the first son ten and a half jars, and then give to the second five full and five empty. Likewise you will give to the third, and there will be an equal division of the three brothers, both in oil and in glass. XIII. The solution: in the first station, then, there were two; in the second, four; in the third, eight; in the fourth, sixteen; in the fifth, thirty-two; in the sixth, sixty-four; in the seventh, one hundred twenty-eight; in the eighth, two hundred fifty-six; in the ninth, five hundred twelve; in the tenth, one thousand twenty-four; in the eleventh, two thousand forty-eight; in the twelfth, four thousand ninety-six; in the fourteenth, sixteen thousand three hundred eighty-four. In the fifteenth, thirty-two thousand seven hundred sixty-eight, and so on. XIV.

The solution: the ox makes no footprint at all in the last furrow, because it itself goes before the plough, and the plough follows it. For however many footprints it here, going before, fixes in the tilled earth, so many does the plough, following after in the tilling, undo. Therefore none of its footprints is found in the last furrow. XV. The solution: from one end of the field three; from the other three, which make six furrow-turnings. XVI. The solution: the first, who asked that two be given to him, had four oxen. But indeed the one who was asked had eight. The one asked gave the petitioner two, and each of them had six. For he who had first received gave back two to the first giver, who had six, and had eight, which is double of four; and to that one there remained four, which is the single amount from eight.

XVII. The solution: first of all, I and my sister would enter the boat and cross over; and the river having been crossed, I would put my sister off the boat and lead the boat back to the bank. Then indeed the sisters of the two men — namely of those who had remained on the shore — would enter. These women therefore having disembarked from the boat, my sister would enter and lead the boat back to us. As she disembarked outside, the two brothers would enter the boat and come across. Then one of them, together with his sister, having entered the boat, would cross over to us. But I and that one who had sailed, my sister remaining outside, would come across.

And us being carried to the shores, let any one of those two women lead the boat back across, and my sister, taken up with her, would come across to us together alike. And he whose sister had remained across, having entered the boat, would lead her back with himself. And so the crossing would be accomplished, with no contagion defiling. XVIII. The solution: in like manner, I would first lead across the goat and leave behind the wolf and the cabbage. Then next I would come and carry over the wolf: and the wolf having been put out, I would lead back across the goat taken up in the boat; and the goat having been put out, I would carry the cabbage across; and again I would row back, and having taken up the goat would lead it across. And by so doing the rowing will have been made wholesome, without the abyss of any tearing. XIX. The solution.

In the same order also as above: first the two children would enter and cross over; and one of them would lead the boat back. Then the mother, having entered the boat, would cross over. Next her son would lead the boat back. This being crossed, his brother, having entered the boat, both would cross over, and again one of them would lead the boat back to the father. This being led back, the son standing outside, the father would cross over; and again the son who had crossed before, having entered the boat, would lead it back to his brother; and now, it being led back, let them both enter and cross over. With such a rowing device beneath, the voyage will have been accomplished, perhaps without shipwreck. XX. The solution: likewise, as above, first the two children would have crossed, and one of them would lead the boat back;

into which the father, having entered, would cross over; and that child who had first crossed with his brother would lead the boat back to the bank; into which his brother, having again entered, both would come across; and one of them therefore having gone out, the other would lead the boat back to the mother; into which the mother, having entered, would come across; as she went out, her son, who had before crossed with the father, having again entered the boat, would lead it back across to his brother; into which both, having entered, would come across, and the crossing would be accomplished with no one dreading shipwreck. XXI. The solution: the field itself has two hundred feet in length, and a hundred feet in width. Multiply twice five from the two hundred; they make forty. Then divide the hundred by four. The fourth part of a hundred is twenty-five.

Whether therefore forty taken twenty-five times, or twenty-five taken forty times, they fill up the number of a thousand. So many sheep, therefore, can be placed there.