Alcuin: Propositiones

Limax fuit ab hierundine invitatus ad prandium infra leucam unam. In die autem non potuit plus quam unam unicam pedis ambulare. Dicat, qui velit, in quot diebus ad idem prandium ipse limax perambulat?

A slug (limax) was invited by Herundina to a luncheon within one leuca. But in a day he could not walk more than one single foot. Let him who wishes say: in how many days will the snail himself perambulate to that same luncheon?

II. PROPOSITIO DE VIRO AMBULANTE IN VIA.

CHAPTER 2. PROPOSITION CONCERNING A MAN WALKING ON THE ROAD.

Quidam vir ambulans per viam vidit sibi alios homines obviantes, et dixit eis: Volebam, ut fuissetis alii tantum, quanti estis; et medietas medietatis; et hujus numeri medietas; tunc una mecum fuissetis. Dicat, qui velit, quanti fuerunt, qui in primis ab illo visi sunt?

A certain man walking along the road saw other people meeting him, and said to them: I wished that you had been only as many as you are, and a half of a half, and half of this number; then you would have been one with me. Let him who wishes say how many there were who were first seen by him?

III. PROPOSITIO DE DUOBUS PROFICISCENTIBUS.

3. PROPOSITION CONCERNING TWO WHO SET OUT.

Duo homines ambulantes per viam, videntesque ciconias, dixerunt inter se: Quot sunt? Qui conferentes numerum dixerunt: Si essent aliae tantae; et ter tantae et medietas tertii, adjectis duobus, C essent. Dicat, qui potest, quantae fuerunt, quae imprimis ab illis visae sunt?

Two men walking along the road, and seeing storks, said to one another: How many are there? Those, comparing the number, said: If there were as many again; and three times as many and the half of a third, with two having been added, they would be 100. Let him who can say how many there were that were first seen by them.

IV. PROPOSITIO DE HOMINE ET EQUIS.

CHAPTER 4. PROPOSITION CONCERNING A MAN AND HORSES.

Quidam homo vidit equos pascentes in campo, optavit dicens: Utinam essetis mei, et essetis alii tantum, et medietas medietatis; certe gloriarer super equos C. Discernat, qui vult, quot equis imprimis vidt ille homo pascentes?

A certain man saw horses grazing in a field, and wished, saying: Would that you were mine, and that you were twice as many, and a half of a half; certainly I would boast over 100 horses. Let him who wishes discern how many horses that man first saw grazing?

V. PROPOSITIO DE EMPTORE DENARIORUM.

5. PROPOSITION CONCERNING THE BUYER OF DENARIUS COINS.

Dixit quidam emptor: Volo de denariis C porcos emere; sic tamen, ut verres X denariis emantur; scrofa autem V denariis; duo vero porcelli denario uno. Dicat, qui intelligit, quot verres, quot scorfae, quotve porcelli esse debeant, ut in neutris numerus nec superabundet, nec minuatur?

A certain buyer said: I wish to buy pigs with the denarii of 100; yet so that boars be bought for 10 denarii each; a sow for 5 denarii; and two piglets for one denarius. Let him who understands say how many boars, how many sows, and how many piglets ought there to be, so that in neither number there be an excess nor a deficiency?

VI. PROPOSITIO DE DUOBUS NEGOTIATORIBUS C SOLIDOS HABENTIBUS.

6. PROPOSITIO DE DUOBUS NEGOTIATORIBUS 100 SOLIDOS HABENTIBUS.

Fuerunt duo negatiatores, habentes C solidos communes, quibus emerunt porcos. Emerunt autem in solidis duobus porcos V, volentes eos saginare, atque iterum venundare, et in solidis lucrum facere. Cumque vidissent tempus non esse ad saginandos porcos, et ipsi eos non valuissent tempore hiemali pascere, tentavere venundando, si potuissent, lucrum facere, sed non poterunt; quia non valebant eos amplius venundare, nisi ut empti fuerant, id est, ut de V porcis duos solidos acciperent.

There were two merchants, having 100 common solidi, with which they bought pigs. Moreover they bought 5 pigs for 2 solidi, intending to fatten them and then to vend them again, and to make a profit in solidi. And when they saw that there was no time to fatten the pigs, and that they themselves were not able to pasture them in the winter season, they tried, by selling, if they could, to make a profit, but they could not; because they were not able to sell them except as they had been bought, that is, to receive 2 solidi for the 5 pigs.

Cum hoc conspexissent, dixerunt ad invicem: Dividamus eos. Dividentes autem et vedentes, sicut emerant, facerunt lucrum. Dicat, qui valet, imprimis quot porci fuerunt; et dividat ac vendat et lucrum faciat, quod facere de simul venditis non valuit.

When they had seen this, they said to one another: Let us divide them. Dividing, moreover, and selling, as they had bought them, they made a profit. Let him who is able first say how many pigs there were; and let him divide and sell and make the profit which he was not able to make by selling them together.

VII. PROPOSITIO DE DISCO PENSANTE LIBRAS XXX.

CHAPTER 7. PROPOSITION CONCERNING A WEIGHING-DISK OF 30 POUNDS.

Est discus qui pensat libras XXX sive solidos DC, habens in se aurum, argentum, aurichalcum, et stannum. Quantum habet auri, ter tantum habet argenti. Quantum argenti, ter aurichalci.

The disk which weighs 30 pounds or 600 solidi, containing in itself gold, silver, aurichalcum, and tin. As much gold as it has, it has three times as much silver. As much silver, three times aurichalcum.

Quantum aurichalci, ter tantum stanni. Dicat, qui potest, quantum in unaquaque specie pensat?

As much aurichalcum; three times as much stannum (tin). Let him who can say how much each kind weighs.

VIII. PROPOSITIO DE CUPA.

CHAPTER 8. PROPOSITION CONCERNING THE CASK.

Est cupa una, quae C metretis impletur capientibus singulis modia tria; habens fistulas III. Ex numero modiorum tertia pars et VI per unam fistulam currit: per alteram tertia pars sola: per tertiam sexta tantum. Dicat nunc, qui vult, quot sextarii per unamquamque, fistulam cucurissent.

There is one cask, which is filled with 100 metretis, each holding three modii; having 3 pipes. From the number of modii a third part and 6 run through one pipe: through another a third part alone: through the third only a sixth. Let him now say, who wishes, how many sextarii would have run through each pipe.

IX. PROPOSITIO DE SAGO.

CHAPTER 9. PROPOSITION CONCERNING SAGO.

Habeo sagum habentem in longitudine cubitos C, et in latitudine LXXX. Volo exinde per portiones sagulos facere, ita ut unaquaeque portio habeat in longitudine cubitos V, et in latitudine cubitos IIII. Dic, rogo, sapiens, quot saguli exinde fieri possint?

I have a sagum measuring in length 100 cubits, and in breadth 80 cubits. I wish thence, by portions, to make little cloaks, so that each portion shall have in length 5 cubits, and in breadth 4 cubits. Tell, I ask, wise one, how many little cloaks can be made from it?

X. PROPOSITIO DE LINTEO.

10. PROPOSITIO DE LINEN CLOTH.

Habeo linteamen unum longum cubitorum LX, latum cubitorum XL. Volo ex eo portiones facere, ita ut unaquaeque portio habeat in longitudine cubitos senos, et in latitudine quaternos, ut sufficiat ad tunicam consuendam. Dicat, qui vult, quot tunicae exinde fieri possint?

I have one linteamen, length 60 cubits, breadth 40 cubits. I wish from it to make portions, so that each portion shall have in length 6 cubits, and in breadth 4 cubits, so as to suffice for sewing a tunic. Let him who will say how many tunics can be made from it.

XI. PROPOSITIO DE DUOBUS HOMINIBUS SORORES ACCIPIENTIBUS.

11. PROPOSITION CONCERNING TWO MEN TAKING SISTERS.

Si duo homines ad invicem, alter alterius sororem in conjugium sumpserit; dic, rogo, qua propinquitate filii eorum pertineant?

If two men to one another have taken each the sister of the other in marriage; say, I ask, by what propinquity are their children related?

XII. PROPOSITIO DE QUODAM PATREFAMILIAS ET TRIBUS FILIIS EJUS.

12. PROPOSITION CONCERNING A CERTAIN PATERFAMILIAS AND HIS THREE SONS.

Quidam paterfamilias moriens dimisit haereditatem tribus filiis suis, XXX ampullas vitreas, quarum decem fuerunt plenae oleo. Aliae decem dimidiae. Tertiae decem vacuae.

A certain paterfamilias, on dying, bequeathed an inheritance to his three sons: 30 glass ampullae, of which ten were full of oil. Another ten were half-full. The third ten were empty.

Dividat, qui potest, oleum et ampullas, ut unicuique eorum de tribus filiis aequaliter obveniat tam de vitro, quam de oleo.

Let him divide, if he can, the oil and the ampullas, so that to each of the three sons there be an equal share both of the glass and of the oil.

XIII. PROPOSITIO DE REGE.

CHAPTER 13. PROPOSITION CONCERNING THE KING.

Quidam rex jussit famulo suo colligere de XXX villis exercitum, eo modo ut ex unaquaque villa tot homines sumeret quotquot illuc adduxisset. Ipse tamen ad villam primam solus venit; ad secundam cum altero; jam ad tertiam tres venerunt. Dicat, qui potest, quot homines fuissent collecti de XXX villis.

A certain king ordered his servant to gather an army from 30 villages, in such a way that from each village he should take as many men as he had brought thither. He himself, however, came to the first village alone; to the second with one other; already to the third three had come. Let him who can say how many men would have been gathered from the 30 villages.

XIV. PROPOSITIO DE BOVE.

14. PROPOSITION ABOUT THE OX.

Bos qui tota die arat, quot vestigia faciat in ultima riga?

The ox which ploughs all day, how many footprints does it make in the last furrow?

XV. PROPOSITIO DE HOMINE.

15. PROPOSITIO DE HOMINE.

Quaero a te ut dicas mihi quot rigas factas habeat homo in agro suo, quando de utroque capite campi tres versuras factas habuerit?

I ask you to tell me how many furrows a man has made in his field, when from each end of the field he has made three turnings?

XVI. PROPOSITIO DE DUOBUS HOMINIBUS BOVES DUCENTIBUS.

16. PROPOSITION CONCERNING TWO MEN LEADING OXEN.

Duo homines ducebant boves per viam, e quibus unus alteri dixit: Da mihi boves duos; et habeo tot boves quot et tu habes. At ille ait: Da mihi et tu duos boves, et habeo duplum quam tu habes. Dicat qui vult, quot boves fuerunt, quot unusquisque habuit.

Two men were leading oxen along the road, of whom one said to the other: "Give me two oxen, and I will have as many oxen as you have." But the other said: "Give me two oxen too, and I will have twice as many as you have." Let anyone who wishes say how many oxen there were, and how many each had.

XVII. PROPOSITIO DE TRIBUS FRATRIBUS SINGULAS HABENTIBUS SORORES.

17. PROPOSITION CONCERNING THREE BROTHERS EACH HAVING A SISTER.

Tres fratres erant qui singulas sorores habebant, et fluvium transire debebant (erat enim unicuique illorum concupiscientia in sorore proximi sui), qui venientes ad fluvium non invenerunt nisi parvam naviculam, in qua non potuerunt amplius nisi duo ex illis transire. Dicat, qui potest, qualiter fluvium transierunt, ne una quidem earum ex ipsis maculata sit?

Three brothers were who each had individual sisters, and they had to cross a river (for to each of them there was a concupiscence toward the sister of his neighbor); who, coming to the river, found nothing but a small boat in which no more than two of them could cross. Let him who can say how they crossed the river so that not even one of the sisters was defiled by them.

XVIII. PROPOSITIO DE HOMINE ET CAPRA ET LUPO.

18. PROPOSITION OF THE MAN AND THE GOAT AND THE WOLF.

Homo quidam debebat ultra flavium transferre lupum, capram, et fasciculum cauli. Et non potuit aliam navem invenire, nisi quae duos tantum ex ipsis ferre valebat. Praeceptum itaque ei fuerat ut omnia haec ultra illaesa omnino transferret.

A certain man had to carry across the river a wolf, a goat, and a bundle of cabbage. Et non potuit aliam navem invenire, nisi quae duos tantum ex ipsis ferre valebat. He could not find any other boat except one that was able to carry only two of them at a time. Praeceptum itaque ei fuerat ut omnia haec ultra illaesa omnino transferret.

Dicat, qui potest, quomodo eis illaesis transire potuit.

Let him tell, who can, how he could carry them across unharmed.

XIX. PROPOSITIO DE VIRO ET MULIERE PONDERANTIBUS.

19. PROPOSITION CONCERNING A MAN AND A WOMAN WEIGHING.

De viro et muliere, quorum uterque pondus habebat plaustri onusti, duos habentes infantes inter utrosque plaustrali pondere pensantes fluvium transire debuerunt. Navem invenerunt quae non poterat ferre plus nisi unum pondus plaustri. Transfretari faciat, qui se putat posse, ne navis mergatur.

Concerning a man and a woman, each of whom weighed as much as a loaded wagon, each bearing two infants whose weight counted toward the wagon's burden, they had to cross a river. They found a boat that could not bear more than one wagon's weight. Let him attempt the ferry who thinks he can, lest the ship be sunk.

XX. PROPOSITIO DE HIRTIIS.

20. PROPOSITION CONCERNING THE HIRTIIS.

De hirtiis masculo et femina habentibus duos natos libram ponderantibus, flumen transire volentibus.

Concerning the Hirtian man and woman, who have two children weighing a pound, wishing to cross a river.

XXI. PROPOSITIO DE CAMPO ET OVIBUS IN EO LOCANDIS.

21. PROPOSITION ABOUT A FIELD AND PLACING SHEEP IN IT.

Est campus qui habet in longitudine pedes CC, et in latitudine pedes C. Volo ibidem mittere oves; sic tamen ut unaquaeque ovis habet in longo pedes V, et in lato pedes IV. Dicat, rogo, qui valet, quot oves ibidem locari possint?

There is a field which has in length 200 feet, and in breadth 100 feet. I wish to put sheep there; so however that each sheep has in length 5 feet, and in width 4 feet. Let him, I ask, who is able, say how many sheep can be placed there?

I. Sequitur solutio de limace.

1. Here follows the solution concerning the slug.

In leuca una sunt mille quingenti passus; VII pedes, ¨X¨C unicae. Quot unicae, tot dies fuerunt, qui faciunt annos CCXLVI, et dies CCX.

In one leuca there are 1500 paces; 7 feet, ¨X¨C unicae. How many unicae, so many days there were, which make 246 years and 210 days.

II. Solutio de eadem propositione

2. Solution on the same proposition

Qui imprimis ab illo visi sunt, fuerunt XXXVI. Alii tantum LXXII. Medietas medietatis XVIII.

Those who first were seen by him were 36. Others only 72.
Half of a half, 18.

Et hujus numeri medietas sunt VIIII. Dic ergo sic: LXXII et XVIII fiunt XC. Adde VIIII, fiunt XCVIIII. Adde loquentem et habebis C.

And the halves of this number are 9. Say therefore thus: 72 and 18 make 90. Add 9, they make 99. Add the speaker and you will have 100.

III. Solutio de ciconiis

3. Solution concerning the storks

XXVIII et XVIII, et tertio sic; fiunt LXXXIIII. Et medietas tertii fiunt XIIII. Sunt in totum XCVIII.

28 and 18, and the third thus; they make 84. And the half of the third makes 14. They are in all 98.

Adjectis duobus, C apparent.

With two added, C appear.

IV. Solutio de equis.

4. Solution concerning horses.

XL equi erant, qui pascebant. Alii tantum fiunt LXXX. Medietas medietatis hujus, id est, XX, si addatur, fiunt C.

40 horses were grazing. Others amount only to 80. The half of the half of this, that is, 20, if added, make 100.

V. Solutio de emptore.

V. Solution concerning the buyer.

Fac VIIII scorfas et unum verrem in quinquaginta quinque denariis; et LXXX porcellos in XL. Ecce porci XC. In residuis V denariis, fac porcellos X, et habebis centenarium numerum in utriusque.

Make 9 sows and one boar for 55 denarii; and 80 piglets for 40. Behold 90 pigs. With the remaining 5 denarii, make 10 piglets, and you will have a hundred head in all of the two.

VI. Solutio de porcis.

6. Solution concerning the pigs.

Imprimis CCL porci erant, qui C solidis sunt comparati, sicut supra dictum est, in duobus solidis V porcos: quia sive quinquaies quinos, sive quinquies L dixeris, CCL numerabis. Quibus divisis unus tulit CXXV, alter similiter. Unus vendidit deteriores tres semper in solido; alter meliores duos in solido.

First, there were 250 pigs, which were bought for 100 solidi, as was said above, at 2 solidi for 5 pigs: for whether you say five times five, or five times 50, you will count 250. Divided between them, one took 125, the other likewise. One sold the poorer ones at 3 for a solidus; the other the better at 2 for a solidus.

Sic evenit, ut is qui deteriores vendidit, de CXX porcis XL solidos est consecutus. Qui vero meliores, LX solidos est consecutus; quia de inferioribus XXX semper in X solidis; de melioribus vinginti autem in X solidis sunt venundati: et remanserunt utriusque V porci, ex quibus ad lucrum IIII solidos et duos denarios facere potueunt.

Thus it came about that he who sold the inferior pigs received 40 solidi from 120 pigs. But he who sold the better ones received 60 solidi; for of the inferior ones 30 are always sold for 10 solidi, and of the better 20 for 10 solidi are vendued: and there remained of each 5 pigs, from which they could make a profit of 4 solidi and 2 denarii.

VII. Solutio.

7. Solution.

Aurum pensat uncias novem: argentum ter uncias VIIII, id est, libras duas et tres uncias. Aurichalcum pensat ter libras duas et III uncias, id est, libras VI et VIIII uncias. Stannum pensat ter libras VI, et ter uncias VIIII, hoc est, libras XX, et III uncias.

Gold weighs 9 ounces: silver weighs three times 9 ounces, that is, 2 pounds and 3 ounces. Aurichalcum weighs three times 2 pounds and 3 ounces, that is, 6 pounds and 9 ounces. Tin weighs three times 6 pounds, and three times 9 ounces, that is, 20 pounds, and 3 ounces.

VIIII unciae, et II librae cum III unciis: et VI librae cum VIIII unciis: et XX librae cum III unciis adunatae, XXX libras efficiunt.

9 ounces, and 2 pounds with 3 ounces: and 6 pounds with 9 ounces: and 20 pounds with 3 ounces, added together, make 30 pounds.

Item aliter ad solidum.

Also, otherwise, to the solidus.

Aurum pensat solidos argenteos XV. Argentum ter XV. id est, XLV. Aurichalcum ter XLV, id est, CXXV. Stannum ter CXXXV, hoc est, CCCCV.

Gold weighs 15 silver solidi. Silver three times 15, that is, 45. Aurichalcum three times 45, that is, 125. Tin three times 135, that is, 405.

Junge CCCCV, et CXXXC: et XLV: et XV; et invenies DC, qui sunt librae XXX.

Join 405, and 135; and 45; and 15; and
you will find 600, which are 30 pounds.

VIII. Solutio.

8. Solution.

Per primam fistulam ¨I¨I¨I DC sextarii cucurrerunt. Per secundum ¨I¨I CCCC. Per tertiam ¨I CC.

Through the first pipe ¨I¨I¨I, 600 sextarii ran. Through the second ¨I¨I, 400 ran. Through the third ¨I, 200 ran.

IX. Solutio.

9. Solution.

De quadringentis octogesima pars V sunt; et centesima IIII. Sive ergo octuagies V, sive centies IIII duxeris, semper CCCC invenies. Tot sagi erunt.

Of 480 the eightieth part is 5; and the hundredth is 4. Therefore whether you multiply eighty by 5, or a hundred by 4, you will always find 400. There will be that many sacks.

X. Solutio.

10. Solution.

Decima pars sexagenarii VI sunt. Decima vero quadragenarii IIII sunt. Sive ergo decimam sexagenarii, sive decimam quadragenarii decies miseris, centum portiones VI cubitorum longas; et IIIIcubitorum latas invenies.

The tenth part of a sexagenary is 6. But the tenth part of a quadragenary is 4. If therefore you multiply the tenth of the sexagenary, or the tenth of the quadragenary, by ten, you will find a hundred portions 6 cubits long and 4 cubits wide.

XI. Solutio ejusdem.

11. Solution of the same.

Verbi gratia: si ego accipiam sororem socii mei, et ille meam, et ex nobis procreentur filii; ego denique sum patruus filii sororis meae; et illa amita filii mei. Et ea propinquitate sibi invecem pertinent.

For example: if I take the sister of my companion, and he takes mine, and from us children are born; I, then, am the patruus (paternal uncle) of the son of my sister; and she the amita (paternal aunt) of my son. And by that kinship they are related to one another.

XII. Solutio.

12. Solution.

Tres igitur sunt filii, et XXX ampullae. Ampullarum autem quaedam X sunt plenae, et X mediae, et X vacue. Duc ter decies; fiunt XXX.

There are therefore three sons, and 30 ampullae. Now some of the ampullae are 10 full, and 10 half‑full, and 10 empty. Take three times ten; they become 30.

Unicuique filio veniunt X ampullae in portionem. Divide autem per tertiam partem, hoc est, da primo filio X semis ampullas, ac deinde da secundo V plenas et V vacuas. Similiter dabis tertio, et erit trium aequa germanorum divisio tam in oleo, quam in vitro.

Each son receives 10 ampullae as his portion. Divide, however, into a third part; that is, give to the first son 10 half‑ampullae, and then give to the second 5 full and 5 empty. You will give likewise to the third, and there will be an equal division among the three brothers both in oil and in the glass.

XIII. Solutio.

13. Solution.

In prima igitur mansione duo fuerunt; in secunda IIII, in tertia VIII, in quarta XVI, in quinta XXXII, in sexta LXIIII, in septima CXXVIII, in octava CCLVI, in nona DXII, in decima ¨I XXIIII, in undecima ¨I¨I XLVIII, in duodecima ¨I¨I¨I¨I XCVI, in quarta decima ¨X¨V¨I CCCLXXXIIII. In quinta decima ¨X¨X¨X¨I¨I DCCLXVIII, etc.

In the first dwelling, therefore, there were 2; in the second, 4, in the third, 8, in the fourth, 16,
in the fifth, 32, in the sixth, 64, in the seventh, 128, in the eighth, 256, in the ninth, 512, in
the tenth, 1024, in the eleventh, 2048, in the twelfth, 4096, in the fourteenth, 16384. In the fifteenth, 32768, etc.

XIV. Solutio.

14. Solution.

Nullum omnino vestigium facit bos in ultima riga, eo quod ipse praecedit aratrum, et hunc aratrum sequitur. Quotquot enim hic praecedendo in exculta terra vestigia figit, tot ille subsequens excolendo resolvit. Propterea illius nullum reperitur vestigium in ultima riga.

The ox makes no trace at all in the last furrow, because he himself goes before the plough, and that plough follows him. For as many footprints as he presses into the cultivated earth by going first, so many the following plough, by cultivating, effaces. Therefore no trace of the former is found in the last furrow.

XV. Solutio.

15. Solution.

Ex uno capite campi III. Ex altero III, quae faciunt rigas versuras VI.

From one head of the field 3. From the other 3, which make six turned furrows.

XVI. Solutio.

16. Solution.

Prior, qui dari sibi duos rogavit, boves habebat IIII. At vero, qui rogabatur, habebat VIII. Dedit quippe rogatus postulanti duos, et habuerunt uterque sex.

The first, who asked that two be given to him, had 4 oxen. But indeed the one who was asked had 8. For, being asked, he gave two to the petitioner, and each had six.

Qui enim prius acceperat, reddidit duos danti priori, qui habebat sex, et habuit VIII, quod est duplum a quatuor, e illi remanserunt IIII, quod est simplum ab VIII.

For he who had received first returned two to the one who had given earlier; that one, who had six, now had 8, which is the double of four; to him remained 4, which is the simple from 8.

XVII. Solutio.

17. Solution.

Primo omninum ego et soror mea introissemus in navem et transfretassemus ultra; transfretatoque fluvio dimisissem sororem meam de nave, et reduxissem navem ad ripam. Tunc vero introissent sorores duorum virorum, illorum videlicet, qui ad littus remanserant. Illis igitur feminis navi egressis, soror mea, intraret, navemque reduceret ad nos.

First of all my sister and I had entered the ship and had crossed over to the far side; and with the river crossed I had sent my sister from the ship, and had brought the ship back to the bank. Then indeed the sisters of two men entered, namely those who had remained on the shore. Therefore, when those women had disembarked from the ship, my sister would enter and would bring the ship back to us.

Illa egrediente foras, duo in navem fratres intrassent, ultraque venissent. Tunc unus ex illis una cum sorore sua navem ingressi ad nos transfretassent. Ego autem et ille, qui navigaverat, sorore mea remanente foras, ultra venissemus.

She, going forth outside, two brothers entered into the ship, and came across; then one of them, having entered the ship together with his sister, crossed over to us. But I and he, who had sailed, with my sister remaining outside, had come across.

Nosque ad littora vectos, una ex illis duabus quaelibet mulieribus, ultra navem reduceret, sororque mea secum recepta pariter ad nos ultra venissent. Et ille, cujus soror ultra remanserat, navem ingressus eam secum reduceret. Et fieret expleta transvectio nullo maculante contagio.

And we, carried to the shores, one of those two would lead any of the women back across the ship, and my sister, received along with her, would likewise have come to us across. And he, whose sister had remained beyond, having boarded the ship would bring her back with him. And the crossing, once completed, would take place with no contagion staining it.

XVIII. Solutio.

18. Solution.

Simili namque tenore ducerem prius capram et dimitterem foris lupum et caulum. Tum deinde venirem, lupumque transferrem: lupoque foris misso capram navi receptam ultra reducerem; capramque foris missam caulum transveherem ultra; atque iterum remigassem, capramque assumptam ultra duxissem. Sicque faciendo facta erit remigatio salubris, absque voragine lacerationis.

For by a like tenor I would first lead the goat and set the wolf and the cabbage outside. Then I would return, and transfer the wolf; and with the wolf sent outside I would bring back the goat, taken into the ship, to the farther side; and with the goat sent outside I would convey the cabbage across to the farther side; and again I would row back, and would have led the goat taken aboard across. And thus by doing this the remigatio will have been made salutary, without the gulf of laceration.

XIX. Solutio.

19. Solution.

Eodem quoque ordine, ut superius. Prius intrassent duo infantes et transissent: unusque ex illis reduceret navem. Tunc mater navem ingressa transisset.

In the same order also, as above. First two infants would enter and cross: and one of them would bring the ship back. Then the mother, having entered the ship, would cross.

Deinde filius ejus reduceret navem. Qua transvecta frater illius navim ingressus ambo ultra transissent, rursusque unus ex illis a patrem reduceret navem. Qua reducta, filio foris stante, pater transiret: rursusque filius, qui ante transierat, ingressus navim eamque ad fratrem reduceret: jamque reductam ingrediantur ambo et transeant.

Then his son would bring back the ship. After it had been carried across, that brother, having entered the ship, both would have crossed to the farther side, and again one of them would be brought back the ship by the father. When it had been brought back, with the son standing outside, the father would cross over; and again the son, who had crossed before, entering the ship would bring it back to his brother: and now, both entering the ship when it had been brought back, they would cross.

Tali subremigante ingenio erit expleta navigatio forsitan sine naufragio.

With such an under-rowing ingenuity propelling,
perhaps the voyage will be completed without shipwreck.

XX. Solutio.

20. Solution.

Similiter, ut superius, transissent prius duo infantes, et unus ex illis navem reduceret; in quam pater ingressus ultra transisset; et ille infans, qui prius cum fratre transierat, navim ad ripam reduceret, in quam frater illius rursus ingressus ambo ultra venissent; unusque propterea ex illis foras egressus; et alter ad matrem reduceret navim: in quam mater ingressa ultra venisset: qua egrediente foras, filius ejus, qui ante cum patre transierat, navim rursus ingressus eam ad fratrem ultra reduceret; in quam ambo ingressi ultra venissent, et fieret expleta transvectio nullo formidante naufragio.

Similarly, as above, let two infants first have crossed, and let one of them bring the ship back; into which the father, having entered, would then have crossed to the farther side; and that infant, who had before crossed with his brother, would bring the ship to the bank, into which his brother, having again entered, both would have come to the farther side; and one of them therefore would have gone ashore; and the other would bring the ship to the mother: into which the mother, having entered, would have gone to the farther side: when she, going out, the son of hers, who had before crossed with the father, having again entered the ship would bring it to his brother on the farther side; into which both having entered would have come to the farther side, and the ferrying would be accomplished with no one fearing shipwreck.

XXI. Solutio.

21. Solution.

Ipse campus habet in longitudine pedes CC. Et in latitudine pedes C. Duc bis quinquenos de CC, fiunt XL. At deinde C divide per IIII. Quarta pars centenarii XXV. Sive ergo XL vicies quinquies; sive XXV quadragies ducti, millenarium implent numerum.

The field itself has in length 200 feet. And in breadth 100 feet. Take from 200 the fivefolds (i.e., divide by 5), you get 40. Then divide 100 by 4. A fourth part of the hundred is 25. Therefore either 40 taken twenty‑five times, or 25 taken forty times, when multiplied, fill the number 1,000.

Tot ergo ibidem oves collocari possunt.

Therefore that many sheep can be placed there.