It seems Dirty Rotten Kitty decided to go on a spree. We last saw him at Crown Tourney in May. He was sitting on the table in the back of the room after copping a feel off Zsof and he disappeared at some point between then and the time we got home and noticed he was gone.
It was very sad- Kenna moped for days and every now and then broke out in a wail of "I miss Kitty". He almost cancelled his Chicago trip because he didn't want to go alone.
I figured since I'd first discovered Kitty rooting around the day old meat section at Krogers, he may have gone back to his old haunts. After dropping Kenna at the airport, Caitlin and I hit every Kroger between Columbus and home. In the 8th Kroger (of course the one that was actually closest to our house) we caught up with him, wandering the aisles with a MacGyver DVD in hand, looking for common household chemicals that can be made into explosives. He looked suspiciously clean and plump, but I didn't ask questions. I grabbed him up, stuffed him in a box with a bottle of Crown Royal and a cigar to take to Kenna and shipped his hiny off to Chicago.
Kenna was very glad to get his little buddy back. I know the news said it was tornadoes in the suburbs the next day, but I have my own theories.
Kenna did ask why I put an empty bottle and cigar stub in the box, though.
Monday, June 9, 2008
Monday, February 11, 2008
The Astrolabe by James E Morrison
First, if you're interested at all in astrolabes, this is the single best, most informative, most comprehensive book I've found. It's clear, concise and well illustrated. I'm very tempted to use the chapter on stereographic projections in my GIS classes.
It doesn't have many illustrations of historic instruments, but that's not it's focus. You can find those in the Time Museum or Western Astrolabes book and get more nuts and bolts info about the production process from Scientific Instruments of Elizabethan England. But for creating and using an astrolabe, you can't find a better reference- UNLESS you really want to use period processes. He doesn't even try to show how it was done, dismissing it with "Readers interested in the mathematical approach used in the Middle Ages are referred to Thomson for a thorough treatment"- that would be the translation of de Plana Spera. As I keep telling my daughter, life gets boring if you don't have anything to long for. I'll start saving the (argh!) $150 for that next.
The contents (with the exception of using trig instead of geometry to create the layout) is exceptional. The actual book is less so. I plan on using this heavily and I don't think it's going to hold up to that kind of use. The paperback cover is glued to the front and back pages to try to give it some stability, but the glue is already losing its hold. The pages themselves are glued in rather than bound. I know there wasn't a real choice in that, but it's a thick book and it's not going to hold up well- think of the Calvin and Hobbes collectors' editions. The paper is already yellowing and I just got it last week. I *know* these choices had to be made to keep the cost of the book down to something that wouldn't make people scream, but I'm going to take it to Kinkos and have it drilled so I can capture it inside a 3 ring binder. I'm not going to risk losing any of the pages. It has enough white space to make that feasible and still have room left over for notes.
It doesn't have many illustrations of historic instruments, but that's not it's focus. You can find those in the Time Museum or Western Astrolabes book and get more nuts and bolts info about the production process from Scientific Instruments of Elizabethan England. But for creating and using an astrolabe, you can't find a better reference- UNLESS you really want to use period processes. He doesn't even try to show how it was done, dismissing it with "Readers interested in the mathematical approach used in the Middle Ages are referred to Thomson for a thorough treatment"- that would be the translation of de Plana Spera. As I keep telling my daughter, life gets boring if you don't have anything to long for. I'll start saving the (argh!) $150 for that next.
The contents (with the exception of using trig instead of geometry to create the layout) is exceptional. The actual book is less so. I plan on using this heavily and I don't think it's going to hold up to that kind of use. The paperback cover is glued to the front and back pages to try to give it some stability, but the glue is already losing its hold. The pages themselves are glued in rather than bound. I know there wasn't a real choice in that, but it's a thick book and it's not going to hold up well- think of the Calvin and Hobbes collectors' editions. The paper is already yellowing and I just got it last week. I *know* these choices had to be made to keep the cost of the book down to something that wouldn't make people scream, but I'm going to take it to Kinkos and have it drilled so I can capture it inside a 3 ring binder. I'm not going to risk losing any of the pages. It has enough white space to make that feasible and still have room left over for notes.
Wednesday, January 16, 2008
Alcuin's Problems to Sharpen the Young Mind
Oh, this is just too much fun to pass up. Call it the SAT-10(th Century)
Alcuin was a student of Egbert (who was a student of Bede), Charlemagne's chief advisor, organizer of his university and intellectual equal. He was personally involved with educating Charlemagne's children and courtiers. He came up with a list of math problems all his students should be able to solve (actually, he stole them from Bede). They look distressingly familiar....
The latin isn’t that hard to translate, so I went to PROPOSITIONES ALCUINI DOCTORIS CAROLI MAGNI IMPERATORIS AD ACUENDOS JUVENES
These are only the first 21 questions. There are 57 total.
About the units:
Alcuin's students would have known the original units, so my translation has modern equivalents. That'll give us a closer feel to what it was like to sit through a Carolingian lecture.
The original latin units:
The denominations of money were the libra, solidus and denarius. I’m using penny, dime and dollar.
A leuca (the predecessor of league) is 1500 paces- about a mile and a half.
The units of weight were the libra (now pound) and uncial (now ounce) with 12 unical to a libra.
A pace is about a foot.
One metretis is about 9 gallons. One sextarius is a pint.
One cubit is 1.5 feet.
1. A limace (A snail)
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 leech invited a snail to lunch a mile and a half away. But he snail can only walk an inch a day. Tell me, if you will, how long will he have to walk for his meal?
2. Viro ambulante in via (A guy was walking down the street)
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?
When a guy walking down the street met others coming toward him, he said "I wish there were others with you, as many as you are, plus a quarter of the sum that would be plus half of that last amount. Then with me as well as thee there would be 100 all together". Tell me, if you will, many people did he see on the road [probably crossing rapidly to the other side as soon as he started talking]
It took me a couple of seconds to figure this out, but I used algebra (2x+1/2x+1/4x+1=100). I don't think that they had access to that kind of symbology. I *think* they would have broken it down into quarters in their minds (or on a scratch board)- 4/4 + 4/4 + 2/4 + 1/4 = 100 - 1
11/4 = 99
1/4 = 9
and everybody knows 1/4 of 36 is 9.
That's the beauty of thinking in 360 degrees- it's easy to simplify things into fractions. I’m beginning to suspect they visualized all their problems as pie charts. .
3. Duobus proficiscentibus visis ciconiis (Two Travelers See Storks)
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 were walking down the road and saw storks. The first guy asked said “How many are there?” After talking it over, they decided there were all totaled one part and one part again and half of a third of those parts plus 2 more there would be 100. Tell me, if you can, how many storks did they see? [I wonder what the latin is for “pink elephants”]
Alcuin writes half of a third, which is an interesting way of putting it because a sixth was just as accessible. I think he may have been getting cute here because I can’t see an advantage into breaking the pie chart down that way. 6/6 + 6/6 + (6/6 + 6/6)/3 + 2 = 100
It never occurred to me before, but I think they must have been balancing equations, at least implicitly.
4. Homine at equis in campo pascentibus (A man watches horses graze in the field)
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 man watches horses graze in a field, wishing “If only they were mine and they were the same number again and half of half of that number, I would have 100 horses to brag about.” Discover, if you will, how many horses does the man see in the pasture?
5. Emptore denariorum (The customer and the dollar)
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?
The customer said “I have 100 dollars to spend on pigs. If a total of 10 dollars will buy a boar, 5 dollars a sow, and 2 piglets for 1 dollar, tell me, who that understands, how many boars, how many sows and how many piglets must I buy in order that the number of pigs equals the number of dollars?
6. Duobus negotiatoribus C solidos communis habentibus (Two dealers have 100 dimes)
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. 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.
There were two dealers who, between them, had 100 dimes. They used the money to buy pigs. They bought 2 pigs for 5 dimes each, intending to fatten them up, then sell them again, increasing their money. However, it was the wrong season for fattening pigs and they didn’t want to feed them through winter because they would only be able to sell them for as much as they bought them for the next year and not making any profit. The decided to divide the pigs and sell them, each making a profit on the 5 dimes. Tell me, observant one, how did they divide them? How could they, by dividing them, gain a profit? Tell me how many they sold and how did they make the profit [Two words- Jimmy Dean].
I have to put in the solution from the Mathematical Gazette because no one in the right mind would ever figure this out.
Solution. Firstly there were 250 pigs bought with 100 shillings at the
above mentioned rate, for five 50s are 250. On division each merchant
had 125. One sold the poorer quality pigs at three for a shilling; the
other sold the better quality pigs at two for a shilling. The one who sold
the poorer pigs received 40 shillings for 120 pigs; the one who sold the
better quality pigs received 60 shillings for 120 pigs. There then
remained 5 of each sort of pig, from which which they could make a
profit of 4 shillings and 2 pence.
7. Disco pensante libras XXX (The dish that weight 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. Quantum aurichalci, ter tantum stanni. Dicat, qui potest, quantum in unaquaque specie pensat?
Now we discuss the dish that weighs 30 pounds, made of gold, silver, brass and tin. It has gold, three times as much silver as gold. It has brass, three times as much as of silver. It has tin, three times as much as of brass. Tell me, if you can, how much of each type is there?
8. Cupa (Barrel)
[Oddly, enough, cupa also means dancing girl or tavern wench]
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 a barrel with 100 metretis capacity, filled by three openings. One opening will carry one third and one sixth of the total flow. One opening will carry on third of the flow. One opening will carry one sixth of the flow. Tell me, if you will, how many sextarii flow through each opening.
9. Sago (Cloak)
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 think that cloak would fit Minimal…
There is a cloak 100 cubits long and 80 wide. I wish to make from this cloak smaller cloaks each with a length of 50 cubits and a width of 30 cubits. Say, my rational inquirer, how many small cloaks can be made?
10. Linteo (Linen)
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 a piece of linen 60 cubits long and 40 cubits wide. I want to sew tunics 6 cubits wide and 4 cubits long. Tell me, if you will, how many tunics can I make? [With the way I sew, one- but it won’t get finished before 12th Night.]
11. Duobus hominibus sorores accipienibus (Two men who receive each other’s sisters)
Si duo homines ad invicem, alter alterius sororem in conjugium sumpserit; dic, rogo, qua propinquitate filii eorum pertineant?
If two men mutually marry each other’s sister, I ask what is the relationship of the sons? [Insert obligatory Aethelmearc joke].
12. Quodam paterfamilias et tribus filiis Rius (The father 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. Dividat, qui potest, oleum et ampullas, ut unicuique eorum de tribus filiis aequaliter obveniat tam de vitro, quam de oleo.
A father was dying and gave his three sons 30 glass jars. 10 were full of oil, 10 were half full and the third 10 were empty. Tell me, if you can, how can the oil and jars be divided equally among the sons?
13. Rege et de eius exercitu (The king and his army)
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.
Once there was a king who ordered his slave to collect from his 30 estates an army so that each estate would give a total of however many men there had been collected up until then. He went to the first estate and took one. At the second estate he took two. At the third estate he took four. Tell me, if you can, how many men he collected from the villas.
[Alcuin thought he was so smart, but he got the answer wrong :-P]
14. Bove (Ox)
Bos qui tota die arat, quot vestigia faciat in ultima riga?
An ox ploughs all day, how many footprints does he leave in the last furrow?
15. Homine (Man)
Quaero a te ut dicas mihi quot rigas factas habeat homo in agro suo, quando de utroque capite campi tres versuras factas habuerit?
One said to me “I saw a man plow his field. In what direction was he facing after the third turn on each side?”
That was my translation, but the answer is 7.
16. Duobus hominibus boves ducentibus (Two men lead an ox)
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 leading oxen down a road. One man asks the other “Give me two oxen, then I’ll have as many as you”. The other responds “Then you give me two oxen and I’ll end up with twice as much as you have”. Tell me, if you will, how many oxen there were and how many each man had.
17. Tribus fratribus singulus habentibus sorores (Three brothers each had 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?
There were three brothers each with a sister who needed to cross a river (each knew that the others wanted his sister [I think this goes along with Problem 11- either that or fratribus translates to friend, but that’s not as interesting]). They found a boat which could only carry two. Tell me, if you can, how to transit the river without a sister being dishonored.
18. Homine capram et lupum (Man, Goat and 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. Dicat, qui potest, quomodo eis illaesis transire potuit.
A man needed to transfer a goat, a wolf and a bunch of cabbages over a river. He could obtain a boat that would carry two and he could not leave any two behind. He had been instructed to transfer all across the river. Tell me, if you will, in what manner did he accomplish this transit.
19. Viro et muliere ponderantibus (A husband and his wife are weighty)
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.
A husband and a wife each weighed as much as a laden cart. They had two children, both of whom weighed as much as the cart. They desired to transit a river. The came upon a boat which could hold only the weight of a cart. How can they all get across, do you think, without the boat sinking?
21. Hirtiis (The hairy ones)
De hirtiis masculo et femina habentibus duos natos libram ponderantibus, flumen transire volentibus.
A hairy male and female have 2 babies who each weigh a pound. They wish transport over a river.
I think something got lost over the years.
21. Campo et ovibus in eo locandis (the field in which sheep are located)
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 is 200 paces long and 100 paces wide. Someone wishes to release sheep into the field in such a way that each sheep has an area 5 paces long and 4 paces wide. Tell me, smartypants, if you wish, how many sheep is it possible to locate there?
* Problems to Sharpen the Young
John Hadley; David Singmaster
The Mathematical Gazette, Vol. 76, No. 475, The Use of the History of Mathematics in the Teaching of Mathematics. (Mar., 1992), pp. 102-126.
http://links.jstor.org/sici?sici=0025-5572%28199203%292%3A76%3A475%3C102%3APTSTY%3E2.0.CO%3B2-W
Alcuin was a student of Egbert (who was a student of Bede), Charlemagne's chief advisor, organizer of his university and intellectual equal. He was personally involved with educating Charlemagne's children and courtiers. He came up with a list of math problems all his students should be able to solve (actually, he stole them from Bede). They look distressingly familiar....
The latin isn’t that hard to translate, so I went to PROPOSITIONES ALCUINI DOCTORIS CAROLI MAGNI IMPERATORIS AD ACUENDOS JUVENES
These are only the first 21 questions. There are 57 total.
About the units:
Alcuin's students would have known the original units, so my translation has modern equivalents. That'll give us a closer feel to what it was like to sit through a Carolingian lecture.
The original latin units:
The denominations of money were the libra, solidus and denarius. I’m using penny, dime and dollar.
A leuca (the predecessor of league) is 1500 paces- about a mile and a half.
The units of weight were the libra (now pound) and uncial (now ounce) with 12 unical to a libra.
A pace is about a foot.
One metretis is about 9 gallons. One sextarius is a pint.
One cubit is 1.5 feet.
1. A limace (A snail)
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 leech invited a snail to lunch a mile and a half away. But he snail can only walk an inch a day. Tell me, if you will, how long will he have to walk for his meal?
2. Viro ambulante in via (A guy was walking down the street)
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?
When a guy walking down the street met others coming toward him, he said "I wish there were others with you, as many as you are, plus a quarter of the sum that would be plus half of that last amount. Then with me as well as thee there would be 100 all together". Tell me, if you will, many people did he see on the road [probably crossing rapidly to the other side as soon as he started talking]
It took me a couple of seconds to figure this out, but I used algebra (2x+1/2x+1/4x+1=100). I don't think that they had access to that kind of symbology. I *think* they would have broken it down into quarters in their minds (or on a scratch board)- 4/4 + 4/4 + 2/4 + 1/4 = 100 - 1
11/4 = 99
1/4 = 9
and everybody knows 1/4 of 36 is 9.
That's the beauty of thinking in 360 degrees- it's easy to simplify things into fractions. I’m beginning to suspect they visualized all their problems as pie charts. .
3. Duobus proficiscentibus visis ciconiis (Two Travelers See Storks)
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 were walking down the road and saw storks. The first guy asked said “How many are there?” After talking it over, they decided there were all totaled one part and one part again and half of a third of those parts plus 2 more there would be 100. Tell me, if you can, how many storks did they see? [I wonder what the latin is for “pink elephants”]
Alcuin writes half of a third, which is an interesting way of putting it because a sixth was just as accessible. I think he may have been getting cute here because I can’t see an advantage into breaking the pie chart down that way. 6/6 + 6/6 + (6/6 + 6/6)/3 + 2 = 100
It never occurred to me before, but I think they must have been balancing equations, at least implicitly.
4. Homine at equis in campo pascentibus (A man watches horses graze in the field)
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 man watches horses graze in a field, wishing “If only they were mine and they were the same number again and half of half of that number, I would have 100 horses to brag about.” Discover, if you will, how many horses does the man see in the pasture?
5. Emptore denariorum (The customer and the dollar)
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?
The customer said “I have 100 dollars to spend on pigs. If a total of 10 dollars will buy a boar, 5 dollars a sow, and 2 piglets for 1 dollar, tell me, who that understands, how many boars, how many sows and how many piglets must I buy in order that the number of pigs equals the number of dollars?
6. Duobus negotiatoribus C solidos communis habentibus (Two dealers have 100 dimes)
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. 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.
There were two dealers who, between them, had 100 dimes. They used the money to buy pigs. They bought 2 pigs for 5 dimes each, intending to fatten them up, then sell them again, increasing their money. However, it was the wrong season for fattening pigs and they didn’t want to feed them through winter because they would only be able to sell them for as much as they bought them for the next year and not making any profit. The decided to divide the pigs and sell them, each making a profit on the 5 dimes. Tell me, observant one, how did they divide them? How could they, by dividing them, gain a profit? Tell me how many they sold and how did they make the profit [Two words- Jimmy Dean].
I have to put in the solution from the Mathematical Gazette because no one in the right mind would ever figure this out.
Solution. Firstly there were 250 pigs bought with 100 shillings at the
above mentioned rate, for five 50s are 250. On division each merchant
had 125. One sold the poorer quality pigs at three for a shilling; the
other sold the better quality pigs at two for a shilling. The one who sold
the poorer pigs received 40 shillings for 120 pigs; the one who sold the
better quality pigs received 60 shillings for 120 pigs. There then
remained 5 of each sort of pig, from which which they could make a
profit of 4 shillings and 2 pence.
7. Disco pensante libras XXX (The dish that weight 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. Quantum aurichalci, ter tantum stanni. Dicat, qui potest, quantum in unaquaque specie pensat?
Now we discuss the dish that weighs 30 pounds, made of gold, silver, brass and tin. It has gold, three times as much silver as gold. It has brass, three times as much as of silver. It has tin, three times as much as of brass. Tell me, if you can, how much of each type is there?
8. Cupa (Barrel)
[Oddly, enough, cupa also means dancing girl or tavern wench]
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 a barrel with 100 metretis capacity, filled by three openings. One opening will carry one third and one sixth of the total flow. One opening will carry on third of the flow. One opening will carry one sixth of the flow. Tell me, if you will, how many sextarii flow through each opening.
9. Sago (Cloak)
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 think that cloak would fit Minimal…
There is a cloak 100 cubits long and 80 wide. I wish to make from this cloak smaller cloaks each with a length of 50 cubits and a width of 30 cubits. Say, my rational inquirer, how many small cloaks can be made?
10. Linteo (Linen)
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 a piece of linen 60 cubits long and 40 cubits wide. I want to sew tunics 6 cubits wide and 4 cubits long. Tell me, if you will, how many tunics can I make? [With the way I sew, one- but it won’t get finished before 12th Night.]
11. Duobus hominibus sorores accipienibus (Two men who receive each other’s sisters)
Si duo homines ad invicem, alter alterius sororem in conjugium sumpserit; dic, rogo, qua propinquitate filii eorum pertineant?
If two men mutually marry each other’s sister, I ask what is the relationship of the sons? [Insert obligatory Aethelmearc joke].
12. Quodam paterfamilias et tribus filiis Rius (The father 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. Dividat, qui potest, oleum et ampullas, ut unicuique eorum de tribus filiis aequaliter obveniat tam de vitro, quam de oleo.
A father was dying and gave his three sons 30 glass jars. 10 were full of oil, 10 were half full and the third 10 were empty. Tell me, if you can, how can the oil and jars be divided equally among the sons?
13. Rege et de eius exercitu (The king and his army)
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.
Once there was a king who ordered his slave to collect from his 30 estates an army so that each estate would give a total of however many men there had been collected up until then. He went to the first estate and took one. At the second estate he took two. At the third estate he took four. Tell me, if you can, how many men he collected from the villas.
[Alcuin thought he was so smart, but he got the answer wrong :-P]
14. Bove (Ox)
Bos qui tota die arat, quot vestigia faciat in ultima riga?
An ox ploughs all day, how many footprints does he leave in the last furrow?
15. Homine (Man)
Quaero a te ut dicas mihi quot rigas factas habeat homo in agro suo, quando de utroque capite campi tres versuras factas habuerit?
One said to me “I saw a man plow his field. In what direction was he facing after the third turn on each side?”
That was my translation, but the answer is 7.
16. Duobus hominibus boves ducentibus (Two men lead an ox)
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 leading oxen down a road. One man asks the other “Give me two oxen, then I’ll have as many as you”. The other responds “Then you give me two oxen and I’ll end up with twice as much as you have”. Tell me, if you will, how many oxen there were and how many each man had.
17. Tribus fratribus singulus habentibus sorores (Three brothers each had 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?
There were three brothers each with a sister who needed to cross a river (each knew that the others wanted his sister [I think this goes along with Problem 11- either that or fratribus translates to friend, but that’s not as interesting]). They found a boat which could only carry two. Tell me, if you can, how to transit the river without a sister being dishonored.
18. Homine capram et lupum (Man, Goat and 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. Dicat, qui potest, quomodo eis illaesis transire potuit.
A man needed to transfer a goat, a wolf and a bunch of cabbages over a river. He could obtain a boat that would carry two and he could not leave any two behind. He had been instructed to transfer all across the river. Tell me, if you will, in what manner did he accomplish this transit.
19. Viro et muliere ponderantibus (A husband and his wife are weighty)
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.
A husband and a wife each weighed as much as a laden cart. They had two children, both of whom weighed as much as the cart. They desired to transit a river. The came upon a boat which could hold only the weight of a cart. How can they all get across, do you think, without the boat sinking?
21. Hirtiis (The hairy ones)
De hirtiis masculo et femina habentibus duos natos libram ponderantibus, flumen transire volentibus.
A hairy male and female have 2 babies who each weigh a pound. They wish transport over a river.
I think something got lost over the years.
21. Campo et ovibus in eo locandis (the field in which sheep are located)
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 is 200 paces long and 100 paces wide. Someone wishes to release sheep into the field in such a way that each sheep has an area 5 paces long and 4 paces wide. Tell me, smartypants, if you wish, how many sheep is it possible to locate there?
* Problems to Sharpen the Young
John Hadley; David Singmaster
The Mathematical Gazette, Vol. 76, No. 475, The Use of the History of Mathematics in the Teaching of Mathematics. (Mar., 1992), pp. 102-126.
http://links.jstor.org/sici?sici=0025-5572%28199203%292%3A76%3A475%3C102%3APTSTY%3E2.0.CO%3B2-W
Wednesday, January 9, 2008
Jim Morrison's The Astrolabe
I got an email from Richard Paselk, who runs one of the best early astronomical instrument websites out there :
However, great news for the astrolabe maker! Jim Morrison has just released his long awaited book on the astrolabe. As a user of his preliminary edition I can tell you it is THE book to get for making astrolabes and using them! You can find info on it at his web site:
http://www.astrolabes.org/theastrolabe.htm
He said Jim Morrison included information about period methods of astrolabe construction, so I'll be getting this with my next paycheck.
However, great news for the astrolabe maker! Jim Morrison has just released his long awaited book on the astrolabe. As a user of his preliminary edition I can tell you it is THE book to get for making astrolabes and using them! You can find info on it at his web site:
http://www.astrolabes.org/theastrolabe.htm
He said Jim Morrison included information about period methods of astrolabe construction, so I'll be getting this with my next paycheck.
Monday, January 7, 2008
Of course one of my worst failings is that I have the attention span of a 2 year old. I can't- CANNOT- look something up in a hard copy dictionary or encyclopedia because I always get side tracked running down tangents. I discovered this again when I finally got access to JSTOR. I started at the beginning of Speculum and downloaded every article that had to do with medieval physical sciences. And Charlemagne, because I'm really fascinated by the Carolingian Renaissance. And Bede- now there was a guy who could think. And the whole Anglo-Saxon period is pretty cool. And the authors in Greek and Roman science who influenced medieval philosophers. And the writers that influenced them. Did you know Otto Neugebauer translated astronomy texts from Babylonian? And there was a bishop who started out skeptical of astrology but got into it as a forecasting tool when the world didn't end when he thought it would? And copyists ended their work with a prayer or jingle about finally being done with the book..... How can you not stop and read about all that stuff? I have a tremendous amount of sympathy for poor Faust.
On the other hand, I now have enough reading material to keep me happy for years. Unfortunately, the only time I get to read is before I go to bed at night and I like to read on my PDA- it's back lit and doesn't wake up the dag-blasted flying monkeys. I have a PDF reader on it, but it doesn't work so well on magazine articles- the print is too small to read no matter how I format it. I'm translating articles a few at a time from PDF to Word docs, but it's not something that can be automated, so I'm having to choose carefully what I want to read first. It's killing me!
I'm also having to try to narrow down my area of research to something reasonable for a persona. That's really difficult because there are fascinating things at every period. I love Carolingian, and they had more from the Romans than a lot of people realize, but they didn't have Ptolomy. I'm drawn to Elizabethan instruments, but they're past some of the truly remarkable changes in thinking that came during the 14th Century. I look best in Tudor garb, but my calligraphy and drawing abilities cry out for uneducated barbarian peasant. I think I may end up with 15 different personas, which is just another way of saying I'm too undisciplined to make a choice.
But one of the first things I'm going to do is draw up a list of instruments and manuscripts that would be reasonable in a well stocked scholar's possession in each century. I may come up with a different persona to go with each list.
On the other hand, I now have enough reading material to keep me happy for years. Unfortunately, the only time I get to read is before I go to bed at night and I like to read on my PDA- it's back lit and doesn't wake up the dag-blasted flying monkeys. I have a PDF reader on it, but it doesn't work so well on magazine articles- the print is too small to read no matter how I format it. I'm translating articles a few at a time from PDF to Word docs, but it's not something that can be automated, so I'm having to choose carefully what I want to read first. It's killing me!
I'm also having to try to narrow down my area of research to something reasonable for a persona. That's really difficult because there are fascinating things at every period. I love Carolingian, and they had more from the Romans than a lot of people realize, but they didn't have Ptolomy. I'm drawn to Elizabethan instruments, but they're past some of the truly remarkable changes in thinking that came during the 14th Century. I look best in Tudor garb, but my calligraphy and drawing abilities cry out for uneducated barbarian peasant. I think I may end up with 15 different personas, which is just another way of saying I'm too undisciplined to make a choice.
But one of the first things I'm going to do is draw up a list of instruments and manuscripts that would be reasonable in a well stocked scholar's possession in each century. I may come up with a different persona to go with each list.
Thursday, December 27, 2007
Western Astrolabes
My husband, who I love, gave me Western Astrolabes by Roderick and Marjorie Webster for Christmas. Its part of the Historic Scientific Instruments of the Adler Planetarium and Astronomy Museum. It says astrolabes, but it covers quadrants and mariner's astrolabes as well. It's split into 5 parts.
The Introduction: A Cross Cultural and Social Perspective gave a fair introduction to the overall development. There are very nice redrawn illustrations from (I think) woodcuts that are going to to useful in future classes, especially if I teach one on surveying. There are also illustrations on how to use it to aid in architectural drawings. The text leaves something to be desired, though. I disagree with their weak assertion that astrolabes were used for astrology and not much else. They mention Levi ben Gerson and Walcher, both of whom used the astrolabe for observations, but then swing into Brahe's denunciation of it as too crude an instrument for his use. It's like saying observations taken with McDonald's 36" telescope aren't valid because the 105" is next door. So? The 105" is better for some things, but the 36" can still be used to do some darned fine science. Just because something was inadequate at the end of the 16th C doesn't mean it wasn't extremely useful earlier.
The next section is a technical introduction. When they say introduction, they mean it, but I think that's probably fair because they wrote an earlier book* that probably covered all that. I'm on a search for it now. The problem with most astrolabe books (IMO) is they assume 20th century knowledge and drawing instruments. I want to know how they did it in period! I'm working on getting together a web site to detail this now, but reading medieval latin gives me a headache. I'm looking for the right translations, but I haven't found them yet. I can put something together from Vitruvius, but I'd like something later, please.
The third section describes the astrolabes. It's lovely, although not as detailed as Epact. They give tables(!) of the info found on each instrument they describe, which is going to help with my compilation of period bright star names and symbols. The text descriptions give enough info to tell what you're looking at, but not enough to recreate it. I suspect Dirty Rotten Kitty will be having his way with Chicago next summer. Kenna wants to go there to look at leaky old Frank Lloyd Wright houses anyway. We'll have to confiscate the BB gun before DRK sees all those windows, though.
The Quadrant section is skimpy and mostly post period, but quadrants are pretty straight forward, so I can't complain.
The Mariner Astrolabe section is short- 2 post period instruments.
The Appendices are nice- a catalog of stars found on the instruments and a lovely bibliography.
It was a good book, but if I had to have just one, I think I might go for Scientific Instruments in Elizabethan England. That's all early 16th C to early 17th C, but there's enough discussion to let me trace back some. Epact is still my favorite place. It seems like I have to reach some of the most interesting info from cached pages, but OMG does it have info on there. I'm holding out high hopes for the History and Practice of Ancient Astronomy by James Evans, which my parents got me this Christmas. And I *really* want to go to Adler now and scrawl notes all over my nice shiny Christmas present.
If anyone is interested, this is the astolabe kit you want. Or email me and I'll send you a PDF of my paper Chaucer kit, but the only plate at the moment is for Portsmouth, Ohio.
* 1.Title: Astrolabe Kit and the Astrolabe: Some Notes on Its History, Construction and Use (Review)
Author(s): J. L. Berggren
Author(s) of Work: Roderick S. Webster
Source: Technology and Culture, Vol. 18, No. 1. (Jan., 1977), pp. 151-153.
Stable URL: http://links.jstor.org/sici?sici=0040-165X%28197701%2918%3A1%3C151%3AAKATAS%3E2.0.CO%3B2-Q
The Introduction: A Cross Cultural and Social Perspective gave a fair introduction to the overall development. There are very nice redrawn illustrations from (I think) woodcuts that are going to to useful in future classes, especially if I teach one on surveying. There are also illustrations on how to use it to aid in architectural drawings. The text leaves something to be desired, though. I disagree with their weak assertion that astrolabes were used for astrology and not much else. They mention Levi ben Gerson and Walcher, both of whom used the astrolabe for observations, but then swing into Brahe's denunciation of it as too crude an instrument for his use. It's like saying observations taken with McDonald's 36" telescope aren't valid because the 105" is next door. So? The 105" is better for some things, but the 36" can still be used to do some darned fine science. Just because something was inadequate at the end of the 16th C doesn't mean it wasn't extremely useful earlier.
The next section is a technical introduction. When they say introduction, they mean it, but I think that's probably fair because they wrote an earlier book* that probably covered all that. I'm on a search for it now. The problem with most astrolabe books (IMO) is they assume 20th century knowledge and drawing instruments. I want to know how they did it in period! I'm working on getting together a web site to detail this now, but reading medieval latin gives me a headache. I'm looking for the right translations, but I haven't found them yet. I can put something together from Vitruvius, but I'd like something later, please.
The third section describes the astrolabes. It's lovely, although not as detailed as Epact. They give tables(!) of the info found on each instrument they describe, which is going to help with my compilation of period bright star names and symbols. The text descriptions give enough info to tell what you're looking at, but not enough to recreate it. I suspect Dirty Rotten Kitty will be having his way with Chicago next summer. Kenna wants to go there to look at leaky old Frank Lloyd Wright houses anyway. We'll have to confiscate the BB gun before DRK sees all those windows, though.
The Quadrant section is skimpy and mostly post period, but quadrants are pretty straight forward, so I can't complain.
The Mariner Astrolabe section is short- 2 post period instruments.
The Appendices are nice- a catalog of stars found on the instruments and a lovely bibliography.
It was a good book, but if I had to have just one, I think I might go for Scientific Instruments in Elizabethan England. That's all early 16th C to early 17th C, but there's enough discussion to let me trace back some. Epact is still my favorite place. It seems like I have to reach some of the most interesting info from cached pages, but OMG does it have info on there. I'm holding out high hopes for the History and Practice of Ancient Astronomy by James Evans, which my parents got me this Christmas. And I *really* want to go to Adler now and scrawl notes all over my nice shiny Christmas present.
If anyone is interested, this is the astolabe kit you want. Or email me and I'll send you a PDF of my paper Chaucer kit, but the only plate at the moment is for Portsmouth, Ohio.
* 1.Title: Astrolabe Kit and the Astrolabe: Some Notes on Its History, Construction and Use (Review)
Author(s): J. L. Berggren
Author(s) of Work: Roderick S. Webster
Source: Technology and Culture, Vol. 18, No. 1. (Jan., 1977), pp. 151-153.
Stable URL: http://links.jstor.org/sici?sici=0040-165X%28197701%2918%3A1%3C151%3AAKATAS%3E2.0.CO%3B2-Q
Flying Monkey Minions
(This is an early mug shot)
I started this blog to keep track of my astrolabe research, so of course my first post will have nothing to do with that.
Dirty Rotten kitty is a member of the family and travels with us wherever we go- mostly because the FBI wants him under close supervision at all times. He's sold the Grand Canyon to BFI for waste disposal, harassed the sprites at Taliesin West, hijacked the USS Salem, incited a counter-revolution at Fanuiel Hall, upgraded the descriptions in the Museum of Natural History, put up an exit sign over the back door of the Alamo.... You get the picture.
I was ignoring work at thinkgeek.com before Christmas and found the screaming monkey slingshots. I *knew* I had to give one of these things to someone I loved. But if I gave one to Kenna, Caitlin would pout and if I gave one to Caitlin, Kenna would liberate it. So I bought 2 to keep the violence to a minimum on Christmas morning.
Unfortunately, I got the monkeys smuggled into the house on Christmas Eve, planning on hanging them from the angel on top of the tree, but the monkeys had other ideas. Once they found out they were entering the lair of Dirty Rotten Kitty, they immediately swore fealty to him. You know how flying monkeys are- once they swear fealty, that's it. So I found an old sweat sock, stuffed the monkeys into it, hung it with the other stockings and waited for Christmas morning. Dirty Rotten Kitty was thrilled with his new flying monkey minions. It was what he'd always wanted, next to explosives and fissile material, of course. The monkeys are all excited- every time I sat down on Christmas Day, they mysteriously launched themselves toward me, screaming merrily. Now they all live together in the Cat Cave under the monkey reading lamp on Kenna's side of the bed, plotting world domination, forging $1 lottery scratch off tickets and singing along with Mike Oldfield CDs.
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