P.Mich.inv. 4966 / Recto
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Record Details
- Inventory Number
- P.Mich.inv. 4966
- Processing Number
- 2357
- Section/Side
- 10 parts ("fragments"); Recto
- Image Side
- Recto
Background and Physical Properties
- Publ./Side
- Recto; Verso is blank
- Material
- Pap
- Size
- Total size: more than 200 x 23.5 cm.
- Frg. I, a (col. i-ii): 13 x 6 cm;
- frg. I, b (col. ii-iv): 12.5 x 8 cm;
- frg. II: 7.5 x 11 cm;
- frg. III, a (col. i-ii): 10.5 x 14 cm, with a tab about 3 x 2.5 cm at the upper left;
- frg. III, b (col. ii-iii): 14.5 x 15 cm;
- frg. III, c (col. iii-iv): 24 x 15 cm;
- frg. III, d (col. iv-v): 33.5 x 22 cm;
- frg. III, e (col. v): 18 x 20 cm
- frg. III, f (col. v-vi): 18.5 x 23.5 cm;
- frg. III, g (col. vi-vii): 20 x 23.5 cm;
- frg. III, h (col. vii): 8 x 13 cm;
- frg. IIII, i (col. vii): 8 x 22.5 cm;
- frg. IV: 7 x 10.5 cm;
- frg. V: 6 x 5.5 cm;
- frg. VI: 3 x 7 cm;
- frg. VII: 2.5 x 3 cm;
- frg. VIII: 4 x 3.5 cm;
- frg. IX: 3.5 x 7 cm;
- frg. X: 3 x 4 cm
- Items
- 19
- Lines
- Frg. I , col. i - 5; col. ii - 6; col. iii - 6; col. iv - 1;
- Frg. II - 11;
- Frg. III, col. i - 11; col. ii - 15; col. iii - 12; col. iv - 18; col. v - 17; col. vi - 21; col. vii - 21;
- Frg. IV = 9;
- Frg. V = 4;
- Frg. VI = 7;
- Frg. VII = 3;
- Frg. VIII = 4;
- Frg. IX = 6;
- Frg. X = 3
- Mounted
- Yes; + 1 folder which has small scraps with no writing.
- Negative
- Yes
- Conservation Status
- Broken off at the left, right, and bottom. Many holes and tears, particularly at the bottom of the fragments; the papyrus was folded, and the pieces which were one on top of the other show the same outline
- Palaeographic Description
- Unusually large, well-formed, but irregular hand, in extremely broad columns; except in the first part of the papyrus (frg. I, i - frg. III, ii), where the hand is not so large and the columns are narrower - the general effect being of greater irregularity and cramping (Robbins)
- Status
- published
- Library
- Ann Arbor
Contents
- Date
- (Early) IInd century A.D.
- Origin
- Unknown
- Provenance
- Unknown
- Acquisition
- purchased in 1928 through the British Museum
- Language
- Greek
- Genre
- Literary: mathematical
- Author
- Unknown
- Type of Text/Title of Work
- Table of Fractions and Arithmetical Problems
- Content
- Table of fractions with prime denominators, followed by arithmetical problems concerning conversions between silver and copper money and between wheat standards, as well as calculations of carriage charges and of interests
- Subject Headings
- Debt.
- Interest.
- Greek literature.
- Mathematics, Greek.
- Weights and measures, Ancient.
- Mathematics, Ancient--Study and teaching.
- Freight and freightage.
- Translation
- (Frg. I, col. i)
- 1/23 = 1/23
- [2/23] = 1/12 + 1/276
- [3/23] = 1/10 + 1/46 + 1/115
- [4/23 = 1/6] + 1/138
- [5/23 = 1/6 + 1/23] + 1/138
- [- - -]
- (Frg. I, col. ii)
- 12/29 = 1/4 + 1/8 + 1/29 + 1/232
- 13/29 = 1/3 + 1/10 + 1/87 + 1/290
- 14/29 = 1//4 + 1/5 + 1/58 + 1/116 + 1/145
- 15/29 = 1/2 + 1/58
- 16/29 = [1/2] + 1/29 + 1/58
- [17/29 = 1/2 + 1/12] + 1/348
- (Frg. I, col. III: too fragmentary to be translated: table of 1/31?)
- (Frg. I, col. IV: too fragmentary to be translated)
- (Frg. II: too fragmentary to be translated)
- (Frg. III, i: too fragmentary to be translated)
- (Frg. III, col. ii)
- To what number [may you add] 15, subtract 20, [add 25], and subtract 30; it will be 100 artabai of wheat? As is needful, add together the added artabai, = 400; add together also the subtracted artabai, [= 50]; from these subtract the added artabai, [40, =] 10 artabai; add to the 100 artabai of wheat, = 110; this is the number. Proof: Add 15, = 125, [and] subtract 20; remainder, 105. Add [25] = 130; subtract 30; remainder, the given 100 artabai of wheat.
- (Frg. III, col. iii)
- (Reduce to silver drachmai:) 43 talents 2000 (drachmai) of copper, on the basis of one stater to 1200 copper (drachmai). As is needful, reduce the 43 talents 2000 (drachmai) to myriads of drachmai; take 1/1200 of these = 216 2/3 staters; multiply [these] by 4 (since the stater contains 4 drachmai), = [866] drachmai 4 obols.
- (Reduce to silver drachmai:) 5800 copper drachmai, on the basis of one stater to 1200 copper drachmai. As is needful, I took 1/1200 of the given 5800 drachmai, = 4 1/2 1/3 staters; multiply these by 4 (since the stater contains 4 drachmai), = 19 drachmai 2 obols.
- (Reduce to copper drachmai:) 3 talents 2000 (drachmai) of silver, on the basis of one stater to 1200 copper drachmai. As is needful, reduce the [3 talents 2000 drachmai to] myriads [of drachmai, = 20,000; of these take 1/4,] because the stater [contains 4 drachmai; = 5000] staters; [multiply these into the] 1200 copper [drachmai, = 6,000,000] copper [drachhmai, = 1000 talents of] copper.
- (Frg. III, col. iv)
- Of 100 loads, some are at the rate of 10 drachmai, some at 20 drachmai, and some at 20 drachmai, and some at 30 drachmai; the whole price is 2500 drachmai. [As] is needful I multiplied the 100 loads x 30 drachmai = 3000; from this subtract the given 2500 drachmai; remainder, 500 drachmai. And subtract the 10 drachmai from the 30 drachmai; remainder [20]. I took 1/20 of the remainder, 500 drachmai, = 24 with remainder of 20; and subtract the 20 drachmai from the 30 drachmai; remainder 10 of the original. And one operates with 20 because it is necessary to subtract the 10 drachmai from 30; remainder, 20 once = 20; take besides 20 (since I took 1/20 of the 500 drachmai and it was 24 with remainder of 20), = 40; take 1/10 of this (because you have subtracted 20 drachmai from 30 drachmai and 10 were left), = 4; = 1, since [4 x 1] = 4, for [I took 1/4] of [4]; and subtract 1 from 24 (since I took 1/20 of the [remainder], 500 drachmai, and it was [24 with remainder of 20]); 23 is left; so many are the loads at [10] drachmai, = 230 drachmai. Put above the [10 drachmai] the 23 loads and below them the 230 drachmai. I took [10 from the] 20 drachmai, = [10, and] 1/10 of the 40 drachmai, [= 4; and] put [above the 20 drachmai the] 4 loads, [and under the 20 drachmai the] 80 [drachmai;] add 4 [+ 23 = 27]; subtract [230 drachmai from] 2[500 drachmai, = 2270 drachmai; and subtract 80 drachmai from 22]70 drachmai = 21[90 drachmai; and subtract the 27 loads from the 100 loads = 73; add 23 + 4 + 73 = 100, and] add [270 + 80 + 2190 drachmai = 2500 drachmai]
- (Frg. III, col. v)
- Add 1/2 to 2 = 2 1/2; and [add] the [2/3] represented by the 20 stadia to the 3 schoeni = 3 2/3; reduce the 2 1/2 to halves; it is 5; and the 3 2/3 to halves; it is 7 1/3, [of which] 1/5 = 1 1/5 1/6 1/10; (this multiplied) into 4 1/2 artabai = 6 [1/2 1/10 and] (multiplied) into the 60 artabai = 396, 1/100 of which = 3 1/2 1/4 1/5 1/100. So much they pay for 60 artabai over the distance of 3 schoeni 20 stadia.
- The freight on [100 artabai] for 3 schoeni is 3 1/2 [artabai]; according to schedule there is a freight charge of 60 artabai; how many artabai will be delivered over a distance of 4 schoeni? As is needful I took [1/3] of the 4 schoeni [=] 1 1/3; (this multiplied) into the 3 1/2 artabai = 4 [2/3, (which reduced) to] thirds = 14; and the 60 artabai (reduced) to thirds = 180, 1/14 [of which] = 12 1/2 1/3 1/42; (this multiplied) into 100 artabai = 1285 2/3 1/21; so many (artabai) will be delivered for 60 artabai [over a distance of] 4 schoeni.
- (Frg. III, col. vi)
- The freight on 100 artabai is [5 artabai]; given a charge of 80 artabai how many artabai in the principal? As is needful, I took [1/5 of] 80 = 16; (multiplied) into 100 = 1600; so many (artabai) in the principal.
- The freight on 100 artabai is 5 artabai; what is the freight on the whole principal amount of 1000 artabai? As is needful I multiplied [5] x 1000 = 5000, of which 1/100 = 50; so many (artabai) are the freight.
- The width of a field is [2 1/2] schoenia; what is the length that there may be 20 arourai? As is needful, reduce the [2 1/2 schoenia to] halves, = 5; and the 20 arourai to halves, = 40; of which 1/5 = 8; this is the length. Proof: Multiply the 2 1/2 schoenia of the width x [the 8] of the length [and it will be] 20.
- (Frg. III, col. vii)
- The [monthly] interest [on 100 drachmai is 2 drachmai 3 obols]; let 4 months pass, and the whole sum [be ] 1100 drachmai; what is the [principal? Multiply] the 2 drachmai 3 obols into the [4 months,] = 10; into 1100 = 11000, [of which 1/100 =] 110 drachmai; I took 1/110 of 11,000 = 100; this is the interest on 1000 (drachmai); so much is the principal.
- The [monthly] interest [on 100 drachmai] is 2 drachmai 3 obols; let 4 months pass, and the principal be 200 drachmai; what is the interest? [Multiply] the 2 drachmai 3 obols into the [4] months = 10; into 200 = 2000, of which 1/100 = 20; so much is the interest.
- There are artabai of wheat [measured] by the 36-choinix measure; how many will there be by the 40-choinix measure? [Multiply] the 30 artabai of wheat by the 36 [=] 1080, of which 1/40 = 27. [There are so many by the 40-]choinix measure.
- (Frg. IV: too fragmentary to be translated)
- (Frg. V)
- [- - -] the mile contains [8(?)] stadia [- - - the stadium] contains 4 [1/6] geometrical schoenia [- - -] and the mile contains [- - -]
- (Frg. VI-X too fragmentary to be translated)
- Papyri.info Link
- https://papyri.info/apis/michigan.apis.2357
Information on Publications
- Publications
-
Series and Volume Editor Year Pg/Nr Photo SB Preferred Citation Corrections PMich III Winter JG 1936 145 Winter JG, PMich III, 145, 1936 -- Frg. I, col. i, l. 2: read [ton] 13, 1/3 1/10 1/8[7] 1/290, instead of [ton] 13, 1/3 1/15 [1/29 1/8]7 1/4[35]; - frg. III, col. iv, l. 10: read elabo(n) t[o] 1/20 ton (drachmon) 500, instead of elab[o]n 1/20 ton [loip(on)] (drachmon) 500; - frg. III, col. v, l. 14: perhaps t]as kephal(aiou), instead of ] . u . [k]ephal(ai--); - frg. III, col. vii, l. 4: read toisautai ke[phalaiou], instead of toisautai me[non 4]; - frg. IX, l. 2: read ] . (drachm--) 4 (ob. 4) (ob. 1/2) p[; l. 6: read (drachm--) 4?] (ob. 4) (ob. 1/2) [; - frg. X, l. 1: read ] . (drachm--?) 4 (ob. 4) (ob. 1/2), instead of ] . . de (drachmai) [. (PH) Frg. I, col. i, l. 2: read [ton] 13, 1/3 1/10 1/8[7] 1/290, instead of [ton] 13, 1/3 1/15 [1/29 1/8]7 1/4[35]; - frg. III, col. iv, l. 10: read elabo(n) t[o] 1/20 ton (drachmon) 500, instead of elab[o]n 1/20 ton [loip(on)] (drachmon) 500; - frg. III, col. v, l. 14: perhaps t]as kephal(aiou), instead of ] . u . [k]ephal(ai--); - frg. III, col. vii, l. 4: read toisautai ke[phalaiou], instead of toisautai me[non 4]; - frg. IX, l. 2: read ] . (drachm--) 4 (ob. 4) (ob. 1/2) p[; l. 6: read (drachm--) 4?] (ob. 4) (ob. 1/2) [; - frg. X, l. 1: read ] . (drachm--?) 4 (ob. 4) (ob. 1/2), instead of ] . . de (drachmai) [. (PH)
Information on Publications--Bibliography
- Bibliography
- Pack[2] 2309; B. Boyaval, ZPE 23 (1976) p. 196; B. Boyaval, ZPE 31 (1978) p. 121-122; W. Knorr, Historia Mathematica 9 (1982) p. 133-171, with pl. I (frg. I)
Availability/System Requirements
- Scanner Initials
- CT
- Date Scanned
- 7/23/1996
- Institution
- s
Cataloging
- Cataloger
- PH
- Year Begin
- 100
- Year End
- 199
Technical Details
- Image Size
- 2981 x 1981
- File Size
- 786 KB
- Record
- 2357
- Link to this Item
-
https://quod.lib.umich.edu/a/apis/x-2357/4966f3.tif
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Related Links
IIIF
- Manifest
-
https://quod.lib.umich.edu/cgi/i/image/api/manifest/apis:2357:4966F3.TIF
Cite this Item
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- Full citation
-
"P.Mich.inv. 4966; Recto." In the digital collection Advanced Papyrological Information System (APIS UM). https://quod.lib.umich.edu/a/apis/x-2357/4966f3.tif. University of Michigan Library Digital Collections. Accessed March 30, 2024.