541:
thirds, then the beads in the three slots for fractions of 1/12 of an uncia cannot show all values from 1/12 of an uncia to 11/12 of an uncia. In particular, it would not be possible to represent 1/12, 2/12 and 5/12. Furthermore, this arrangement would allow for seemingly unnecessary values of 13/12, 14/12 and 17/12. Even more significant, it is logically impossible for there to be a rational progression of arrangements of the beads in step with unit increasing values of twelfths. Likewise, if each of the beads in the lower slot is assumed to have a value of 1/6 of an uncia, there is again an irregular series of values available to the user, no possible value of 1/12 and an extraneous value of 13/12. It is only by employing a value of 1/12 for each of the beads in the lower slot that all values of twelfths from 1/12 to 11/12 can be represented and in a logical ternary, binary, binary progression for the slots from bottom to top. This can be best appreciated by reference to the figure below. Alternative usages of the beads in the lower slot
40:
168:| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |O| |O| |O| |O| |O| |O| |O| |O| |X| CCC|ƆƆƆ CC|ƆƆ C|Ɔ C X I Ө | | --- --- --- --- --- --- --- --- S |O| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |O| |O| |O| |O| |O| |O| |O| |O| Ɔ |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| |O| | | |O| |O| |O| |O| |O| |O| |O| |O| 2 |O| |O| |O|
274:
549:
integer portion allowed for direct transcription from and to the written Roman numerals. No matter what the true usage was, what cannot be denied by the very format of the abacus is that if not yet proven, these instruments provide very strong arguments in favour of far greater facility with practical mathematics known and practised by the Romans in this authors view.
331:
209:. The first column was either a single slot with 4 beads or 3 slots with one, one and two beads respectively top to bottom. In either case, three symbols were included beside the single slot version or one symbol per slot for the three slot version. Many measures were aggregated by twelfths. Thus the Roman pound ('libra'), consisted of 12 ounces (
186:
without the horizontal bar beside the top slot, a backwards C beside the middle slot and a 2 symbol beside the bottom slot, depending on the example abacus and the source which could be
Friedlein, Menninger or Ifrah. These latter two slots are for mixed-base math, a development unique to the Roman hand abacus described in following sections.
537:
for the top and upper middle, the symbols for the semuncia and sicilicus could also indicate the value of the slot itself and since there is only one bead in each, would be the value of the bead also. This would allow the symbols for all three of these slots to represent the slot value without involving any contradictions.
545:
symbols. To complete the known possibilities, in one example found by this author, the first and second columns were transposed. It would not be unremarkable if the makers of these instruments produced output with minor differences, since the vast number of variations in modern calculators provide a compelling example.
548:
What can be deduced from these Roman abacuses, is the undeniable proof that Romans were using a device that exhibited a decimal, place-value system, and the inferred knowledge of a zero value as represented by a column with no beads in a counted position. Furthermore, the biquinary-like nature of the
544:
It can be argued that the beads in this first column could have been used as originally believed and widely stated, i.e. as ½, ¼ and ⅓ and ⅔, completely independently of each other. However this is more difficult to support in the case where this first column is a single slot with the three inscribed
164:
hand abacus shown here as a reconstruction contains seven longer and seven shorter grooves used for whole number counting, the former having up to four beads in each, and the latter having just one. The rightmost two grooves were for fractional counting. The abacus was made of a metal plate where the
390:
This is however even more strongly supported by
Gottfried Friedlein in the table at the end of the book which summarizes the use of a very extensive set of alternative formats for different values including that of fractions. In the entry in this table numbered 14 referring back to (Zu) 48, he lists
536:
If this symbol refers to the total value of the slot (i.e. 1/72 of an as), then each of the two counters can only have a value of half this or 1/144 of an as or 1/12 of an uncia. This then suggests that these two counters did in fact count twelfths of an uncia and not thirds of an uncia. Likewise,
540:
A further argument which suggests the lower slot represents twelfths rather than thirds of an uncia is best described by the figure above. The diagram above assumes for ease that one is using fractions of an uncia as a unit value equal to one. If the beads in the lower slot of column I represent
185:
The upper slots contained a single bead, while the lower slots contained four beads, the only exceptions being the two rightmost columns, column 2 marked Ө and column 1 with three symbols down the side of a single slot or beside three separate slots with Ɛ, 3 or S or a symbol like the £ sign but
120:, give us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more. Above all, it has preserved the fact of the
338:
The first column was arranged either as a single slot with three different symbols or as three separate slots with one, one and two beads or counters respectively and a distinct symbol for each slot. It is most likely that the rightmost slot or slots were used to enumerate fractions of an
447:
is the symbol resembling a capital "S", but he also includes the symbol that resembles a numeral three with horizontal line at the top, the whole rotated 180 degrees. It is these two symbols that appear on samples of abacus in different museums. The symbol for the
375:. These two characters are to be found in the table of Roman fractions on page 75 of Graham Flegg's book. Finally, the last or lower character is most similar but not identical to the character in Flegg's table to denote 1/144 of an
34:
A reconstruction of a Roman hand abacus, made by the RGZ Museum in Mainz, 1977. The original is bronze and is held by the
Bibliothèque nationale de France, in Paris. This example is missing many counter
530:
This results in two opposing interpretations of this slot, that of
Friedlein and that of many other experts such as Ifrah, and Menninger who propose the one and two thirds usage.
552:
The reconstruction of a Roman hand abacus in the
Cabinet, supports this. The replica Roman hand abacus at, shown alone here, plus the description of a Roman abacus on page 23 of
347:. The upper character in this slot (or the top slot where the rightmost column is three separate slots) is the character most closely resembling that used to denote a
148:
have been used since ancient times. With one bead above and four below the bar, the systematic configuration of the Roman abacus is comparable to the modern
171:
The lower groove marked I indicates units, X tens, and so on up to millions. The beads in the upper shorter grooves denote fives (five units, five tens,
39:
182:
Computations are made by means of beads which it is believed would have been slid up and down the grooves to indicate the value of each column.
467:. However, he stated specifically in the penultimate sentence of section 32 on page 23, the two beads in the bottom slot each have a value of /
775:
747:
31:
843:
628:
202:
838:
722:
676:
600:
317:
124:
counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called
651:
Die
Zahlzeichen und das elementare rechnen der Griechen und Römer und des Christlichen Abendlandes vom 7. bis 13. Jahrhundert
284:
789:
761:
97:
459:, which resembles very closely a cursive digit 2. Now, as stated by Friedlein, this symbol indicates the value of /
507:
respectively. This contradicts all existing documents that state this lower slot was used to count thirds of an
189:
The longer slot with five beads below the Ө position allowed for the counting of 1/12 of a whole unit called an
452:
is that found on the abacus and resembles a large right single quotation mark spanning the entire line height.
299:
176:
295:
594:
58:, a portable, but less capable, base-10 version of earlier abacuses like those that were used by the
693:
112:
with unattached counters or pebbles. The
Etruscan cameo and the Greek predecessors, such as the
783:
755:
718:
672:
624:
582:
833:
109:
206:
113:
91:
554:
Die
Zahlzeichen und das elementare Rechnen der Griechen und Römer und des christlichen
827:
808:
83:
48:
257:
161:
105:
669:
The
Universal History of Numbers: From Prehistory to the Invention of the Computer
165:
beads ran in slots. The size was such that it could fit in a modern shirt pocket.
104:
For more extensive and complicated calculations, such as those involved in Roman
117:
63:
17:
30:
87:
586:
252:
330:
79:
75:
256:, the principal copper coin in Roman currency, was also divided into 12
86:. It greatly reduced the time needed to perform the basic operations of
215:
145:
149:
59:
54:
302:. Statements consisting only of original research should be removed.
239:. There were however other measures in common use - for example the
443:
as used on the Roman bronze abacus, "auf dem chernan abacus". The
343:
and these were, from top to bottom, 1/2 s, 1/4 s and 1/12 s of an
329:
220:
152:, although the soroban was historically derived from the suanpan.
38:
29:
431:). Of prime importance, he specifically notes the formats of the
260:. Again, the abacus was ideally suited for counting currency.
74:
The Roman abacus was the first portable calculating device for
621:
Number Words and Number
Symbols: A Cultural History of Numbers
267:
363:. Likewise, the next character is that used to indicate a
573:
Sugden, Keith F. (Fall 1981). "A History of the Abacus".
291:
108:, there was, in addition to the hand abacus, a true
235:, the standard furrow length when plowing, was 120
213:) (1 uncia = 28 grams). A measure of volume,
43:Velser's reconstruction of Roman abacus (ca. 1600)
334:Alternative usages of the beads in the lower slot
471:. This would allow this slot to represent only /
8:
219:, consisted of 12 heminae (1 hemina = 0.273
778:. Archived from the original on 2012-03-26.
750:. Archived from the original on 2016-03-27.
644:
642:
640:
614:
612:
610:
556:provides further evidence of such devices.
201:are derived), making the abacus useful for
455:The most important symbol is that for the
140:are thus little stones (used as counters).
318:Learn how and when to remove this message
807:Stephenson, Stephen K. (July 7, 2010),
662:
660:
565:
781:
753:
748:"Abacus-Online-Museum of Jörn Lütjens"
592:
533:There is however a third possibility.
737:des Médailles, Bibliothèque nationale
7:
623:. Dover Publications. p. 315.
715:Numbers: Their History and Meaning
383:, which is the same as 1/12 of an
136:means 'pebble' or 'gravel stone';
25:
231:) (1 uncia = 2.43 cm). The
272:
359:or 1/24 of the base unit, the
193:(from which the English words
144:Both the Roman abacus and the
1:
813:, IEEE Global History Network
649:Friedlein, Gottfried (1869).
599:: CS1 maint: date and year (
575:Accounting Historians Journal
776:"Replica Roman Hand Abacus"
298:the claims made and adding
860:
391:different symbols for the
844:Ancient Roman mathematics
788:: CS1 maint: unfit URL (
760:: CS1 maint: unfit URL (
619:Menninger, Karl (2013) .
839:Ancient Roman technology
671:. Vol. 1. Harvill.
177:bi-quinary coded decimal
694:"The Roman Hand-Abacus"
667:Ifrah, Georges (1998).
587:10.2308/0148-4184.8.2.1
227:), was 12 inches (
27:Base-10 portable abacus
713:Flegg, Graham (1984).
653:(in German). Erlangen.
503:with two beads) of an
335:
142:
44:
36:
371:, which is 1/4 of an
333:
102:
42:
33:
179:place value system.
128:, the Romans called
692:Stephenson, Steve.
483:with one bead) or /
223:). The Roman foot (
355:denotes 1/2 of an
351:or 1/24. The name
336:
283:possibly contains
45:
37:
810:Ancient Computers
328:
327:
320:
285:original research
264:Symbols and usage
132:. The Latin word
82:, and presumably
16:(Redirected from
851:
820:
819:
818:
794:
793:
787:
779:
772:
766:
765:
759:
751:
744:
738:
735:
729:
728:
710:
704:
703:
701:
700:
689:
683:
682:
664:
655:
654:
646:
635:
634:
616:
605:
604:
598:
590:
570:
323:
316:
312:
309:
303:
300:inline citations
276:
275:
268:
150:Japanese soroban
21:
18:Roman arithmetic
859:
858:
854:
853:
852:
850:
849:
848:
824:
823:
816:
814:
806:
803:
801:Further reading
798:
797:
780:
774:
773:
769:
752:
746:
745:
741:
736:
732:
725:
712:
711:
707:
698:
696:
691:
690:
686:
679:
666:
665:
658:
648:
647:
638:
631:
618:
617:
608:
591:
572:
571:
567:
562:
522:
518:
514:
502:
498:
494:
490:
486:
482:
478:
474:
470:
462:
430:
422:
417:dimidia sextula
414:
406:
398:
381:dimidio sextula
324:
313:
307:
304:
289:
277:
273:
266:
169:
158:
146:Chinese suanpan
110:reckoning board
72:
28:
23:
22:
15:
12:
11:
5:
857:
855:
847:
846:
841:
836:
826:
825:
822:
821:
802:
799:
796:
795:
767:
739:
730:
723:
705:
684:
677:
656:
636:
630:978-0486319773
629:
606:
564:
563:
561:
558:
520:
516:
512:
500:
496:
492:
488:
484:
480:
476:
472:
468:
460:
428:
420:
412:
404:
396:
367:or 1/48 of an
326:
325:
280:
278:
271:
265:
262:
207:Roman currency
203:Roman measures
167:
157:
154:
114:Salamis Tablet
98:Karl Menninger
92:Roman numerals
84:tax collectors
71:
68:
51:developed the
49:Ancient Romans
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
856:
845:
842:
840:
837:
835:
832:
831:
829:
812:
811:
805:
804:
800:
791:
785:
777:
771:
768:
763:
757:
749:
743:
740:
734:
731:
726:
724:0-14-022564-1
720:
716:
709:
706:
695:
688:
685:
680:
678:1-86046-324-X
674:
670:
663:
661:
657:
652:
645:
643:
641:
637:
632:
626:
622:
615:
613:
611:
607:
602:
596:
588:
584:
580:
576:
569:
566:
559:
557:
555:
550:
546:
542:
538:
534:
531:
528:
526:
510:
506:
466:
458:
453:
451:
446:
442:
438:
434:
426:
418:
410:
402:
394:
388:
386:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
342:
332:
322:
319:
311:
301:
297:
293:
287:
286:
281:This section
279:
270:
269:
263:
261:
259:
255:
254:
248:
246:
242:
238:
234:
230:
226:
222:
218:
217:
212:
208:
204:
200:
196:
192:
187:
183:
180:
178:
175:resembling a
174:
166:
163:
155:
153:
151:
147:
141:
139:
135:
131:
127:
123:
119:
115:
111:
107:
101:
99:
95:
93:
89:
85:
81:
77:
69:
67:
65:
61:
57:
56:
50:
41:
32:
19:
815:, retrieved
809:
770:
742:
733:
714:
708:
697:. Retrieved
687:
668:
650:
620:
595:cite journal
578:
574:
568:
553:
551:
547:
543:
539:
535:
532:
529:
524:
508:
504:
464:
456:
454:
449:
444:
440:
436:
432:
424:
416:
408:
400:
392:
389:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
337:
314:
305:
282:
251:
249:
244:
240:
236:
232:
228:
224:
214:
210:
198:
194:
190:
188:
184:
181:
172:
170:
159:
143:
137:
133:
129:
125:
121:
106:land surveys
103:
96:
73:
52:
46:
717:. Penguin.
581:(2): 1–22.
423:), and the
118:Darius Vase
64:Babylonians
53:Roman hand
828:Categories
817:2011-07-02
699:2007-07-04
560:References
425:scriptulum
308:March 2024
292:improve it
162:Late Roman
122:unattached
88:arithmetic
450:sicilicus
437:sicilicus
401:sicilicus
365:sicilicus
296:verifying
241:sextarius
80:merchants
76:engineers
784:cite web
756:cite web
445:semuncia
433:semuncia
393:semuncia
353:semuncia
349:semuncia
243:was two
116:and the
511:(i.e. /
487:(i.e. /
475:(i.e. /
457:sextula
441:sextula
415:), the
409:sextula
407:), the
399:), the
290:Please
245:heminae
216:congius
138:calculi
130:calculi
126:psephoi
100:said:
834:Abacus
721:
675:
627:
523:of an
463:of an
379:, the
258:unciae
229:unciae
221:litres
211:unciae
173:etc.),
156:Layout
90:using
70:Origin
60:Greeks
55:abacus
35:beads.
515:and /
509:uncia
505:uncia
385:uncia
373:uncia
357:uncia
345:uncia
341:uncia
237:pedes
233:actus
199:ounce
191:uncia
790:link
762:link
719:ISBN
673:ISBN
625:ISBN
601:link
439:and
250:The
205:and
197:and
195:inch
160:The
134:calx
62:and
47:The
583:doi
519:× /
499:× /
495:= /
491:× /
479:× /
429:288
421:144
294:by
225:pes
830::
786:}}
782:{{
758:}}
754:{{
659:^
639:^
609:^
597:}}
593:{{
577:.
527:.
525:As
521:12
501:12
493:12
485:36
481:12
473:72
469:72
465:As
461:72
435:,
427:(/
419:(/
413:72
411:(/
405:48
403:(/
397:24
395:(/
387:.
377:As
369:As
361:As
253:as
247:.
94:.
78:,
66:.
792:)
764:)
727:.
702:.
681:.
633:.
603:)
589:.
585::
579:8
517:3
513:3
497:3
489:6
477:6
321:)
315:(
310:)
306:(
288:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
