napier’s bones and genaille-lucas’s rods
TRANSCRIPT
Napier’s bonesand Genaille-Lucas’s rods
Denis Roegel∗
20 May 2015(last version: 17 March 2021)
Abstract
This note presents a survey of Napier’s bones and of their more “auto-matic” evolution, the Genaille-Lucas rods. Generalizations of both to otherbases are also shown.
Contents1 Napier’s bones 2
1.1 Napier’s original bones . . . . . . . . . . . . . . . . . . . . . . . 21.2 Napier’s bones in other bases . . . . . . . . . . . . . . . . . . . . 8
2 The Genaille-Lucas rods 122.1 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 The original multiplication rods . . . . . . . . . . . . . . 132.1.2 Multiplication rods in other bases . . . . . . . . . . . . . 17
2.2 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 The original division rods . . . . . . . . . . . . . . . . . 212.2.2 Division rods in other bases . . . . . . . . . . . . . . . . 24
2.3 Financial rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
∗Denis Roegel, LORIA, BP 239, 54506 Vandœuvre-lès-Nancy cedex, France,[email protected]
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1 Napier’s bones
1.1 Napier’s original bonesIn his “Rabdology” published in 1617 under the title Rabdologiæ, seu numeratio-nis per virgulas libri duo [26], John Napier (1550–1617) introduced his famous“bones.” Three years after the seminal work on logarithms [25, 33], here was amuch more practical tool for everyday calculations.
Napier’s bones were aimed at facilitating multiplications and divisions. Abasic set of “bones” is made of 11 rods, one for each integer from 0 to 9, and anadditional one serving as an index. More rods can be used, and several basic setscan be combined at will.
Figure 1 shows the basic set of bones. Each bone from 0 to 9 carries themultiples of the corresponding value. Units and tens are separated by a diagonal.For instance, the multiples of 7 are 7, 1/4, 2/1, 2/8, 3/5, etc. The diagonals will beused to transfer carries naturally. The rods actually have four faces, so that eachrod actually carries four sets of multiples, often making it unnecessary to use rodsfrom another set to complement the basic ones.
Figure 2 shows how these rods are used for a multiplication. If we wish tocompute 4×1897, we take the rods 1, 8, 9, and 7, and we put them next to eachother, at the right of the special index rod. On line 4 (marked in yellow), we canread the result. The digits can be read from the right and each diagonal strip mustbe evaluated to one digit. In that example, the rightmost digit is therefore 8, thencomes 2+ 6 = 8 (where we intentionally used the same color), then 3+ 2 = 5,then 3+ 4 = 7, and the result is therefore 7588. In general, there will be carriesfrom one strip to the next, but not in this simple example.
Figure 3 shows a more complex multiplication, involving several identicalrods, as well as carries. Upper left blank triangles correspond to the value 0,and we represent them as Napier did.
In order to multiply a number by another one greater than 9, one computes thepartial products with the bones, and the addition of these products is performedby hand.
Napier’s bones can also be used for division. The same rods are used formultiplying a number and for dividing by that number. So, if we want to dividea number by 4257, we will use the four rods 4, 2, 5, and 7 (figure 4). Let’s forinstance divide 123456 by 4257. We start with the first five digits of the number,12345, as this is the smallest prefix greater than the divider. And using the bones,we seek the greatest multiple of 4257 not larger than 12345. It is the secondmultiple, 8514. We then subtract 8514 to 12345, yielding 3831, and since we haveused the second multiple, we know that the first digit of the quotient is 2. Then,we add the next digit from the initial number, yielding 38316. To this we subtractthe largest possible multiple of 4257, namely the ninth multiple, 38313, and the
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remainder is 3. Consequently, we have found that 123456 = 29×4257+3. Eventhough this looks complex to our 21st century’s eyes, we have to realize that thebones made it essentially easy to provide the multiples not only of 0 to 9, but ofany number, with the need of only a few additions of carries here and there. Thehuman operator did not need to know much more than very limited additions.
Napier also provided two additional rods for their use in the extraction ofsquare and cubic roots which are not described here. Finally, he devised a way tofacilitate the multiplication of multi-digit numbers. His tool, the promptuarium,prealigns the various partial products, so that one then merely has to add digits indiagonals, not only for one partial product, but for all of them at the same time.We will also not describe these extensions here, but perhaps in another article.
Napier’s bones were used in several “machines”, in particular in Schickard’scalculating machine (1623), as an aid for the computer. The carries were neverperformed automatically. A number of constructions, such as those of Schott(1668) and Leupold (1727), have made it easier to manipulate the rods, but withoutsubstantially improving the original design.
The main problem of Napier’s bones is of course that the multiples are notgiven in their final form, and there remains the need to perform additional ad-ditions and carries. Performing the additions can be made (presumably) slightlyeasier if the figures to be added are laid out vertically, and “slanted” rods for thatpurpose have been manufactured in the 19th century. Another partial solution wasthat of Raussain (1738) who used various colors in order to facilitate the align-ments, but it was still necessary to add the carries [19]. An effective and practicalsolution, however, was only made available around 1884 by Genaille and Lucas.
Napier’s bones, as well as the solution of Genaille and Lucas, are still occa-sionally used nowadays as educational tools [34].
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0
0
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9
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Figure 1: The Napier bones. There is one rod for every integer from 0 to 9, andan index rod to the left.
4
1
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Figure 2: A calculation example: 4 × 1897 = /4 + 3/2 + 3/6 + 2/8/ =/7/5/8/8/= 7588.
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Figure 3: Another calculation example: 7 × 3877019 = /2/1 + 5/6 +4/9 + 4/9/0/7 + 6/3/ = /2/6/10/13/9/0/13/3/ = /2/7/1/3/9/1/3/3/ =27139133.
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Figure 4: When dividing by 4257, the rods 4, 2, 5, and 7 are used, and then theyprovide the various multiples needed to perform the division (see main text).
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1.2 Napier’s bones in other basesNapier’s bones can be extended to other bases. Figures 5, 6, 7, 8, 9, and 10 showthe bones for bases 2, 3, 5, 8, 11, and 16. Figures 11 and 12 show computationsin base 7 and 16.
0
0
1
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1
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Figure 5: Napier’s bones in base 2.
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Figure 6: Napier’s bones in base 3.
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Figure 7: Napier’s bones in base 5.
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Figure 8: Napier’s bones in base 8 (octal).
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Figure 9: Napier’s bones in base 11.
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75
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Figure 10: Napier’s bones in base 16 (hexadecimal).
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Figure 11: A computation in base 7.
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B1E
623
7
A
232
A4B
F41
D
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023
72D
932
A37
B3C
C41
D46
E4B
F
5
I
Figure 12: A computation in base 16 (only the 5th multiple is shown).
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2 The Genaille-Lucas rodsHenri Genaille (2 October 1856 Briaucourt, Haute-Saône–16 May 1903) was anengineer of the French railways.1 He invented a number of arithmetic construc-tions and several of them are kept in the collections of the CNAM in Paris [1]. Atthe end of the 1880s, he constructed for instance a piano arithmétique, a machinefor finding the primes of the form 2n −1 [41, pp. 493–494].
Around 1878 [6], he came up with a new computation method which waspresented at the 7th meeting of the Association française pour l’avancement dessciences in 1878 in Paris. At that time, Genaille apparently did not yet have theidea of using rods, but the underlying concepts were already there. In particular,one commentator, Laisant2, observed that the “lines” drawn on Genaille’s boardcould be replaced by grooves and that a small ball could be led through it, andits path could then automatically let appear the result of a computation, and in acertain way, the computation would be performed by gravity.
Genaille’s method was then probably improved and his new rods were men-tioned by the French mathematician Édouard Lucas (1842–1891) [2, 3] at the 1884meeting of the Association française pour l’avancement des sciences in Blois [21].In 1885, Genaille and Lucas then published four sets of calculating rods.3 The firstset was a set of Napier bones [12]. The three other sets were Genaille’s inventionsand were aimed at simplifying multiplications and divisions [10, 11, 9].4 Theywere highlighted as being “exact” and “instantaneous.” According to a reviewpublished in 1885 [5], the boxes of rods were prepared for the 1889 universalexhibition in Paris. Lucas briefly mentioned these rods in his posthumous thirdvolume of Récréations mathématiques [22], where he stated that the idea of therods was Genaille’s, but that Lucas added the three other faces.
The main motivation of the Genaille-Lucas rods is to dispense of the carries inNapier’s bones. Genaille and Lucas basically made two sets of rods, one for themultiplication, and another for the division, in which both operations can be donesmoothlessly. The operator never performs any addition and is not aware of anycarries. This is possible because the manipulation of carries is made so discrete,
1In the 1883 list of members of the Association française pour l’avancement des sciences, heis listed as “Ingénieur civil au bureau central des chemins de fer de l’État,” and his address was“16 rue Saint-Étienne” in Tours. In 1878, he was engineer in Lyon. Of Genaille’s life we knowvery little [30, p. 14].
2Probably Charles-Ange Laisant (1841–1920).3The exact chronology of the publication of these rods is often given incorrectly. Some authors,
for instance Williams [42, 43, 44], write that Lucas presented a problem on arithmetic at the 1885meeting of the Association française pour l’avancement des sciences, when in fact he alreadymentioned the existence of Genaille’s rods at the previous meeting in 1884. The presentation ofthe problem must therefore have occurred earlier, perhaps already in 1878. Boxed sets of the rodswere published in 1885, but they have possibly been displayed at the 1884 meeting.
4For the original instructions which came with these rods, see Weiss’s articles [38, 39].
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so natural, that one feels that there are no carries at all! The division set also hada more specialized form for financial calculations.
2.1 Multiplication2.1.1 The original multiplication rods
In order to (apparently) dispense of the carries, Genaille and Lucas have actuallypacked more information into each cell. Consider for instance Napier’s bone 4,and its seventh multiple. It contains the two figures 2 (tens) and 8 (units) (seefigure 13). But if this is part of a wider product, there may be figures anywherebetween 0 and 6 coming in and added to the units. In other words, the actualunits may be 8, but also 9, 0, 1, 2, 3, and 4. In the last four cases, the tenswill no longer be 2, but 3. All these cases are possible and are made explicitin a cell of the Genaille-Lucas rods. Figure 14 shows three cells, and how theyencode the various possibilities. A cell contains a numerical representation of thevalues of its possible units (for instance 8, 9, 0, 1, 2, 3, and 4), and a graphicalrepresentation of the possible values of the tens. For each value of the units, thereis a corresponding value of the tens. The first cell in figure 14 shows that if thereis no incoming carry (0), the units will be 8, and the tens will be 2 (4× 7 = 28).The second cell is identical to the first, but shows what happens if the incomingcarry is 2, the units will then be 0, and the tens will be 3 (4× 7+ 2 = 30). Thethird cell illustrates that there is a relationship between cells (n,m) and (m,n) inthat one is a extension (or truncation) of the other. The third cell is here shorter,because there are only four possible incoming carries.
28
4
7
I
Figure 13: The information contained in one cell of Napier’s bones.
Each cell contains one or two triangles. It can never contain more than two tri-angles, because the transition from the first triangle to the second one correspondsto the passage of the units from 9 to 0, and there can be only one such transitionin a cell.5
5The main features of Genaille’s rods for multiplication were however already given by KarlSchönbichler in 1850, see the account by Weiss [38]. But Schönbichler apparently didn’t go so faras to market the new rods.
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8901234
0123456
0123456
outputcarry
inputcarry
7
ruler 4
8901234
0123456
0123456
outputcarry
inputcarry
7
ruler 4
8901
0123
0123
outputcarry
inputcarry
4
ruler 7
Figure 14: A cell (left and middle) and its conjunct (right).
Figure 15 shows the entire set of Genaille-Lucas rods for multiplication. Thereare ten rods giving all possibilities when multiplying the rod’s value by 2, 3, upto9. Genaille and Lucas started with 2, as 1 is not really needed.
Figures 16 and 17 show how the rods are actually used and how effective thecalculation is. In order to compute 3× 567439, we take the rods 5, 6, 7, 4, 3,and 9, as we would have done with Napier’s bones. Then, on the row marked 3on the left (index) rod, we start on the first line at the right, here marked with ablack wedge. This gives the first digit, 7. We continue from right to left, merelybe following the triangles. We always stay in the same triangle, and move fromthe base (right) to the tip (left). So, 7 leads to 1, which leads to 3, which leadsto 2, which leads to 0, which leads to 7, which leads to 1. And we are done!The result is 1702317. No apparent carries. Of course, the truth is that whenmoving from the first to second rod, we have added 2, by taking the last entry ofthe second cell, and similarly with the other transitions, but all this took place verynaturally. We did not actually perform additions, we took the paths correspondingto the required additions. It is possible to consider improvements to these rods,one of them being to hide the figures of a rod which will never be used. Thiscan be achieved if the rods are allowed to overlap and carry holes, as illustratedby Weiss [38]. It is also possible to write all the rods on cylinders and to showonly a portion of each cylinder, thereby basically providing ten rods at each place.Such a “calculating machine” has been rebuilt by Valéry Monnier in 2011. Severalvariant constructions are possible.
For multiplying by numbers containing several digits, another constructionhas been proposed [21], [22, p. 82], but it is still cumbersome, and Genaille andLucas had planned to construct an electrical calculating machine (in 1884!) forsolving these problems [7], a machine that unfortunately never came into being.6
6A revival of this early idea was the machine patented by Nicoladze in 1928 and which wasbasically an electrical wiring of Genaille’s triangles [29, 28].
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The machine was to compute the product of two numbers of 10 digits, and wasgoing to cost less than 20 francs [22, p. 83]. In 1894, Genaille presented anothermachine for general multiplications, but it is not based on the Genaille rods [8].
01012012301234012345012345601234567012345678
023345456756789678901789012389012345901234567
145678890101234234567456789067890123890123456
267901234556789890123123456745678901789012345
389234678901234456789890123423456789678901234
401567012356789012345567890101234567567890123
523890456701234678901234567889012345456789012
645123890156789234567901234567890123345678901
767456234501234890123678901245678901234567890
889789678956789456789345678923456789123456789
9×2
3
4
5
6
7
8
9
01012012301234012345012345601234567012345678
Figure 15: The Genaille-Lucas rods.
15
01567012356789012345567890101234567567890123
523890456701234678901234567889012345456789012
645123890156789234567901234567890123345678901
789234678901234456789890123423456789678901234
467901234556789890123123456745678901789012345
389789678956789456789345678923456789123456789
9×2
3
4
5
6
7
8
9
01012012301234012345012345601234567012345678
Figure 16: A computation example.
16
567
5890
6123
7234
4901
3789
9×
3012
Figure 17: The same example, with only the relevant row.
2.1.2 Multiplication rods in other bases
The Genaille-Lucas multiplying rods can easily be extended to other bases, andtheir properties remain the same. In particular, there are still at most two trianglesper cell, but the graphical representation of triangles obviously becomes inade-quate for higher bases, as it may be difficult to separate the triangles. Figure 18shows the Genaille-Lucas rods in base 6, figure 19 shows them in base 16, wherethey really become difficult to use. Figure 20 shows the details of two base-16(hexadecimal) cells, the second one with a very thin triangle. We could of courseadopt a different representation, but the pure Genaille-Lucas representation obvi-ously reaches its limits. Figures 21 and 22 illustrate two multiplications in base16.
01012012301234
023345450150123
145012234545012
201345012334501
323012450123450
445345234512345
5×2
3
4
5
01012012301234
Figure 18: The Genaille-Lucas rods in base 6.
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0101201230123401234501234560123456701234567801234567890123456789A0123456789AB0123456789ABC0123456789ABCD0123456789ABCDE
0233454567567896789AB789ABCD89ABCDEF9ABCDEF01ABCDEF0123BCDEF012345CDEF01234567DEF0123456789EF0123456789ABF0123456789ABCD
14567889ABABCDECDEF01EF012340123456723456789A456789ABCD6789ABCDEF089ABCDEF0123ABCDEF0123456CDEF0123456789EF0123456789ABC
2679ABCDEFF012323456756789AB89ABCDEFBCDEF0123EF01234567123456789AB456789ABCDEF789ABCDEF0123ABCDEF01234567DEF0123456789AB
389CDE01234567889ABCDCDEF01201234567456789ABC89ABCDEF01CDEF01234560123456789AB456789ABCDEF089ABCDEF012345CDEF0123456789A
4ABF0145679ABCDEF0123345678989ABCDEFDEF01234523456789AB789ABCDEF01CDEF01234567123456789ABCD6789ABCDEF0123BCDEF0123456789
5CD23489ABEF012456789ABCDEF0012345676789ABCDECDEF01234523456789ABC89ABCDEF0123EF0123456789A456789ABCDEF01ABCDEF012345678
6EF567CDEF34567ABCDEF123456789ABCDEFF012345676789ABCDEFDEF01234567456789ABCDEFBCDEF0123456723456789ABCDEF9ABCDEF01234567
723BCD4567DEF016789ABF01234589ABCDEF123456789ABCDEF01233456789ABCDCDEF0123456756789ABCDEF01EF0123456789AB789ABCDEF012345
945EF089AB23456CDEF016789ABC01234567ABCDEF012456789ABCDEF01234567889ABCDEF012323456789ABCDECDEF01234567896789ABCDEF01234
A67123CDEF789AB234567DEF012389ABCDEF3456789ABEF012345679ABCDEF0123456789ABCDEFF0123456789ABABCDEF0123456756789ABCDEF0123
B894560123CDEF089ABCD456789A01234567CDEF0123489ABCDEF01456789ABCDE0123456789ABCDEF01234567889ABCDEF012345456789ABCDEF012
CAB789456712345EF0123BCDEF0189ABCDEF56789ABCD23456789ABF0123456789CDEF012345679ABCDEF0123456789ABCDEF01233456789ABCDEF01
DCDABC89AB6789A456789234567801234567EF0123456CDEF012345ABCDEF0123489ABCDEF01236789ABCDEF012456789ABCDEF0123456789ABCDEF0
EEFDEFCDEFBCDEFABCDEF9ABCDEF89ABCDEF789ABCDEF6789ABCDEF56789ABCDEF456789ABCDEF3456789ABCDEF23456789ABCDEF123456789ABCDEF
F×2
3
4
5
6
7
8
9
A
B
C
D
E
F
0101201230123401234501234560123456701234567801234567890123456789A0123456789AB0123456789ABC0123456789ABCD0123456789ABCDE
Figure 19: The Genaille-Lucas rods in base 16.
18
CDEF
0123
0123
outputcarry
inputcarry
4
ruler 7
F0123456789
0123456789A
0123456789A
outputcarry
inputcarry
B
ruler D
Figure 20: Two cells in base 16.
0123456
0789ABCD
1EF01234
26789ABC
ADEF0123
B56789AB
3CDEF012
4456789A
C3456789
5ABCDEF0
61234567
789ABCDE
8F012345
9×
7
0123456
Figure 21: Multiplying 012AB34C56789 by 7 in base 16.
19
6789ABCDEF012
E456789ABCDEF0
4ABCDEF0123456
223456789ABCDE
AF0123456789AB
B789ABCDEF0123
3456789ABCDEF0
4CDEF012345678
CEF0123456789A
6BCDEF01234567
756789ABCDEF01
9×
D
0123456789ABC
Figure 22: Multiplying E42AB34C679 by D in base 16.
20
2.2 DivisionNapier’s bones made it possible to divide numbers, and the rods were basicallyused to provide multiples of the divider, so that the operator didn’t have to memo-rize them. However, it was still necessary to find the largest multiple, to fulfill itscalculation (add the carries), and to subtract it from a prefix of the number to bedivided.
2.2.1 The original division rods
Genaille and Lucas devised a new set of rods which did not automate all of thesetasks, but provided a simple solution in a special case, namely that of dividingby a one-digit number. Contrary to Napier’s bones where the bones chosen cor-responded to the divider, with Genaille-Lucas’s rods, one selects the rods corre-sponding to the number to be divided. An index then provides an entry to the(one-digit) divider.
Cells now contain all the possible quotients and remainders. For instance,figure 23 shows the cell for dividing . . . 7 by 6. Since the previous remainder issmaller than 6, this cell encodes the divisions of 07 by 6, 17 by 6, 27 by 6, etc.,until 57 by 6. These six different divisions are shown by the six entry points atthe left of the cell. And for each division, the line leads to the correspondingremainder. In the example shown, dividing 17 by 6 results in a quotient of 2 (inblue) and a remainder of 5 (in red). The remainder will become the entry point ofthe cell located at its right. All cells are encoded similarly.
1
2
4
6
7
9
0
1
2
3
4
5
quotient remainder
6
ruler 7
Figure 23: The structure of a cell.
21
Figures 24 and 25 show computations examples and figure 26 shows the entireset of rods for division.
In figure 24, we divide 536027 by 4, and the entry point is the first line ofthe leftmost cell corresponding to the divider 4. It is the first line, because theincoming (left) remainder is naturally 0. Then, one has merely to follow the lines,and the result of the division can readily be read as 134006, with a remainderof 3: 536027 = 4× 134006+ 3. Additionnal figures can be obtained by addingadditional 0 rods.
1
3
6
8
50
3
5
8
31
4
6
9
60
2
5
7
00
3
5
8
21
4
6
9
7
4
0123
Figure 24: A computation example.
0
1
3
4
5
6
7
8
9
70
1
2
3
4
6
7
8
9
40
1
2
3
4
5
6
8
9
20
1
2
3
5
6
7
8
9
51
2
3
4
5
6
7
8
9
90
1
2
3
4
5
6
7
9
10
1
2
3
4
5
6
7
8
00
1
2
3
4
5
6
7
9
10
1
2
3
4
6
7
8
9
4
9
012345678
Figure 25: Another computation example.
22
0
5
0
3
6
0
2
5
7
0
2
4
6
8
0
1
3
5
6
8
0
1
2
4
5
7
8
0
1
2
3
5
6
7
8
0
1
2
3
4
5
6
7
8
00
5
0
3
7
0
2
5
7
0
2
4
6
8
0
1
3
5
6
8
0
1
3
4
5
7
8
0
1
2
3
5
6
7
8
0
1
2
3
4
5
6
7
9
11
6
0
4
7
0
3
5
8
0
2
4
6
8
0
2
3
5
7
8
0
1
3
4
6
7
8
0
1
2
4
5
6
7
9
0
1
2
3
4
5
6
8
9
21
6
1
4
7
0
3
5
8
0
2
4
6
8
0
2
3
5
7
8
0
1
3
4
6
7
9
0
1
2
4
5
6
7
9
0
1
2
3
4
5
7
8
9
32
7
1
4
8
1
3
6
8
0
2
4
6
8
0
2
4
5
7
9
0
2
3
4
6
7
9
0
1
3
4
5
6
8
9
0
1
2
3
4
6
7
8
9
42
7
1
5
8
1
3
6
8
1
3
5
7
9
0
2
4
5
7
9
0
2
3
5
6
7
9
0
1
3
4
5
6
8
9
0
1
2
3
5
6
7
8
9
53
8
2
5
8
1
4
6
9
1
3
5
7
9
1
2
4
6
7
9
0
2
3
5
6
8
9
0
2
3
4
5
7
8
9
0
1
2
4
5
6
7
8
9
63
8
2
5
9
1
4
6
9
1
3
5
7
9
1
2
4
6
7
9
1
2
3
5
6
8
9
0
2
3
4
5
7
8
9
0
1
3
4
5
6
7
8
9
74
9
2
6
9
2
4
7
9
1
3
5
7
9
1
3
4
6
8
9
1
2
4
5
6
8
9
1
2
3
4
6
7
8
9
0
2
3
4
5
6
7
8
9
84
9
3
6
9
2
4
7
9
1
3
5
7
9
1
3
4
6
8
9
1
2
4
5
7
8
9
1
2
3
4
6
7
8
9
1
2
3
4
5
6
7
8
9
9
2
3
4
5
6
7
8
9
01012012301234012345012345601234567012345678
Figure 26: The complete set of rods for division.
23
2.2.2 Division rods in other bases
Likewise, the division rods can be extended to other bases, and figure 27 shows therods for base 8. Figure 28 shows three examples using the base 16 rods, but theyare admittedly not that practical. Figure 29 show the structure of a hexadecimal(base 16) cell, and figure 30 shows a hexadecimal division restricted to one rowof the rods.
2
6
1
4
7
1
3
5
7
1
2
4
5
7
0
2
3
4
6
7
0
1
3
4
5
6
7
51
5
1
3
6
0
2
4
6
0
2
3
5
7
0
1
3
4
5
7
0
1
2
3
5
6
7
33
7
2
4
7
1
3
5
7
1
2
4
6
7
1
2
3
5
6
7
0
2
3
4
5
6
7
60
4
0
2
5
0
2
4
6
0
1
3
4
6
0
1
2
4
5
6
0
1
2
3
4
5
6
01
5
0
3
6
0
2
4
6
0
2
3
5
6
0
1
3
4
5
7
0
1
2
3
4
6
7
23
7
2
5
7
1
3
5
7
1
3
4
6
7
1
2
3
5
6
7
1
2
3
4
5
6
7
7
2
3
4
5
6
7
010120123012340123450123456
Figure 27: The division rods for base 8.
24
2
A
1
7
C
1
5
9
D
1
4
7
A
D
0
3
6
8
B
E
0
3
5
7
9
C
E
0
2
4
6
8
A
C
E
0
2
4
5
7
9
B
D
E
0
2
3
5
6
8
A
B
D
E
0
1
3
4
6
7
9
A
C
D
F
0
1
3
4
5
7
8
9
B
C
D
F
0
1
2
4
5
6
7
9
A
B
C
D
F
0
1
2
3
4
6
7
8
9
A
B
C
E
F
0
1
2
3
4
5
6
7
8
9
B
C
D
E
F
51
9
1
6
B
0
4
8
C
0
3
7
A
D
0
3
5
8
B
D
0
2
5
7
9
B
E
0
2
4
6
8
A
C
E
0
2
3
5
7
9
B
C
E
0
1
3
5
6
8
9
B
D
E
0
1
3
4
6
7
9
A
B
D
E
0
1
2
4
5
6
8
9
A
C
D
E
0
1
2
3
5
6
7
8
A
B
C
D
F
0
1
2
3
4
5
7
8
9
A
B
C
D
F
0
1
2
3
4
5
6
7
8
9
A
B
D
E
F
33
B
2
7
C
1
5
9
D
1
4
7
A
E
1
3
6
9
B
E
0
3
5
7
A
C
E
0
2
4
6
8
A
C
E
0
2
4
6
7
9
B
D
E
0
2
3
5
7
8
A
B
D
F
0
2
3
4
6
7
9
A
C
D
F
0
1
3
4
5
7
8
9
B
C
D
F
0
1
2
4
5
6
7
9
A
B
C
E
F
0
1
2
3
5
6
7
8
9
A
B
D
E
F
0
1
2
3
4
5
6
7
8
A
B
C
D
E
F
60
8
0
5
A
0
4
8
C
0
3
6
9
C
0
2
5
8
A
D
0
2
4
6
9
B
D
0
2
4
6
8
A
C
E
0
1
3
5
7
8
A
C
E
0
1
3
4
6
8
9
B
C
E
0
1
2
4
5
7
8
A
B
D
E
0
1
2
4
5
6
8
9
A
C
D
E
0
1
2
3
4
6
7
8
9
B
C
D
E
0
1
2
3
4
5
6
8
9
A
B
C
D
E
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
01
9
0
6
B
0
4
8
C
0
3
6
A
D
0
3
5
8
B
D
0
2
4
7
9
B
E
0
2
4
6
8
A
C
E
0
2
3
5
7
9
A
C
E
0
1
3
5
6
8
9
B
D
E
0
1
3
4
6
7
8
A
B
D
E
0
1
2
4
5
6
8
9
A
C
D
E
0
1
2
3
5
6
7
8
A
B
C
D
E
0
1
2
3
4
5
7
8
9
A
B
C
D
F
0
1
2
3
4
5
6
7
8
9
A
B
C
E
F
23
B
2
7
D
1
5
9
D
1
4
7
B
E
1
3
6
9
B
E
1
3
5
7
A
C
E
0
2
4
6
8
A
C
E
0
2
4
6
7
9
B
D
F
0
2
3
5
7
8
A
B
D
F
0
2
3
5
6
7
9
A
C
D
F
0
1
3
4
5
7
8
9
B
C
D
F
0
1
3
4
5
6
7
9
A
B
C
E
F
0
1
2
3
5
6
7
8
9
A
B
D
E
F
0
1
2
3
4
5
6
7
9
A
B
C
D
E
F
7
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0101201230123401234501234560123456701234567801234567890123456789A0123456789AB0123456789ABC0123456789ABCD0123456789ABCDE
2
A
1
7
C
1
5
9
D
1
4
7
A
D
0
3
6
8
B
E
0
3
5
7
9
C
E
0
2
4
6
8
A
C
E
0
2
4
5
7
9
B
D
E
0
2
3
5
6
8
A
B
D
E
0
1
3
4
6
7
9
A
C
D
F
0
1
3
4
5
7
8
9
B
C
D
F
0
1
2
4
5
6
7
9
A
B
C
D
F
0
1
2
3
4
6
7
8
9
A
B
C
E
F
0
1
2
3
4
5
6
7
8
9
B
C
D
E
F
51
9
1
6
B
0
4
8
C
0
3
7
A
D
0
3
5
8
B
D
0
2
5
7
9
B
E
0
2
4
6
8
A
C
E
0
2
3
5
7
9
B
C
E
0
1
3
5
6
8
9
B
D
E
0
1
3
4
6
7
9
A
B
D
E
0
1
2
4
5
6
8
9
A
C
D
E
0
1
2
3
5
6
7
8
A
B
C
D
F
0
1
2
3
4
5
7
8
9
A
B
C
D
F
0
1
2
3
4
5
6
7
8
9
A
B
D
E
F
33
B
2
7
C
1
5
9
D
1
4
7
A
E
1
3
6
9
B
E
0
3
5
7
A
C
E
0
2
4
6
8
A
C
E
0
2
4
6
7
9
B
D
E
0
2
3
5
7
8
A
B
D
F
0
2
3
4
6
7
9
A
C
D
F
0
1
3
4
5
7
8
9
B
C
D
F
0
1
2
4
5
6
7
9
A
B
C
E
F
0
1
2
3
5
6
7
8
9
A
B
D
E
F
0
1
2
3
4
5
6
7
8
A
B
C
D
E
F
60
8
0
5
A
0
4
8
C
0
3
6
9
C
0
2
5
8
A
D
0
2
4
6
9
B
D
0
2
4
6
8
A
C
E
0
1
3
5
7
8
A
C
E
0
1
3
4
6
8
9
B
C
E
0
1
2
4
5
7
8
A
B
D
E
0
1
2
4
5
6
8
9
A
C
D
E
0
1
2
3
4
6
7
8
9
B
C
D
E
0
1
2
3
4
5
6
8
9
A
B
C
D
E
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
01
9
0
6
B
0
4
8
C
0
3
6
A
D
0
3
5
8
B
D
0
2
4
7
9
B
E
0
2
4
6
8
A
C
E
0
2
3
5
7
9
A
C
E
0
1
3
5
6
8
9
B
D
E
0
1
3
4
6
7
8
A
B
D
E
0
1
2
4
5
6
8
9
A
C
D
E
0
1
2
3
5
6
7
8
A
B
C
D
E
0
1
2
3
4
5
7
8
9
A
B
C
D
F
0
1
2
3
4
5
6
7
8
9
A
B
C
E
F
23
B
2
7
D
1
5
9
D
1
4
7
B
E
1
3
6
9
B
E
1
3
5
7
A
C
E
0
2
4
6
8
A
C
E
0
2
4
6
7
9
B
D
F
0
2
3
5
7
8
A
B
D
F
0
2
3
5
6
7
9
A
C
D
F
0
1
3
4
5
7
8
9
B
C
D
F
0
1
3
4
5
6
7
9
A
B
C
E
F
0
1
2
3
5
6
7
8
9
A
B
D
E
F
0
1
2
3
4
5
6
7
9
A
B
C
D
E
F
75
D
3
8
E
2
6
A
E
2
5
8
B
E
1
4
7
9
C
F
1
3
6
8
A
C
F
1
3
5
7
9
B
D
F
1
2
4
6
8
A
B
D
F
1
2
4
5
7
9
A
C
D
F
0
2
3
5
6
8
9
B
C
E
F
0
2
3
4
6
7
8
A
B
C
E
F
0
2
3
4
5
6
8
9
A
B
D
E
F
0
1
3
4
5
6
7
8
9
B
C
D
E
F
0
1
2
3
4
6
7
8
9
A
B
C
D
E
F
A5
D
3
9
E
2
6
A
E
2
5
8
B
F
1
4
7
9
C
F
1
3
6
8
A
D
F
1
3
5
7
9
B
D
F
1
3
4
6
8
A
B
D
F
1
2
4
5
7
9
A
C
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F
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8
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B
C
E
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0
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A
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A
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E
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0
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D
E
F
0
1
2
3
5
6
7
8
9
A
B
C
D
E
F
B6
E
4
9
E
3
7
B
F
2
5
8
C
F
2
4
7
A
C
F
1
4
6
8
A
D
F
1
3
5
7
9
B
D
F
1
3
4
6
8
A
C
D
F
1
2
4
6
7
9
A
C
E
F
1
2
4
5
6
8
9
B
C
E
F
1
2
3
5
6
7
9
A
B
D
E
F
0
2
3
4
5
7
8
9
A
C
D
E
F
0
2
3
4
5
6
7
8
A
B
C
D
E
F
0
1
2
4
5
6
7
8
9
A
B
C
D
E
F
C
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0101201230123401234501234560123456701234567801234567890123456789A0123456789AB0123456789ABC0123456789ABCD0123456789ABCDE
0
8
0
5
B
0
4
8
C
0
3
6
9
D
0
2
5
8
A
D
0
2
4
7
9
B
D
0
2
4
6
8
A
C
E
0
1
3
5
7
9
A
C
E
0
1
3
4
6
8
9
B
C
E
0
1
3
4
5
7
8
A
B
D
E
0
1
2
4
5
6
8
9
A
C
D
E
0
1
2
3
5
6
7
8
9
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C
D
E
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1
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3
4
5
6
8
9
A
B
C
D
E
0
1
2
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5
6
7
8
9
A
B
C
D
F
13
B
2
7
D
1
5
9
D
1
4
7
B
E
1
3
6
9
B
E
1
3
5
7
A
C
E
0
2
4
6
8
A
C
E
0
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4
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7
9
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D
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7
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A
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D
F
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3
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6
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9
A
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C
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1
3
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9
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1
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9
A
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75
D
3
8
E
2
6
A
E
2
5
8
B
E
1
4
7
9
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F
1
3
6
8
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C
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3
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7
9
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D
F
1
2
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6
8
A
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7
9
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C
D
F
0
2
3
5
6
8
9
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C
E
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0
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7
8
A
B
C
E
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A1
9
0
6
B
0
4
8
C
0
3
6
A
D
0
3
5
8
B
D
0
2
4
7
9
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8
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9
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27
F
5
A
F
3
7
B
F
3
6
9
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F
2
5
7
A
D
F
2
4
6
9
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9
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9
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C
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8
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9
A
B
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1
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1
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A
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A
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F2
A
1
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C
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A
D
0
3
6
8
B
E
0
3
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7
9
C
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8
A
C
E
0
2
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9
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C
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55
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E
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F
1
4
7
9
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F
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A
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1
3
5
7
9
B
D
F
1
3
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A
B
C
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F
0
2
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E
F
0
1
3
4
5
6
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9
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C
D
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F
0
1
2
3
5
6
7
8
9
A
B
C
D
E
F
B3
B
2
7
D
1
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9
D
1
4
7
B
E
1
3
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9
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E
1
3
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2
4
6
8
A
C
E
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2
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6
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8
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1
3
4
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7
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9
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C
D
F
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1
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E
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1
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76
E
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7
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7
8
A
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1
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6
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8
9
A
B
C
D
E
F
D0
8
0
5
B
0
4
8
C
0
3
6
9
D
0
2
5
8
A
D
0
2
4
7
9
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D
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A
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E
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C
D
E
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1
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8
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A
B
C
D
E
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1
2
3
4
5
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7
8
9
A
B
C
D
F
14
C
3
8
D
2
6
A
E
1
5
8
B
E
1
4
6
9
C
E
1
3
5
8
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C
F
1
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9
B
D
F
1
2
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A
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D
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F
0101201230123401234501234560123456701234567801234567890123456789A0123456789AB0123456789ABC0123456789ABCD0123456789ABCDE
Figure 28: Three examples in base 16.
25
1
3
6
9
B
E
0
1
2
3
4
5
quotient remainder
6
ruler 7
Figure 29: A division cell in base 16.
1
5
9
D
50
4
8
C
31
5
9
D
60
4
8
C
00
4
8
C
21
5
9
D
72
6
A
E
A2
6
A
E
B3
7
B
F
C
4
0123
Figure 30: A division in base 16 showing only the row used.
26
2.3 Financial rodsIn financial computations, the assumption is sometimes that the commercial yearonly contains 360 days, and that the daily rate is defined as the yearly rate dividedby 360. This is of course an approximation, but it is a good approximation forsmall rates.
These assumptions make it very easy to adapt the division rods for financialcomputations. In fact, the financial rods are merely division rods restricted to thedividers 2, 4, 6, 8, 9 and 12 (see figure 31). The last strip, for the divider 12, isconstructed like the other ones, but for clarity some lines are dashed and we haverepresented them as in the set originally published. Now, how are these rods usedand how do they relate to rates?
If we have a yearly rate of 3%, and start with an amount of 1234 euros, thenthe daily rate is naturally
1234× 3100
360=
123412
1000.
Thus, we see that dividing by 12 will correspond to a rate of 3%. Similarly, wewould find that the dividers 9, 8, 6 and 4 correspond to rates of 4%, 4.5%, 6%and 9%. These rates were written on the index ruler. The rate corresponding todividing by 2 is 18% and was not written for lack of space.
In the original instructions (see their reproduction in Weiss’ article [39]), theexample of an initial sum of 52300 francs with a rate of 4.5% is taken, and therods yield the value 06537, which needs to be divided by 1000, resulting in adaily rate of 6 francs and 537 thousandths. These instructions also mention thatthe rate of 5%, although not provided by the rods, can easily be obtained by firstcomputing the daily rate at 4%, and adding to the value found its quarter, whichcan be obtained by the row for 9%. For instance, if we wish to have the daily rateof 52300 francs at 5%, we would first obtain 5.811 francs (4%), and the quarterof this value is 1.452 francs (1%), which added to 5.811 yields 7.263 francs (5%).With the above definition of a daily rate, percentages can be added.
Examples of computations are shown in figures 32 and 33. In the first case, westart with the sum of 536027 euros at a rate of 9% and obtain a value of 134006,that is 134 euros and 6 thousandth for one day. In the second case, we start with7611459040 euros at 3%, and obtain a daily rate of 634288.253 euros.
27
0
5
0
2
5
7
0
1
3
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6
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0
1
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6
7
8
0
1
2
3
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6
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0
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5
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8
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00
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6
7
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1
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0
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5
5
6
7
8
9
11
6
0
3
5
8
0
2
3
5
7
8
0
1
2
4
5
6
7
9
0
1
2
3
4
5
6
8
9
0
1
1
2
3
4
5
6
6
7
8
9
21
6
0
3
5
8
0
2
3
5
7
8
0
1
2
4
5
6
7
9
0
1
2
3
4
5
7
8
9
0
1
1
2
3
4
5
6
6
7
8
9
32
7
1
3
6
8
0
2
4
5
7
9
0
1
3
4
5
6
8
9
0
1
2
3
4
6
7
8
9
0
1
2
2
3
4
5
6
7
7
8
9
42
7
1
3
6
8
0
2
4
5
7
9
0
1
3
4
5
6
8
9
0
1
2
3
5
6
7
8
9
0
1
2
2
3
4
5
6
7
7
8
9
53
8
1
4
6
9
1
2
4
6
7
9
0
2
3
4
5
7
8
9
0
1
2
4
5
6
7
8
9
0
1
2
3
3
4
5
6
7
8
8
9
63
8
1
4
6
9
1
2
4
6
7
9
0
2
3
4
5
7
8
9
0
1
3
4
5
6
7
8
9
0
1
2
3
3
4
5
6
7
8
8
9
74
9
2
4
7
9
1
3
4
6
8
9
1
2
3
4
6
7
8
9
0
2
3
4
5
6
7
8
9
0
1
2
3
4
4
5
6
7
8
9
9
84
9
2
4
7
9
1
3
4
6
8
9
1
2
3
4
6
7
8
9
1
2
3
4
5
6
7
8
9
0
1
2
3
4
4
5
6
7
8
9
9
912
14
9 %16
6 %18
41/2%19
4 %112
3 %
Figure 31: The financial rods.
28
2
7
1
3
6
8
0
2
4
5
7
9
0
1
3
4
5
6
8
9
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6
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8
9
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1
2
2
3
4
5
6
7
7
8
9
51
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5
8
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3
5
7
8
0
1
2
4
5
6
7
9
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2
3
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5
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8
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0
1
1
2
3
4
5
6
6
7
8
9
33
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1
4
6
9
1
2
4
6
7
9
0
2
3
4
5
7
8
9
0
1
2
4
5
6
7
8
9
0
1
2
3
3
4
5
6
7
8
8
9
60
5
0
2
5
7
0
1
3
5
6
8
0
1
2
3
5
6
7
8
0
1
2
3
4
5
6
7
8
0
0
1
2
3
4
5
5
6
7
8
9
01
6
0
3
5
8
0
2
3
5
7
8
0
1
2
4
5
6
7
9
0
1
2
3
4
5
6
8
9
0
1
1
2
3
4
5
6
6
7
8
9
23
8
1
4
6
9
1
2
4
6
7
9
0
2
3
4
5
7
8
9
0
1
3
4
5
6
7
8
9
0
1
2
3
3
4
5
6
7
8
8
9
712
14
9 %16
6 %18
41/2%19
4 %112
3 %
Figure 32: An example of computation.
29
0
1
2
3
3
4
5
6
7
8
8
9
70
1
2
3
3
4
5
6
7
8
8
9
60
0
1
2
3
4
5
5
6
7
8
9
10
0
1
2
3
4
5
5
6
7
8
9
10
1
2
2
3
4
5
6
7
7
8
9
40
1
2
2
3
4
5
6
7
7
8
9
50
1
2
3
4
4
5
6
7
8
9
9
90
0
1
2
3
4
5
5
6
7
8
9
00
1
2
2
3
4
5
6
7
7
8
9
40
0
1
2
3
4
5
5
6
7
8
9
0112
3 %
Figure 33: A computation shown in isolation.
30
References[1] Conservatoire national des arts et métiers. Catalogue du musée, section A :
instruments et machines à calculer. Paris: CNAM, 1942.
[2] Anne-Marie Décaillot. L’arithméticien Édouard Lucas (1842–1891) :théorie et instrumentation. Revue d’histoire des mathématiques, 4:191–236,1998.
[3] Anne-Marie Décaillot-Laulagnet. Édouard Lucas (1842–1891) : leparcours original d’un scientifique français dans la deuxième moitié duXIXe siècle. PhD thesis, Université René Descartes-Paris V, 1999.
[4] Martin Gardner. Knotted doughnuts and other mathematicalentertainments. New York: W. H. Freeman, 1986. [see chapter 7].
[5] Les appareils à calculs exacts et instantanés pour simplifier la multiplicationet la division, inventés par M. Henri Genaille, et perfectionnés parM. Édouard Lucas. Nouvelles annales de mathématiques, 4 (3rdseries):516–519, November 1885.
[6] Henri Genaille. Sur une nouvelle machine à calculer. In Associationfrançaise pour l’avancement des sciences : Compte rendu de la 7e session,Paris, 1878, pages 181–182. Paris: Association française pourl’avancement des sciences, 1879.
[7] Henri Genaille. Machine à calculer. In Association française pourl’avancement des sciences : Compte rendu de la 12e session, Rouen, 1883,pages 177–178. Paris: Association française pour l’avancement dessciences, 1884.
[8] Henri Genaille. Le calculateur Henri Genaille. In Association françaisepour l’avancement des sciences : Compte rendu de la 23e session, Caen,1894, pages 272–276. Paris: Association française pour l’avancement dessciences, 1895.
[9] Henri Genaille and Édouard Lucas. Les réglettes financières, appareils àcalculs exacts et instantanés pour simplifier les calculs financiers etcommerciaux. Paris: Librairie classique Eugène Belin, 1885. [set of actualrods with instructions].
[10] Henri Genaille and Édouard Lucas. Les réglettes multiplicatrices, appareilsà calculs exacts et instantanés pour simplifier la multiplication et ladivision. Paris: Librairie classique Eugène Belin, 1885. [set of actual rodswith instructions].
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[11] Henri Genaille and Édouard Lucas. Les réglettes multisectrices, appareils àcalculs exacts et instantanés pour simplifier la division. Paris: Librairieclassique Eugène Belin, 1885. [set of actual rods with instructions].
[12] Henri Genaille and Édouard Lucas. Les réglettes népériennes, joujouxcalculateurs ayant pour but de simplifier l’étude et de faciliter la pratiquedes opérations de l’arithmétique. Paris: Librairie classique Eugène Belin,1885. [set of actual rods with instructions].
[13] Charles-Édouard Guillaume. Les baguettes de Napier. La Nature,916:37–38, 1891.
[14] Charles-Édouard Guillaume. Les réglettes multiplicatrices. La Nature,962:355–356, 1891.
[15] Louis Frédéric Gustave Jacob. Le calcul mécanique. Paris: Octave Doin etfils, 1911. [Genaille’s rods are briefly described on page 8].
[16] B. R. Jones. Genaille’s rods: an ingenious improvement on Napier’s. TheMathematical Gazette, 48(363):17–22, February 1964.
[17] Philipp S. Jones. Tangible arithmetic I: Napier’s and Genaille’s rods. TheMathematics Teacher, 47(7):482–487, November 1954.
[18] Harold Daniel Larsen. Genaille’s rods. The American MathematicalMonthly, 60(2):141, 1953. [very brief note].
[19] Françoise Launay. Antoine Raussain (1706–1786), un maître demathématiques estimé de D’Alembert. Recherches sur Diderot et surl’Encyclopédie, 47:315–323, 2012.
[20] Eugène Lemaire. Exposition publique de machines à calculer anciennes etmodernes. Bulletin de la Société d’encouragement pour l’industrienationale, 132:608–644, septembre-octobre 1920. [pp. 610, 614 and 625 onGenaille’s rods].
[21] Édouard Lucas. Le calcul et les machines à calculer. In Associationfrançaise pour l’avancement des sciences : Compte rendu de la 13e session,Blois, 1884, première partie, pages 111–141. Paris: Association françaisepour l’avancement des sciences, 1885. [pp. 139–141 on Genaille].
[22] Édouard Lucas. Récréations mathématiques, volume 3. Paris:Gauthier-Villars, 1893. [pages 81–83 on Genaille’s rods].
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[23] Jean Marguin. Histoire des instruments et machines à calculer : Troissiècles de mécanique pensante, 1642–1942. Paris: Hermann, 1994. [seepages 30–34 for Napier’s bones, and pages 35–37 for Genaille-Lucas’srods].
[24] Rudolf Mehmke. Numerisches Rechnen. In Enzyklopädie dermathematischen Wissenschaften mit Einschluss ihrer Anwendungen,volume 1-2, pages 938–1079. Leipzig: B. G. Teubner, 1900–1904.[Mehmke’s chapter was published in 1902. A French translation by Mauriced’Ocagne was published in the Encyclopédie des sciences mathématiquespures et appliquées, tome 1, volume 4, 1906–1911, fascicules 2 (1908) and3 (1909), pages 196–452, under the title “Calculs numériques.” TheGenaille-Lucas rulers are briefly mentioned on pages 956–957.].
[25] John Napier. Mirifici logarithmorum canonis descriptio. Edinburgh:Andrew Hart, 1614.
[26] John Napier. Rabdologiæ, seu numerationis per virgulas libri duo.Edinburgh: Andrew Hart, 1617.
[27] John Napier. Rabdology. Cambridge, Massachusetts: The MIT Press, 1990.[English translation of [26]].
[28] Georges Nicoladze. Arithmomètre à multiplication directe purementélectrique. Comptes rendus hebdomadaires des séances de l’Académie dessciences, 186:123–124, 1928.
[29] Georges Nicoladze. Arithmomètre électrique pour faire directement lesmultiplications et divisions (French patent 647.778), 1928.
[30] Maurice d’Ocagne. Le calcul simplifié par les procédés mécaniques etgraphiques. Paris: Gauthier-Villars, 1905. [second edition, pages 14–18 onGenaille’s rods].
[31] Maurice d’Ocagne. Le calcul simplifié: graphical and mechanical methodsfor simplifying calculation. Cambridge, MA: MIT Press, 1986.[Translation of the 1928 French edition, see pages 7–8 on Genaille’s rods.].
[32] Brian Randell, editor. The origins of digital computers: Selected papers.New York: Springer-Verlag, 1982.
[33] Denis Roegel. Napier’s ideal construction of the logarithms. Technicalreport, LORIA, Nancy, 2010.
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[34] Carl R. Seaquist, Padmanabhan Seshaiyer, and Dianne Crowley.Calculation across cultures and history. Texas College MathematicsJournal, 1(1):15–31, 2005.
[35] René Taton and Jean-Paul Flad. Le calcul mécanique, volume 367 of Quesais-je ? Paris: Presses Universitaires de France, 1963. [pp. 15–18 oncomputing rods, including Genaille’s].
[36] Walther von Dyck. Katalog mathematischer undmathematisch-physikalischer Modelle, Apparate und Instrumente.München: C. Wolf, 1892. [brief description of Genaille-Lucas’s rulers onpage 146].
[37] Stephan Weiss. Die Multipliziertafeln, ihre Ausgestaltung undVerwendung, 2003. http://www.mechrech.info.
[38] Stephan Weiss. Die Multiplizierstäbe von Genaille und Lucas —Beschreibung und Vergleich mit den Rechenstäben von Neper, 2004.http://www.mechrech.info.
[39] Stephan Weiss. Die Dividierstäbe und die Finanzrechenstäbe von Genailleund Lucas, 2005. http://www.mechrech.info.
[40] Stephan Weiss. Die Rechenstäbe von Neper, ihre Varianten undNachfolger, 2007. http://www.mechrech.info.
[41] Hugh C. Williams and Jeffrey Outlaw Shallit. Factoring integers beforecomputers. In Walter Gautschi, editor, Mathematics of Computation 50thAnniversary Symposium, Vancouver, British Columbia, August 9–13, 1993,volume 48 of Proceedings of Symposia in Applied Mathematics, pages481–531. American Mathematical Society, 1994.
[42] Michael Roy Williams. From Napier to Lucas: The use of Napier’s bonesin calculating instruments. IEEE Annals of the history of computing,5(3):279–296, July 1983.
[43] Michael Roy Williams. Early calculation. In William Aspray, editor,Computing before computers, chapter 1, pages 3–58. Ames, Iowa: IowaState University Press, 1990. [Genaille’s rods are described on pages20–23].
[44] Michael Roy Williams. A history of computing technology. Los Alamitos:IEEE Computer Society Press, 1997. [second edition, Genaille’s rods aredescribed on pp. 93–96].
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