The Dozenal Society of America A Scale Edward Brooks, Ph.D.∗

he mode of reckoning by twelves and measures, as twelve to a , or dozens, may be supposed to have twelve to a ; and is still very gener- T had its origin in the observation of ally employed in wholesale business, extend- the celestial phenomena, there being twelve ing to the second and even to the third term months or lunations commonly reckoned in of the progression. Thus, twelve dozen, or a solar year. The Romans likewise adopted 144, make the of the northern the same number to mark the subdivisions of nations, or the gross of traders; and twelve their unit of measure or of weight. The scale times this again, or 1728, make the double or appears also in our subdivisions of weights great gross.

A Duodecimal Scale

s already explained, any number system. Second, in naming numbers by any A may be made the basis of a system of system, we give independent names up to the numeration and notation. The decimal basis base, and then reckon by groups, using the is a mere accident, and in some respects an simple names to number the groups. Bearing unfortunate one, both for science and art. The these principles in mind, we are ready to un- duodecimal basis would have been greatly su- derstand Numeration, Notation, and the Fun- perior, giving greater simplicity to the science, damental Rules in Duodecimal Arithmetic. and facilitating its various applications. In Numeration.—In naming numbers by this chapter it will be explained how arith- the duodecimal system, we would first name metic might have been developed upon a the simple numbers from one to eleven, and duodecimal basis. then, adding one more unit, form a group, and In order to make the matter clear, I call name this group twelve. We would then, as attention to two or three principles of numera- in the decimal system, use these first names tion and notation. First, the bases of numera- to number the groups. Naming numbers in tion and notation should be the same; that is, this way, we would have the simple names, if we write numbers in a duodecimal system, one, two, three, etc., up to twelve. Continuing we should also name numbers by a duodecimal from twelve, we would have one and twelve,

∗Originally published in The Philosophy of Arithmetic as Developed from the Three Funda- mental Processes of Sythesis, Analysis, and Comparison, published by Sower, Potts & Company, Philadelphia, 1104 (1876.), by the same author. It is made available in this newly typeset, standalone format by the Dozenal Society of America, 11EX (2014.) (http://www.dozenal.org). Text courtesy of Google Books.

1 two and twelve, three and twelve, etc., up to cording to a duodecimal system could be eas- twelve and twelve, which we would call two ily applied. Were we actually forming such a twelves. Passing on from this we would have system, the simplest method would be to in- two twelves and one, two twelves and two, etc., troduce only a few new names for the smaller to three twelves, and so on until we reach groups, and then take the names of the higher twelve twelves, when we would form a new groups of the decimal system, with perhaps a group containing twelve twelves, and give this slight modification in their orthography and new group a new name, as gross, and then pronunciation, to name the higher groups of employ the first simple names again to num- the new scale. Thus, million, billion, etc., ber the gross. In this way we would continue could be used to name the new groups with- grouping by twelves, and giving a new name out any confusion, as they do not indicate to each group, as in the decimal scale by tens, any definite number of units to the mind, but as far as is necessary. merely so many collections of smaller collec- These names, in the duodecimal system, tions. Indeed, even the word thousand, with a would naturally become abbreviated by use, modification of its orthography, say thousun, as the corresponding names in the decimal might be used to represent a collection of system. Thus, as in the decimal system ten twelve groups, each containing a gross, with- was changed to teen, we may suppose twelve out any confusion of ideas. Their etymolog- to be changed to teel, and omitting the “and” ical formation would not be an objection of as in the common system, we would count any particular force, as no one in using them one-teel, two-teel, thir-teel, four-teel, fif-teel, thinks of their primary signification. These six-teel, etc., up to eleven-teel. Two-twelves terms are not suggested as the best, but as might be changed into two-tel, or twen-tel, cor- the simplest in making the transition from responding to two-ty or twenty, and we would the old to the new system. It will also be no- continue to count twentel-one, twentel-two, ticed that our departure in the decimal scale etc. Three-twelves might be contracted into from the principle of the system, by using the three-tel or thirtel, corresponding to three-ty or terms eleven and twelve, would facilitate the thirty of the decimal system; four-twelves to adoption of a duodecimal system. fourtel, five twelves to fiftel, etc., up to a gross. To illustrate the subject more fully, let us Proceeding in the same manner, a collection adopt the names suggested, and apply them to of twelve gross would need a new name, and the scale. Naming numbers according to the thus on to the higher groups of the scale. method explained, we would have the names In this manner, the names of numbers ac- as indicated in the following series:

one oneteel twentel-one one gross and one two twoteel twentel-two one gross and two three thlrteel twentel-eight two gross and five four fourteel twentel-eleven six gross and seven five fifteel thirtel-one ten gross and eight six sixteel fortel-two eleven gross am nine seven seventeel fiftel-six one thousun eight eighteel sixtel-eight one thousun and five nine nineteel seventel-nine one thousun four gross ten tenteel tentel-ten and seven eleven eleventeel eleventel-eleven two thousun seven gross twelve twentel one gross and fortel-one

2 Notation.—The writing of numbers by the character Φ and eleven by Π. the duodecimal system would be an immediate These characters, with the zero, would be outgrowth of the method of naming numbers combined to represent numbers in the duodec- in this system. As in the decimal system of imal scale in the same manner as the nine notation, it would be necessary to employ a digits represent numbers in the decimal scale. number of characters one less than the number Thus, twelve would be represented by 10, sig- of units in the base, besides the character for nifying one of the groups containing twelve; 11 nothing. Since the group contains twelve units, would represent one and twelve, or oneteel; 12 the number of significant characters would be would represent two and twelve, or twoteel; 13 eleven—two more than in the decimal system. would represent thirteel; 14, fourteel; 15, fif- For these characters we should use the nine teel, etc. Continuing thus, 20 would represent digits of the decimal system, and then intro- two twelves, or twentel; 21, twentel-one; 23, duce new characters for the numbers ten and twentel-three, etc. The notation of numbers eleven. To illustrate, we will represent ten by up to a thousun may be indicated as follows:—

one, 1 twelve, 10 thirtel, 30 two, 2 oneteel, 11 thirtel-two, 32 three, 3 twoteel, 12 thirtel-five, 35 etc., etc. twentel, 20 thirtel-ten, 3Φ nine, 9 twentel-one, 21 thirtel-eleven, 3Π ten, Φ twentel-ten, 2Φ one gross, 100 eleven, Π twentel-eleven, 2Π one thousun, 1000

Extending the series as explained above, we should have the following notation table:— Trillyuns. Gross of Billyuns. Twelves of Billyuns. Billyuns. Gross of Millyuns. Twelves of Millyuns. Millyuns. Gross of Thousuns. Twelves of Thousuns. Thousuns. Gross. Twelves. Units. 8 5 Π 4 6 Π 8 5 7 Φ 3 6 5 From the explanation given it is clearly quired, as all of its methods and principles seen that a system of duodecimal arithmetic would remain unchanged. Indeed, so readily might be easily developed, and readily learned could the change be made, that in view of the and reduced to practice. Employing the names great advantages of the system, one is almost which I have indicated, or others similar to ready to believe that the time will come when them, the change from the decimal to the scientific men will turn their attention seri- duodecimal system would be much less diffi- ously to the matter and endeavor to effect the cult than has usually been supposed. It would change. be necessary to learn the method of naming Fundamental Operations.—In order and writing numbers, which we have seen is to show how readily the transition could be very simple, and a new addition and multipli- made, I will present the method of operation cation table, from which we could readily de- in the fundamental rules. We would proceed rive the elementary differences and quotients. first to form an addition table containing the The rest of the science would be readily ac- elementary sums, which, as in the decimal

3 system, we would commit to memory. From “twelve times,” as in our present system there this we could readily derive the elementary is no need of extending beyond “ten times.” differences used in subtraction. Such a table From this table of elementary products, we is given on page 6. can readily derive the table of elementary quo- By means of this table we tients as we do in the decimal system. Such a can readily find the sum or dif- Operation table will be found on page 6. ference of numbers expressed 487Π It will be interesting to notice several pe- in the duodecimal system. To 5Φ38 culiarities of this table, similar to those of illustrate, required the sum of 63Π7 the decimal system. As the column of “five 487Π, 5Φ38, 63Π7, Φ856. The Φ856 times” ends alternately in 5 and 0, making it solution of this would be as 23748 so easily learned by children, so the column of follows: Adding the column “six times” in the duodecimal table will end of units, 6 units and 7 units are 11 units, alternately in 6 and 0. In our present table and 8 units are 19 units, and Π units are the sum of the two terms of each product in 28 units, or 2 twelves and 8 units; writing the column of “nine times” equals nine, so the units, and carrying 2 to the column of in the duodecimal table, the sum of the two twelves, we have 2 twelves and 5 twelves are terms of each product in the column of “eleven 7 twelves, and Π twelves are 16 twelves, and times” equals eleven. We also notice that each 3 twelves are 19 twelves, and 7 twelves are 24 product in the column of “twelve times” ends twelves, or 2 gross and 4 twelves; writing the in 0, as does each product in the column of twelves, and carrying 2 to the third column, “ten times” of our present table. we have 2 gross and 8 gross are Φ gross, and By means of the multipli- 3 gross are 11 gross, and Φ gross are 1Π gross, cation table we can readily Operation and 8 gross are 27 gross, or 2 thousuns and find the product or quotient 34Φ8 7 gross; 2 thousuns and Φ thousuns are 10 of numbers expressed in the 3Π7 thousuns, and 6 thousuns are 16 thousuns, and duodecimal scale. To illus- 31Φ28 5 thousuns are 1Π thousuns, and 4 thousuns trate multiplication, let it be 4Π594 are 23 thousuns; hence the amount is 23748. required to find the product of 14280 To illustrate subtraction 54Φ8 by 3Π7. We would solve 1953768 let it be required to find the Operation this as follows: Using the first difference between 6428 and 6428 term of the multiplier, 7 times 8 are 48, 7 2564. We would solve this as 2564 times Φ are 5Φ, and 4 are 62, 7 times 4 are follows: Subtracting 4 units 3Φ84 24 and 6 are 2Φ, 7 times 5 are 2Π and 2 are from 8 units we have 4 units 31, making the first partial product 31Φ28; remaining; we cannot take 6 twelves from 2 multiplying by Π we have Π times 8 are 74, twelves, so we add 10 twelves and have 12 Π times Φ are 92 and 7 are 99, Π times 4 are twelves; 6 twelves from 12 twelves leaves 8 38 and 9 are 45, Π times 5 are 47 and 4 are twelves; carrying 1 to 5 we have 6 gross; we 4Π; 3 times 8 are 20, 3 times Φ are 26 and 2 cannot take 6 gross from 4 gross; adding 10 are 28, 3 times 4 are 10 and 2 are 12, 3 times as before we have 6 gross from 14 gross leaves 5 are 13 and 1 are 14. Adding up the partial Φ gross; adding 1 thousun to 2 thousuns, we products, we have as the complete product, have 3 thousuns from 6 thousuns leaves 3 1953768. thousuns; hence the remainder is 3Φ84. To illustrate division, let it be required to In order to multiply and divide, we first find the quotient of 1953768 divided by 3Π7. form a multiplication table similar to that now We would solve this as follows: We find that used in the decimal system, and commit it to the divisor is contained in the first four terms memory. This table need not extend beyond of the dividend 5 times, and multiplying 3Π7

4 by 5 we have 179Π; subtracting this from the dividing by 2 times 3 or 6, we find the second dividend we have a remainder, 174; bringing term of the root to be 4; completing the di- down the next figure of the dividend and pro- visor and multiplying 64 by 4, we have 214; ceeding as before, we have for the quotient subtracting and bringing down, we have 3Π01, 54Φ8. and dividing by 2 times 34, or 68, we have 7 for the figure of the root; completing the Operation divisor and multiplying it by 7, we have 3Π01, 3Π7)1953768)54Π8 which leaves no remainder. 179Π The above tables and calculations seem 1747 awkward to one who is familiar with the deci- 13Φ4 mal system; but it should be remembered that 3636 337Φ a beginner would learn the addition and mul- 2788 tiplication tables and the calculations based 2788 on them, just as readily as he now learns them in the decimal system. The practical value of such a system, in addition to what has already The method of finding been said, may be seen in the calculation of the square or cube root of Operation interest, the rules for which would be greatly a number expressed in the Π·53·01(347 simplified on account of the relation of the duodecimal scale is similar 9 number of months in a year (12) to the base, to that used in the decimal 64)253 and also of the relation of the rate to the same, scale, as may be shown by 214 which would be some 8% or 9% ; that is, 8 or an example. Thus, find the 687)3Π01 9 per gross. I hope to be able in a few years square root of Π5301. The 3Π01 to publish a small work in which the whole greatest square in Π is 9; science of arithmetic shall be developed on subtracting and bringing down a period, and the duodecimal basis.

This text is reproduced verbatim from the chapter (1876.). This text is in the public domain and was and some introductory material as found in The Phi- obtained from Google Books. It has been typeset en- losophy of Arithmetic as Developed from tirely anew and standalone by the Dozenal Society of the Three Fundamental Processes of Sythe- America, 11EX (2014.) (http://www.dozenal.org), sis, Analysis, and Comparison, which was pub- using the LATEX document preparation system and lished by Sower, Potts & Company, Philadelphia, 1104 associated packages.

5 Addition Table 2+1=3 3+1=4 4+1=5 5+1=6 6+1=7 7+1=8 8+1=9 9+1=Φ Φ+1=Π Π+1=10 10+1=11 2+2=4 3+2=5 4+2=6 5+2=7 6+2=8 7+2=9 8+2=Φ 9+2=Π Φ+2=10 Π+2=11 10+2=12 2 + 3 = 5 3 + 3 = 6 4 + 3 = 7 5 + 3 = 8 6 + 3 = 9 7 + 3 = Φ 8 + 3 = Π 9 + 3 = 10 Φ + 3 = 11 Π + 3 = 12 10 + 3 = 13 2 + 4 = 6 3 + 4 = 7 4 + 4 = 8 5 + 4 = 9 6 + 4 = Φ 7 + 4 = Π 8 + 4 = 10 9 + 4 = 11 Φ + 4 = 12 Π + 4 = 13 10 + 4 = 14 2 + 5 = 7 3 + 5 = 8 4 + 5 = 9 5 + 5 = Φ 6 + 5 = Π 7 + 5 = 10 8 + 5 = 11 9 + 5 = 12 Φ + 5 = 13 Π + 5 = 14 10 + 5 = 15 2 + 6 = 8 3 + 6 = 9 4 + 6 = Φ 5 + 6 = Π 6 + 6 = 10 7 + 6 = 11 8 + 6 = 12 9 + 6 = 13 Φ + 6 = 14 Π + 6 = 15 10 + 6 = 16 2 + 7 = 9 3 + 7 = Φ 4 + 7 = Π 5 + 7 = 10 6 + 7 = 11 7 + 7 = 12 8 + 7 = 13 9 + 7 = 14 Φ + 7 = 15 Π + 7 = 16 10 + 7 = 17 2 + 8 = Φ 3 + 8 = Π 4 + 8 = 10 5 + 8 = 11 6 + 8 = 12 7 + 8 = 13 8 + 8 = 14 9 + 8 = 15 Φ + 8 = 16 Π + 8 = 17 10 + 8 = 18 2 + 9 = Π 3 + 9 = 10 4 + 9 = 11 5 + 9 = 12 6 + 9 = 13 7 + 9 = 14 8 + 9 = 15 9 + 9 = 16 Φ + 9 = 17 Π + 9 = 18 10 + 9 = 19 2 + Φ = 10 3 + Φ = 11 4 + Φ = 12 5 + Φ = 13 6 + Φ = 14 7 + Φ = 15 8 + Φ = 16 9 + Φ = 17 Φ + Φ = 18 Π + Φ = 19 10 + Φ = 1Φ 2 + Π = 11 3 + Π = 12 4 + Π = 13 5 + Π = 14 6 + Π = 15 7 + Π = 16 8 + Π = 17 9 + Π = 18 Φ + Π = 19 Π + Π = 1Φ 10 + Π = 1Π 2 + 10 = 12 3 + 10 = 13 4 + 10 = 14 5 + 10 = 15 6 + 10 = 16 7 + 10 = 17 8 + 10 = 18 9 + 10 = 19 Φ + 10 = 1Φ Π + 10 = 1Π 10 + 10 = 20 6

Multiplication Table 2 × 1 = 2 3 × 1 = 3 4 × 1 = 4 5 × 1 = 5 6 × 1 = 6 7 × 1 = 7 8 × 1 = 8 9 × 1 = 9 Φ × 1 = Φ Π × 1 = Π 10 × 1 = 10 2 × 2 = 4 3 × 2 = 6 4 × 2 = 8 5 × 2 = Φ 6 × 2 = 10 7 × 2 = 12 8 × 2 = 14 9 × 2 = 16 Φ × 2 = 18 Π × 2 = 1Φ 10 × 2 = 20 2 × 3 = 6 3 × 3 = 9 4 × 3 = 10 5 × 3 = 13 6 × 3 = 16 7 × 3 = 19 8 × 3 = 20 9 × 3 = 23 Φ × 3 = 26 Π × 3 = 29 10 × 3 = 30 2 × 4 = 8 3 × 4 = 10 4 × 4 = 14 5 × 4 = 18 6 × 4 = 20 7 × 4 = 24 8 × 4 = 28 9 × 4 = 30 Φ × 4 = 34 Π × 4 = 38 10 × 4 = 40 2 × 5 = Φ 3 × 5 = 13 4 × 5 = 18 5 × 5 = 21 6 × 5 = 26 7 × 5 = 2Π 8 × 5 = 34 9 × 5 = 39 Φ × 5 = 42 Π × 5 = 47 10 × 5 = 50 2 × 6 = 10 3 × 6 = 16 4 × 6 = 20 5 × 6 = 26 6 × 6 = 30 7 × 6 = 36 8 × 6 = 40 9 × 6 = 46 Φ × 6 = 50 Π × 6 = 56 10 × 6 = 60 2 × 7 = 12 3 × 7 = 19 4 × 7 = 24 5 × 7 = 2Π 6 × 7 = 36 7 × 7 = 41 8 × 7 = 48 9 × 7 = 53 Φ × 7 = 5Φ Π × 7 = 65 10 × 7 = 70 2 × 8 = 14 3 × 8 = 20 4 × 8 = 28 5 × 8 = 34 6 × 8 = 40 7 × 8 = 48 8 × 8 = 54 9 × 8 = 60 Φ × 8 = 68 Π × 8 = 74 10 × 8 = 80 2 × 9 = 16 3 × 9 = 23 4 × 9 = 30 5 × 9 = 39 6 × 9 = 46 7 × 9 = 53 8 × 9 = 60 9 × 9 = 69 Φ × 9 = 76 Π × 9 = 83 10 × 9 = 90 2 × Φ = 18 3 × Φ = 26 4 × Φ = 34 5 × Φ = 42 6 × Φ = 50 7 × Φ = 5Φ 8 × Φ = 68 9 × Φ = 76 Φ × Φ = 84 Π × Φ = 92 10 × Φ = Φ0 2 × Π = 1Φ 3 × Π = 29 4 × Π = 38 5 × Π = 47 6 × Π = 56 7 × Π = 65 8 × Π = 74 9 × Π = 83 Φ × Π = 92 Π × Π = Φ1 10 × Π = Π0 2 × 10 = 20 3 × 10 = 30 4 × 10 = 40 5 × 10 = 50 6 × 10 = 60 7 × 10 = 70 8 × 10 = 80 9 × 10 = 90 Φ × 10 = Φ0 Π × 10 = Π0 10 × 10 = 100