“Introduction to Arithmetic Book 2” by Nicomachus (c. 100 AD)

“Introduction to Arithmetic Book 2” by Nicomachus (c. 100 AD)

  • An element is the smallest thing that can be analyzed
    • letters are elements of speech & writing
    • sounds are elements of melody
    • 4 elements: fire, water, air & earth –> elements of the universe
  • Equality is the elementary principle of relative numbers
    • 1 & 2 are the elements of absolute numbers
    • Take 3 terms of any relation as long as the 1st & 2nd numbers have the same ratio as the 2nd & 3rd (e.g. 1:2:4)
    • subtract 1st term from 2nd term (2-1=1)
      • add first term twice to new term (2×1+1=3)
        • subtract this from the 3rd (4-3=1) –> will have primitive ratio
  • Take even-times even series (1,2,4,8,16,32…)
    • next line add first 2 together & then multiply by 2 (ad infinitum)
      • diagonals are triples
        • if you do this starting with multiples of 3, new diagonals will be quadruples
    • 1 2 4 8 16 32
      3 6 12 24 48
      9 18 36 72
      27 54 108
      81 162
      243
  • Symbols for numbers don’t show up in nature, so we can just as well use something countable like α (alpha)
    • e.g. α=1, αα=2, ααα=3, etc.
    • Unity is the starting point of a line, just as a point or an alpha multiplied by a point or an alpha will return a point or an alpha
      • unity is an element without dimension
        • Dimensions = 1= line, 2=surface, 3=solid
  • A point without dimension isn’t a line, but the element of one
  • A line is not a surface but an element of one
  • A surface is not a solid but an element of one
  • With numbers, unity is the start of a line, a linear number are the start of a plane number & a plane number is the start of a solid
    • Most original & elementary form of plane number is a triangle
    • Even rectilinear numbers can be made of triangles but a triangle can’t be decomposed any further
  • With an equilateral triangle, numbers are 3,6,10,15,21,28…
    • α = 1 point
    •  α
    • αα = 3 points (1+2)
    • α
    • αα
    • ααα = 6 points (3+3)
    • α
    • αα
    • ααα
    • αααα = 10 points (6+4)
    • α
    • αα
    • ααα
    • αααα
    • ααααα = 15 points (10+5)
      • increase the triangle but adding increasing numbers to the base
  • A square uses 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
    • α = 1 point
    • αα
    • αα = 4 points (1+3)
    • ααα
    • ααα
    • ααα = 9 points (4+5)
    • αααα
    • αααα
    • αααα
    • αααα = 16 points (9+7)
    • ααααα
    • ααααα
    • ααααα
    • ααααα
    • ααααα = 25 points (16+9)
      • each time square is enlarged, it’s done so by 2 from the previous increase
  • With a pentagon – 1, 5, 12, 22,
    • α = 1 point
    • α
    • αα
    • αα = 5 points (1+4)
    • α
    • αα
    • ααα
    • ααα
    • ααα = 12 points (5+7)
    • α
    • αα
    • ααα
    • αααα
    • αααα
    • αααα
    • αααα = 22 points (12+10)
      • each time pentagon is enlarged, it’s done so by 3 from the previous increase
  • Hexagonal, heptagonal… shapes can be constructed as such…
Shape Point Size 1 Size 2 Size 3 Size 4 Size 5 Size 6 Size 7 Size 8 Size 9
Triangle 1 3 6 10 15 21 28 36 45 55
Add to enlarge –> 2 3 4 5 6 7 8 9 10
Square 1 4 9 16 25 36 49 64 81 100
Add to enlarge –> 3 5 7 9 11 13 15 17 19
Pentagon 1 5 12 22 35 51 70 92 117 145
Add to enlarge –> 4 7 10 13 16 19 22 25 28
Hexagon 1 6 15 28 45 66 91 120 153 190
Add to enlarge –> 5 9 13 17 21 25 29 33 37
Heptagon 1 7 18 34 55 81 112 148 189 235
Add to enlarge –> 6 11 16 21 26 31 36 41 46
  • Every square can be divided into 2 triangles
    • Every square number is the sum of consecutive triangular numbers
      • adding a triangle to a square makes it a pentagon
      • adding a triangle to a pentagon makes it a hexagon
        • Because a triangle makes a polygon a higher type, it’s the element of polygons
    • A solid number – one whose units form a solid shap
    • 1st appearance is a pyramid with a wide base narrowing to a sharp apex
      • 4 sides are triangles with a square base
      • can be pentagonal, hexagonal…
    • triangular pyramids are made by adding each successive square number number to previous sum of: 1, 4, 9, 16…
      • 1 + 4 =5
      • 5+9 = 14
      • 14+16=30…
    • Truncated pyramids are where the tip doesn’t taper all the way to the top
  • Cubes are multiplied= equal height, length & width
    • e.g. 1x1x1=1, 2x2x2=8, 3x3x3=27
    • When dimensions aren’t equal they are scalene or wedges
      • paralellepipedons are where 2 of the numbers are the same but not the 3rd
    • Heteromecic number – representation is quadrilinear & quadrangular but the sides aren’t equal
      • e.g. 1×2=2, 2×3=6, 3×4=12, 4×5=20
    • if using “sameness” as 1 & “otherness” as 2 (odd v. even)
      • squares tend to 1 & sameness because they’re the sum of odd numbers
      • heteromecic tend to 2 & otherness because they’re the sum of even numbers
    • multiply odd numbers together make an odd number
    • multiply even numbers together make an even number
    • multiply an odd number with an even number will make an even number
    • Bricks – square only on 1 side = e.g. 8x8x2
    • Wedges – 3 unequal sides = e.g. 1x2x3
  • Going further with equality & inequality, odd & even, square & heteromecic gives us some more properties of numbers
    • objects are made from warring & opposite elements
    • Harmony arises from opposites because it is the unification of the diverse & reconciliation of the contrary-minded
    • 2 series compared – squares & heteromecic
      • Squares 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225
        Hetermecic 2 6 12 20 30 42 56 72 90 110 132 156 182 210 240
      • ratios are increased: 1:2, 2:3, 3:4…
      • Compare 9:12:16 –> 9×16 = 12×12 = 144,   16:20:25 –> 16×25=20×20= 400
      • Subtracting a side from its square makes it heteromecic
        • 3×3=9 –> 3×2=6, 9×9=81 –> 9×8=72
      • Add a square with the next heteromecic & a heteromecic with the next square & you’ll get a tringle
        • (1+6) + (4+12) = 25,  (9+20) + (16+30) = 75
      • Add or subtract a side from a square & make it heteromecic
        • One is odd & one is even
        • Odds – look at the series of multiples (e.g. triples)
          • 1,3,9,27,81 –> every other one is a square
  • Proportion – a combination of 2 or more ratios or relations
    • e.g. 1:2:4 – proportion of doubling
    • e.g. 1:2:3 – continued proportion of quantity
  • Oldest proportions are arithmetic, geometric & harmonic
    • Arithmetic – preserves differences, not ratios
      • 10-8 = 6-4 BUT 10:8 ≠ 6:4
    • Geometric – preserves ratios but not differences
      • 100:10 = 10:1 BUT 100-10 ≠ 10-1
      • With even times evens (1,2,4,8,16…) all have a doubling proportion
      • in general, the product of 2 squares is another square BUT
        • Product of a heteromecic & a square isn’t a square
      • Product of 2 cubes is another cube
      • Product of 2 evens is an even & a product of 2 odds is an odd bu even x odd is even
    • Harmonic – ratio of largest term to smallest equaling the difference of the largest from the mean to the difference of the mean to the smallest
      • e.g. (for all further examples of anything A is largest, B is middle & C is smallest) – A, B, C
        • (A-B):(B+C)=A:C
        • w/ 6:4:3 – (6-4):(4-3) = 6:3
  • Musical ratios are harmonic
    • elementary 4:3 –> diatessaron
    • 3:2 –> diapente
    • 6:2 –> diapason
    • 4:1 –> di-diapason
      • It’s called harmonic because a cube has 12 sides, 8 angles & 6 faces – 6:4:3
  • Arithmetic mean = (A+B)/N
    • e.g. 40 & 10 –> (40+10)/2 = 25
  • Geometric mean = √(AxB)
    • e.g. √(40×10) = 20
  • Harmonic mean = [(A-B)*B]/(A+B) + B
    • e.g. [(40-10)x10]/(40+10) + 10 = 16
  • 4th type of proportion – Subcontrary
    • AxB = (B-C)(BxC)
    • 6:5:3 –> 6×5 = (5-3)x(5×3)
  • 5th type of proportion
    • AB = 2AC
    • 5:4:2 –> 5×4 = 2x5x2
  • 6th type of proportion
    • A:B = (A-B):(B-C)
    • 6:4:1 –> 6:4 = (6-4):(4-1)
  • 7th type of proportion
    • A:C = (A-C):(A-B)
    • 9:8:6 –> 9:6 = (9-6):(9-8)
  • 8th type of proportion
    • A:C = (A-C):(B-C)
    • 9:7:6 –> 9:6 = (9-6):(7:6)
  • 9th type of proportion
    • A:C ≥ (A-C):(B-C)
    • 7:6:4 –> 7:4 ≥ (7-4):(6-4)
  • 10th type of proportion
    • (B-C) < (A-B)
    • 8:5:3 –> (5-3) < (8-5)
  • Perfect proportion – 3 dimension with 10 proportions
    • 4 terms – A B C D
    • A & D are products of 3 numbers
    • B & C are the arithmetic & harmonic means of A & D
    • 12:9:8:6
      • arithmetic mean – (16+2)/2 = 9 (B)
      • harmonic mean – ((12-6)x6)/18 + 6 = 8 (C)

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