 “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)