“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 (21=1)
 add first term twice to new term (2×1+1=3)
 subtract this from the 3rd (43=1) –> will have primitive ratio
 add first term twice to new term (2×1+1=3)
 Take eventimes 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
 diagonals are triples

1 2 4 8 16 32 3 6 12 24 48 9 18 36 72 27 54 108 81 162 243
 next line add first 2 together & then multiply by 2 (ad infinitum)
 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
 unity is an element without dimension
 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
 Every square number is the sum of consecutive triangular numbers
 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 contraryminded
 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
 108 = 64 BUT 10:8 ≠ 6:4
 Geometric – preserves ratios but not differences
 100:10 = 10:1 BUT 10010 ≠ 101
 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
 (AB):(B+C)=A:C
 w/ 6:4:3 – (64):(43) = 6:3
 e.g. (for all further examples of anything A is largest, B is middle & C is smallest) – A, B, C
 Arithmetic – preserves differences, not ratios
 Musical ratios are harmonic
 elementary 4:3 –> diatessaron
 3:2 –> diapente
 6:2 –> diapason
 4:1 –> didiapason
 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 = [(AB)*B]/(A+B) + B
 e.g. [(4010)x10]/(40+10) + 10 = 16
 4th type of proportion – Subcontrary
 AxB = (BC)(BxC)
 6:5:3 –> 6×5 = (53)x(5×3)
 5th type of proportion
 AB = 2AC
 5:4:2 –> 5×4 = 2x5x2
 6th type of proportion
 A:B = (AB):(BC)
 6:4:1 –> 6:4 = (64):(41)
 7th type of proportion
 A:C = (AC):(AB)
 9:8:6 –> 9:6 = (96):(98)
 8th type of proportion
 A:C = (AC):(BC)
 9:7:6 –> 9:6 = (96):(7:6)
 9th type of proportion
 A:C ≥ (AC):(BC)
 7:6:4 –> 7:4 ≥ (74):(64)
 10th type of proportion
 (BC) < (AB)
 8:5:3 –> (53) < (85)
 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 – ((126)x6)/18 + 6 = 8 (C)