“Introduction to Arithmetic Book 1” by Nicomachus (c. 100 AD)

• Ancients defined Philosophy as the love of wisdom & those who had any knowledge in a field was “wise”
• Pythagoras restricted the term “wise” to one who had knowledge & understanding of reality
• Wisdom was knowledge or science of truth in real things that never change in the absolute
• Material things change & immaterial things never change – quality, quantity, size, relations, time, space – concepts, not measurements
• Apply wisdom to that
• Immaterial things are eternal & never change but things that are born & are destroyed go through constant change
• Timaeus asked – What is always the same – never born or never dies? What is always becoming & never remaining
• One is had by a mental process with reasoning & the other is always the same
• If we look for happiness, it’s solely accomplished by Philosophy for wisdom & truth. It’s necessary to distinguish & systematize things (qualities)
• Unified & continuous things (qualities) are magnitudes & discontinuous things (quantities) are multitudes
• Multitudes begin at a root & never stop
• Magnitudes can’t bring division process to an end & proceed to infinity
• Science refers to limited things – not ceaseless (multitudes)
• Quantitative terms – relative to other things (double, greater, smaller, etc) not like qualitative (odd, even, perfect). We should know arithmetic to do an investigation of quantity
• Arithmetic is about quantity of what rests. Astronomy is a science about what is in motion & revolution
• Without these 2, we can’t really know truth & therefor can’t have wisdom & can’t philosophize
• Knowledge of math allows us to know the nature of things in part or whole. They give us clearer understanding
• Plato said – with every diagram, schematic, system & law, nature ought to appear to one who studies it correctly.
• To do otherwise & stumble across the truth would require pure luck because only these can provide a clear path to truth
• Socrates’s interlocutor in the Republic says that math shows itself as useful – arithmetic for counting, geometry for sieges & partition of land, music for entertainment & worship, & astronomy for farming & navigation
• Socrates answered – you think I think these things are useless, but it’s by the truth alone that we can understand the universe
• Where do we begin in our quest for truth?
• Arithmetic is master & other sciences are off-shoots
• It needs to exist before geometry (need counting, multiplication, etc)
• Music also needs arithmetic to understand harmonies & harmonic ratios
• Astronomy needs geometry to understand the motions of stars & celestial bodies
• It makes sense to start with the mother of all forms of math.
• Everything dealing with numbers has a pattern or a sketch. With numbers, they have a true & eternal essence as of artistic plan, along with time, motion, heavens, stars, etc.
• Makes sense that there’s a scientific pattern because everything with a real harmony has opposites
• Of these things, you can figure out the quality & quantity of them
• Even numbers can be divided into 2 pars without a factional remainder
• Odd numbers cannot
• Even-times even numbers – take a number, divide it by 2 & it still has an even number. Do this all the way down to unity (1). 64/2= 32. 32/2=16. 16/2=8. 8/2=4. 4/2=2. 2/1=1.
• When an odd number shows up (e.g. 7) it will not be able to be divided by 2.
• If no. terms of an even-times even number in a series is even, then the product of the extremes will equal the product of the means. E.g. 1×64=2×32=4×16=8×8
• 9 – Even-times odd numbers have an even multiple but not like even-times even
• it has clear division but halves aren’t divisibly even
• E.g. 6, 10, 14, 18 – halves of these numbers are indivisible by 2
• 18/2= 9, which is not divisible by 2
• Number may be even but its halves are odd
• Greatest extreme alone is divisible
• Odd-times even numbers – single mean between 2 extremes
• Can be divided by 2 & by 2 again but not all the way to 1
• (e.g. 24, 28, 40) – subdivision never ends with unity (1).
• Has qualities of former 2 but not all
• E.g. 24 -> 4×6, 2×12, 6×4 (each one of these is divisible by 2) but 8×3 is not divisible by 2.
• It is a mixture of both kinds
• Starting with 3, odd numbers are 3, 5, 7, 9, 11, 13, 17, 19…
• Starting with 3, even-times even numbers are 4, 8, 16, 32, 64, 128, 256…
• Multiply an odd number times an even-times even number & you’ll get an odd-times even number
• e.g. 3×4=12, 3×8=24, 3×16=28, 3×32=96
• e.g. 5×4=20, 5×8=40, 5×16=80, 5×32=160
• Any combination in a series will lead to an odd-times even number
• 3 types of odd numbers
• Prime – can only be divided by itself & 1: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
• Composite – has fractional part beyond self & 1
• 9 = 3×3, 15=3×5, 21=3×7, 27=3×9
• These are secondary because they can use anther multiple besides itself & 1
• These are composite because they are made up of 2 other numbers
• 13 – Numbers derived from primes or composited but can’t be reduced to unity in comparison to each other
• 9:25 = each one is a composite, i.e. multiple of 2 other numbers but cannot be reduced beyond this ratio
• Use Eratosthenes’s sieve by taking all numbers in a series & begin at 2 to find numbers that are composites of 2. Then take 3 & find all numbers that are composites of three.
• You will be able to find which numbers are prime number this way
• Even numbers can be superabundant, deficient & intermediary
• Superabundant numbers – have over & above factors belonging to it, overstepping symmetry
• E.g. 24 = 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1
• These all add up to more than 24
• Perfect numbers – neither make their parts greater than themselves added together, nor shows self greater than its parts, always equaling to the sum of its parts
• 6 & 28
• 6= 1×6, 2×3, 3×2, 6×1 –> 1/6 + 1/3 + 1/2 = 1,  –> 3 + 2 + 1 = 6
• 28 = 1×28, 2×14, 4×7, 7×4, 14×2, 28×1 –> 1/28 + 1/14 + 1/7 + 1/4 + 1/2 = 1 –> 14+7+4+2+1 = 28
• Only 6 is a perfect number in 0-9.
• Only 28 is a perfect number in 10-99
• Only 496 is a perfect number in 100-999
• Only 8128 is a perfect number in 1000-9999
• Last digit alternates between 6 & 8
• Even-times even number & prime
• for p = 2:   21(22 − 1) = 6
• for p = 3:   22(23 − 1) = 28
• for p = 5:   24(25 − 1) = 496
• for p = 7:   26(27 − 1) = 8128
• We’ve talked about absolute quantity, now let’s talk about relative equality & inequality
• everything is either equal or unequal to something else
• 100 = 100, 10 = 10, 2 = 2, 1 = 1
• Equal has no other words to explain itself
• Unequal has many ways to portray itself (father – son, teacher – student)
• 5 different kinds of inequalities – multiples, superparticular, superpartient, multiple superparticular, multiple superpartient
• Multiple is greater than the original 2:1 is double, 3:1 is triple, etc
• submultiple is smaller than the original – 1 is subdouble of 2, 1 is subdouble of 3, etc.
• All evens are doubles
• Superparticular – a number that contains in itself the whole of the other number
• Comparing by ratios – 3:2, 6:4, 9:6 (all the same by multiples
• you can compare multiples by rows & columns:
•  1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100
• Superpartient – a ratio with a larger number than the smaller by more than 1.
• E.g. 5:3, 7:4, without being able to be reduced (4:2 = 2:1)
• Superparticulars & superpartients can be found with successions of even & odds
• Starting with 3 & compare only odds against all numbers (5:3, 7:4, 9:5)
• You can multiply both sides by a factor with similar results
• 5:3 –> 10:6 –> 15:9, etc.
• Multiple superparticulars are when the larger number is more than twice the other number
• take smaller number, multiply by at least 2 & add 1 more
• 5:2 –> 2×2 + 1 = 5 –> 5:2
• 7:3 –> 3×2 + 1 = 7 –> 7:3
• works in multiples & retains same qualities
• Multiple superpartient – when a number contains the whole of another number plus more than 1
• 8:3 –> 3×2 + 2 = 8 –> 8:3
• 11:4 –> 4×2 + 3 = 11 –> 11:4
• This is a good place to start by defining terms so we can start using them in our analysis
• Nature shows up using these terms & we must be able to recognize & use them
• Includes invariable & inviolable laws of nature

1. 1. 