“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)
 Timaeus asked – What is always the same – never born or never dies? What is always becoming & never remaining
 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
 Arithmetic is about quantity of what rests. Astronomy is a science about what is in motion & revolution
 Where do we begin in our quest for truth?
 Arithmetic is master & other sciences are offshoots
 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
 Makes sense that there’s a scientific pattern because everything with a real harmony has opposites
 Even numbers can be divided into 2 pars without a factional remainder
 Odd numbers cannot
 Eventimes 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 eventimes 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
 When an odd number shows up (e.g. 7) it will not be able to be divided by 2.
 9 – Eventimes odd numbers have an even multiple but not like eventimes 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
 Oddtimes 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, eventimes even numbers are 4, 8, 16, 32, 64, 128, 256…
 Multiply an odd number times an eventimes even number & you’ll get an oddtimes 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 oddtimes even number
 Multiply an odd number times an eventimes even number & you’ll get an oddtimes even number
 Can be divided by 2 & by 2 again but not all the way to 1
 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
 Superabundant numbers – have over & above factors belonging to it, overstepping symmetry
 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 09.
 Only 28 is a perfect number in 1099
 Only 496 is a perfect number in 100999
 Only 8128 is a perfect number in 10009999
 Last digit alternates between 6 & 8
 Eventimes even number & prime
 for p = 2: 2^{1}(2^{2} − 1) = 6
 for p = 3: 2^{2}(2^{3} − 1) = 28
 for p = 5: 2^{4}(2^{5} − 1) = 496
 for p = 7: 2^{6}(2^{7} − 1) = 8128
 6 & 28
 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
 everything is either equal or unequal to something else
 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
 take smaller number, multiply by at least 2 & add 1 more
 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