return to Vignettes of Ancient Mathematics
Note on translations: the words ‘arithmos’ (plural: arithmoi) and ‘plêthos’ (plêthê) are left merely transliterated. Both have something of the sense of ‘number’, although ‘arithmos’ is more common outside Euclid's Elements for an ordinary cardinal number (not relevant in the Elements). However, in the Elements, a ‘plêthos’ is any collection that can be put into 1-1 correspondence with another collection, while an ‘arithmos’ is specifically a ‘plêthos’ of units (see def. 2). And so, we find plêthê being put into 1-1 correspondence with the units in an arithmos. I think it better to conceive the two notions within the practice of the Elements.
Abbreviations and Symbols used in the notes
Some basic notions and assumptions
Prop. 1: When unequal arithmoi are displayed, and the smaller is repeatedly taken away in turn from the larger, if the remaining arithmos never measures out the one before it, until a unit is left from it, the initial arithmoi will be prime in relation to one another.
Corollary: From this, in fact, it is obvious that if an arithmos measures two arithmoi, it will measure the largest common measure of them, which it was required to prove.
Prop. 2: Given arithmoi that are not prime in relation to one another, to find the largest common measure of them.
Prop. 3: Given three arithmoi not prime in relation to one another, to find the largest common measure of them.
Prop. 4: Every arithmos smaller than any arithmos is either a part or parts.
Prop. 5: If an arithmos is a part of an arithmos, and another is the same part of another, then together it will be the same part of it together that the one is of the one.
Prop. 6: If an arithmos is parts of an arithmos, and another the same parts of another, then together both will be the same part of together both that one is of one of them.
Prop. 7: If an arithmos is a part of an arithmos, just the part that an arithmos taken away is of an arithmos taken away, the remainder will also be the same part of the remainder what the whole is of the whole.
Prop. 8: If an arithmos is parts of an arithmos, just the parts that an arithmos taken away is of an arithmos taken away, the remainder will also be the same parts of the remainder what the whole is of the whole.
Prop. 9: If an arithmos is a part of an arithmos, and another is the same part of another, then alternately (alternando) the part or parts that the first is of the third is the same part or same parts as the second is of the fourth.
Prop. 10: If an arithmos is parts or an arithmos, and another is the same parts of another, then alternately (alternando) the parts which he first is of the third will be the same parts, or the same part, as the second of the fourth.
Prop. 11: If it is: as a whole to a whole, so what's taken away to what's taken away, the remainder to the remainder will be as whole to whole.
Prop. 12: If however-many arithmoi are in-ratio, it will be: as one of the leaders to one of the followers, so all the leaders to all the followers
Prop. 13: If four arithmoi are in-ratio, they will also be alternando in-ratio.
Prop. 14: If however-many arithmoi and others equal to them in plêthos, by being taken in pairs and in the same ratio, then they will be ex aequali in the same ratio.
Prop. 15: If a unit measures an arithmos equal-times as another arithmos measures some other arithmos, then, alternando, the unit will measure the third arithmos equal-times as the second arithmos the fourth.
Prop. 16: If two arithmoi by multiplying one another make certain arithmoi, those from them will be equal to one another.
Prop. 17: If an arithmos by multiplying two arithmoi makes some arithmoi, they will have the same ratio as those multiplied.
Prop. 18: If two arithmoi by multiplying some arithmos make certain arithmoi, those that come about from them will have the same ratio as those multiplied.
Prop. 19: If four arithmoi are in-ratio, the arithmos that comes about from the first and fourth will be equal to the arithmos that comes about from the second and third; if the arithmos from the first and fourth is equal to that from the second and third, the four arithmoi will be in-ratio.
Prop. 20 (vulgo): If three arithmoi are in-ratio, that under the extremes is equal to that from the middle, and if that under the extremes is equal to that from the middle, the three arithmoi are in-ratio.
Prop. 20: The smallest arithmoi of those having the same ratio measure equal-times those having the same ratio, the larger the larger and the smaller the smaller.
Prop. 22 (vulgo, where Prop. 20 is labelled '21') If there are three arithmoi and others equal to them in plêthos, taken two by two and in the same ratio, and the proportion of them is arranged, then they will be in the same ratio from an equal (ex aequali).
Prop. 21: Primes in relation to one another are the smallest of those having the same ratio as they.
Prop. 22: The least arithmoi of those having the same ratio as they are prime in relation to one another.
Prop. 23: If two arithmoi are prime in relation to one another, the arithmos measuring one of them will be prime in relation to the remaining one.
Prop. 24: If two arithmoi are prime in relation to some arithmos, the arithmos that comes about from them will also be prime in relation to the same arithmos.
Prop. 25: If two arithmoi are prime in relation to one another, the arithmos that comes about from one of them will be prime in relation to the remaining one.
Prop. 26: If two arithmoi are prime in relation to two arithmoi, both yo each, the arithmoi that come about from them will also be prime in relation to one another.
Prop. 27: If two arithmoi are prime in relation to one another and each by multiplying itself makes some arithmos, the arithmoi that come about from them will be prime in relation to one another, and if the initial arithmoi by multiplying those that came about make some arithmoi, those will be prime to one another [and this keeps occurring with the last arithmoi].
Prop. 28: If two arithmoi are prime in relation to one another, together the arithmos of them will also be prime in relation to each of them, and if together the arithmos of them is prime in relation to one of them, then the initial arithmoi will be prime in relation to one another.
Prop. 29: Every prime arithmos is prime in relation to every arithmos which it does not measure.
Prop. 30: If two arithmoi, by multiplying one another make some arithmos, and some prime arithmos measures the arithmos that comes about from them, then it will measure one of the initial arithmoi.
Prop. 31: Every composite arithmos is measured by some prime arithmos.
Prop. 32: Every arithmos either is prime or is measured by some prime.
Prop. 33: Given however many arithmoi, to find the smallest of those having the same ratio as they.
Prop. 34: Given two arithmoi, to find the smallest arithmos that they measure.
Prop. 35: If two arithmoi measure some arithmos, the smallest measured by them will also measure it.
Prop. 36: Given three arithmoi, to find the smallest arithmos that they measure.
Prop. 37: If an arithmos is measured by some arithmos, the measured arithmos will have a part homonymous to the measure.
Prop. 38: If an arithmos has any part, it will be measured by an arithmos homonymous with the part.
Prop. 39: To find an arithmos which will be the least arithmos having given parts.
Abbreviations and symbols used in the notes
u = unit , if more than one is needed, we can write u', u'', u''', ..., so long as it is understood that units are not ordered. Similarly, since arithmoi are units collected, there may be many threes, and these can be distinguished, if needed, 3', 3'', ....
I use {a', a'', ...} for a plêthos. Nothing sophisticated should be understood by the brackets, e.g., as marking a set, a modern notion. They simply determine the boundaries of a list of things grouped together. For example, an arithmos b might break up (be partitioned) into two units, u' and u'', which we could write {u', u''}. It is an issue never raised in Euclid, but part of the modern criticism of Euclid, that in any partition, all of {a', a'', ...} are disjoint, that is, no unit in any arithmos of {a', a'', ...} is in another other arithmos in the plêthos. Keep in mind also that since an arithmos is a plêthos, {u',u'',u'''} is a three, while {{u',u'',u'''}, {u'''',u''''',u''''''}} is two threes, which we could rearrange as {{u',u''},{u''',u''''},{u''''',u''''''}} or three twos, or as six separate units, {{u'},{u''},{u'''},{u''''},{u'''''},{u''''''}}, or again as a single list, a six arithmos, {u',u'',u''',u'''',u''''',u''''''}. This is not doing anything to the objects except rearranging the divisions of the list. I don't know whether this constitutes a philosophy of number for the ancient world, but it is implicit in how Euclid speaks about plêthê and arithmoi.
We need to distinguish between when a < b and when the units of a are also units of b (a ⊂ b), where there is a one-one-correspondence betwenn the units of a and the units of b (a = b) and and where they are identical (a IS b) similarly for operations '+' and '-'. These will be Compos[{a', a'', ...},b] and Remain[a,b;c], i.e. when b is removedfrom a what remains is c. For convience, it will usually be preferable to have a function, Comp[{a', a'', ...}] will be the arithmos that results from treating the units of {a', a'', ...} as forming a plêthos. Comp[{{u',u''},{u''',u''''},{u''''',u''''''}}] = {u',u'',u''',u'''',u''''',u''''''}.
{a',a'', ...} =_{plêthos} {b',b'', ...} == there is a 1-to-one correspondence between the arithmoi in the list a',a'', ...} and the arithmoi in the list {a',a'', ...}, or colloquially, the one plêthos is equal (in plêthos) to the other plêthos.
a = b, where a, b are arithmoi, and a is {u',u'',...} and b is {u''',u'''',...} can be understood as {{u'},{u''},...} =_{plêthos} {{u'''},{u''''},...}
These four are almost equivalent primitive notions:
Partition[a,b; {a_{b}',a_{b}'',...}] == Compos[{a_{b}', a_{b}'', ...}; b] & for each a_{b} in {a_{b}', a_{b}'', ...}: a_{b} = a. If a is a unit, we will still write the partition as {u_{b}', u_{b}'', ...}
Meas[a,b] == a measures b
Part[a,b] == a < b & Meas[a,b] == a is a part of b (as Mueller (PDMEE, ) points out, the condition that a < b may, in usage, amount to a ≤ b)
Mult[b,a] == b is a part of a
In the case where a does not measure b
Parts[a,b] == ~Part[a,b], note, as Euclid uses it, there can be no requirement that a < b (see vii 9, 10).
M-Remain[a,b;c] == ~Meas[a,b] & ∃n Partition[{a_{n}', a_{n}'', ...}; n] & n ⊂ b & Remov(b,n;c) & c < n (note: each a_{n} ⊂ b)
If Meas[a,b], then b may be partitioned into units or arithmoi equal to a
These are equivalent primitive but undefined notions:
Eq-Meas[a,b;c,d] == a measures b as many times as c measures d. Note that the times is not an arithmos, although it will correspond to an arithmos, once N-Mult is introduced.
Eq-Part[a,b;c,d] == a is the same part of b as c measures d
Eq-Mult[b,a;d,c] == b is the same multiple of a as d is of c
Two basic notions:
LCMeas[a,b, ...;m] == m is the largest common measure of a,b, ...
SCMult[a,b,...;m] == m is the smallest common multiple of a,b, ...
The first is also an undefined notion but will be equivalent to the second defined notion.
NT-Meas[n-t;a,b] == a measures b n-times == there is a partition: Partition[a,b; {a_{b}',a_{b}'',...}], where {a_{b}',a_{b}'',...} gets 'counted' out n-times, but this only gets a rigorous meaning as N-Mult[n,a;b]. Note that n-t is not an arithmos, nor even a plêthos, but is represented in N-Mult by an arithmos. It is the times of something that is repeated, once, twice, etc.
N-Mult[a,b;c] == there are partitions: Partition[b,c; {b_{c}',b_{c}'',...}] & Partition[u,a; {u'_{a},u''_{a},...}] &{b_{c}',b_{c}'',...} =_{plêthos}{u'_{a},u''_{a},...}].
R-Prime[a,b] == ∀n: Meas[n,a] & Meas[n,b] ⇒ n = u
S-Part[a,b;c,d] == a < b & Eq-Meas[a,b;c,d]
It is unclear which of the following three is Euclid's conception of 'same parts'. All seem consistent with his arguments, and by vii 22 they are all three equivalent. Where it doesn't matter, I shall write S-Parts[a,b;c,d]
S-PartsE[a,b;c,d] == ~Eq-Meas[a,b;c,d] & ∃h,k Eq-Meas[h,a;k,c] & Eq-Meas[h,b;k,d]
S-PartsA[a,b;c,d] == S-PartsE[a,b;c,d] & ∀h,k Eq-Meas[h,a;k,c] ⟷ Eq-Meas[h,b;k,d]
S-PartsLCM[a,b;c,d] == where LCMeas[a,b;m] & LCMeas[c,d;n], & ~Eq-Meas[a,b;c,d] & Eq-Meas[m,a;n,c] & Eq-Meas[m,b;n,d]
a ⊂ b & b ⊂ c ⇒ a ⊂ c
a = b & b = c ⇒ a = c a = b ⟺ b = a
Meas(a,b) & Meas(b,c) ⇒ Meas(a,c) Meas(a,a)
{a',a'', ...} =_{plêthos} {b',b'', ...} & {b',b'', ...} =_{plêthos} {c',c'', ...} ⇒ {a',a'', ...} =_{plêthos} {c',c'', ...}
{a',a'', ...} =_{plêthos} {b',b'', ...} ⟺ {b',b'', ...} =_{plêthos} {a',a'', ...}
Meas[a,b] ⟺ there are {a_{b}',a_{b}'',...}: Partition[a,b; {a_{b}',a_{b}'',...}]
Meas[a,b] & Meas[b,c] ⇒ Meas[a,c] (but this follows from transitivity of ⊂ and =, and the equivalence of Meas with the existence of a Partition)
Meas[a,b] & Meas[b,a] ⇒ a = b
∀n,u (u is a unit & n is an arithmos ⇒ ~Meas[n,u]
Meas[u,a]
a = b ⟹ Meas[a,a]
Meas[a,b] & Meas[b,c] ⟹ Meas[a,c] (used in vii 1)
Meas[a,b] iff there is a partition such that Partition[a,b; {a_{b}',a_{b}'',...}]
a = c - b & Meas[m,a] and Meas[m,c] ⇒ Meas[m,b]
Eq-Meas[a,b;c,d] iff there is a partition Partition[a,b; {a_{b}',a_{b}'',...}] and Partition[c,d; {c_{d}',c_{d}'',...}] and {a_{b}',a_{b}'',...} =_{plêthos} {c_{d}',c_{d}'',...}. This is used frequently and may just be the definition of Eq-Meas.
It trivially follows that: Eq-Meas[a,b;c,d] & Eq-Meas[c,d;e,f] ⟹ Eq-Meas[a,b;e,f],
proof: Let there be partitions such that
Partition[a,b; {a_{b}',a_{b}'',...}] and Partition[c,d; {c_{d}',c_{d}'',...}] and Partition[e,f; {e_{f}',e_{f}'',...}], where {a_{b}',a_{b}'',...} =_{plêthos} {c_{d}',c_{d}'',...} and {c_{d}',c_{d}'',...} =_{plêthos}{e_{f}',e_{f}'',...}. By transitivity, {a_{b}',a_{b}'',...} =_{plêthos} {e_{f}',e_{f}'',...}
So too, Eq-Meas[a,b;c,d] ⟺ Eq-Meas[c,d;a,b]
S-Part[e,f;g,h] ⟹ S-Parts[g,h;e,f] (trivial from the definitions and vii 4)
S-Parts[e,f;g,h] ⟹ S-Parts[g,h;e,f] (trivial from the definitions and vii 4)
S-Part[e,f;g,h] & S-Part[g,h;j,k] ⟹ S-Part[e,f;j,k] (trivial from the definitions and transitivity for < and Eq-Meas)
S-Parts[e,f;g,h] & S-Parts[g,h;j,k] ⟹ S-Parts[e,f;j,k] (not trivial, and probably just assumed, although it is proved for proportions of magnitudes, v 11. It seems to be first used, implicitly, in vii 10.)
~Meas[a,b] & a < b ⟹ there is a c such that Remain[a,b;c]
Eq-Meas[u,a;u,b] ⟺ a = b