[Maths-Education] Ratio and Proportion

[David Pimm]

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I think part of the historical confusion comes from these seeming to be the
same sort of things (compounded by the nearly set phrase of 'ratio and
proportion', as if to be said in a single breath). Why, on the surface,
would there be a different word for 'the equality of ratios' when there
isn't a different word for the equality of whole numbers or fractions, say
- though as I write this, I realise there is the word 'congruent' for
equality of (certain) geometric figures.

I think one problem arises from the tendency to make simple nouns from
unsimple things - the phrase should be 'in proportion' rather than a noun
'a proportion' (making objects out of relationships). So, four quantities
A, B, C, D are in proportion (one, two or three quantities can't be in
proportion, because we son;t know which one), when A:B :: C:D. (I suppose
for Anne's example this could be extended to six quantities A, B, C, D, E,
F are in [common, the same, equal, help] proportion if
A:B :: C:D :: E:F)

I used the old geometric symbol :: rather than = as Geoff did to point up
how the relationship of proportionality (another noun in need to
specification?) is often read 'is as' rather than simply 'is'. That 'as'
signifies for me the second level of comparison and the double colon of the
notation also signals a double comparison.

The phrase 'the proportion of goldfish' is itself a nonsense as you cannot
have a proportion of a single thing: there is always (he asserted rashly) a
double comparison, a comparison of comparions, if you like). But in this
compressed usage here it is completely tacit. I agree with Geoff: 'The
proportion of goldfish is 6 out of 24' implicitly means "the ratio of
goldfish compared with all fish in this particular tank is as the ratio of
6 to 24" - or am I simply saving the phenomena?. And writing 6:24 as the
fraction 6/24 compounds things further.

I think it is an instance of how a simple little word can reveal quite
another way of thinking about the things: 'the square on the hypotenuse'
(geometric figure related to another geometric figure versus 'the square of
the hypotenuse' (arithmetical operation carried out on numerical measure of
length) provides another.

David Fowler's account of pre-Eudoxan ratio theory (The Mathematics of
Plato's Academy) reconstructs an independent definition of 'the ratio of A
to B' (based on anthyphairesis, "continued subtraction in turn"),
independent, that is, of any definition of proportion, whereas Eudoxus'
theory (in Euclid book 5) is really a theory of proportion.

But I think another compounding problem is that 'everyday usage' cares
little for mathematical distinctions or desires to keep things straight
(hence the 'part to whole' observation might be empirically accurate with
respect to common usage - I don't have the Collins COBUILD English
dictionary to hand, but it would be interesting to look there and see what
usages they managed to cull from the large database of actual useage).

David


David Pimm
Dept of Secondary Education,
Faculty of Education,
University of Alberta,
Edmonton, AB
Canada T6G 2G5

David.Pimm@ualberta.ca

tel: (780)-492-0150
sec: (780)-492-0148
fax: (780)-492-0162
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I think part of the historical confusion comes from these seeming to be
the same sort of things (compounded by the nearly set phrase of 'ratio
and proportion', as if to be said in a single breath). Why, on the
surface, would there be a different word for 'the equality of ratios'
when there isn't a different word for the equality of whole numbers or
fractions, say - though as I write this, I realise there is the word
'congruent' for equality of (certain) geometric figures.


I think one problem arises from the tendency to make simple nouns from
unsimple things - the phrase should be 'in proportion' rather than a
noun 'a proportion' (making objects out of relationships). So, four
quantities A, B, C, D are in proportion (one, two or three quantities
can't be in proportion, because we son;t know which one), when A:B ::
C:D. (I suppose for Anne's example this could be extended to six
quantities A, B, C, D, E, F are in [common, the same, equal,
<italic>help</italic>] proportion if 

A:B :: C:D :: E:F) 


I used the old geometric symbol :: rather than = as Geoff did to point
up how the relationship of proportionality (another noun in need to
specification?) is often read 'is as' rather than simply 'is'. That
'as' signifies for me the second level of comparison and the double
colon of the notation also signals a double comparison. 


The phrase 'the proportion of goldfish' is itself a nonsense as you
cannot have a proportion of a single thing: there is
<italic>always</italic> (he asserted rashly) a double comparison, a
comparison of comparions, if you like). But in this compressed usage
here it is completely tacit. I agree with Geoff: 'The proportion of
goldfish is 6 out of 24' implicitly means "the ratio of goldfish
compared with all fish in this particular tank is as the ratio of 6 to
24" - or am I simply saving the phenomena?. And writing 6:24 as the
fraction 6/24 compounds things further.


I think it is an instance of how a simple little word can reveal quite
another way of thinking about the things: 'the square <bold>on</bold>
the hypotenuse' (geometric figure related to another geometric figure
versus 'the square <bold>of</bold> the hypotenuse' (arithmetical
operation carried out on numerical measure of length) provides
another.


David Fowler's account of pre-Eudoxan ratio theory (<italic>The
Mathematics of Plato's Academy) </italic>reconstructs an independent
definition of 'the ratio of A to B' (based on
<italic>anthyphairesis</italic>, "continued subtraction in turn"),
independent, that is, of any definition of proportion, whereas Eudoxus'
theory (in Euclid book 5) is really a theory of proportion.


But I think another compounding problem is that 'everyday usage' cares
little for mathematical distinctions or desires to keep things straight
(hence the 'part to whole' observation might be empirically accurate
with respect to common usage - I don't have the Collins COBUILD English
dictionary to hand, but it would be interesting to look there and see
what usages they managed to cull from the large database of actual
useage).


David

David Pimm

Dept of Secondary Education,

Faculty of Education,

University of Alberta,

Edmonton, AB

Canada T6G 2G5


David.Pimm@ualberta.ca


tel: (780)-492-0150

sec: (780)-492-0148

fax: (780)-492-0162

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