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mixi.mo@virgin.net
science forum beginner
Joined: 22 Jun 2006
Posts: 9
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 Posted: Thu Jun 22, 2006 5:54 pm Post subject:
typographic symbols, geometry, maths puzzle
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I have an interest in typography and geometry and
wondered if there was any link between the development
of the shapes of some letters, thru geometry, triganometry
and mathematics.
I was wondering whether it was possible to calculate
sin, cos, tan without a calculator, if so how, and if the
following values were given to these early Greek letters,
is it still possible to calculate those angles?
A = 1
B = 2
Γ = 3
Δ = 4
E = 5
H = 8
Θ = 9
I = 10
K = 20
Λ = 30
M = 40
N = 50
Ξ = 60
O = 70
P = 100
Σ = 200
T = 300
Y = 400
Φ = 500
X = 600
Ψ = 700
Ω = 800
and if so, how?
Thank you,
Mixi |
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##minty
science forum beginner
Joined: 22 Jun 2006
Posts: 2
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| Posted: Thu Jun 22, 2006 6:51 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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mixi.mo@virgin.net wrote:
| Quote: | I have an interest in typography and geometry and
wondered if there was any link between the development
of the shapes of some letters, thru geometry, triganometry
and mathematics.
I was wondering whether it was possible to calculate
sin, cos, tan without a calculator, if so how, and if the
|
SOH CAH TOA
S sine
O opposite - side opposite angle of interest
H hypoteneuse - the longest side in a right angled triangle
C cosine
A adjacent - side next to the angle of interest which is not the
hypoteneuse
T tangent
/|c
/ C|
5/ |4
/ _|
a /A__3__|_|b
sinA = opp/hyp = 4/5
cosA = adj/hyp = 3/5
tanA = opp/adj = 4/3
sinC = opp/hyp = 3/5
cosA = adj/hyp = 4/5
tanA = opp/adj = 3/4
##minty...
| Quote: | following values were given to these early Greek letters,
is it still possible to calculate those angles?
A = 1
B = 2
Γ = 3
Δ = 4
E = 5
H = 8
Θ = 9
I = 10
K = 20
Λ = 30
M = 40
N = 50
Ξ = 60
O = 70
P = 100
Σ = 200
T = 300
Y = 400
Φ = 500
X = 600
Ψ = 700
Ω = 800
and if so, how?
Thank you,
Mixi |
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kenney@cix.compulink.co.u
science forum beginner
Joined: 23 Jun 2006
Posts: 2
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| Posted: Fri Jun 23, 2006 11:15 am Post subject:
Re: typographic symbols, geometry, maths puzzle
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In article
<1150998872.192032.153580@b68g2000cwa.googlegroups.com>,
mixi.mo@virgin.net () wrote:
| Quote: | I was wondering whether it was possible to calculate
sin, cos, tan without a calculator, if so how, and if the
following values were given to these early Greek letters,
is it still possible to calculate those angles?
|
You are obviously unaware of the history of maths. Sine etc date
back to Pythagorus and can be calculated geometrically. They can
be calculated arithmetically by the difference method. I can not
remember the exact method but that was because when I was at
school we used books of log, sin, cos, tan and some I forget
tables.
Number systems are arbitrary ways of representing the actual
number. Actual mathematical operations are not affected except by
ease of use. Positional notation was adopted because it made some
operations easier to learn.
Ken Young |
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mixi.mo@virgin.net
science forum beginner
Joined: 22 Jun 2006
Posts: 9
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| Posted: Fri Jun 23, 2006 12:41 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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Minty wrote:-
Ah but thats cheating! I was looking at the
geometry of the letterforms/symbols and traced
them back through their history, - I wondered if
the characters themselves used to be mnemonic devices
for (mostly) illiterate artisans, craftsmen and
masons. For example, can two or more lettershapes
be combined to formulate a 3rd?
Φ = 500
I = 10
O = 70
how does Φ represent pi using those numbers?
I + Ξ = 10+60 = 70, although E = 5,
what is the equasion or formula (if any) that
led to E ? ( Ξ still represents 3 in China)
How was pythagoras' formula written in old
Greek, and is there any way of calculating the
angles using those ancient letter symbols?
Ken Young wrote:-
| Quote: | You are obviously unaware of the history of maths. Sine etc date
back to Pythagorus and can be calculated geometrically. They can
be calculated arithmetically by the difference method. I can not
remember the exact method but that was because when I was at
school we used books of log, sin, cos, tan and some I forget
tables.
|
Yes!, but I'd really like to know how the ancient Greeks
wrote it in ancient Greek! not everyone could write or read
and people have been planning, building, farming many thousands
of years,
| Quote: | Number systems are arbitrary ways of representing the actual
number. Actual mathematical operations are not affected except by
ease of use. Positional notation was adopted because it made some
operations easier to learn.
|
Typographically, those symbols also once represented objects or
ideas, I just had an idea that maybe the letters of ancient
Greece also held some vestiges of old formulae ? any ideas?
6,7, 80 & 90 are missing (6 may have been represented by Z)
Mixi |
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Larry Lard
science forum Guru Wannabe
Joined: 09 May 2005
Posts: 166
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| Posted: Fri Jun 23, 2006 1:24 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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mixi.mo@virgin.net wrote:
| Quote: | Typographically, those symbols also once represented objects or
ideas, I just had an idea that maybe the letters of ancient
Greece also held some vestiges of old formulae ?
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But the Greeks got their letters from the Phoenicians, who aren't
remembered for their mathematics.
--
Larry Lard
Replies to group please |
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Dave L. Renfro
science forum Guru
Joined: 29 Apr 2005
Posts: 570
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| Posted: Fri Jun 23, 2006 3:32 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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kenney@cix.compulink.co.uk wrote (to another poster):
| Quote: | You are obviously unaware of the history of maths.
Sine etc date back to Pythagorus and can be calculated
geometrically. They can be calculated arithmetically
by the difference method. I can not remember the exact
method but that was because when I was at school we
used books of log, sin, cos, tan and some I forget
tables.
|
I don't know what you mean by the difference method,
but if it's (linear) interpolation, that's not going
to do what you said it does. Interpolation was used
to estimate values in a table at one higher level of
graduation than they're given for. Usually, this means
that if the input values are 0.00 to 9.99 for every
hundredth, for example, then you can use linear
interpolation to estimate what the value would be
for something like 6.853.
I don't know what you mean by forgetting the tables.
No one had to memorize the tables when I was in school.
Dave L. Renfro |
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Tom McDonald
science forum beginner
Joined: 14 Feb 2006
Posts: 13
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| Posted: Fri Jun 23, 2006 5:17 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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mixi.mo@virgin.net wrote:
| Quote: | I have an interest in typography and geometry
|
Do you have an interest in archaeology?
Does this topic relate in any way to archaeology?
If it does, please make the connection.
If it does not, please take s.a. out of the distribution list.
Thank you.
<snip> |
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IE J
science forum beginner
Joined: 23 Jun 2006
Posts: 1
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| Posted: Fri Jun 23, 2006 5:51 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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"Dave L. Renfro" <renfr1dl@cmich.edu> wrote in message
news:1151076724.435474.227370@u72g2000cwu.googlegroups.com...
| Quote: | kenney@cix.compulink.co.uk wrote (to another poster):
You are obviously unaware of the history of maths.
Sine etc date back to Pythagorus and can be calculated
geometrically. They can be calculated arithmetically
by the difference method. I can not remember the exact
method but that was because when I was at school we
used books of log, sin, cos, tan and some I forget
tables.
I don't know what you mean by the difference method,
but if it's (linear) interpolation, that's not going
to do what you said it does. Interpolation was used
to estimate values in a table at one higher level of
graduation than they're given for. Usually, this means
that if the input values are 0.00 to 9.99 for every
hundredth, for example, then you can use linear
interpolation to estimate what the value would be
for something like 6.853.
I don't know what you mean by forgetting the tables.
No one had to memorize the tables when I was in school.
Dave L. Renfro
|
Good God Dave,
while it's interesting to read about sine in sci.archaeology short before I
will send a drawing of how to calculate where the Vinland's voyager went,
I am on the other hand deeply distressed with the fact shown that you, and
maybe many others, haven't had enough Math education in ordinary school to
learn that sine by no means has anything to do with interpolation.
The aritmetic proof for the sine you can read in tables are so easily shown
that I have to tell you that I have seen it done for ordinary 15-16 year old
schoolboys and schoolgirls in what we here in Sweden calls 'Grundskolan'.
Grundskolan is the school from 1st to 9th class. It's not supposed that the
children, teenagers, should know the proof by heart, but it's supposed that
they should know enough to understand how it's calculated and why it's a
very handy thing to know how to use in real life situations, such as for
example building etc.
Inger E
> |
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kenney@cix.compulink.co.u
science forum beginner
Joined: 23 Jun 2006
Posts: 2
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| Posted: Fri Jun 23, 2006 7:13 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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In article
<1151076724.435474.227370@u72g2000cwu.googlegroups.com>,
renfr1dl@cmich.edu (Dave L. Renfro) wrote:
| Quote: | I don't know what you mean by the difference method,
but if it's (linear) interpolation, that's not going
to do what you said it does.
|
It is a matter of calculating the difference between two values
of an equation until you get constant differences between values
and then working back to get the next value. It could need third
or fourth order differences. For a fuller explanation try
googling on "Difference Engine". Effectively it meant you needed
a skilled mathematician to reduce an equation to something that
could be done by addition and subtraction.
| Quote: | I don't know what you mean by forgetting the tables.
No one had to memorize the tables when I was in school.
|
I forgot the methods, we were taught how to calculate
logarithms. Of course I also learned how to use a slide rule.
Ken Young |
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IEJ
science forum beginner
Joined: 23 Jun 2006
Posts: 2
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| Posted: Fri Jun 23, 2006 7:51 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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<kenney@cix.compulink.co.uk> wrote in message
news:hq6dnYHKzrH2oAHZRVnytw@pipex.net...
| Quote: | In article
1151076724.435474.227370@u72g2000cwu.googlegroups.com>,
renfr1dl@cmich.edu (Dave L. Renfro) wrote:
I don't know what you mean by the difference method,
but if it's (linear) interpolation, that's not going
to do what you said it does.
It is a matter of calculating the difference between two values
of an equation until you get constant differences between values
and then working back to get the next value. It could need third
or fourth order differences. For a fuller explanation try
googling on "Difference Engine". Effectively it meant you needed
a skilled mathematician to reduce an equation to something that
could be done by addition and subtraction.
I don't know what you mean by forgetting the tables.
No one had to memorize the tables when I was in school.
I forgot the methods, we were taught how to calculate
logarithms. Of course I also learned how to use a slide rule.
Ken Young
|
Ken it's much easier than that.
" The sine of an angle is the ratio of the length of the opposite side to
the length of the hypotenuse. "
http://en.wikipedia.org/wiki/Trigonometric_function
for those who aren't familiar with what a hypotenuse is, a hypotenuse is the
longest side of a right-angled triangle. In other words the side opposite
the right angle.
Now this is needed information to understand Pythagorean Theorem, the
theorem that Pythagora's once presented. Since there are more than 60
different proofs on this page, I think it would be possible for
non-mathematicians at least to find one of all to understand this type of
math easier.
http://www.cut-the-knot.org/pythagoras/index.shtml
Inger E
.. |
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Dave L. Renfro
science forum Guru
Joined: 29 Apr 2005
Posts: 570
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| Posted: Fri Jun 23, 2006 7:57 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
|
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Dave L. Renfro wrote:
| Quote: | I don't know what you mean by the difference method,
but if it's (linear) interpolation, that's not going
to do what you said it does.
It is a matter of calculating the difference between two values
of an equation until you get constant differences between values
and then working back to get the next value. It could need third
or fourth order differences. For a fuller explanation try
googling on "Difference Engine". Effectively it meant you needed
a skilled mathematician to reduce an equation to something that
could be done by addition and subtraction.
|
kenney@cix.compulink.co.uk wrote:
O-K, I found some relevant hits ... thanks.
http://www.google.com/search?q=Difference-Engine+trigonometry
Dave L. Renfro |
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Dave L. Renfro
science forum Guru
Joined: 29 Apr 2005
Posts: 570
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| Posted: Fri Jun 23, 2006 8:04 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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IE J wrote (in part):
| Quote: | Good God Dave,
while it's interesting to read about sine in sci.archaeology
short before I will send a drawing of how to calculate where
the Vinland's voyager went, I am on the other hand deeply
distressed with the fact shown that you, and maybe many
others, haven't had enough Math education in ordinary
school to learn that sine by no means has anything to
do with interpolation.
|
I was talking about interpolation for tables of numerical
values, whether these are trig. values, logarithm values,
statistical tables, etc.
http://genealogy.math.ndsu.nodak.edu/html/id.phtml?id=1072
Dave L. Renfro |
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mixi.mo@virgin.net
science forum beginner
Joined: 22 Jun 2006
Posts: 9
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| Posted: Fri Jun 23, 2006 8:36 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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Larry Lard wrote:
| Quote: | But the Greeks got their letters from the Phoenicians, who aren't
remembered for their mathematics.
|
See your point, but go back further in time and all letters
are derived from pictures of things or depict cardinal quantities,
the Babylonians are mostly remembered for their astronomy,
the Greeks used Babylonian abstract system but with an
adapted version of their own alphabetic system. Greece was
the intermediary between East and West, and the classical
Greek alphabet has different values from the old Greece.
(from what I've read)
I was comparing developments in letter styles, and it looks
confusing because some shapes which I would have thought
would develop into a familiar graphic actually turns out
to be something else. Some letters which were originally
used disappear but arnt replaced. Some letters seemed to
spring from nowhere although they are related to an ancient
graphic which is now represented by something else. Its
almost as if there were 2 waves in the development of letter
shapes or their use, so there is some hint that the ancient
one survived but was over written by something new.
I'm sure that the confusion lay at the using of the same shape
to represent both sounds and number. I simply wanted to
play around with the idea that if you forget the association
between a letter and its sound, some shapes still have some
kind of value associated to them and means to convey ideas,
e.g Chinese combines graphics into single ideograms, but not
necessarily their sounds.
....I was wondering if geometry was involved in the numerical
values associated with the old letters, if so and how?....
Tom McDonald relied:-
| Quote: | Do you have an interest in archaeology?
Does this topic relate in any way to archaeology?
|
| Quote: | If it does, please make the connection.
|
| Quote: | If it does not, please take s.a. out of the distribution list.
|
I read somewhere that the first example of the use of a
right angle triangle in architecture was at Mohanjo Jaro
in the Indus Valley, am I wrong? are there any other places
in the world that used it too? were there any earlier?
thanks
Mixi |
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##minty
science forum beginner
Joined: 22 Jun 2006
Posts: 2
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| Posted: Fri Jun 23, 2006 8:56 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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mixi.mo@virgin.net wrote:
| Quote: | Minty wrote:-
SOH CAH TOA
Ah but thats cheating! I was looking at the
geometry of the letterforms/symbols and traced
them back through their history, - I wondered if
the characters themselves used to be mnemonic devices
for (mostly) illiterate artisans, craftsmen and
masons. For example, can two or more lettershapes
be combined to formulate a 3rd?
Φ = 500
I = 10
O = 70
how does Φ represent pi using those numbers?
|
They'd use the pi letter of course ;)
More seriously, perhaps you ought to look into number series,
arithmetic and geometric progressions, Maclaurin's series, Taylor's
theorem, and other such thing, even pythagorus, fibonacci and stuff.
Then binomial expansion etc... Sine, cosine and tangents have number
series expansions, so you can calculate something to however much
precision you fancy. It's all high school stuff.
##minty...
| Quote: |
I + Ξ = 10+60 = 70, although E = 5,
what is the equasion or formula (if any) that
led to E ? ( Ξ still represents 3 in China)
How was pythagoras' formula written in old
Greek, and is there any way of calculating the
angles using those ancient letter symbols?
Ken Young wrote:-
You are obviously unaware of the history of maths. Sine etc date
back to Pythagorus and can be calculated geometrically. They can
be calculated arithmetically by the difference method. I can not
remember the exact method but that was because when I was at
school we used books of log, sin, cos, tan and some I forget
tables.
Yes!, but I'd really like to know how the ancient Greeks
wrote it in ancient Greek! not everyone could write or read
and people have been planning, building, farming many thousands
of years,
Number systems are arbitrary ways of representing the actual
number. Actual mathematical operations are not affected except by
ease of use. Positional notation was adopted because it made some
operations easier to learn.
Typographically, those symbols also once represented objects or
ideas, I just had an idea that maybe the letters of ancient
Greece also held some vestiges of old formulae ? any ideas?
6,7, 80 & 90 are missing (6 may have been represented by Z)
Mixi |
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| Back to top |
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IEJ
science forum beginner
Joined: 23 Jun 2006
Posts: 2
|
| Posted: Fri Jun 23, 2006 11:30 pm Post subject:
Re: typographic symbols, geometry, maths puzzle
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|
"##minty" <nibblenot@lycos.com> wrote in message
news:1151096187.519371.216870@p79g2000cwp.googlegroups.com...
mixi.mo@virgin.net wrote:
| Quote: | Minty wrote:-
SOH CAH TOA
Ah but thats cheating! I was looking at the
geometry of the letterforms/symbols and traced
them back through their history, - I wondered if
the characters themselves used to be mnemonic devices
for (mostly) illiterate artisans, craftsmen and
masons. For example, can two or more lettershapes
be combined to formulate a 3rd?
? = 500
I = 10
O = 70
how does ? represent pi using those numbers?
|
They'd use the pi letter of course ;)
More seriously, perhaps you ought to look into number series,
arithmetic and geometric progressions, Maclaurin's series, Taylor's
theorem, and other such thing, even pythagorus, fibonacci and stuff.
Then binomial expansion etc... Sine, cosine and tangents have number
series expansions, so you can calculate something to however much
precision you fancy. It's all high school stuff.
##minty...
IEJ: and you as well as I still remember what one once learnt......
Inger E |
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