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Hannibal lecter

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Registered: 2016-02-11

Posts: 392

from napier book

Besides this, there is another rule for accuracy ; that is to say, when an unknown or incommensurable quantity is included between numerical limits not differing by many
units.
Thus if the diameter of a circle contain 497
parts, since it is not possible to ascertain precisely
of how many parts the circumference consists, the
more experienced, in accordance with the views of
Archimedes, have enclosed it within limits, namely
1562 and 1561. Again, if the side of a square
contain 1000 parts, the diagonal will be the square
root of the number 2000000. Since this is an in- commensurable number, we seek for its limits by
extraction of the square root, namely 1415 the
greater limit and 1414 the less limit, or more
accurately 1414(604/2828) greater, and 1414 (604/2829)

I searched a lot but I didn't find these what is 1562 greater limit and 1561 less limit
also what is 1414(604/2828) greater, and 1414 (604/2829)

if the side of square contain 1000 parts the diagonal will be the square root of number 2000?
isn't that should be 1 * square root of 2
why he put six zeros?
that book is talking about accuracy right now
also what is these numbers the 1414? and the 1561

Last edited by Hannibal lecter (2023-06-23 05:11:32)

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Wisdom is a tree which grows in the heart and fruits on the tongue

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

also allow me to quote this :-

Arithmetical progressions : 1, 2, 3, 4, 5, 6, 7,
&c. ; or 2, 4, 6, 8, 10, 12, 14, 16,
Geometrical progressions: 1, 2, 4, 8, 16, 32, 64,
or 243,81, 27, 9, 3, 1.
in these progressions we require accuracy and ease in
working. Accuracy is obtained by taking large numbers
for a basis ; but large numbers are most easily made from
small by adding cyphers,

Thus instead of 100000, which the less experienced make the greatest sine, the more learned
put 10.000.000, whereby the difference of all sines
is better expressed. Wherefore also we use the
same for radius and for the greatest of our geometrical proportional.

what does he put? he just increase the zeros it's mean he changed the number is that accuracy?

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

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Registered: 2010-06-20

Posts: 10,621

Re: from napier book

Please give a reference for this book.

B

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

this is the book shared bia google drive

https://drive.google.com/file/d/1eYeHtD4H-hnJdYAVTAxtSB_DY4zkhpS7/view?usp=sharing

I don't know if you can download it I found it on web
it's a translated version for the original book of john napier

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

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Registered: 2010-06-20

Posts: 10,621

Re: from napier book

I've got the book.  What page are you asking about?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

it's page 9 he is talking about 1562 and 1561

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

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Registered: 2010-06-20

Posts: 10,621

Re: from napier book

Ok found it. Napier's language is not modern so it's tricky to follow what he is talking about.  He seems to be talking in general terms about how accurately you might want to calculate something.  So he uses examples where he leaves off the least significant digits without seriously affecting the quality of the result.  In the Archimedes example he has a diameter divided into 497 parts ( =N say) so he reasons that the circumference should be divided into  pi times N parts.  If you take pi as 3.141 you get an answer that lies between  1562 and 1561.

Bob

ps. If you want to learn about logarithms (maybe after sines!)  then look at MIF to get more up to date language.

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

Ok found it. Napier's language is not modern so it's tricky to follow what he is talking about.  He seems to be talking in general terms about how accurately you might want to calculate something.  So he uses examples where he leaves off the least significant digits without seriously affecting the quality of the result.  In the Archimedes example he has a diameter divided into 497 parts ( =N say) so he reasons that the circumference should be divided into  pi times N parts.  If you take pi as 3.141 you get an answer that lies between  1562 and 1561.

Bob

ps. If you want to learn about logarithms (maybe after sines!)  then look at MIF to get more up to date language.

when I do pi * 497 I find it's equal to 1561.37154883 to 1562
also in the book he is talking about a table which is that table? where can I find it

mr bob this book translation by WILLIAM RAE MACDONALD,
is there another  better translation that understandable by students give me any another translation please

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: from napier book

Napier is credited with the invention of logarithms. He wrote that book in Latin and Macdonald translated it into English. But the style and language is Napier's. 

The table is a list of logarithmic values. Before the invention of computers and calculators, logarithms were used to enable calculations without the need for long multiplication and division.  When I was at school we were given such a table and taught how to use them.

There's a page about him in Wiki: https://en.wikipedia.org/wiki/John_Napier

If you want to learn about logs then I recommend the MIF page:
https://www.mathsisfun.com/algebra/logarithms.html

Hope that helps,

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

Napier is credited with the invention of logarithms. He wrote that book in Latin and Macdonald translated it into English. But the style and language is Napier's. 

The table is a list of logarithmic values. Before the invention of computers and calculators, logarithms were used to enable calculations without the need for long multiplication and division.  When I was at school we were given such a table and taught how to use them.

There's a page about him in Wiki: https://en.wikipedia.org/wiki/John_Napier

If you want to learn about logs then I recommend the MIF page:
https://www.mathsisfun.com/algebra/logarithms.html

Hope that helps,

Bob

I didn't found the table in MIF can you send it to me here please

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Wisdom is a tree which grows in the heart and fruits on the tongue

Jai Ganesh

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Registered: 2005-06-28

Posts: 48,401

Re: from napier book

Here's the link.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

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Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

Here's the link.

yes but there is a table where is that table the book is talking about table of logarithm

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Wisdom is a tree which grows in the heart and fruits on the tongue

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

Napier is credited with the invention of logarithms. He wrote that book in Latin and Macdonald translated it into English. But the style and language is Napier's. 

The table is a list of logarithmic values. Before the invention of computers and calculators, logarithms were used to enable calculations without the need for long multiplication and division.  When I was at school we were given such a table and taught how to use them.

There's a page about him in Wiki: https://en.wikipedia.org/wiki/John_Napier

If you want to learn about logs then I recommend the MIF page:
https://www.mathsisfun.com/algebra/logarithms.html

Hope that helps,

Bob

Mr bob where can I found that table it's not there in MIF and tutorial for how to use it

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Wisdom is a tree which grows in the heart and fruits on the tongue

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: from napier book

Try:

https://jscholarship.library.jhu.edu/bi … 623049.pdf

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Hannibal lecter

Member

Registered: 2016-02-11

Posts: 392

Re: from napier book

Try:

https://jscholarship.library.jhu.edu/bi … 623049.pdf

Bob

Mr Bob it's a useful book thank you so much

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Wisdom is a tree which grows in the heart and fruits on the tongue