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You complain to me of the difficulty
you find in retaining what you learn of Euclid.
Ideas rise all along in your mind in correspondence
to the terms expressions, & the nor are you insensible convinced
to the necessity wherewith each inclusion follows
from it's premises.
But that conviction
founded on
vanishes with it
terms presented under no
other than a particular
aspect
Merits a place among
the great benefactor
of human kind
The of the Author's
character, will
to the eye of him that takes the
pains to look for them
make penetrate their appearance
through a composition
even on this subject
ently so un
Inacuracy
of B.D.
Prop. XLV
It is demanded to make a Parallelogram equal to any
right line figure. Euclid has drawn one of
4 sides only, it is necessary to divide that
the figure should be divided into triangles.
Euclid has told you only to draw B.D. which
does individe that individual right lined figure into
triangles. Now was the assumed figure
a Triangle it would not be necessary to
draw any lines for it could be as well
made equal to this triangle as to
the sum of the two divisions of it.
Again if the learner should make any general
Idea from the being had to draw B.D. which
are situated at two angles of the figure and
on that account imagine that in all cases he was
to draw a line from one angle to another
he in the case of this triangle could
not divide by so doing for that
line drawn would coincide with one
of the boundaries.
By this Prop one learns how to make
a parallelogram equal to any number
of Triangles if therefore you meet
with a figure divided into triangles
you can make a parallelogram equal
to all these divisions {parts} taken
together and if any
figure can be divided into triangles
this may equal any .
NB. The consequence conclusion A Young man
much consonant in lines and figures
being and who had thought he understood
the 3 first books of Euclid at
least being desired to make
a parallelogram equal to a
given right lined figure which
had more than 4 sides
could no confessed he could not
do it for that he could tell where
to draw B.D. adding at the same
time that if he knew the in figure
had but four sides he could do
it and thence concluded that
the expression of any right lined figure
must be faintly it being only demonstrated with respect to four sided figures. Simpsons Euclid he had before him all the time.
At your age or thereabouts I too was read reading the
6 first books of Euclid, & understanding stood him no
better. At this somewhat maturer stage of my
faculties, I will attack him once more, &of if I
conquer it will be your's to pursue the victory.
My pursuits you,
into my affection, lead me into other very distant regions.
If I understand him his notions, it must be by superadding
a few others of my own+. As If those notions arise,
I will communicate them to you, & then you too
will understand him in the same manner.
+ for his alone I see
more and more are
not sufficient to explain
the case
page break
EXPRESSION
Area confounded
with Boundaries.
Take care you do not confound the notion of the whole contents of a figure, with that of the mere boundaries
because one triangle is equal to another, it does not follow that every or that any of the sides or of the angles of the 1st
is equal to any of the sides or angles of the 2d. I used to be every now & then thinking that the square of a line was always
four times that line, because four such lines are put for the more broad sides.
A striking example that Euclid confounded the figure with
its boundaries is, in the 34th Prog Book 1st "Parallelogramorum
Spatiorum". Example.
Show us Euclid's the demonstrations
given by
Euclid are those that are
more generally
read throughout
the world in general than
those of Payne &
Cowley & probably
ever will be, how
superior soever may
be the method of those
authors, hence we see the reason why it is Euclid's always to be that must be always
quoted
If a man has convinced himself of the truth
of the proposition referred to, & by the demonstration+
given by the author referred to, the reference
will awaken in him the memory of his having done
so: & perhaps in virtue of the operation if done, that
of the method which he took to do so.
Mathematical demonstrations must in the nature of
things rest in some measure upon authority, &
be received upon trust. When if a posterior proposition
I build the demonstration upon a prior
what do I do? I refer for the time to part of some some book
where that prior has been proved. I refer for example
to the 5th prop. of the 1st book
of Euclid. He therefore who is satisfied for the moment
or the truth of the prior by this reference, is
for so long satisfied of two things. 1st that I have
mentioned a proposition as demonstrated by
Euclid, that really is demonstrated by Euclid[+]
[+] 2dly that Euclid's
demonstration is
a true one.
For so long therefore
his conviction of the
truth of the posterior
proposition rests upon
the authorities of 2
persons, mine & Euclid's.
It may be said, that
his persuasion is formed
on this reference,
but upon his own particular internal
persuasion of
truth of the prior
proportion: & this
may often be the case:
if it is not always
for if it came there would be no use in these
references, nor would they be ever made.
CLASSIFICAT
prefat
in the world's eye
Euclid having been master of our predecessors & our masters,
will be the master of our successors & of our Scholars & our success
It is now too late to think of establishing any
other Elements. Authority the mistress empress too often
the tyrant of mankind will govern vindicate {claim} her influence {still have her influence}
over that very class of men that piques itself most in being independent
of her power. In vain would
others seek to substitute any <add>their mistakes to his by any superiority </add> improve upon Euclid of perspicuity, in
degree of superiority exemplified in any under
or if any topic, whether of perspicuity of procedure
or of method.
Nothing that any one
can do for the clearing
of these paths of this
science which he Euclid has
trodden will bear
any proportion to what
he has done
The reason is, whatever is said in Geometry
must be spoken to all nations
what Frenchman or what
German will abandon
that authority [of] Euclid for that of Payne
or Cowley?
[+] This Prop: might be a corollary
to the preceding in which it is shewn how
to make a parallelogram of a given line
and angle equal to any triangle and this
is only making a such joined together
as a right lined Figure is divided into.
Qu: is it necessary first to shew that
any rightlined figure may be divided into triangles
at least the simple one Euclid & or his adorer
Simpson should have done it.</p>
As for
their
no other elements that
can be written will supersede
the necessity of learning these.
Identifier:  JB/135/030/002 "JB/" can not be assigned to a declared number type with value 135.



135 

030 
euclid prefat: 

002 
classificat. prefat 

text sheet 
2 

recto 

jeremy bentham; samuel bentham 
[[watermarks::gr [with crown] [britannia motif]]] 

46148 
