1. If it should be necessary to insert any new Propositions.
Those of Euclid should still retain their former names, as they
are refered to be those names from Books of Mechanics &c.
Use of it
2. By the assistance of the Cards described Method II.
The Order of Prop: and Cor: may be changed so as to
place all those together which relate to the several
genera & species, and By this means each parcel
may be made equally perfect. For Ex: if after having
divided Triangles and Parallelograms into 2 Parcels
you find that for Triangles as well as Parallelograms
upon the same bases & between the same parallels
are proved equal, But that Triangles only are if on equal bases
proved equal, then you perceive that Parallelograms
should be proved equal under like circumstances.
of Different Authors
3. Consider the different orders Geometerical Truths have been
disposed in, by different Authors, and the different ends they
had in view in [purposes they have sought to compass by] such
The order necessary to be assumed for the purpose of demonstrating
the several Propositions is one thing.
Species of Order
The order which would be most proper convenient & methodical
for the purpose of teaching them (were the business only
to apprehend and retain them) would be another.
The order in which they occurred to the Inventor, probably
was a third, certainly it was different from the first.
We have therefore (unless the two latter coincide) three
different species of Orders 1st the order of Demonstration.
2d the order of conception & retension 3dly the order of
The 1st and only state in which Geometry has been
transmitted to us by the Antients is a state far advanced
towards perfection. I mean relatively according to the
order of demonstration, we see it not in this Science
in its first rude essays as we do those for example of
Natural History & Legislation
It will appear very plain that Euclid could never
have first thought of his Propositions in the order in
which he has placed them, but it is to be supposed he
he placed them in their present order for the purpose
Order of Instruction
The order of Simple or Authoritative Illustration[+] is the
same with the Order of enquiry. It is in its Nature for Instruction
to be determined by the Order of Enquiry: for how should a man who had it
in other respects in his choice think if answering questions, but in
the order in which they were asked.
[+] as it may be termed
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sir samuel bentham
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