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Observations
Prop XIV & XV
Book III
There is the same no more<add> the same reason for joining 14 & 15 Prop. Book 3d
being divided than together as there is for joining
The several Cases of the 7 & 8th Prop. into one.
Term.
double meaning
The word Term is made use of by several of the Editors
in 2 very different senses, in one they mean a "Term of expression" in the Logical sense
in the other in the Mathematical sense a Term or boundary of a line Vid. De Chalis preface to 5th Book.
Cunn's 3d Prop
5th Book.
Word Each
The meaning of the Word each seems to be understood by
neither Euclid or any of his Editors, "έκαστος" is the expression in
the Greek and the Translators share all of them translated
it litteraly into "each" without Considering the meaning of
it but, prejudiced as in any many other cases to
the Infallibility to Euclid because what he says
in generalisation as if that was a reason he should
be infallible, a remarkable example of the bad
effect of making use of this word each in as are
appropriate its real one, is to be found in 3d Prop
5the Book of Cunn's Edition. He says "If the 1st to"
"the same Multiple of the 2d as the 3d of the 4th, and there be"
taken Equimultiples of the 1st & 3d. Then will each
of the Magnitudes taken he had by the and Equimultiple be Equimultiples of the 2 & 5th.
2d and 4th.
Each in its litteral sense in this case is the same
as either, if so it let either be put into its room then will it
be "Then will either of the Magnitudes taken be Equimultiple
of the 2d & 4th but this is not the time there are in this instance
2 words which together but this the opposite meaning to
that he intends to convey for it is determined which of the
two magnitudes taken shall be equimultiple of the 2d & which of the 4th but supposing the word
each did determine which
of the 2 magnit. you should
take the expression sentence would
then lead you into an error
for he says each is an
equimult. of the 2 and 4th
when he means that
therefore ifin he did expect
that each should determine
some one he saves could expect
any body to understand that
is equal to one, when he tells you it is to
both
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On the Use which
Euclid makes
of Lines and
Letters
Words not defined
Antecedent}
Consequent}
Magnitude
Relation of Magnitude
Rank
Excess
Simplitude
In treating of Propositions of Quantities Euclid makes use of
Lines as examples and to distinguish the line or lines part
or parts of lines, which he means to speak of at one intent
from those which he means not to speak of at the
same instant he fixes to letters at the extremities
of each part line or part of a line. For example suppose
he meant if of the line A.B. which is divided into
A—3 parts, he meant to speak of the middle division
C—he would fix the letters CD and call it the part C
D—D. Now CD the letters C.D. of themselves did give
B—you no idea of the part he mentions, but you
know he means by putting these letters to refer you
to the line divided in question. When you look at
that and have found the letter CD. you see in the that
their position denotes the middle division. how much
sooner might a it have been known that he meant
the Middle division if he had said so! for then the
trouble of referring to the figure (by which you probably
lost your place) would have been saved besides that
of finding out the letters position of those letters with
respect to the figure.
Magnitudes
instead of
Lines
Better It would be better to express make use of
the more general term of magnitude than to be
confined to the idea of a line and if it was necessary
to give an example in Numbers or if possible in
letters standing for number in their common order.
Equimultiples
Euclid uses the Word Equimultiple but without
defining it.
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