The Philosophy of Mathematics, With Special Reference to the Elements of Geometry And the Infinitesimal Method
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In our eReader you can find the full English version of the book. Read The Philosophy of Mathematics, With Special Reference to the Elements of Geometry And the Infinitesimal Method Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book The Philosophy of Mathematics, With Special Reference to the Elements of Geometry And the Infinitesimal Method
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To quickly assess the difficulty of the text, read a short excerpt:
. + « is, according to a well-known algebraic formula, equal to n{n + 1) 1 2 ' f|(1 + 2 + 3...
xi n) = H 2 B n k {n h -\- k) ~H' 2 _B H ,,H(H+/.)_B 2 H .,H.^+H^ 7« 78 THE PHILOSOPHY OF MATHEMATICS.
Now, if S be the sum of the rectangles, we shall have However small h may be made, or however great, in other words, the number of rectangles may be con- ceived to be, the two variables S and its value will be equal to each other. Hence, as has been demonstrated, their limits are equal. But the limit... of S is the area B H' + H it of the triangle ABC, and the limit of — X ■ H 2 is — X — , or . That is, the area of the triangle H 2' 2 ^ A B C is one-half the product of its base by its alti- tude.
Now, it may be clearly shown how it was that Pas- cal, as well as Roberval and others, started from a false hypothesis or first principle, and yet arrived at a perfectly correct conclusion. He committed an error first in supposing that the sum of the rectangles would ultimately be equal to the area of the triangle; he committed another error, in the second place, in sup- posing that he could reject indefinitely small quanti- ties without making any difference in the result; and these two errors, being opposite and equal, just exactly neutralized each other.
What to read after The Philosophy of Mathematics, With Special Reference to the Elements of Geometry And the Infinitesimal Method?
You can find similar books in the "Read Also" column, or choose other free books by Bledsoe Albert Taylor to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
. + « is, according to a well-known algebraic formula, equal to n{n + 1) 1 2 ' f|(1 + 2 + 3...
xi n) = H 2 B n k {n h -\- k) ~H' 2 _B H ,,H(H+/.)_B 2 H .,H.^+H^ 7« 78 THE PHILOSOPHY OF MATHEMATICS.
Now, if S be the sum of the rectangles, we shall have However small h may be made, or however great, in other words, the number of rectangles may be con- ceived to be, the two variables S and its value will be equal to each other. Hence, as has been demonstrated, their limits are equal. But the limit... of S is the area B H' + H it of the triangle ABC, and the limit of — X ■ H 2 is — X — , or . That is, the area of the triangle H 2' 2 ^ A B C is one-half the product of its base by its alti- tude.
Now, it may be clearly shown how it was that Pas- cal, as well as Roberval and others, started from a false hypothesis or first principle, and yet arrived at a perfectly correct conclusion. He committed an error first in supposing that the sum of the rectangles would ultimately be equal to the area of the triangle; he committed another error, in the second place, in sup- posing that he could reject indefinitely small quanti- ties without making any difference in the result; and these two errors, being opposite and equal, just exactly neutralized each other.
What to read after The Philosophy of Mathematics, With Special Reference to the Elements of Geometry And the Infinitesimal Method?
You can find similar books in the "Read Also" column, or choose other free books by Bledsoe Albert Taylor to read onlineMoreLess
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