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" A greater magnitude is said to be a multiple of a less, when the greater is measured by the less ; that is, ' when the greater contains the less a certain number of times exactly.' 3. ' Ratio is a mutual relation of two magnitudes of the same kind to... "
Pantologia. A new (cabinet) cyclopędia, by J.M. Good, O. Gregory, and N ... - Page 89
by John Mason Good - 1813
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The first six books of the Elements of Euclid, and propositions i.-xxi. of ...

Euclides - 1885
...greater. in. Eatio is the mutual relation of two magnitudes of the same kind with respect to quantity. iv. Magnitudes are said to have a ratio to one another...when the less can be multiplied so as to exceed the greater. These definitions require explanation, especially Def . m., . which has the fault of conveying...
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The Harpur Euclid: An Edition of Euclid's Elements

Edward Mann Langley, W. Seys Phillips - 1890 - 515 pages
...part* or 'measure' of the greater ; (2) the greater is called a ' multiple' of the less. 3. ' Ratio ' is a mutual relation of two magnitudes of the same kind to one another in respect of quantity. We have given the definition in its usual form ; but the word ' quantity ' is misleading. De Morgan...
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Musical Pitch and the Measurement of Intervals Among the Ancient Greeks

Charles William Leverett Johnson - Music theory - 1896 - 76 pages
...notes. 'Katio is defined by Euclid in the following words (Euclid, Elements, v. def. 3) : " Batió is a mutual relation of two magnitudes of the same kind to one another in respect of quantity," or rather of " quantnplicity." It is immaterial which of the two magnitudes first receives the attention...
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Proceedings of the Edinburgh Mathematical Society, Volumes 15-16

Edinburgh Mathematical Society - Electronic journals - 1897
...text of Euclid there may be said to be two definitions, of which the first is " ratio is the (or a) relation of two magnitudes of the same kind to one another in respect of quantuplicity." It may be remarked that this definition has a rather curious history. Barrow, a most...
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The School World, Volume 3

Education - 1901
...compass is incomplete, and on p. 290 we have the remarkable statement that "Euclid's definition that 'magnitudes are said to have a ratio to one another...when the less can be multiplied so as to exceed the greater' is only an indirect way of stating that two magnitudes have a ratio when, and only when, they...
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Ellen: Or, Whisperings of an Old Pine, Volume 1

Joseph Battell - Force and energy - 1903
...denominator or standard of measure. But it is done both ways. " In the first definition, that ratio is a mutual relation of two magnitudes of the same kind, to one another, in respect of quantity, the ratio between the same two quantities or other equally proportional quantities cannot vary ; and...
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School Science and Mathematics, Volume 8

Education - 1908
...Vol. XIII, 1907. p. 392. Archimedean postulate is hidden in one of the definitions (Def. 4, Bk. V.)- "Magnitudes are said to have a ratio to one another,...less can be multiplied so as to exceed the other." As long as an argument involves assumptions which are not explicitly set forth in the mind, the logic...
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The Mathematics of Measurement: A Critical History

John James Roche, ROCHE - Mathematics - 1998 - 330 pages
...ratio as a relationship between two quantitative terms and expressed it as such. For Euclid108, 'Ratio is a mutual relation of two magnitudes of the same kind to one another in respect of quantity'. Ratio belonged, therefore, to the category of relation and was not a simple quantity or a single number....
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The American Mathematical Monthly: The Official Journal of the ..., Volume 22

Mathematicians - 1915
...exceeded." Euclid in his Elements (Bk. V, Def. 4) gives the postulate in the form of a definition: "Magnitudes are said to have a ratio to one another,...less can be multiplied so as to exceed the other." The Method of Archimedes, a book formerly thought to be irretrievably lost, but fortunately discovered...
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The Calendar of Owens college, Manchester

Manchester univ - 1877
...a given line in a given point and bisecting the circumference of a given circle. 9. Magnitudes arc said to have a ratio to one another when the less can be multiplied so as to exceed the greater. Explain this. Prove that equal magnitudes have the same ratio to the same magnitude and conversely....
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