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" The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple... "
Elements of Geometry, Geometrical Analysis, and Plane Trigonometry: With an ... - Page 458
by Sir John Leslie - 1809 - 493 pages
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The Contents of the Fifth and Sixth Books of Euclid

Euclid, Micaiah John Muller Hill - 1900 - 165 pages
...following manner : — The first of four magnitudes is said to have the same ratio to the second, as the third has to the fourth, when any equimultiples...being taken, and any equimultiples whatsoever of the second and fourth being taken; if the multiple of the first be less than that of the second, the multiple...
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The Contents of the Fifth and Sixth Books of Euclid

Euclid, Micaiah John Muller Hill - 1900 - 190 pages
...Art. 67. Def. 13. EUCLID'S TEST FOR EQUAL RATIOS. Euclid states this Test in the following manner: — The first of four magnitudes is said to have the same ratio to the second, as the third has to the fourth, when any equimultiples whatsoever of the first and third being taken,...
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The Teaching of Elementary Mathematics

David Eugene Smith - 1902 - 418 pages
...Teaching. One has but to read the Euclidean definition of equal ratios to be assured of this fact : " The first of four magnitudes is said to have the same...being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the...
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Ellen: Or, Whisperings of an Old Pine, Volume 1

Joseph Battell - 1903 - 722 pages
...must contain 2, 3 X 3, or 9 times. See Axiom 6. PROPOSITION IV. ' If the first of four magnitudes has the same ratio to the second which the third has to the fourth, and if any equimultiple whatever is taken of the first and third, and any whatever of the second and...
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First Stage Practical Plane and Solid Geometry

G. F. Burn - 1903 - 272 pages
...is the equality of two ratios. Four quantities are therefore in true proportion when the first has the same ratio to the second which the third has to the fourth ; thus, since 3 : 6 and 1 : 2 are equal ratios, we have a true proportion in 3 : 6 : : 1 : 2 (read...
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Report of the ... Meeting of the Australasian Association for the ..., Volume 10

Australasian Association for the Advancement of Science. Meeting - 1905 - 786 pages
...geometrically as sound as Euclid's, and infinitely more natural. I need only quote it to enforce the point: " The first of four magnitudes is said to have the same...the second which the third has to the fourth when, if the first be divided into any number whatever of equal parts, and the third be divided into the...
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Report of the ... Meeting, Volume 10

ANZAAS (Association) - 1905 - 782 pages
...geometrically as sound as Euclid's, and infinitely more natural. I need only quote it to enforce the point : " The first of four magnitudes is said to have the same...the second which the third has to the fourth when, if the first be divided into any number whatever of equal parts, and the third be divided into the...
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Books 3-9

Euclid - 1908 - 456 pages
...unnecessary. Simson added, in place of Theon's corollary, the following : " Likewise, if the first has the same ratio to the second which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth...
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The Teaching of Mathematics in Secondary Schools

Arthur Schultze - 1912 - 398 pages
...has only to read Euclid's definition* to be convinced that it is utterly unfit for young students. * "The first of four magnitudes is said to have the...being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that The fact that the arithmetical view...
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The Development of Logic

William Calvert Kneale, Martha Kneale - 1962 - 802 pages
...of mathematics. The essence of it is a new definition of the equality of ratios between magnitudes : 'The first of four magnitudes is said to have the same ratio to the second that the third has to the fourth, when any equimultiples whatever of the first and the third being...
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