G.268 unit measure number — Heath
VII.15
If an
unit measure any
number , and another number measure any other number the same number of times,
alternately also, the unit will measure the third number the same number of times that the second measures the fourth.
For let the unit A measure any number / B C / , and let another number D measure any other number / E F / the same number of times;
I say that, alternately also, the unit A measures the number D the same number of times that / B C / measures / E F / .
A
B
G
H
C
D
E
K
L
F
For, since the unit A measures the number / B C / the same number of times that D measures / E F / ,
therefore, as many units as there are in / B C / , so many numbers equal to D are there in / E F / also.
Let / B C / be divided into the units in it, / B G / , / G H / , / H C / , and / E F / into the numbers / E K / , / K L / , / L F / equal to D .
Thus the multitude of / B G / , / G H / , / H C / will be equal to the multitude of / E K / , / K L / , / L F / .
And, since the units / B G / , / G H / , / H C / are equal to one another, and the numbers / E K / , / K L / , / L F / are also equal to one another, while the multitude of the units / B G / , / G H / , / H C / is equal to the multitude of the numbers / E K / , / K L / , / L F / ,
therefore, as the unit / B G / is to the number / E K / , so will the unit / G H / be to the number / K L / , and the unit / H C / to the number / L F / .
Therefore also, as one of the
antecedents is to one of the consequents, so will all the antecedents be to all the consequents;
[G.265] therefore, as the unit
/ B G / is to the number
/ E K / , so is
/ B C / to
/ E F / .
But the unit / B G / is equal to the unit A , and the number / E K / to the number D .
Therefore, as the unit A is to the number D , so is / B C / to / E F / .
Therefore the unit A measures the number D the same number of times that / B C / measures / E F / .
Q. E. D.
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"G.170" [label="G.170", tooltip="G.170 (V.def.2): The greater is amultiple of the less when it ismeasured by the less.", fillcolor="#224422", URL="../G.170/index.html", target="_top"];
"G.176" [label="G.176", tooltip="G.176 (V.def.5): Magnitudes are said tobe in the same ratio, the first to the second and the third to the fourth, when, if anyequimultiples whatever be taken of the first and third, and any equimul...", fillcolor="#224422", URL="../G.176/index.html", target="_top"];
"G.180" [label="G.180", tooltip="G.180 (V.def.6): Let magnitudes which have thesame ratio be calledproportional.", fillcolor="#224422", URL="../G.180/index.html", target="_top"];
"G.181" [label="G.181", tooltip="G.181 (V.def.11): The termcorresponding magnitudes is used of antecedents in relation to antecedents, and of consequents in relation to consequents.", fillcolor="#224422", URL="../G.181/index.html", target="_top"];
"G.194" [label="G.194", tooltip="G.194 (V.def.12): Alternate ratio means taking theantecedent in relation to the antecedent and the consequent in relation to the consequent.", fillcolor="#224422", URL="../G.194/index.html", target="_top"];
"G.247" [label="G.247", tooltip="G.247 (VII.def.1): Anunit is that by virtue of which each of the things that exist is called one.", fillcolor="#224422", URL="../G.247/index.html", target="_top"];
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"G.254" [label="G.254", tooltip="G.254 (VII.def.4): Butparts when it does notmeasure it.", fillcolor="#224422", URL="../G.254/index.html", target="_top"];
"G.256" [label="G.256", tooltip="G.256 (VII.def.5): The greaternumber is amultiple of the less when it ismeasured by the less.", fillcolor="#224422", URL="../G.256/index.html", target="_top"];
"G.257" [label="G.257", tooltip="G.257 (VII.5): If a number be a part of a number, and another be the same part of another, the sum is the same part of the sum that the one is of the one.", fillcolor="#222244", URL="../G.257/index.html", target="_top"];
"G.258" [label="G.258", tooltip="G.258 (VII.6): If a number be parts of a number, and another be the same parts of another, the sum is the same parts of the sum that the one is of the one.", fillcolor="#222244", URL="../G.258/index.html", target="_top"];
"G.263" [label="G.263", tooltip="G.263 (VII.def.20): Numbers areproportional when the first is thesame multiple, or thesame part, or thesame parts, of the second that the third is of the fourth.", fillcolor="#224422", URL="../G.263/index.html", target="_top"];
"G.265" [label="G.265", tooltip="G.265 (VII.12): If there be as many numbers as we please in proportion, then, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents.", fillcolor="#222244", URL="../G.265/index.html", target="_top"];
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"G.268" -> "G.169";
"G.176" -> "G.170";
"G.257" -> "G.170";
"G.180" -> "G.176";
"G.263" -> "G.180";
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"G.265" -> "G.181";
"G.268" -> "G.181";
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"G.268" -> "G.247";
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"G.256" -> "G.248";
"G.257" -> "G.248";
"G.258" -> "G.248";
"G.265" -> "G.248";
"G.268" -> "G.248";
"G.254" -> "G.253";
"G.256" -> "G.253";
"G.257" -> "G.253";
"G.258" -> "G.253";
"G.263" -> "G.253";
"G.265" -> "G.253";
"G.258" -> "G.254";
"G.263" -> "G.254";
"G.265" -> "G.254";
"G.257" -> "G.256";
"G.263" -> "G.256";
"G.258" -> "G.257";
"G.265" -> "G.257";
"G.265" -> "G.258";
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Required for
G.270 G.272 G.273 G.275 G.278 G.279 G.280 G.281 G.285 G.289 G.290 G.292 G.293 G.294 G.295 G.296 G.297 G.298 G.299 G.301 G.303 G.304 G.305 G.306 G.307 G.309 G.312 G.314 G.315 G.316 G.317 G.321 G.322 G.323 G.324 G.325 G.326 G.327 G.328 G.329 G.330 G.331 G.332 G.333 G.334 G.335 G.337 G.338 G.339 G.340 G.342 G.343 G.344 G.345 G.346 G.347 G.348 G.349 G.364 G.370 G.381 G.383 G.384 G.385 G.389 G.390 G.396 G.397 G.398 G.399 G.400 G.401 G.402 G.403 G.404 G.405 G.406 G.407 G.408 G.409 G.410 G.411 G.412 G.413 G.414 G.415 G.416 G.417 G.418 G.419 G.420 G.421 G.423 G.425 G.427 G.429 G.431 G.433 G.434 G.435 G.436 G.437 G.438 G.439 G.440 G.441 G.442 G.443 G.444 G.445 G.446 G.447 G.448 G.449 G.450 G.451 G.452 G.453 G.457 G.458 G.459 G.460 G.461 G.462 G.463 G.464 G.465 G.466 G.467 G.468 G.470 G.472 G.474 G.476 G.477 G.478 G.479 G.480 G.481 G.482 G.483 G.484 G.485 G.486 G.487 G.488 G.489 G.490 G.491 G.492 G.493 G.494 G.495 G.496 G.497 G.498 G.499 G.500 G.584 G.589 G.597 G.600 G.601
