G.189 ratios ratio
V.11
Ratios which are the same with the
same ratio are also the same with one another.
ratios_ratio ()
prove as A is to B , so is E to F
A
B
G
L
C
D
H
M
E
F
K
N
assert As A is to B , so let C be to D , and, as C is to D , so let E be to F
assert I say that, as A is to B , so is E to F
construct Of
A ,
C ,
E let
equimultiples G ,
H ,
K be taken, and of
B ,
D ,
F other, chance, equimultiples
L ,
M ,
N .
assert Then since, as A is to B , so is C to D , and of A , C equimultiples G , H have been taken, and of B , D other, chance, equimultiples L , M ,
assert if G is in excess of L , H is also in excess of M , if = , = , and if less, less
assert Since, as C is to D , so is E to F , and of C , E equimultiples H , K have been taken, and of D , F other, chance, equimultiples M , N → therefore, if H is in excess of M , K is also in excess of N , if = , = , and if less, less
assert We saw that, if H was in excess of M , G was also in excess of L
assert If = , =
assert if less, less
assert So that, in addition, if G is in excess of L , K is also in excess of N , if = , = , and if less, less
assert G , K are equimultiples of A , E , while L , N are other, chance, equimultiples of B , F
∴ as A is to B , so is E to F
Dependency Graph
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"G.2" [label="G.2", tooltip="G.2 (I.def.2): A line [ A B ] is a projection of a relationship of two points A, B.", fillcolor="#224422", URL="../G.2/index.html", target="_top"];
"G.3" [label="G.3", tooltip="G.3 (I.def.3): The extremities of a segment / A B / are points A, B.", fillcolor="#224422", URL="../G.3/index.html", target="_top"];
"G.8" [label="G.8", tooltip="G.8 (I.def.13): A boundary is the extremity of anything.", fillcolor="#224422", URL="../G.8/index.html", target="_top"];
"G.9" [label="G.9", tooltip="G.9 (I.def.14): A figure is that which is contained by any boundary or boundaries.", fillcolor="#224422", URL="../G.9/index.html", target="_top"];
"G.169" [label="G.169", tooltip="G.169 (V.def.1): A magnitude is apart of a magnitude, the less of the greater, when it measures the greater.", fillcolor="#224422", URL="../G.169/index.html", target="_top"];
"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.173" [label="G.173", tooltip="G.173 (V.def.3): Aratio is a sort of relation in respect of size between two magnitudes of the same kind.", fillcolor="#224422", URL="../G.173/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"];
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"G.2" -> "G.1";
"G.3" -> "G.1";
"G.3" -> "G.2";
"G.8" -> "G.3";
"G.9" -> "G.8";
"G.189" -> "G.9";
"G.170" -> "G.169";
"G.176" -> "G.170";
"G.189" -> "G.170";
"G.189" -> "G.173";
"G.189" -> "G.176";
}
Required for
G.195 G.199 G.201 G.206 G.207 G.208 G.210 G.211 G.212 G.213 G.214 G.215 G.216 G.218 G.219 G.220 G.221 G.222 G.223 G.225 G.226 G.227 G.228 G.229 G.231 G.232 G.234 G.235 G.236 G.237 G.238 G.239 G.240 G.241 G.243 G.244 G.245 G.377 G.378 G.381 G.382 G.383 G.384 G.385 G.386 G.389 G.390 G.393 G.394 G.395 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.501 G.520 G.528 G.536 G.541 G.542 G.543 G.544 G.546 G.547 G.550 G.551 G.552 G.554 G.555 G.556 G.557 G.558 G.559 G.560 G.563 G.565 G.570 G.571 G.572 G.573 G.577 G.578 G.579 G.580 G.582 G.583 G.584 G.586 G.587 G.588 G.589 G.592 G.597 G.600 G.601
