MATHEMAT1CS.-PLANE GEOMETRY. 



[BOOK vi. raor. L 



B)A(M 

 mli 



to find tlie greatest magnitude that 

 will measure both. Let A be the 

 neater of the two magnitudes, and 

 from it take the greatest possible mul- 

 tiple of B (via., mH), leaving a re- 

 mainder C, Ian, of course-, than B. 

 In like manner take from B the greatest 

 possible multiple of C (vis., wC), leav- 

 ing a remainder 1>, lea than C. In 

 like manner take from C the greatest 

 possible multiple of I) (viz., /<D), leav- 

 ing a rem.iin.u-r K leas than 1) ; and 

 so on, as til the margin : the greatest 

 common measure will bo that re- 

 mainder which exactly measures the 

 preceding one : for instance, if K measure 1) so that 

 yE D, then K ia the greatest common measure of 

 AandB. 



For every common measure of A and B, as it measures 

 B, will measure mB ; consequently, as it at the same 

 time measures A, it will measure A mB, by last pro- 

 position ; that is, it will measure C. This being the 

 case, it must also measure nC, and therefore, last pro- 

 position, it must measure B nC ; that is, it must 

 measure D : for similar reasons it must measure E ; and 

 so on. Consequently every measure of A and B also 

 measures each of the remainders C, D, E, <fcc. ; and as 

 these remainders become less and less, it follows that 

 that must be the greatest common measure which is ex- 

 actly equal to the last of these remainders, and at which 

 the operation terminates. But if the operation never 

 terminate, the diminishing series of remainders being 

 continued without end, then, as there is no last remain- 

 der, there can be no common measure at all in other 

 words, the magnitudes A, B will be incommensurable. 



PROPOSITION XXIII. THEORKM. 



If one magnitude contain another, and leave a re- 

 mainder, such that the greater of the two magnitudes is 

 to the less, as the less is to that remainder, then the two 

 magnitudes will be incommensurable. Let A, B be the 

 two magnitudes, such that the greater A contains the 

 lens B, m times, leaving a remainder C ; that is, such 

 that A mB = C ; then if A : B : : B : C, the magnitudes 

 A, B will be incammfnsurable. 



For let C, D, E, <tc., be the successive remainders in 

 the operation for finding the common measure (Prop. 

 XXII.) Then C cannot measure B, for then B would 



measure A, so that there would not be any rrmaimlir 

 \\\.) ; but C is contained as often in B as B i.i 

 contained in A (Def. VI.) Let mB be the greatest 

 multiple of It which is contained in A, and take .(', an 

 equimultiple of C : then (Prop. VIII.) A : 1! : : ./.I! : wC, 

 and (1'rops. XIV. and XI.) A : B : : A mB : B mC : 

 but by hypothesis A wiB =- C, and B mC D : 

 therefore A : B : : C : D ; and since A : B : : B : 0, .-. 

 (Prop. II.)B:0::C:D; hence D cannot measure C, 

 inasmuch as C cannot measure B, (Prop. XII.) 



Let now nC be the greatest multiple of C in B, and 

 take iD, an equimultiple of D : then, from what is 

 proved above, C : D : : D : E ; hence E cannot measure 

 D, inasmuch as D, as just proved, cannot measure C. 

 And the reasoning is the same for every successive re- 

 mainder ; so that no remainder can ever measure the 

 preceding remainder ; and therefore the operation for the 

 common measure can never terminate ; that is, the two 

 magnitudes A, Bare incommensurable. .', if one magnitude 

 die. Q. E. D. 



The operation explained in Prop. XXII. is that 

 actually performed on a pair of numbers, when the object 

 is to ascertain whether those numbers have a common 

 measure, and to discover the grtateitt common measure. 



In this arithmetical process, should a remainder ever 

 become 1, we conclude that no common measure of the 

 two numbers exists ; because m Arithmetic, as remark ul 

 at page 571, 1 is not regarded as an arithmetical common 

 measure. When the two numbers have no factor in 

 common, a unit-remainder must always occur to apprise 

 us of the fact, after a finite number of steps of the work ; 

 since the remainders all whole numbers go on con- 

 tinually diminishing. But in magnitudes not susceptible 

 of numerical representation, the operation referred to 

 could not be practically applied ^as long as any re- 

 mainder occurred, so long must the work be continued ; 

 and therefore, in the case of incommensurable magni- 

 tudes, it would be endless, evjn if we could practically 

 carry forward the steps. But the proposition just estab- 

 lished, furnishes a geometrical test of incommensurability 

 that may be readily appealed to, as will be seun in tlio 

 proposition respecting the side and diagonal of a square, 

 at the end of the Sixth Book- It will also bo shown 

 hereafter, by aid of the present theorem, that if a line 

 be divided, as in Prop. XL, Book II., the two parts of 

 that lino will be incommensurable ; and therefore that it 

 would be quite impossible to express both by numerical 

 values of their lengths. 



CHAPTER VI. 

 ELEMENTS OF EUCLID. BOOK VL 



DEFINITIONS. 



IP 



Similar rectilineal 

 figures are those which 

 have the several angles 

 in one equal to those 

 in the other, each to 

 each, and the sides 

 about the equal angles that is, which include the equal 

 angles, proportionals. 



n. 



Two sides of one figure are said to be reciprocally pro- 

 portional to two sides of another, when one of the sides 

 of thu first is to one of the sides of the second as the re- 

 maining side of the second is to the remaining side of 

 the first. 



in. 



A straight line is said to be cut in extreme and mean 

 ratio, when the whole is to the greater segment as the 

 greater segment is to the less. 



IV. 



The altitude pf any figure is the straight line drawn 



from its vertex perpendicular to the base, or opposite 



side, and terminating in that side, or the side prolonged. 



Thu the perpendicular A D, drawn from the vertex A to the baio 



1M', is thu altitude of the triangle ABC, m 1'rupuMtiun Vill. iu 

 thu book. 



PROPOSITION L -^THEORKM. 

 Triangles and parallelograms of the tame altitude are to one 



another as (heir bases. 



Let the triangles A B C, A J) K, and the parallelograms 

 E C, F D, have the same altitude : then as the base 1 1 ( ' 



is to the base 1)K, 



i A F so is the triangle 



A B C to the tri- 

 angle AD K, and 

 the i>arallelograin 

 E C to the paral- 

 lelogram F D. 



Produce I: K 

 both ways t<> II 

 and L, ami make 

 111!, (i II., <tc., 

 any number of 



U O B 





