PROP. XVII.

19. THEOREM. If there be three magnitudes, and other three, and if the first have a greater ratio to the second, in the former set, than the first has to the second, in the latter; and if, also, the second have to the third, in the former set, a greater ratio. than the second has to the third, in the latter then shall the first have a greater ratio to the third, in the former set, than the first has to the third, in the latter.

;

Let A, B, C, be three magnitudes, and D, E, F, three other magnitudes: If (A: B) be greater than (D: E), and (B:C) greater than (E: F), then is (A:C)>(D: F).

For let G be a magnitude such that (G:C) :: (E:F);

.. (hyp. and E. 10. 5.) B>G;

(E. 8. 5.), (A:G) > (A: B).

Again, let H be a magnitude such that (H:G) :: (D:E);

.. (hyp. and E. 13. 5.) (H:G)<(A: B):
Much more then is (H:G)<(A:G);
(E. 10. 5.), A>H;

... (E. 8. 5), (A:C)>(H:C):

But (hyp. and E. 22. 5.), (H:C):: (D: F);
.. (A:C) > (D: F).

U

PROP. XVIII.

20. THEOREM. If there be three magnitudes, and other three, and if the first have to the second, in the former set, a greater ratio than the second has to the third, in the latter; and if, also, the second have to the third, in the former set, a greater ratio than the first has to the second, in the latter; then shall the first have to the third, in the former set, a greater ratio, than the first has to the third, in the latter.

Let A, B, C, be three magnitudes, and D, E, F, three other magnitudes: If (A: B) be greater than (E:F), and (B:C) greater than (D: E), then is (A:C)>(D:F).

For let G be a magnitude such that (G:C) :: (D:E);

.. (hyp. and E. 10. 5.) B>G;

(E. 8. 5.), (A:G)>(A:B):

Again, let H be a magnitude such that (H:G) :: (E:F);

.. (hyp. and E. 13. 5.), (H:G)<(A:G);
.. (E. 10. 5.), A>H;

.. (E. 8. 5.), (A: C) > (H:C);

But (hyp. and E. 23. 5.) (H:C):: (D:F);

S. (A:C)>(D:F).

PROP. XIX.

propor

21. THEOREM. If three magnitudes be tionals, the two extremes are, together, greater than the double of the mean.

Let A, B, C, be three magnitudes which are proportionals: Then A+C>2B.

For (hyp. and E. 6. def. B. 5.), (A: B) :: (B: C); .. (E. 25. 5.) A+C>B+B

i. e. A+C>2B.

22. COR. An arithmetic mean proportional, between two given magnitudes, is greater than a geometric mean proportional between the same two magnitudes.

PROP. XX.

23. THEOREM. If there be two sets of magnitudes, the one geometric, and the other arithmetic, proportionals, and if the two first magnitudes be the same in both, any other magnitude in the former set, shall be greater than the corresponding magnitude in the latter..

Let the magnitudes A, B, C, D, E, &c. be geometric proportionals, and let the magnitudes A,

B, c, d, e, &c. be arithmetic proportionals; then is C>c, D>d, E>e, and so on.

For, first, let A be the least magnitude, in each series;

.. (S. 15. 5. cor.) C-B>B-A:

But, from the property of arithmetic proportion,

B-A=c-B;

.. C-B>c-B;

... C> c.

c;

In the

Again, (S. 15. 5. cor.) DC > C — B, and as hath been shewn, CB > c- B ord ... D-C>d-c; and C>c; ..D>d. same manner it may be shewn that E>c, and so on. Secondly, let A be the greatest magnitude in each series:

Then (S. 15. 5. cor.) A-B> B-C;

But, from the property of arithmetic proportion,

A-B=B-c;

.. B-c > B-C

.. C> c.

Again, (S. 15. 5. cor.) B-C>C-D; and it has been shewn that C>c; much more then is B-c >C-D:

But B-c-c-d;

.. c-d>C-D;

..D>d:

And, in the same manner, it may be shewn that, in this case, also, E>e, and so on.

24. COR. The two first magnitudes, in both

the sets, being the same, if the second of the geometric proportionals be greater than the second of the arithmetic proportionals, then, much more, will every other magnitude, in the former set, be greater than the corresponding magnitude in the latter.

PROP. XXI.

25. THEOREM. If there be two series of magnitudes, the one arithmetically proportional, the other geometrically proportional, but each having the same magnitude for its first term, and if the last term of the arithmetic series be not less than the last term of the geometric series, any other term of the former series shall be greater than the corresponding term in the latter.

Let the magnitudes A, B, C, D, E, &c. Q, be geometric proportionals; and let the magnitudes A, b, c, d, e, &c. q, be arithmetic proportionals; then if q be not less than Q, b> B, c> C, d>D, and so on.

For (S. 20. 5. and cor.) if B be equal to b, or greater than b, Q>q; which is contrary to the hypothesis; ..b>B:

Again, in the two series B, C, D, &c. Q, b, c, d, &c, q, let b, which has been shewn to be greater than B, be supposed to become equal to B, and q to remain of a magnitude not less than Q; then it is